Grundlehren der mathematischen Wissenschaften 334

Grundlehren dermathematischen WissenschaftenA Series of Comprehensive Studies in Mathematics

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334

ABC

Edoardo Sernesi

Deformations of Algebraic

Schemes

Department of Mathematics

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Preface

In one sense, deformation theory is as old as algebraic geometry itself: this is becauseall algebro-geometric objects can be “deformed” by suitably varying the coefficientsof their defining equations, and this has of course always been known by the classicalgeometers. Nevertheless, a correct understanding of what “deforming” means leadsinto the technically most difficult parts of our discipline. It is fair to say that suchtechnical obstacles have had a vast impact on the crisis of the classical languageand on the development of the modern one, based on the theory of schemes and oncohomological methods.

The modern point of view originates from the seminal work of Kodaira andSpencer on small deformations of complex analytic manifolds and from its forma-lization and translation into the language of schemes given by Grothendieck. I willnot recount the history of the subject here since good surveys already exist (e.g. [27],[138], [145], [168]). Today, while this area is rapidly developing, a self-containedtext covering the basic results of what we can call “classical deformation theory”seems to be missing. Moreover, a number of technicalities and “well-known” factsare scattered in a vast literature as folklore, sometimes with proofs available only inthe complex analytic category. This book is an attempt to fill such a gap, at least par-tially. More precisely, it aims at giving an account with complete proofs of the resultsand techniques which are needed to understand the local deformation theory of alge-braic schemes over an algebraically closed field, thus providing the tools needed, forexample, in the local study of Hilbert schemes and moduli problems. The existingmonographs, like [14], [93], [105], [109], [124], [163], [175], [176], [184], all aimat goals different from the above.

For these reasons my approach has been to work exclusively in the category oflocally noetherian schemes over a fixed algebraically closed field k, to avoid switch-ing back and forth between the algebraic and the analytic category. I tried to make thetext self-contained as much as possible, but without forgetting that all the technicalideas and prerequisites can be found in [3] and [2]: therefore the reader is advisedto keep a copy of them to hand while reading this text. In any case a good fami-liarity with [84] and with a standard text in commutative algebra like [48] or [127]

VI Preface

will be generally sufficient; the classical [167] and [190] will be also useful. A goodacquaintance with homological algebra is assumed throughout.

One of the difficulties of writing about this subject is that it needs a great num-ber of technical results, which make it hard to maintain a proper balance betweengenerality and understandability. In order to overcome this problem I tried to keepthe technicalities to a minimum, and I introduced the main deformation problems inan elementary fashion in Chapter 1; they are then reconsidered as functors of Artinrings in Chapter 2, where the main results of the theory are proved. The first twochapters therefore give a self-contained treatment of formal deformation theory viathe “classical” approach; cotangent complexes and functors are not introduced, northe method of differential graded Lie algebras. Another chapter treats in more detailthe most important deformation functors, with the single exception of vector bun-dles; this was motivated by reasons of space and because good monographs on thesubject are already available (e.g. [90], [118], [59]). Although they are not the centralissue of the book, I considered it necessary to include a chapter on Hilbert schemesand Quot schemes, since it would be impossible to give meaningful examples andapplications without them, and because of the lack of an appropriate reference. De-formation theory is closely tied with classical algebraic geometry because some ofthe issues which had remained controversial and unclear in the old language havefound a natural explanation using the methods discussed here. I have included a sec-tion on plane curves which gives a good illustration of this point.

Unfortunately, important topics and results have been omitted because of lackof space, energy and competence. In particular, I did not include the construction ofany global moduli spaces/stacks, which would have taken me too far from the maintheme.

The book is organized in the following way. Chapter 1 starts with a concise treat-ment of algebra extensions which are fundamental in deformation theory. It thendiscusses locally trivial infinitesimal deformations of algebraic schemes. Chapter 2deals with “functors of Artin rings”, the abstract tool for the study of formal defor-mation theory. The main result of this theory is Schlessinger’s theorem. A section onobstruction theory, an elementary but crucial technical point, is included. We discussthe relation between formal and algebraic deformations and the algebraization prob-lem. This part is not entirely self-contained since Artin’s algebraization theorem isnot proved in general and the approximation theorem is only stated. The last sectionexplains the role of automorphisms and the related notion of “isotriviality”. Chap-ter 3 is an introduction to the most important deformation problems. By applyingSchlessinger’s theorem to them, we derive the existence of formal (semi)universaldeformations. Many examples are discussed in detail so that all the basic principlesof deformation theory become visible. This chapter can be used as a reference forseveral standard facts of deformation theory, and it can be also helpful in supplement-ing the study of the more abstract Chapter 2. Chapter 4 is devoted to the constructionand general properties of Hilbert schemes, Quot schemes and their variants, the “flagHilbert schemes”. It ends with a section on plane curves, where the main propertiesof Severi varieties are discussed. My approach to the proof of existence of nodalcurves with any number of nodes uses multiple point schemes and is apparently new.

Preface VII

In the Appendices I have collected several topics which are well known and standardbut I felt it would be convenient for the reader to have them available here.

Acknowledgements. Firstly, I would like to express my deepest gratitude toD. Lieberman, who introduced me to the study of deformations of complex mani-folds a long time ago. More recently, R. Hartshorne, after a careful reading of aprevious draft of this book, made a number of comments and suggestions which sig-nificantly contributed in improving it. I warmly thank him for his generous help. Formany extremely useful remarks on another draft of the book I am also indebted toM. Brion. I gratefully acknowledge comments and suggestions from other colleaguesand students, in particular: L. Badescu, I. Bauer, A. Bruno, M. Gonzalez, A. Lopez,M. Manetti, A. Molina Rojas, D. Tossici, A. Verra, A. Vistoli.

I would also like to thank L. Caporaso, F. Catanese, C. Ciliberto, L. Ein,D. Laksov and H. Lange for encouragement and support which have been of greathelp.

Contents

Terminology and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Infinitesimal deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.2 The module ExA(R, I ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.3 Extensions of schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Locally trivial deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.1 Generalities on deformations . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2.2 Infinitesimal deformations of nonsingular affine schemes . . . 231.2.3 Extending automorphisms of deformations . . . . . . . . . . . . . . . 261.2.4 First-order locally trivial deformations . . . . . . . . . . . . . . . . . . 291.2.5 Higher-order deformations – obstructions . . . . . . . . . . . . . . . . 32

2 Formal deformation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1 Obstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Functors of Artin rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3 The theorem of Schlessinger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.4 The local moduli functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.4.2 Obstruction spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.4.3 Algebraic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.5 Formal versus algebraic deformations . . . . . . . . . . . . . . . . . . . . . . . . . . 752.6 Automorphisms and prorepresentability . . . . . . . . . . . . . . . . . . . . . . . . 89

2.6.1 The automorphism functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.6.2 Isotriviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

X Contents

3 Examples of deformation functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.1 Affine schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.1.1 First-order deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.1.2 The second cotangent module and obstructions . . . . . . . . . . . 1093.1.3 Comparison with deformations of the nonsingular locus . . . . 1173.1.4 Quotient singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.2 Closed subschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.2.1 The local Hilbert functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.2.2 Obstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.2.3 The forgetful morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.2.4 The local relative Hilbert functor . . . . . . . . . . . . . . . . . . . . . . . 136

3.3 Invertible sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.3.1 The local Picard functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.3.2 Deformations of sections, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413.3.3 Deformations of pairs (X, L) . . . . . . . . . . . . . . . . . . . . . . . . . . 1453.3.4 Deformations of sections, II . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

3.4 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563.4.1 Deformations of a morphism leaving domain and target fixed 1573.4.2 Deformations of a morphism leaving the target fixed . . . . . . 1613.4.3 Morphisms from a nonsingular curve with fixed target . . . . . 1723.4.4 Deformations of a closed embedding . . . . . . . . . . . . . . . . . . . . 1763.4.5 Stability and costability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

4 Hilbert and Quot schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1874.1 Castelnuovo–Mumford regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1874.2 Flatness in the projective case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

4.2.1 Flatness and Hilbert polynomials . . . . . . . . . . . . . . . . . . . . . . . 1944.2.2 Stratifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.2.3 Flattening stratifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.3 Hilbert schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2064.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2064.3.2 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2074.3.3 Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2094.3.4 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

4.4 Quot schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2194.4.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2194.4.2 Local properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

4.5 Flag Hilbert schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.5.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.5.2 Local properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

4.6 Examples and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2354.6.1 Complete intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2354.6.2 An obstructed nonsingular curve in IP3 . . . . . . . . . . . . . . . . . 2374.6.3 An obstructed (nonreduced) scheme . . . . . . . . . . . . . . . . . . . . 2384.6.4 Relative grassmannians and projective bundles . . . . . . . . . . . 240

Contents XI

4.6.5 Hilbert schemes of points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2474.6.6 Schemes of morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2494.6.7 Focal loci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

4.7 Plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2544.7.1 Equisingular infinitesimal deformations . . . . . . . . . . . . . . . . . 2544.7.2 The Severi varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2564.7.3 Nonemptiness of Severi varieties . . . . . . . . . . . . . . . . . . . . . . . 262

A Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

B Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

C Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

D Complete intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305D.1 Regular embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305D.2 Relative complete intersection morphisms . . . . . . . . . . . . . . . . . . . . . . 307

E Functorial language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

Terminology and notation

All rings will be commutative with 1. A ring homomorphism A → B is calledessentially of finite type (e.f.t.) if B is a localization of an A-algebra of finite type.We will also say that B is e.f.t. over A.

We will always denote by k a fixed algebraically closed field. All schemes willbe assumed to be defined over k, locally noetherian and separated, and all algebraicsheaves will be quasi-coherent unless otherwise specified. If X and Y are schemeswe will write X × Y instead of X ×k Y . If S is a scheme and s ∈ S we denote byk(s) = OS,s/mS,s the residue field of S at s.

As is customary, various categories will be denoted by indicating their objectswithin parentheses when it will be clear what the morphisms in the category are. Forinstance (sets), (A-modules), etc. The class of objects of a category C will be denotedby ob(C). The dual of a category C will be denoted by C. Given categories C and D,a contravariant functor F from C to D will be always denoted as a covariant functorF : C → D.

We will consider the following categories of k-algebras:

A = the category of local artinian k-algebras with residuefield k

A = the category of complete local noetherian k-algebraswith residue field k

A∗ = the category of local noetherian k-algebras withresidue field k

(k-algebras) = the category of noetherian k-algebras

Morphisms are unitary k-homomorphisms, which are local in A, A and A∗. For agiven Λ in ob(A∗) we will consider the following:

AΛ = the category of local artinian Λ-algebraswith residue field k

A∗Λ = the category of local noetherian Λ-algebraswith residue field k

2 Terminology and notation

They are subcategories of A and A∗ respectively. If Λ is in ob(A) then we will let

AΛ = the category of complete local noetherian Λ-algebraswith residue field k

which is a subcategory of A. Moreover, we will set:

(schemes) = the category of schemes

(i.e. of locally noetherian separated k-schemes) and

(algschemes) = the category of algebraic schemes

For a given scheme Z we set

(schemes/Z) = the category of Z -schemes(algschemes/Z) = the category of algebraic Z -schemes

hi (X,F) denotes dim[Hi (X,F)] where F is a coherent sheaf on the completescheme X . When no confusion is possible we will sometimes write Hi (F) and hi (F)instead of Hi (X,F) and hi (X,F) respectively.∐

i Xi denotes the disjoint union of the schemes Xi .If E is a vector space or a locally free sheaf we will always denote its dual by E∨.

If V is a k-vector space we will denote by IP(V ) the projective space Proj(Sym(V∨))(where Sym(−) is the symmetric algebra of −): thus the closed points of IP(V ) arethe one-dimensional subspaces of V . Similarly, if E is a locally free sheaf on analgebraic scheme S, the projective bundle associated to E will be defined as

IP(E) = Proj(Sym(E∨))

Note that this definition is dual to the one given in [84], p. 162.For all definitions not explicitly given we will refer to [84].

Introduction

La methode generale consiste toujours a faire desconstructions formelles, ce qui consiste essentiellement afaire de la geometrie algebrique sur un anneau artinien, et aen tirer des conclusions de nature “algebrique” en utilisantles trois theoremes fondamentaux (Grothendieck [71], p. 11).

Deformation theory is a formalization of the Kodaira–Nirenberg–Spencer–Kuranishi(KNSK) approach to the study of small deformations of complex manifolds. Itsmain ideas are clearly outlined in the series of Bourbaki seminar expositions byGrothendieck which go under the name of “Fondements de la Geometrie Algebrique”[2]; in particular they are explained in detail in [72] (see especially page 17), whilethe technical foundations are laid in [71]. The quotation at the top of this page givesa concise description of the method employed.

The first step of this formalization consists in studying infinitesimal deforma-tions, and this is accomplished via the notion of “functor of Artin rings”; the studyof such functors leads to the construction of “formal deformations”. This methodenhances the analogies between the analytic and the algebraic cases, and at the sametime hides some delicate phenomena typical of the algebraic geometrical world.These phenomena become visible when one tries to pass from formal to algebraicdeformations. The techniques of deformation theory have a variety of applicationswhich make them an extremely useful tool, especially in understanding the localstructure of schemes defined by geometrical conditions or by functorial construc-tions.

In this introduction we shall explain in outline the logical structure of deforma-tion theory; for this purpose we will start by outlining the KNSK theory of smalldeformations of compact complex manifolds.

Given a compact complex manifold X , a family of deformations of X is a com-mutative diagram of holomorphic maps between complex manifolds

ξ :X ⊂ X↓ ↓ π

to−→ B

with π proper and smooth (i.e. with everywhere surjective differential), B connectedand where denotes the singleton space. We denote by Xt the fibre π−1(t), t ∈ B.It is a standard fact that, locally on B, X is differentiably a product so that π can beviewed locally as a family of complex structures on the differentiable manifold Xdiff.

4 Introduction

The family ξ is trivial at to if there is a neighbourhood U ⊂ B of to such that wehave π−1(U ) ∼= X ×U analytically.

Kodaira and Spencer started by defining, for every tangent vector ∂∂t ∈ Tto B, the

derivative of the family π along ∂∂t as an element

∂Xt

∂t∈ H1(X, TX )

thus giving a linear mapκ : Tto B → H1(X, TX )

called the Kodaira–Spencer map of the family π . They showed that if π is trivial at tothen κ( ∂

∂t ) = 0 for all ∂∂t ∈ Tto B. Then they investigated the problem of classifying

all small deformations of X , by constructing a “complete family” of deformations ofX . A family ξ as above is called complete if for every other family of deformationsof X :

η :X ⊂ Y↓ ↓ p

mo−→ M

there is an open neighbourhood V ⊂ M and a commutative diagram

X

p−1(V ) → X↓ ↓V → B

inducing an isomorphism p−1(V ) ∼= V ×B X . The family is called universal if it iscomplete and moreover, the morphism V → B is unique locally around mo for eachfamily η as above. Kodaira and Spencer proved that if κ is surjective then the familyξ is complete. The following existence result was then proved:

Theorem 0.0.1 (Kodaira–Nirenberg–Spencer [106]). If H2(X, TX ) = 0 then thereexists a complete family of deformations of X whose Kodaira–Spencer map is anisomorphism. If, moreover, H0(X, TX ) = 0 then such complete family is universal.

Later Kuranishi [111] generalized this result by showing that a complete familyof deformations of X such that κ is an isomorphism exists without assumptions onH2(X, TX ) provided the base B is allowed to be an analytic space.

We want to rephrase everything algebraically as far as possible. Let’s fix an alge-braically closed field k and consider an algebraic k-scheme X . A local deformation,or a local family of deformations, of X is a cartesian diagram

ξ :X → X↓ ↓ π

Spec(k) ⊂ S

where π is a flat morphism, S = Spec(A) where A is a local k-algebra with residuefield k, and X is identified with the fibre over the closed point. If X is nonsingular

Introduction 5

and/or projective we will require π to be smooth and/or projective. We say that ξ is adeformation over Spec(A) or over A. If in particular A is an artinian local k-algebrathen we speak of an infinitesimal deformation.

The notion of local family has the fundamental property of being funtorial. Giventwo deformations of X :

ξ :X → X↓ ↓ π

Spec(k) ⊂ Spec(A)and η :

X → Y↓ ↓ ρ

Spec(k) ⊂ Spec(A)

parametrized by the same Spec(A), an isomorphism ξ ∼= η is defined to be a mor-phism f : X → Y of schemes over Spec(A) inducing the identity on the closedfibre, i.e. such that the following diagram

X

X f−→ Y

Spec(A)

is commutative. Consider the category

A∗ = (noetherian local k-algebras with residue field k)

and its full subcategory

A = (artinian local k-algebras with residue field k)

One defines a covariant functor

DefX : A∗ → (sets)

byDefX (A) = local deformations of X over Spec(A)/(isomorphism)

This is the functor of local deformations of X ; its restriction to A is the functor ofinfinitesimal deformations of X . One may now ask whether DefX is representable,namely if there is a noetherian local k-algebra O and a local deformation

υ :X → X ↓ ↓ p

Spec(k) ⊂ Spec(O)

which is universal, i.e. such that any other local deformation ξ is obtained by pullingback υ under a unique Spec(A)→ Spec(O).

The approach of Grothendieck to this problem was to formalize the method ofKodaira and Spencer, which consists in a formal construction followed by a proof ofconvergence. In the search for the universal deformation υ the formal construction

6 Introduction

corresponds to the construction of the sequence of its restrictions to the truncationsSpec(O/mn+1

O ):

un :X → X n↓ ↓

Spec(k) → Spec(O/mn+1O )

n ≥ 0

These are infinitesimal deformations of X because the rings O/mn+1O are in A. The

sequence u = un can be considered as a formal approximation of υ. It is a specialcase of a formal deformation: more precisely, a formal deformation of X is given bya complete local k-algebra R with residue field k and by a sequence of infinitesimaldeformations

ξn :X → Xn

↓ ↓Spec(k) → Spec(R/mn+1

R )n ≥ 0

such that ξn → ξn−1 under the truncation R/mn+1R → R/mn

R . In our case R = O.The goal of the formal step in deformation theory is the construction of u for a givenX , i.e. of a formal deformation having a suitable universal property which is inheritedfrom the corresponding property of υ, and which we do not need to specify now.

Observe that in trying to perform the formal step we will at best succeed in de-scribing O and not O. Since a formal deformation consists of infinitesimal deforma-tions, for the construction of u we will only need to work with the covariant functor

DefX : A→ (sets)

A covariant functor F : A → (sets) is called a functor of Artin rings. To everycomplete local k-algebra R we can associate a functor of Artin rings h R by

h R(A) = HomA(R, A)

A functor of this form is called prorepresentable. By categorical general nonsenseone shows that a formal deformation ξ defines a morphism of functors (a naturaltransformation) h R → DefX and that this morphism is an isomorphism preciselywhen ξ is universal. Therefore we see that the search for u is a problem of prorep-resentability of DefX . More generally, to every local deformation problem there cor-responds a functor of Artin rings F analogous to DefX ; the task of constructinga formal universal deformation for the given problem consists in showing that Fis prorepresentable, producing the ring R prorepresenting F and the formal uni-versal deformation defining the isomorphism h R → F . This is the scheme of ap-proach to the formal part of every local deformation problem as it was outlinedby Grothendieck. What one needs is to find criteria for the prorepresentability ofa functor of Artin rings; we will also need to consider properties weaker than prorep-resentability (semiuniversality) satisfied by more general classes of functors comingfrom interesting deformation theoretic problems. Necessary and sufficient conditionsof prorepresentability and of semiuniversality are given by Schlessinger’s theorem.

After having solved the problem of existence of a formal universal (or semiuni-versal) deformation by means of necessary and sufficient conditions for its existence

Introduction 7

one still has to decide whether O and υ exist and to find them. To pass from O toO is the analogue of the convergence step in the Kodaira–Spencer theory, and it isa very difficult problem, the algebraization problem. Under reasonably general as-sumptions one shows that there exists a deformation υ over an algebraic local ring(i.e. the henselization of a local k-algebra essentially of finite type) which does notquite represent the functor DefX but at least has a universal (or semiuniversal) associ-ated formal deformation. The further property of representing DefX is not in generalsatisfied by (O, υ), being related to the existence of nontrivial automorphisms of X .This part of the theory is largely due to the work of M. Artin, and based on the notionsof effectivity of a formal deformation and of local finite presentation of a functor, al-ready introduced by Grothendieck. The main technical tool is Artin’s approximationtheorem.

1

Infinitesimal deformations

The purpose of this chapter is to introduce the reader to deformation theory in anelementary and direct fashion. We will be especially interested in first-order defor-mations and obstructions and in giving them appropriate interpretation mostly byelementary Cech cohomology computations. We will start by introducing some al-gebraic tools needed. For other notions used we refer the reader to the appendices.

1.1 Extensions

1.1.1 Generalities

Let A → R be a ring homomorphism. An A-extension of R (or of R by I ) is anexact sequence

(R′, ϕ) : 0→ I → R′ ϕ−→ R→ 0

where R′ is an A-algebra and ϕ is a homomorphism of A-algebras whose kernel Iis an ideal of R′ satisfying I 2 = (0). This condition implies that I has a structure ofR-module. (R′, ϕ) is also called an extension of A-algebras.

If (R′, ϕ) and (R′′, ψ) are A-extensions of R by I , an A-homomorphismξ : R′ → R′′ is called an isomorphism of extensions if the following diagramcommutes:

0→ I → R′ → R → 0‖ ↓ ξ ‖

0→ I → R′′ → R → 0

Such a ξ is necessarily an isomorphism of A-algebras. More generally, given A-extensions (R′, ϕ) and (R′′, ψ) of R, not necessarily having the same kernel, a homo-morphism of A-algebras r : R′ → R′′ such that ψr = ϕ is called a homomorphismof extensions.

The following lemma is immediate.

Lemma 1.1.1. Let (R′, ϕ) be an extension as above. Given an A-algebra B andtwo A-homomorphisms f1, f2 : B → R′ such that ϕ f1 = ϕ f2 the induced map

10 1 Infinitesimal deformations

f2 − f1 : B → I is an A-derivation. In particular, given two homomorphisms ofextensions

r1, r2 : (R′, ϕ)→ (R′′, ψ)

the induced map r2 − r1 : R′ → ker(ψ) is an A-derivation.

The A-extension (R′, ϕ) is called trivial if it has a section, that is, if there existsa homomorphism of A-algebras σ : R → R′ such that ϕσ = 1R . We also say that(R′, ϕ) splits, and we call σ a splitting.

Given an R-module I , a trivial A-extension of R by I can be constructed byconsidering the A-algebra R⊕I whose underlying A-module is R⊕ I and with mul-tiplication defined by:

(r, i)(s, j) = (rs, r j + si)

The first projectionp : R⊕I → R

defines an A-extension of R by I which is trivial: a section q is given byq(r) = (r, 0).

The sections of p can be identified with the A-derivations d : R → I . Indeed, ifwe have a section σ : R→ R⊕I with σ(r) = (r, d(r)) then for all r, r ′ ∈ R:

σ(rr ′) = (rr ′, d(rr ′)) = σ(r)σ (r ′) = (r, d(r))(r ′, d(r ′)) = (rr ′, rd(r ′)+ r ′d(r))

and if a ∈ A then:

σ(ar) = (ar, d(ar)) = aσ(r) = a(r, d(r)) = (ar, ad(r))

hence d : R → I is an A-derivation. Conversely, every A-derivation d : R → Idefines a section σd : R→ R⊕I by σd(r) = (r, d(r)).

Every trivial A-extension (R′, ϕ) of R by I is isomorphic to (R⊕I, p). Ifσ : R→ R′ is a section an isomorphism ξ : R⊕I → R′ is given by:

ξ((r, i)) = σ(r)+ i

and its inverse isξ−1(r ′) = (ϕ(r ′), r ′ − σϕ(r ′))

An A-extension (P, f ) of R will be called versal if for every other A-extension(R′, ϕ) of R there is a homomorphism of extensions r : (P, f ) → (R′, ϕ). IfR = P/I where P is a polynomial algebra over A then

0→ I/I 2 → P/I 2 → R→ 0

is a versal A-extension of R. Therefore, since such a P always exists, we see thatevery A-algebra R has a versal extension.

1.1 Extensions 11

Examples 1.1.2. (i) Every A-extension of A is trivial because by definition it has asection. Therefore it is of the form A⊕V for an A-module V . In particular, if t isan indeterminate the A-extension A[t]/(t2) of A is trivial, and is denoted by A[ε](where ε = t mod (t2) satisfies ε2 = 0). The corresponding exact sequence is:

0→ (ε)→ A[ε] → A→ 0

A[ε] is called the algebra of dual numbers over A.

(ii) Assume that K is a field. If R is a local K -algebra with residue field K aK -extension of R by K is called a small extension of R. Let

(R′, f ) : 0→ (t)→ R′ f−→ R→ 0

be a small K -extension; in other words t ∈ m R′ is annihilated by m R′ so that (t) is aK -vector space of dimension one.

(R′, f ) is trivial if and only if the surjective linear map induced by f :

f1 : m R′

m2R′→ m R

m2R

is not bijective.Indeed for the trivial K -extension

0→ (t)→ R⊕(t)→ R→ 0

we have t ∈ m R⊕(t) \ m2R⊕(t)

, hence the map f1 is not injective because f1(t) = 0.

Conversely, if f1 is not injective then f1(t) = 0; choose a vector subspaceU ⊂ m R′/m2

R′ such that m R′/m2R′ = U ⊕ (t) and let V ⊂ R′ be the subring

generated by U . Then V is a subring mapped isomorphically onto R by f . The in-verse of f|V is a section of f , therefore (R′, f ) is trivial.

For example, it follows from this criterion that the extension of K -algebras

0→ (tn)

(tn+1)→ K [t]

(tn+1)→ K [t]

(tn)→ 0

n ≥ 2, is nontrivial.

(iii) Let K be a field. The K -algebra

K [ε, ε′] := K [t, t ′]/(t, t ′)2

is a K -extension of K [ε] by K in two different ways. The first

0→ (ε′)→ K [ε, ε′] pε−→ K [ε] → 0


is a trivial extension, isomorphic to p∗((K [ε′], p′)):

0→ (ε′) → K [ε] ×K K [ε′] → K [ε] → 0‖ ↓ ↓ p

0→ (ε′) → K [ε′] p′−→ K → 0

The isomorphism is given by

K [ε, ε′] −→ K [ε] ×K K [ε′]a + bε + b′ε′ −→ (a + bε, a + b′ε′)

The second way is by “sum”:

0→ (ε − ε′) → K [ε, ε′] +−→ K [ε] → 0

a + bε + b′ε′ → a + (b + b′)ε

We leave it as an exercise to show that (K [ε, ε′],+) is isomorphic to (K [ε, ε′], pε).

1.1.2 The module ExA(R, I)

Let A→ R be a ring homomorphism. In this subsection we will show how to give anR-module structure to the set of isomorphism classes of extensions of an A-algebraR by a module I , closely following the analogous theory of extensions in an abeliancategory as explained, for example, in Chapter III of [123].

Let (R′, ϕ) be an A-extension of R by I and f : S → R a homomorphism ofA-algebras. We can define an A-extension f ∗(R′, ϕ) of S by I , called the pullbackof (R′, ϕ) by f , in the following way:

f ∗(R′, ϕ) : 0→ I → R′ ×R S → S → 0‖ ↓ ↓ f

(R′, ϕ) : 0→ I → R′ → R → 0

where R′ ×R S denotes the fibred product defined in the usual way.Let λ : I → J be a homomorphism of R-modules. The pushout of (R′, ϕ) by λ is

the A-extension λ∗(R′, ϕ) of R by J defined by the following commutative diagram:

0→ Iα−→ R′ ϕ−→ R → 0

↓ λ ↓ ‖0→ J → R′

∐I J → R → 0

where

R′∐

I

J = R′⊕J

(−α(i), λ(i)), i ∈ I

Definition 1.1.3. For every A-algebra R and for every R-module I we defineExA(R, I ) to be the set of isomorphism classes of A-extensions of R by I . If (R′, ϕ)is such an extension we will denote by [R′, ϕ] ∈ ExA(R, I ) its class.

1.1 Extensions 13

Using the operations of pullback and pushout it is possible to define an R-modulestructure on ExA(R, I ).

If r ∈ R and [R′, ϕ] ∈ ExA(R, I ) we define

r [R′, ϕ] = [r∗(R′, ϕ)]where r : I → I is the multiplication by r .

Given [R′, ϕ], [R′′, ψ] ∈ ExA(R, I ), to define their sum we use the followingdiagram:

0 0 0 ↓ ↓

I ⊕ I I = I ↓ ↓

0 → I → R′ ×R R′′ → R′ → 0‖ ↓ ↓

0 → I → R′′ → R → 0↓ ↓ 0 0 0

which defines an A-extension:

(R′ ×R R′′, ζ ) : 0→ I ⊕ I → R′ ×R R′′ ζ−→ R→ 0

We define[R′, ϕ] + [R′′, ψ] := [δ∗(R′ ×R R′′, ζ )]

where δ : I ⊕ I → I is the “sum homomorphism”: δ(i ⊕ j) = i + j .

Proposition 1.1.4. Let A → R be a ring homomorphism and I an R-module.With the operations defined above ExA(R, I ) is an R-module whose zero elementis [R⊕I, p]. This construction defines a covariant functor:

(R-modules) −→ (R-modules)I −→ ExA(R, I )

( f : I → J ) −→ ( f∗ : ExA(R, I )→ ExA(R, J ))

Proof. Straightforward. It is likewise straightforward to check that if f : R → S is a homomorphism of

A-algebras and I is an S-module, then the operation of pullback induces an applica-tion:

f ∗ : ExA(S, I )→ ExA(R, I )

which is a homomorphism of R-modules.We have the following useful result.

Proposition 1.1.5. Let A be a ring, f : S→ R a homomorphism of A-algebras andlet I be an R-module. Then there is an exact sequence of R-modules:

0→ DerS(R, I )→ DerA(R, I )→ DerA(S, I )⊗S Rρ−→

→ ExS(R, I )v−→ ExA(R, I )

f ∗−→ ExA(S, I )⊗S R


Proof. v is the obvious application sending an S-extension to itself considered as anA-extension. An A-extension

0→ I → R′ ϕ−→ R→ 0

is also an S-extension if and only if there exists f ′ : S→ R such that the triangle

R′ → R ↑

S

commutes, and this is equivalent to saying that f ∗(R′, ϕ) is trivial. This proves theexactness in ExA(R, I ).

The homomorphism ρ is defined by letting ρ(d) = (R⊕I, p) where the structureof S-algebra on R⊕I is given by the homomorphism

s → ( f (s), d(s))

Clearly, vρ = 0. On the other hand, for

(R′, ϕ) : 0→ I → R′ ϕ−→ R → 0↑S

to define an element of ker(v) there must exist an isomorphism of A-algebrasR′ → R⊕I inducing the identity on I and on R. Hence the compositionS→ R′ → R⊕I is of the form

s → ( f (s), d(s))

for some d ∈ DerA(S, I ): therefore the sequence is exact at ExS(R, I ). To prove theexactness at DerA(S, I ) note that ρ(d) = 0 if and only if p : R⊕I → R has a sectionas a homomorphism of S-algebras, if and only if there exists an A-derivation R→ Iwhose restriction to S is d: this proves the assertion. The exactness at DerS(R, I )and DerA(R, I ) is straightforward. Definition 1.1.6. The R-module ExA(R, R) is called the first cotangent module of Rover A and it is denoted by T 1

R/A. In the case A = k we will write T 1R instead of

T 1R/k.

Proposition 1.1.7. Let A → B be an e.f.t. ring homomorphism and let B = P/Jwhere P is a smooth A-algebra. Then for every B-module M we have an exactsequence:

DerA(P, M)→ HomB(J/J 2, M)→ ExA(B, M)→ 0 (1.1)

If A→ B is a smooth homomorphism then ExA(B, M) = 0 for every B-module M.

Proof. We have a natural surjective homomorphism

HomB(J/J 2, M) → ExA(B, M)λ → λ∗(η)

1.1 Extensions 15

whereη : 0→ J/J 2 → P/J 2 → B → 0

The surjectivity follows from the fact that η is versal. The extension λ∗(η) is trivialif and only if we have a commutative diagram

0→ J/J 2 → P/J 2 → B → 0↓ λ ↓ ‖

0→ M → B⊕M → B → 0

if and only if λ extends to an A-derivation D : P/J 2 → M , equivalently to anA-derivation D : P → M . The last assertion is immediate (see Theorem C.9). Corollary 1.1.8. If A→ B is an e.f.t. ring homomorphism and M is a finitely gener-ated B-module then ExA(B, M) is a finitely generated B-module. In particular T 1

B/Ais a finitely generated B-module and we have an exact sequence:

0→ HomB(ΩB/A, M)→ HomB(ΩP/A ⊗P B, M)→

→ HomB(I/I 2, M)→ ExA(B, M)→ 0(1.2)

if B = P/J for a smooth A-algebra P and an ideal J ⊂ P.

Proof. It is a direct consequence of the exact sequence (1.1). 1.1.3 Extensions of schemes

Let X → S be a morphism of schemes. An extension of X/S is a closed immersionX ⊂ X ′, where X ′ is an S-scheme, defined by a sheaf of ideals I ⊂ OX ′ suchthat I2 = 0. It follows that I is, in a natural way, a sheaf of OX -modules, whichcoincides with the conormal sheaf of X ⊂ X ′. To give an extension X ⊂ X ′ of X/Sis equivalent to giving an exact sequence on X :

E : 0→ I → OX ′ϕ−→ OX → 0

where I is an OX -module, ϕ is a homomorphism of OS-algebras and I2 = 0 inOX ′ ; we call E an extension of X/S by I or with kernel I. Two such extensions OX ′and OX ′′ are called isomorphic if there is an OS-homomorphism α : OX ′ → OX ′′inducing the identity on both I and OX . It follows that α must necessarily be anS-isomorphism.

We denote by Ex(X/S, I) the set of isomorphism classes of extensions of X/Swith kernel I. In the case where Spec(B) → Spec(A) is a morphism of affineschemes and I = M we have an obvious identification:

ExA(B, M) = Ex(X/S, I)

If S = Spec(A) is affine we will sometimes write ExA(X, I) instead ofEx(X/Spec(A), I). Exactly as in the affine case one proves that Ex(X/S, I) is aΓ (X,OX )-module with identity element the class of the trivial extension:

0→ I → OX ⊕I → OX → 0


where OX ⊕I is defined as in the affine case (see Section 1.1). The correspondence

I → Ex(X/S, I)

defines a covariant functor from OX -modules to Γ (X,OX )-modules.In deformation theory the case I = OX is the most important one, being related

to first-order deformations. If, more generally, I is a locally free sheaf we get thenotions of ribbon, carpet etc. (see [17]).

Using the fact that the exact sequence (1.2) of page 15 localizes, it is immediateto check that the cotangent module localizes. More specifically, it is straightforwardto show that given a morphism of finite type of schemes f : X → S one can define aquasi-coherent sheaf T 1

X/S on X with the following properties. If U = Spec(A) is an

affine open subset of S and V = Spec(B) is an affine open subset of f −1(U ), then

Γ (V, T 1X/S) = T 1

B/A

It follows from the properties of the cotangent modules that T 1X/S is coherent. T 1

X/S is

called the first cotangent sheaf of X/S. We will write T 1X if S = Spec(k). For future

reference it will be convenient to state the following:

Proposition 1.1.9. (i) If X is an algebraic scheme then T 1X is supported on the sin-

gular locus of X. More generally, if X → S is a morphism of finite type ofalgebraic schemes then T 1

X/S is supported on the locus where X is not smoothover S.

(ii) If we have a closed embedding X ⊂ Y with Y nonsingular, then we have an exactsequence of coherent sheaves on X:

0→ TX → TY |X → NX/Y → T 1X → 0 (1.3)

so that, letting N ′X/Y = ker[NX/Y → T 1X ], we have the short exact sequence

0→ TX → TY |X → N ′X/Y → 0 (1.4)

N ′X/Y is called the equisingular normal sheaf of X in Y .(iii) For every scheme S and morphism of S-schemes f : X → Y we have an exact

sequence of sheaves

0→ TX/Y → TX/S → Hom( f ∗Ω1Y/S,OX )→ T 1

X/Y → T 1X/S → f ∗T 1

Y/S(1.5)

(iv) When S = Spec(k) and f is a closed embedding of algebraic schemes, with Ynonsingular, we have TX/Y = 0 and NX/Y = T 1

X/Y . Moreover, (1.3) is a specialcase of (1.5) in this case.

Proof. (i) Use Proposition 1.1.7.(ii) (1.3) globalizes the exact sequence (1.2).(iii) (1.5) globalizes the exact sequence of Proposition 1.1.5.(iv) Follows from (1.2) and 1.1.5.

1.1 Extensions 17

Note that the first half of the exact sequence (1.5) is the dual of the cotangentsequence of f . For more about (1.5) see also (3.43), page 162. The following is abasic result:

Theorem 1.1.10. Let X → S be a morphism of finite type of algebraic schemes andI a coherent locally free sheaf on X. Assume that X is reduced and S-smooth on adense open subset. Then there is a canonical identification

Ex(X/S, I) = Ext1OX(Ω1

X/S, I)

which to the isomorphism class of an extension of X/S:

E : 0→ I → OX ′ → OX → 0

associates the isomorphism class of the relative conormal sequence of X ⊂ X ′:

cE : 0→ I δ→ (Ω1X ′/S)|X → Ω1

X/S → 0

(which is exact also on the left).

Proof. Suppose given an extension E . Since I is locally free and X is reduced inorder to show that cE is exact on the left it suffices to prove that ker(δ) is torsion,equivalently that cE is exact at every general closed point x of any irreducible com-ponent of X . Since X is smooth over S at x it follows from 1.1.7 that there is anaffine open neighbourhood U of x such that E|U is trivial. From Theorem B.3 we de-duce that the relative conormal sequence of E|U is split exact. Since it coincides withthe restriction of cX ′ to U we see that δ|U is injective; this shows that ker(δ) is tor-sion and cE is exact. Since isomorphic extensions have isomorphic relative cotangentsequences we have a well-defined map

c− : Ex(X/S, I)→ Ext1OX(Ω1

X/S, I)

Now let

η : 0→ I → A p→ Ω1X/S → 0

define an element of Ext1OX(Ω1

X/S, I). Letting d : OX → Ω1X/S be the canonical

derivation, consider the sheaf of OS algebras O = A×Ω1X/S

OX : over an open subset

U ⊂ X we have Γ (U,O) = (a, f ) : p(a) = d( f ) and the multiplication rule is

(a, f )(a′, f ′) = ( f a′ + f ′a, f f ′)

Then we have an exact commutative diagram

0→ I → O → OX → 0‖ ↓ d ↓ d

0→ I → A → Ω1X/S → 0


where one immediately checks that the projection d is an OS-derivation and thereforeit must factor as

O→ Ω1O/OS

⊗O OX → Aand we have an exact commutative diagram

0→ I → O → OX → 0‖ ↓ ↓ dI → Ω1

O/OS⊗O OX → Ω1

X/S → 0‖ ↓ ‖

0→ I → A → Ω1X/S → 0

(1.6)

which implies Ω1O/OS

⊗O OX ∼= A. Therefore, letting eη be the extension given bythe first row of (1.6), we see that ceη = η. Similarly, one shows that ecE = E for any[E] ∈ Ex(X/S, I). Therefore c− and e− are inverse of each other and the conclusionfollows. Corollary 1.1.11. Let X → S be a morphism of finite type of algebraic schemes,smooth on a dense open subset of X. Assume X reduced. Then there is a canonicalisomorphism of coherent sheaves on X:

T 1X/S∼= Ext1

OX(Ω1

X/S,OX )

In particular, if X is a reduced algebraic scheme then

T 1X∼= Ext1

OX(Ω1

X ,OX )

and if, moreover, X = Spec(B0) then

T 1B0∼= Ext1k(ΩB0/k, B0)

Proof. An immediate consequence of the above theorem. A closer analysis of the proof of Theorem 1.1.10 shows that without assuming X

reduced we only have an inclusion

Ext1OX

(Ω1X ,OX ) ⊂ T 1

X

NOTES

1. An alternative approach to the topics treated in this section can be obtained by means ofthe so-called “truncated cotangent complex”, first introduced in [76]. A more general versionof the cotangent complex was introduced in [120] and later incorporated in general theoriesof Andre [4], Quillen [146] and Tate [181]. Here we will just recall the main facts aboutthe truncated cotangent complex, without entering into any details, with the only purpose ofshowing the relation to the notions introduced in this section. For details we refer to [94].

Let A be a ring and R an A-algebra. To every A-extension

η : 0→ I → R′ ϕ−→ R→ 0

1.1 Extensions 19

we associate a complex c•(η) of R-modules (also denoted c•(ϕ)) defined as follows:

c0(η) = ΩR′/A ⊗R′ Rc1(η) = Icn(η) = (0) n = 0, 1

d1 : c1(η)→ c0(η) is the map x → d(x)⊗ 1. In other words c•(η) consists of the first mapin the conormal sequence of ϕ. If r : (R′, ϕ)→ (R′′, ψ) is a homomorphism of A-extensionsthen r induces a homomorphism of complexes

c•(r) : c•(ϕ)→ c•(ψ)

in an obvious way. The following is easy to establish:

• Let r1, r2 : (R′, ϕ) → (R′′, ψ) be two homomorphisms of A-extensions of R. Thenc•(r1) and c•(r2) are homotopic. As an immediate consequence we have:

• if (E, p) and (F, q) are two versal A-extensions of R then the complexes c•(p) and c•(q)are homotopically equivalent.

Definition 1.1.12. Let A be a ring, R an A-algebra and (E, p) a versal A-extension of R. Thehomotopy class of the complex c•(p) is called the (truncated) cotangent complex of R over Aand denoted by T (R/A).

If R = P/I for a polynomial A-algebra P the A-extension

0→ I/I 2 → P/I 2 → R→ 0 (1.7)

is versal and therefore the complex

I/I 2 δ−→ ΩP/A ⊗P R (1.8)

where δ is the map appearing in the conormal sequence of A→ P → R, represents T (R/A).If R is e.f.t. and P is replaced by a smooth A-algebra then (1.7) is again a versal A-extensionand (1.8) again represents T (R/A). From the fact that every A-algebra R can be obtained asthe quotient of a polynomial A-algebra P it follows that the cotangent complex T (R/A) existsfor every A-algebra R.

The cotangent complex can be used to define “upper and lower cotangent functors”, asfollows.

Definition 1.1.13. Let A → B be a ring homomorphism, M a B-module, and let

c• = C1d1−→ C0 represent T (B/A). Then for i = 0, 1 the lower cotangent module of

B over A relative to M is:

Ti (B/A, M) = Hi (c• ⊗B M)

and the upper cotangent module of B over A relative to M is:

T i (B/A, M) = Hi (Hom(c•, M))

Because of the definition of cotangent complex it follows that the cotangent modulesare independent on the choice of the complex c• representing T (B/A), but only depend on


A, B, M . Moreover, one immediately checks that the definition is functorial in M and there-fore we have covariant functors:

Ti (B/A,−) : (B-modules)→ (B-modules) i = 0, 1

andT i (B/A,−) : (B-modules)→ (B-modules) i = 0, 1

One immediately sees that for i = 0 the cotangent functors are:

T0(B/A, M) = ΩB/A ⊗B M

andT 0(B/A, M) = DerA(B, M)

From the extension (1.7) we obtain the exact sequences:

0→ T1(B/A, M)→ I/I 2 ⊗B M → ΩP/A ⊗P M → ΩB/A ⊗B M → 0

and0→ HomB(ΩB/A, M)→ HomB(ΩP/A ⊗P B, M)→

HomB(I/I 2, M)→ T 1(B/A, M)→ 0

If B is e.f.t. then in (1.7) P can be chosen to be a smooth A-algebra; in this case it follows thatTi (B/A, M) and T i (B/A, M) are finitely generated B-modules if M is finitely generated.Moreover, recalling Corollary 1.1.8, we see that we have an identification

T 1(B/A, M) = ExA(B, M)

2. The topics of this section originate from [76]. See also [1], Ch. 0I V , § 18. The proof ofTheorem 1.1.10 has been taken from [17]; see also [66].

1.2 Locally trivial deformations

1.2.1 Generalities on deformations

Let X be an algebraic scheme. A cartesian diagram of morphisms of schemes

η :X → X↓ ↓ π

Spec(k)s−→ S

where π is flat and surjective, and S is connected, is called a family of deforma-tions, or simply a deformation, of X parametrized by S, or over S; we call S and Xrespectively the parameter scheme and the total scheme of the deformation. If S is al-gebraic, for each k-rational point t ∈ S the scheme-theoretic fibre X (t) is also calleda deformation of X . When S = Spec(A) with A in ob(A∗) and s ∈ S is the closedpoint we have a local family of deformations (shortly a local deformation) of X overA. The deformation η will be also denoted by (S, η) or (A, η) when S = Spec(A).

1.2 Locally trivial deformations 21

The local deformation (A, η) is infinitesimal (resp. first-order) if A ∈ ob(A) (resp.A = k[ε]). Given another deformation

ξ :X → Y↓ ↓

Spec(k) → S

of X over S, an isomorphism of η with ξ is an S-isomorphism φ : X → Y inducingthe identity on X , i.e. such that the following diagram is commutative:

X

X φ−→ Y

S

By a pointed scheme we will mean a pair (S, s) where S is a scheme and s ∈ S. If Kis a field we call (S, s) a K -pointed scheme if K ∼= k(s).

Observe that for every X and for every k-pointed scheme (S, s) there exists atleast one family of deformation of X over S, namely the product family:

X → X × S↓ ↓

Spec(k)s−→ S

A deformation of X over S is called trivial if it is isomorphic to the product family.It will be also called a trivial family with fibre X . All fibres over k-rational pointsof a trivial deformation of X parametrized by an algebraic scheme are isomorphicto X . The converse is not true: there are deformations which are not trivial but haveisomorphic fibres over all the k-rational points (see Example 1.2.2(ii) below). Thescheme X is called rigid if every infinitesimal deformation of X over A is trivial forevery A in ob(A).

Given a deformation η of X over S as above and a morphism (S′, s′) → (S, s)of k-pointed schemes there is induced a commutative diagram by base change

X → X ×S S′↓ ↓

Spec(k) → S′

which is clearly a deformation of X over S′. This operation is functorial, in the sensethat it commutes with composition of morphisms and the identity morphism does notchange η. Moreover, it carries isomorphic deformations to isomorphic ones.

An infinitesimal deformation η of X is called locally trivial if every point x ∈ Xhas an open neighbourhood Ux ⊂ X such that

Ux → X|Ux

↓ ↓Spec(k) → S

is a trivial deformation of Ux .


Remark 1.2.1. Let

η :X

j−→ X↓ ↓ π

Spec(k)s−→ S

be a family of deformations of an algebraic scheme X parametrized by an algebraicscheme S and let Z ⊂ X be a proper closed subset. Then

X\Z j−→ X \ j (Z)↓ ↓ π ′

Spec(k)s−→ S

is a family of deformations of X\Z having the same fibres as π over t ∈ S for t = s:thus such fibres are deformations both of X and of X\Z . This shows that the defi-nition of family of deformations given above is somewhat ambiguous unless we as-sume that π is projective or that the deformation is infinitesimal. In what follows wewill restrict to the consideration of deformations of projective schemes and/or of in-finitesimal deformations when discussing the general theory, so that such ambiguitywill be removed; only occasionally will we consider non-infinitesimal deformationsof affine schemes.

Examples 1.2.2. (i) The quadric Q ⊂ A3 of equation xy − t = 0 defines, via theprojection

A3 → A1

(x, y, t) → t

a flat family Q → A1 whose fibres are affine conics. This family is not trivial sincethe fibre Q(0) is singular, hence not isomorphic to the fibres Q(t), t = 0, which arenonsingular.

(ii) Consider, for a given integer m ≥ 0, the rational ruled surface

Fm = IP(OIP1(m)⊕OIP1)

The structural morphism π : Fm → IP1 defines a flat family whose fibres are allisomorphic to IP1; but if m > 0 then π is not a trivial family because Fm ∼= F0 =IP1 × IP1 (see Example B.11(iii)).

(iii) Let 0 ≤ n < m be two distinct nonnegative integers having the same parityand let k= 1

2 (m − n). Consider two copies of A2 × IP1 given asProj(k[t, z, ξ0, ξ1]) =: W and Proj(k[t, z′, ξ ′0, ξ ′1]) =: W ′ (here the rings are gradedwith respect to the variables ξi and ξ ′i ). Letting ξ = ξ1/ξ0 and ξ ′ = ξ ′1/ξ ′0 considerthe open subsets

Spec(k[t, z, ξ ]) ⊂ W, Spec(k[t, z′, ξ ′]) ⊂ W ′

and glue them together along the open subsets

Spec(k[t, z, z−1, ξ ]) ⊂ Spec(k[t, z, ξ ])


andSpec(k[t, z′, z′−1, ξ ′]) ⊂ Spec(k[t, z′, ξ ′])

according to the following rules:

z′ = z−1, ξ ′ = zmξ + t zk (1.9)

This induces a gluing of W and W ′ along

Proj(k[t, z, z−1, ξ0, ξ1]) and Proj(k[t, z′, z′−1, ξ ′0, ξ ′1])Call the resulting scheme W and f :W → A1 = Spec(k[t]) the morphism inducedby the projections. Then f is a flat morphism because it is locally a projection;moreover,

W(0) ∼= Fm

Let W = f −1(A1\0) and f :W → A1\0 the restriction of f .In k[t, t−1, z, ξ ] define

ζ = zkξ − t

tξ

and in k[t, t−1, z′, ξ ′]ζ ′ = ξ ′

t z′m−kξ ′ + t2

It is straightforward to verify that the gluing (1.9) induces the relation

ζ ′ = znζ

This means that we have an isomorphism

W ∼= Fn × (A1\0)compatible with the projections to A1\0. Therefore the family f is trivial, in par-ticular all its fibres are isomorphic to Fn , but the family f is not trivial becauseW(0) ∼= Fm .

(iv) Let f : X → Y be a surjective morphism of algebraic schemes, with Xintegral and Y an irreducible and nonsingular curve. Then f is flat. This is a specialcase of Prop. III.9.7 of [84]. Therefore f defines a family of deformations of any ofits closed fibres.

1.2.2 Infinitesimal deformations of nonsingular affine schemes

We will start by considering infinitesimal deformations of affine schemes. We needthe following:

Lemma 1.2.3. Let Z0 be a closed subscheme of a scheme Z, defined by a sheaf ofnilpotent ideals N ⊂ OZ . If Z0 is affine then Z is affine as well.


Proof. Let r ≥ 2 be the smallest integer such that Nr = (0). Since we have a chainof inclusions

Z ⊃ V (Nr−1) ⊃ V (Nr−2) ⊃ · · · ⊃ V (N ) = Z0

it suffices to prove the assertion in the case r = 2. In this case N is a coherentOZ0 -module, and therefore

H1(Z , N ) = H1(Z0, N ) = 0

Let R0 be the k-algebra such that Z0 = Spec(R0). We have the exact sequence:

0→ H0(Z , N )→ H0(Z ,OZ )→ R0 → 0

Put R = H0(Z ,OZ ) and let Z ′ = Spec(R). We have a commutative diagram:

Zθ−→ Z ′Z0

The sheaf homomorphism θ−1OZ ′ → OZ is clearly injective and θ is a homeomor-phism. It will therefore suffice to prove that θ−1OZ ′ → OZ is surjective.

Let z ∈ Z and f ∈ Γ (U,OZ ) for some affine open neighbourhood U of z. Letf0 = f|U∩Z0 . It is possible to find ϕ0, ψ0 ∈ R0 such that f0 = ϕ0

ψ0, ψ0(z) = 0

and ψ0 = 0 on Z0\U , because Z0 is affine. Let ψ ∈ R be such that ψ|Z0 = ψ0 (itexists by the surjectivity of R → R0). Then ψ(z) = 0 and ψ = 0 on Z\U . Thereexists n 0 such that ψn f =: g ∈ R (it suffices to cover Z with affines). Thenf = g

ψn ∈ θ−1OZ ′ . Let B0 be a k-algebra, and let X0 = Spec(B0). Consider an infinitesimal defor-

mation of X0 parametrized by Spec(A), where A is in ob(A). By definition this is acartesian diagram

X0 → X↓ ↓

Spec(k) → Spec(A)

where X is a scheme flat over Spec(A). By Lemma 1.2.3 X is necessarily affine.Therefore, equivalently, we can talk about an infinitesimal deformation of B0 over Aas a cartesian diagram of k-algebras:

B → B0↑ ↑A → k

(1.10)

with A → B flat. Note that to give this diagram is the same as to give A → Bflat and a k-isomorphism B ⊗A k → B0. We will sometimes abbreviate by callingA→ B the deformation.

Given another deformation A → B ′ of B0 over A, an isomorphism of deforma-tions of A→ B to A→ B ′ is a homomorphism ϕ : B → B ′ of A-algebras inducinga commutative diagram:


B0

Bϕ−→ B ′

A

It follows from Lemma A.4 that such a ϕ is an isomorphism.An infinitesimal deformation of B0 over A is trivial if it is isomorphic to the

product deformationB0 ⊗k A → B0↑ ↑A → k

The k-algebra B0 is called rigid if Spec(B0) is rigid.

Theorem 1.2.4. Every smooth k-algebra is rigid. In particular, every affine nonsin-gular algebraic variety is rigid.

Proof. Suppose k → B0 is smooth, and suppose given a first-order deformation ofB0:

η0 :B → B0↑ f ↑

k[ε] → k

Consider the commutative diagram:

B → B0↑ f ↑

k[ε] → B0[ε]where B0[ε] = B0⊗k[ε]. Since f is smooth (because flat with smooth fibre, see [84],ch. III, Th. 10.2) and the right vertical morphism is a k[ε]-extension, by Theorem C.9there exists a k[ε]-homomorphism φ : B → B0[ε] making the diagram

B → B0↑ f ↑

k[ε] → B0[ε]commutative. Therefore φ is an isomorphism of deformations and η0 is trivial.

Consider more generally a deformation of B0

η :B → B0↑ f ↑A → k

parametrized by A in ob(A). To show that η is trivial we proceed by induction ond = dimk(A). The case d = 2 has been already proved; assume d ≥ 3 and let

0→ (t)→ A→ A′ → 0


be a small extension. Consider the commutative diagram:

B → B ⊗A A′ ∼= B0 ⊗k A′↑ f ↑A → B0 ⊗k A

f is smooth, the upper right isomorphism is by the inductive hypothesis, and the rightvertical homomorphism is an A-extension. By the smoothness of f and by TheoremC.9 we deduce the existence of an A-homomorphism B → B0 ⊗k A which is anisomorphism of deformations. Example 1.2.5. Let λ ∈ k and B0 = k[X, Y ]/(Y 2 − X (X − 1)(X − λ)). If λ = 0, 1then B0 is a smooth k-algebra, being the coordinate ring of a nonsingular plane cubiccurve. By Theorem 1.2.4, B0 is rigid. On the other hand, the elementary theory ofelliptic curves (see [84]) shows that the following flat family of affine curves

Speck[X, Y ]/(Y 2 − X (X − 1)(X − (λ+ t)))↓

Spec(k[t])is not trivial around the origin t = 0 so that it defines a nontrivial (non-infinitesimal)deformation of B0. This example shows that by studying infinitesimal deformationsof affine schemes we are losing some information. In this specific case we will seethat this information is recovered by considering the infinitesimal deformations ofthe projective closure of Spec(B0) (see Corollary 2.6.6, page 94).

1.2.3 Extending automorphisms of deformations

In deformation theory it is very important to have a good control of automorphismsof deformations and of their extendability properties. We will now begin to introducesuch matters and to recall some terminology. In Section 2.6 we will consider theseproblems again in general, and we will relate them with the property of “prorepre-sentability”. Let’s start with a basic lemma.

Lemma 1.2.6. Let B0 be a k-algebra, and

e : 0→ (t)→ A→ A→ 0

a small extension in A. Then there is a canonical isomorphism of groups:automorphisms of the trivial deformation B0 ⊗k A

inducing the identity on B0 ⊗k A

→ Derk(B0, B0)

In particular the group on the left is abelian.

Proof. Every automorphism θ : B0 ⊗k A → B0 ⊗k A belonging to the first groupmust be A-linear and induce the identity mod t . Therefore:

θ(x) = x + tdx


where d : B0 ⊗k A→ B0 is a A-derivation (Lemma 1.1.1). But

Der A(B0 ⊗k A, B0) = HomB0⊗k A(ΩB0⊗k A/ A, B0)

= HomB0(ΩB0/k, B0) = Derk(B0, B0)

By sending θ → d we define the correspondence of the statement. Since θ is deter-mined by d the correspondence is one to one. Clearly the identity corresponds to thezero derivation. If we compose two automorphisms:

B0 ⊗k Aθ−→ B0 ⊗k A

σ−→ B0 ⊗k A

where θ(x) = x + tdx , σ(x) = x + tδx , we obtain:

σ(θ(x)) = θ(x)+ tδ(θ(x)) = x + tdx + t (δx + tδ(dx)) = x + t (dx + δx)

therefore the correspondence is a group isomorphism. Recall the following well-known definition.

Definition 1.2.7. Let G be a group acting on a set T and let

π : G × T → T

be the map defining the action. T is called a homogeneous space under (the actionof) G if π is transitive, i.e. if

π(G × t) = T

for some (equivalently for any) t ∈ T (i.e. if there is only one orbit). The action iscalled free if for every point t ∈ T the stabilizer Gt = g ∈ G : gt = t is trivial,i.e. gt = t implies g = 1G for all t ∈ T . If the action is both transitive and free thenT is called a principal homogeneous space (or a torsor) under (the action of) G.

To an action π : G × T → T we can associate the map:

p : G × T → T × T

(g, t) → (gt, t)

The condition that the action is transitive (resp. free) is equivalent to p being sur-jective (resp. injective); therefore T is a torsor under G if and only if p is bijective.Note that π = pr1 p is determined by p.

More generally, suppose that we have a map of sets f : T → T ′. Then the mapp factors through T ×T ′ T ⊂ T × T if and only if the action π is compatible withf , i.e. if f (t) = f (gt) for all t ∈ T , g ∈ G. As before, the map

p : G × T → T ×T ′ T

is surjective (resp. injective) if and only if the action of G is transitive (resp. free) onall the non-empty fibres of f . In particular, p is bijective if and only if all the non-empty fibres of f are torsors under G. In this case one also says, according to [3],


p. 114, that T over T ′ is a formal principal homogeneous space (or a pseudo-torsor)under G.

Now we come back to deformations and we prove a generalization of Lemma1.2.6.

Lemma 1.2.8. Let B0 be a k-algebra,

e : 0→ (t)→ A→ A→ 0

a small extension in A, A → B a deformation of B0 and A → B = B ⊗ A A theinduced deformation of B0 over A. Let σ : B → B be an automorphism of thedeformation. Then:

(i) If

Autσ (B) :=

automorphisms τ : B → B such that τ ⊗ A A = σ= ∅

then there is a free and transitive action

Derk(B0, B0)× Autσ (B)→ Autσ (B)

defined by(d, τ ) → τ + td

(ii) If B0 is a smooth k-algebra then Autσ (B) = ∅ for any σ .

Proof. (i) Recall that we have a chain of natural identifications

Der A(B, B0) = HomB(ΩB/ A, B0) = HomB0(ΩB/ A ⊗ A k, B0)

= HomB0(ΩB0/k, B0) = Derk(B0, B0)

Therefore the action in the statement is well defined once we consider ad ∈ Derk(B0, B0) as an A-derivation of B into B0. Given any two elementsτ, η ∈ Autσ (B) we have by definition:

qτ = σq = qη

where q : B → B is the projection; hence by Lemma 1.1.1,

η − τ : B → t B0 = ker(q)

is an A-derivation which is 0 if and only if η = τ . This implies that the action is freeand transitive.

(ii) Since B0 is smooth the deformation A→ B is trivial (Theorem 1.2.4), so thatwe have an A-isomorphism B ∼= B0⊗k A; moreover, B0⊗k A is a smooth A-algebra(Proposition C.2(iii)) hence B is A-smooth. Let σ : B → B be any automorphismof the deformation and consider the diagram of A-algebras:


Bq→ B

σ→ B↑ qB

Since ker(q) = t B0 is a square-zero ideal, and since B is A-smooth, we deducethat there is σ : B → B such that qσ = σq. It is immediate to check that σ is anisomorphism and therefore σ ∈ Autσ (B).

In (ii) the condition that B0 is smooth cannot be removed. A simple example isgiven in 2.6.8(i). The extendability of automorphisms of deformations of not neces-sarily affine schemes will be studied in § 2.6.

1.2.4 First-order locally trivial deformations

We will now apply 1.2.6 to first-order deformations of any algebraic variety.

Proposition 1.2.9. Let X be an algebraic variety. There is a 1–1 correspondence:

κ :

isomorphism classes of first order

locally trivial deformations of X

→ H1(X, TX )

called the Kodaira–Spencer correspondence, where TX = Hom(Ω1X ,OX ) =

Derk(OX ,OX ), such that κ(ξ) = 0 if and only if ξ is the trivial deformation class.In particular if X is nonsingular then κ is a 1–1 correspondence

κ :

isomorphism classes of

first-order deformations of X

→ H1(X, TX )

Proof. Given a first-order locally trivial deformation

X → X↓ ↓

Spec(k) → Spec(k[ε])choose an affine open cover U = Ui i∈I of X such that X|Ui is trivial for all i . Foreach index i we therefore have an isomorphism of deformations:

θi : Ui × Spec(k[ε])→ X|Ui

by 1.2.4. Then for each i, j ∈ I

θi j := θ−1i θ j : Ui j × Spec(k[ε])→ Ui j × Spec(k[ε])

is an automorphism of the trivial deformation Ui j × Spec(k[ε]). By Lemma 1.2.6,θi j corresponds to a di j ∈ Γ (Ui j , TX ). Since on each Ui jk we have

θi jθ jkθ−1ik = 1Ui jk×Spec(k[ε]) (1.11)


it follows thatdi j + d jk − dik = 0

i.e. di j is a Cech 1-cocycle and therefore defines an element of H1(X, TX ). It iseasy to check that this element does not depend on the choice of the open cover U .If we have another deformation

X → X ′↓ ↓

Spec(k) → Spec(k[ε])and Φ : X → X ′ is an isomorphism of deformations then for each i ∈ I there isinduced an automorphism:

αi : Ui × Spec(k[ε]) θi−→ X|Ui

Φ|Ui−→ X ′|Ui

θ′−1i−→ Ui × Spec(k[ε])

and therefore a corresponding ai ∈ Γ (Ui , TX ). We have θ ′i αi = Φ|Ui θi and therefore

(θ ′i αi )−1(θ ′jα j ) = θ−1

i Φ−1|Ui j

Φ|Ui j θ j = θ−1i θ j

thusα−1

i θ ′i jα j = θi j

Equivalently:d ′i j + a j − ai = di j

namely, di j and d ′i j are cohomologous, and therefore define the same element of

H1(X, TX ).Conversely, given θ ∈ H1(X, TX ) we can represent it by a Cech 1-cocycle

di j ∈ Z1(U , TX ) with respect to some affine open cover U . To each di j we canassociate an automorphism θi j of the trivial deformation Ui j×Spec(k[ε]) by Lemma1.2.6. They satisfy the identities (1.11). We can therefore use these automorphismsto patch the schemes Ui ×Spec(k[ε]) by the well-known procedure (see [84], p. 69).We obtain a Spec(k[ε])-scheme X which, by construction, defines a locally trivialfirst-order deformation of X . The equivalence between κ(ξ) = 0 and the triviality ofξ is easily proved. The last assertion follows from the first one because all deforma-tions of a nonsingular variety are locally trivial by Theorem 1.2.4. Definition 1.2.10. For every locally trivial first-order deformation ξ of a variety Xthe cohomology class κ(ξ) ∈ H1(X, TX ) is called the Kodaira–Spencer class of ξ .

Let

ξ :X → X↓ ↓ f

Spec(k)s−→ S

(1.12)

be a family of deformations of a nonsingular variety X . By pulling back this familyby morphisms Spec(k[ε]) → S with image s and applying the Kodaira–Spencercorrespondence (Proposition 1.2.9) we define a linear map


κξ : TS,s → H1(X, TX )

also denoted by κ f,s or κX /S,s , which is called the Kodaira–Spencer map of thefamily ξ .

Examples 1.2.11. (i) Let m ≥ 1 and let π : Fm → IP1 be the structural morphismof the rational ruled surface Fm (see B.11(iii)). Then π is not a trivial family but hasa trivial restriction around each closed point s ∈ IP1, thus κπ,s = 0.

(ii) Consider an unramified covering π : X → S of degree n ≥ 2 where X andS are projective nonsingular and irreducible algebraic curves. All fibres of π overthe closed points consist of n distinct points, hence they are all isomorphic. More-over, each such fibre is rigid and unobstructed as an abstract variety. In particularthe Kodaira–Spencer map is zero at each closed point s ∈ S. On the other hand,π−1(U ) is irreducible for each open subset U ⊂ S and therefore the restrictionπU : π−1(U )→ U is a nontrivial family; this follows also from the fact that π doesnot have rational sections.

This example exhibits a phenomenon which is not detected by infinitesimal con-siderations and in some sense opposite to the one described in Example 1.2.5: wecan have a flat projective family of deformations, all of whose geometric fibres areisomorphic, but which is nevertheless nontrivial over every Zariski open subset of thebase. Note that this is different from what happens with the projections Fm → IP1,m ≥ 1 of Example (i), which are nontrivial but have trivial restriction to a Zariskiopen neighbourhood of every point of IP1. See Subsection 2.6.2 for more about this.

(iii) Let 0 ≤ n < m be integers having the same parity, and letk = 1

2 (m − n). Consider the smooth proper morphism f : W → A1 introducedin Example 1.2.2(iii), whose fibres are W(0) ∼= Fm , and W(t) ∼= Fn for t = 0.Recall that the family f is given as the gluing of two copies of A2 × IP1:

W = Proj(k[t, z, ξ0, ξ1]), W ′ = Proj(k[t, z′, ξ ′0, ξ ′1])along Proj(k[t, z, z−1, ξ0, ξ1]) and Proj(k[t, z′, z′−1, ξ ′0, ξ ′1]) according to the rules:

z′ = z−1, ξ ′ = zmξ + t zk

where ξ = ξ1/ξ0 and ξ ′ = ξ ′1/ξ ′0 are nonhomogeneous coordinates on the corre-sponding copies of IP1.

Let’s compute the local Kodaira–Spencer map κ f,0 of f at 0. The image κ f,0(ddt )

is the element of H1(Fm, TFm ) corresponding to the first-order deformation of Fm

obtained by gluing

W0 := Proj(k[ε, z, ξ0, ξ1]) W ′0 := Proj(k[ε, z′, ξ ′0, ξ ′1])along Proj(k[ε, z, z−1, ξ0, ξ1]) and Proj(k[ε, z′, z′−1, ξ ′0, ξ ′1]) according to the rules

z′ = z−1, ξ ′ = zmξ + εzk


By definition we have that κ f,0(ddt ) is the element of H1(U , TFm ), where

U = W0, W ′0, defined by the 1-cocycle corresponding to the vector field on W0∩W ′0zk ∂

∂ξ

According to Example B.11(iii) this element is nonzero; therefore κ f,0 is injective.

Similarly, we can consider a smooth proper family F : Y → Am−1 defined asfollows. Y is the gluing of

Y := Proj(k[t1, . . . , tm−1, z, ξ0, ξ1])and

Y ′ := Proj(k[t1, . . . , tm−1, z′, ξ ′0, ξ ′1])along Proj(k[t1, . . . , tm−1, z, z−1, ξ0, ξ1]) and Proj(k[t1, . . . , tm−1, z′, z′−1, ξ ′0, ξ ′1])according to the rules:

z′ = z−1, ξ ′ = zmξ +m−1∑j=1

t j zj

The morphism F is defined by the projections onto Spec(k[t1, . . . , tm−1]); the fibreof F over 0 is Y(0) ∼= Fm . The computation we just did immediately implies thatthe local Kodaira–Spencer map

κF,0 : T0Am−1 → H1(Fm, TFm )

is an isomorphism.

1.2.5 Higher-order deformations – obstructions

Let X be a nonsingular algebraic variety. Consider a small extension

e : 0→ (t)→ A→ A→ 0

in A and let

ξ :X → X↓ ↓

Spec(k) → Spec(A)

be an infinitesimal deformation of X . A lifting of ξ to A consists in a deformation

ξ :X → X↓ ↓

Spec(k) → Spec( A)


and an isomorphism of deformations

X

X φ−→ X ×Spec( A) Spec(A)

Spec(A)

If we want to study arbitrary infinitesimal deformations, and not only first-order ones,it is important to know whether, given ξ and e, a lifting of ξ to A exists, and howmany there are. Such information can then be used to build an inductive procedure forthe description of infinitesimal deformations. The following proposition addressesthis question.

Proposition 1.2.12. Given A in ob(A) and an infinitesimal deformation ξ of X overA:

(i) To every small extension e of A there is associated an element oξ (e) ∈ H2(X, TX )

called the obstruction to lifting ξ to A, which is 0 if and only if a lifting of ξ toA exists.

(ii) If oξ (e) = 0 then there is a natural transitive action of H 1(X, TX ) on the set ofisomorphism classes of liftings of ξ to A.

(iii) The correspondence e → oξ (e) defines a k-linear map

oξ : Exk(A, k)→ H2(X, TX )

Proof. Let U = Ui i∈I be an affine open cover of X . We have isomorphisms

θi : Ui × Spec(A)→ X|Ui

and consequently, θi j := θ−1i θ j is an automorphism of the trivial deformation

Ui j × Spec(A). Moreover,θi jθ jk = θik (1.13)

on Ui jk × Spec(A). To give a lifting ξ of ξ to A it is necessary and sufficient to givea collection of automorphisms θi j of the trivial deformations Ui j × Spec( A) suchthat

(a) θi j θ jk = θik

(b) θi j restricts to θi j on Ui j × Spec(A)

In fact from such data we will be able to define X by patching the local piecesUi × Spec( A) along the open subsets Ui j × Spec( A) in the usual way. To estab-lish the existence of the collection θi j let’s choose arbitrarily automorphisms θi j satisfying condition (b); they exist by Lemma 1.2.8(ii). Let

θi jk = θi j θ jk θ−1ik

This is an automorphism of the trivial deformation Ui jk×Spec( A). Since by (1.13) itrestricts to the identity on Ui jk×Spec(A), by Lemma 1.2.6 we can identify each θi jk


with a di jk ∈ Γ (Ui jk, TX ) and it is immediate to check that di jk ∈ Z2(U , TX ). Ifwe choose different automorphisms Φi j of the trivial deformations Ui j × Spec( A)satisfying the analogue of condition (b) then

Φi j = θi j + tdi j (1.14)

for some di j ∈ Γ (Ui j , TX ), by Lemma 1.2.8(i). For each i, j, k the automorphism

Φi jΦ jkΦ−1ik

corresponds to the derivation

δi jk = di jk + (di j + d jk − dik)

and therefore we see that the 2-cocycles di jk and δi jk are cohomologous. Theircohomology class

oξ (e) ∈ H2(X, TX )

depends only on ξ and e and is 0 if and only if we can find a collection of automor-phisms Φi j such that δi jk = 0 for all i, j, k ∈ I . In such a case Φi j defines alifting ξ of ξ . This proves (i).

Assume that oξ (e) = 0, i.e. that the lifting ξ of ξ exists. Then we can choosethe collection θi j of automorphisms satisfying conditions (a) and (b) as above, inparticular di jk = 0, all i, j, k. Any other choice of a lifting ξ of ξ to A correspondsto a choice of automorphisms Φi j satisfying (1.14) and the analogue of condition(b). Therefore, for all i, j, k, we have

0 = δi jk = di j + d jk − dik

so that di j ∈ Z1(U , TX ) defines an element d ∈ H1(X, TX ). As before, onechecks that this element depends only on the isomorphism class of ξ ; it follows in astraightforward way that the correspondence (ξ , d) → ξ defines a transitive actionof H1(X, TX ) on the set of isomorphism classes of liftings of ξ to A. This proves(ii).

(iii) is left to the reader. Definition 1.2.13. The deformation ξ is called unobstructed if oξ is the zero map;otherwise ξ is called obstructed. X is unobstructed if every infinitesimal deformationof X is unobstructed; otherwise X is obstructed.

Corollary 1.2.14. A nonsingular variety X is unobstructed if

H2(X, TX ) = (0)

The proof is obvious.

Corollary 1.2.15. A nonsingular variety X is rigid if and only if

H1(X, TX ) = (0)


Proof. The hypothesis implies, by Proposition 1.2.9, that all first-order deformationsof X are trivial; moreover, by Proposition 1.2.12(ii), it implies that every infinitesimaldeformation of X over any A in ob(A) has at most one lifting to any small extensionof A. These two facts together easily give the conclusion. Examples 1.2.16. (i) If X is a projective nonsingular curve of genus g then from theRiemann–Roch theorem it follows that

h1(X, TX ) = 0 if g = 0

1 if g = 13g − 3 if g ≥ 2

and h2(X, TX ) = 0. In particular, projective nonsingular curves are unobstructed.

(ii) If X is a projective, irreducible and nonsingular surface X then

H2(X, TX ) ∼= H0(X,Ω1X ⊗ K X )∨

by Serre duality, and this rarely vanishes. For example, a nonsingular surface of de-gree ≥ 5 in IP3 satisfies H2(X, TX ) = (0), but it is nevertheless unobstructed (seeExample 3.2.11(i)); therefore the sufficient condition of Corollary 1.2.14 is not nec-essary. In general a surface such that H2(X, TX ) = (0) can be obstructed, but explicitexamples are not elementary (see [96], [24], [87]). We will describe a class of suchexamples in Theorem 3.4.26, page 185. In § 2.4 we will show how to construct exam-ples of obstructed 3-folds (see remarks following Proposition 3.4.25, page 185). Thefirst examples of obstructed compact complex manifolds where given in Kodaira–Spencer [107], § 16: they are of the form T × IP1, where T is a two-dimensionalcomplex torus.

(iii) The projective space IPn is rigid for every n ≥ 1. In fact it follows immedi-ately from the Euler sequence:

0→ OIPn → OIPn (1)n+1 → TIPn → 0

that H1(IPn, TIPn ) = 0. Similarly, one shows that finite products

IPn1 × · · · × IPnk

of projective spaces are rigid.

(iv) The ruled surfaces Fm are unobstructed because

h2(Fm, TFm ) = 0

(see (B.13), page 292).

2

Formal deformation theory

In this chapter we develop the theory of “functors of Artin rings”. The main resultof this theory is a theorem of Schlessinger giving necessary and sufficient conditionsfor a functor of Artin rings to have a semiuniversal or a universal formal element.We then apply the functorial machinery to the construction of formal semiuniversal,or universal, deformations, which is the final goal of formal deformation theory, andwe explain the relation between formal and algebraic deformations. In order to makethe arguments easier to follow, these applications are given only in the case of thedeformation functors DefX and Def′X of an algebraic scheme. Only in Chapter 3 willwe apply the general theory to other deformation functors.

2.1 Obstructions

In this section we investigate the notion of formal smoothness in the category A∗ us-ing the language of extensions. The results we prove are crucial for the understandingof obstructions in deformation theory. Our treatment is an expansion of [157]; for amore systematic treatment we refer to [54].

Let Λ ∈ ob(A∗) and µ : Λ→ R be in ob(A∗Λ). The relative obstruction spaceof R/Λ is

o(R/Λ) := ExΛ(R, k)

If Λ = k then o(R/k) is called the (absolute) obstruction space of R and sim-ply denoted by o(R). We say that R is unobstructed (resp. obstructed) over Λ ifo(R/Λ) = (0) (resp. if o(R/Λ) = (0)); R is said to be unobstructed (resp. ob-structed) if o(R) = (0) (resp. if o(R) = (0)). Given a homomorphism f : R → Sin A∗Λ we denote by

o( f/Λ) : o(S/Λ)→ o(R/Λ)

the linear map induced by pullback:

o( f/Λ)([η]) = [ f ∗η] ∈ ExΛ(R, k)

38 2 Formal deformation theory

for all [η] ∈ ExΛ(S, k). Since this definition is functorial we have a contravariantfunctor:

o(−/Λ) : A∗Λ→ (vector spaces/k)

When Λ = k we write o( f ) instead of o( f/k). If µ is such that o(µ) is injective onesometimes says that R is less obstructed than Λ. By applying Proposition 1.1.5 weobtain an exact sequence for each f : R→ S in A∗Λ:

0→ tS/R → tS/Λ→ tR/Λ→ o(S/R)→ o(S/Λ)o( f/Λ)−→ o(R/Λ) (2.1)

In the case Λ = k we obtain the exact sequence:

0→ tS/R → tS → tR → o(S/R)→ o(S)o( f )−→ o(R) (2.2)

which relates the absolute and the relative obstruction spaces.The next result gives a description of o(R/Λ) and an interpretation of formal

smoothness of a Λ-algebra (R, m) in A∗Λ.

Proposition 2.1.1. Assume that Λ is in ob(A).

(i) Let (R, m) be in ob(A∗Λ) and let χ : R → R be the natural homomorphism of R

into its m-adic completion R. Then the induced map:

o(χ/Λ) : o(R/Λ)→ o(R/Λ)

is an isomorphism.(ii) For every (R, m) in ob(A∗Λ) let d = dimk(tR/Λ) and let

R = Λ[[X1, . . . , Xd ]]/J

with J ⊂ (X)2, be a presentation of the m-adic completion R. Then there is anatural isomorphism:

o(R/Λ) ∼= (J/(X)J)∨

In particular, R is unobstructed over Λ if and only if it is a formally smoothΛ-algebra.

Proof. (i) Let

η : 0→ k→ S→ R→ 0

be a small extension; denote by m′ the maximal ideal of S.

Claim: S is complete.Let fn ⊂ S be a Cauchy sequence; then the image sequence fn in R is

Cauchy, hence it converges to a limit which we may assume to be zero, after possibly

2.1 Obstructions 39

subtracting a constant sequence from fn. We have fn ∈ me(n), with limn[e(n)] =∞. For every n we may find gn ∈ m′e(n) lying above fn . The sequence gn in S isCauchy and converges to zero, and fn − gn is a Cauchy sequence in k. Since k iscomplete as an S-module, because it is annihilated by the maximal ideal, fn − gnconverges to a limit f ∈ k. This is also the limit of fn because

fn − f = ( fn − gn − f )+ gn

Therefore S is complete.

If χ∗(η/Λ) is trivial the section induces a homomorphism g : R → S whichfactors through R because S is complete. Hence η is trivial. This proves that o(χ/Λ)is injective.

Given a small Λ-extension of R:

(S, ϕ) : 0→ k→ S→ R→ 0

the map ϕ : S→ R is surjective and ker(ϕ) = k = k. Therefore [S, ϕ] ∈ ExΛ(R, k)and o(χ/Λ)([S, ϕ]) = [S, ϕ]: this means that o(χ/Λ) is also surjective.

(ii) R is a formally smooth Λ-algebra if and only if R is a power series ring overΛ, i.e. if and only if J = (0). Therefore the last assertion follows from the fact thatJ/(X)J = (0) if and only if J = (0), by Nakayama’s lemma.

In order to prove the first assertion we may assume that R is in ob(AΛ), sinceo(R/Λ) = o(R/Λ) by the first part of the proposition. Hence R = Λ[[X ]]/J withJ ⊂ (X)2. The extension of R:

Φ : 0→ J/(X)J → Λ[[X ]]/(X)J → R→ 0

induces by pushouts a homomorphism:

α : (J/(X)J

)∨ → ExΛ(R, k) = o(R)

d −→ [d∗Φ]Letting M be the maximal ideal of Λ[[X ]]/(X)J we have J/(X)J ⊂ M2. If d ∈(J/(X)J )∨ is such that [d∗Φ] = 0 then we have:

Φ : 0→ J/(X)J → Λ[[X ]]/(X)J → R → 0↓ d ↓ h ‖

d∗Φ : 0→ k → A → R → 0

with d∗Φ trivial. From Example 1.1.2(ii) it follows that the generator ε of k in A iscontained in m A\m2

A. Since h(J/(X)J ) ⊂ m2A we deduce that d = 0. It follows that

α is injective.Conversely, given a Λ-extension (A, ϕ) of R by k it is possible to find a lifting:

Λ[[X ]]↓ ϕ

ϕ : A → R


because A is complete (see C.3(ii)). From the fact that ker(ϕ) = k it follows thatker(ϕ) ⊃ (X)J and therefore we have a commutative diagram:

Φ : 0→ J/(X)J → Λ[[X ]]/(X)J → R → 0↓ d ↓ ϕ ‖

(A, ϕ) : 0→ k → A → R → 0

in which d is the map induced by ϕ. It follows that (A, ϕ) = d∗Φ; hence α issurjective. Corollary 2.1.2. For every R in ob(A∗) the following are true:

(i) dimk[o(R)] <∞

(i i) dimk(tR) ≥ dim(R) ≥ dimk(tR) − dimk[o(R)](where dim(R) means Krull dimension of R). In (ii) the first equality holds ifand only if R is formally smooth; the second equality holds if and only if R =k[[X1, . . . , Xd ]]/J , with J ⊂ (X)2 and J generated by a regular sequence.

Proof. We may assume that R is in ob(A); hence R = R = k[[X1, . . . , Xd ]]/J ,with J ⊂ (X)2 and o(R) ∼= (J/(X)J )∨. Then (i) and (ii) follow from the fact thatdimk[J/(X)J ] is the number of elements of a minimal set of generators of J . Remarks 2.1.3. The only formally smooth k-algebra in A is k itself. By 2.1.1(ii)this means that o(k) = (0) and that o(A) = (0) for every A = k in A. The followingare some special cases.

• If A = k⊕V , a trivial extension of k by a vector space V of dimension d, thenA ∼= k[X1, . . . , Xd ]/(X)2, and o(A) = [(X)2/(X)3]∨.

• If A = k[X ]/(X)k then o(A) = [(X)k/(X)k+1]∨. In particular, if A = k[t]/(tn),n ≥ 2, then o(A) = [(tn)/(tn+1)]∨ is one-dimensional; from the proof of2.1.1(ii) it follows immediately that o(A) is generated by the class of the ex-tension:

0→ (tn)/(tn+1)→ k[t]/(tn+1)→ k[t]/(tn)→ 0

We will need the following:

Lemma 2.1.4. (i) Let µ : Λ → R be a homomorphism in A∗. Given a small ex-tension η : B → A in A and a homomorphism ϕ : R → A, the conditionϕ∗(η) ∈ ker(o(µ)) is equivalent to the existence of a commutative diagram:

Aϕ←− R

↑ η ↑ µB ←−

ϕΛ

(2.3)

Moreover, ϕ∗(η) = 0 if and only if there exists ϕ′ : R → B such that theresulting diagram

2.1 Obstructions 41

Aϕ←− R

↑ η ϕ′ ↑ µ

B ←−ϕ

Λ

(2.4)

is commutative.(ii) For every Λ in ob(A) and µ : Λ → R in ob(A∗Λ) there exists A in ob(AΛ)

and a homomorphism p : R → A such that o(p/Λ) : o(A/Λ) → o(R/Λ) issurjective.

Proof. (i) is left to the reader.(ii) We will show something more precise, namely that if pn : R → R/mn+1 is

the natural map, then

o(pn/Λ) : o((R/mn+1)/Λ)→ o(R/Λ)

is surjective for all n 0.Since o(pn/Λ) factors through o(R/Λ), which is isomorphic to o(R/Λ), we may

assume that R is in AΛ. Let’s write:

R = Λ[[X ]]/J

where J = (g1, . . . , gs) ⊂ (X)2. Let n 0 be such that g j /∈ (X)n+2 for allj = 1, . . . , s. Then we have:

R

mn+1= Λ[[X ]](

J, (X)n+1)

and therefore

o((R/mn+1)/Λ) =[

(J, (X)n+1)

((X)J, (X)n+2)

]∨The map o(pn/Λ) is the transpose of

ın : J

(X)J→

[(J, (X)n+1)

((X)J, (X)n+2)

]

induced by the inclusion J ⊂ (J, (X)n+1). From the hypothesis on n it follows thatif γ ∈ J ∩ ((X)J, (X)n+2) then γ ∈ (X)J ; this means that ın is injective, i.e. thato(pn/Λ) is surjective.

The following theorem gives a characterization of formally smooth homomor-phisms in A∗.

Theorem 2.1.5. Let µ : Λ → R be a homomorphism in A∗. The following condi-tions are equivalent:

(i) For every commutative diagram (2.3) with η a small extension in A∗ there existsϕ′ : R→ B such that diagram (2.4) is commutative.


(ii) µ is formally smooth.(iii) dµ : tR → tΛ is surjective and o(µ) is injective.(iv) o(R/Λ) = (0).

Proof. (i)⇒ (i i) is trivial.

(i i) ⇒ (i i i) Let v ∈ tΛ be given as a k-algebra homomorphism Λ → k[ε].The formal smoothness of µ implies the existence of a homomorphism w : R →k[ε] which makes the following diagram commutative:

k ← R↑ ↑

k[ε] ← Λ

and this means that dµ(w) = v . Therefore dµ is surjective.Consider a commutative diagram of k-algebra homomorphisms (2.3) with η a

small extension in A. Then o(ϕ)([η]) ∈ ker(o(µ)). By the formal smoothness of µthere exists ϕ′ : R → B making (2.4) commutative: this implies that o(ϕ)([η]) = 0.Since, by Lemma 2.1.4, ϕ and η can be chosen so that o(ϕ)([η]) is an arbitraryelement of ker(o(µ)), we deduce that o(µ) is injective.

(i i i)⇔ (iv) follows from the exact sequence (2.2).

(i i i)⇒ (i). Consider a diagram (2.3) with η a small extension in A∗. Then

o(ϕ)([η]) ∈ ker(o(µ))

By assumption o(ϕ)([η]) = 0, and therefore there exists ϕ : R → B such thatηϕ = ϕ. It follows that ϕ − ϕµ : Λ→ ker(η) = k is a k-derivation. By assumptionthere exists a k-derivation v : R→ k such that ϕ − ϕµ = dµ(v) = vµ.

Then ϕ′ := ϕ + v : R → B is a k-homomorphism which obviously satisfiesηϕ′ = ϕ. Moreover,

ϕ′µ = (ϕ + v)µ = ϕµ+ vµ = ϕ

and therefore ϕ′ makes (2.4) commutative. In the special case Λ = k we obtain that a k-algebra R in ob(A∗) is unobstructed

if and only if R is formally smooth, a result already proven in 2.1.1. More generally,the theorem says that µ is formally smooth if and only if dµ is surjective and R isless obstructed than Λ. The following corollary is immediate.

Corollary 2.1.6. (i) Let µ : Λ → R be a homomorphism in A∗ such that dµ issurjective and R is formally smooth. Then Λ and µ are formally smooth.

(ii) A homomorphism µ : Λ → R in A∗ is formally etale if and only if dµ isan isomorphism and o(µ) is injective. This happens in particular if dµ is anisomorphism and R is formally smooth.

(iii) If R ∈ ob(A∗) then the natural homomorphism R→ R is formally etale.

2.1 Obstructions 43

In practice it is seldom possible to compute the obstruction map o(µ) explicitlyfor a given µ : Λ→ R in A∗. But for the purpose of studying the formal smoothnessof µ all that counts is to have information about ker[o(µ)]. This can be achievedsomewhat indirectly by means of the following simple result, which turns out to bevery effective in practice.

Proposition 2.1.7. Let µ : Λ→ R be a morphism in A∗. Assume that there exists ak-vector space υ(R/Λ) such that for every homomorphism ϕ : R → A in A∗Λ withA in ob(AΛ) there is a k-linear map

ϕυ : ExΛ(A, k)→ υ(R/Λ)

satisfyingker(ϕυ) = ker[o(ϕ/Λ)]

whereExΛ(A, k)

ϕυ−→ υ(R/Λ)

o(ϕ/Λ)

ExΛ(R, k) = o(R/Λ)

Then there is a natural k-linear inclusion

o(R/Λ) ⊂ υ(R/Λ)

Proof. Choosing ϕ such that o(ϕ/Λ) surjects onto o(R/Λ) we obtain an inclusiono(R/Λ) ⊂ υ(R/Λ) as asserted.

In practice this proposition will be applied as follows. Given ϕ, to give an ele-ment of ExΛ(A, k) is the same as to give a commutative diagram (2.3). Assume thatto each such diagram one associates in a linear way an element of υ(R/Λ) whichvanishes if and only if there is an extension ϕ′ : R → B making (2.4) commutative.Then the proposition applies.

Taking Λ = k we get the following absolute version of the proposition, wherefor the last assertion we apply 2.1.2(ii).

Corollary 2.1.8. Let R be in ob(A∗). Assume that there exists a k-vector space υ(R)such that for every morphism ϕ : R → A in A∗ with A in ob(A) there is a k-linearmap

ϕυ : o(A)→ υ(R)

satisfyingker(ϕυ) = ker[o(ϕ)]

Then there is a natural k-linear inclusion

o(R) ⊂ υ(R)

If υ(R) is finite dimensional then

dimk(tR) ≥ dim(R) ≥ dimk(tR)− dimk[υ(R)]A k-vector space υ(R) satisfying the conditions of 2.1.8 will sometimes be called

an obstruction space for R.


2.2 Functors of Artin rings

A functor of Artin rings is a covariant functor

F : AΛ −→ (sets)

where Λ ∈ ob(A). Let A be in ob(AΛ). An element ξ ∈ F(A) will be called aninfinitesimal deformation of ξ0 ∈ F(k) if ξ → ξ0 under the map F(A)→ F(k); ifA = k[ε] then ξ is called a first-order deformation of ξ0.

Examples of functors of Artin rings are obtained by fixing an R in ob(AΛ) andletting:

h R/Λ(A) = HomAΛ(R, A) for every A in ob(AΛ)

Such a functor is clearly nothing but the restriction to AΛ of a representable functoron AΛ. A functor of Artin rings isomorphic to h R/Λ for some R in ob(AΛ) is calledprorepresentable. In the case Λ = k we write h R instead of h R/k.

Every representable functor h R/Λ, R in ob(AΛ), is a (trivial) example of prorep-resentable functor.

Typically, a prorepresentable functor of Artin rings arises as follows. One con-siders a scheme M and the restriction

Φ : A −→ (sets)

Φ(A) = Hom(Spec(A), M)

of the functor of points

Hom(−, M) : (schemes) −→ (sets)

Φ is a functor of Artin rings; if ϕ : Spec(A)→ M is an element of Φ(A) then ϕis an infinitesimal deformation of the composition

Spec(k)→ Spec(A)ϕ−→ M

where the first morphism corresponds to A → A/m A = k. For a fixed k-rationalpoint m ∈ M , one may consider the subfunctor

F : A −→ (sets)

of Φ defined as follows:

F(A) = Hom(Spec(A), M)m = morphisms Spec(A)→ M

whose image is m

(2.5)

We have F = h R , where R = OM,m , so F is prorepresentable. We will introduceseveral functors of Artin rings which are defined by deformation problems. The hopeis that they are of the form (2.5) for some k-pointed algebraic scheme (M, m) whichhas some geometrical meaning with respect to the deformation problem under con-sideration (a “moduli scheme”). As we will see, this sometimes happens, but most

2.2 Functors of Artin rings 45

of the time it doesn’t. We will thus have to content ourselves to prove that such func-tors have weaker properties. In this section and in the following one we will studyprorepresentability and other properties of functors of Artin rings from an abstractpoint of view. Later on we will consider specific examples of such functors relatedto deformation problems.

A prorepresentable functor F = h R/Λ satisfies the following conditions:

H0) F(k) consists of one element (the canonical quotient R→ R/m R = k)

LetA′ A′′

A(2.6)

be a diagram in AΛ and consider the natural map

α : F(A′ ×A A′′)→ F(A′)×F(A) F(A′′)

induced by the commutative diagram:

F(A′ ×A A′′) → F(A′′)↓ ↓

F(A′) → F(A)

Then

H) (left exactness) For every diagram (2.6) α is bijective (straightforward to check).H f ) F(k[ε]) has a structure of finite-dimensional k-vector space.

In fact

F(k[ε]) = HomΛ−alg(R, k[ε]) = DerΛ(R, k) = tR/Λ

is the relative tangent space of R over Λ (here the Λ-algebra structure of k[ε] isgiven by the composition Λ→ k→ k[ε], (see B.9(vi)).

A property weaker than H satisfied by a prorepresentable functor F is the fol-lowing:

Hε) α is bijective if A = k and A′′ = k[ε].It is interesting to observe that if F is prorepresentable the structure of k-vector

space on F(k[ε]) can be reconstructed in a purely functorial way using only proper-ties H0 and Hε , and without using the prorepresentability explicitly.

Indeed it is an easy exercise to check that the homomorphism:

+ : k[ε] ×k k[ε] −→ k[ε]

(a + bε, a + b′ε) −→ a + (b + b′)ε


induces the sum operation on F(k[ε]) as the composition

F(k[ε])× F(k[ε])→ F(k[ε] ×k k[ε]) F(+)−→ F(k[ε])where the first map is the inverse of α. Associativity is checked using Hε .

The zero is the image of F(k)→ F(k[ε]). The multiplication by a scalar c ∈ kis induced in F(k[ε]) by the morphism

k[ε] −→ k[ε]

a + bε −→ a + (cb)ε

We can therefore state the following:

Lemma 2.2.1. If F is a functor of Artin rings having properties H0 and Hε then theset F(k[ε]) has a structure of k-vector space in a functorial way. This vector spaceis called the tangent space of the functor F, and is denoted by tF . If F = h R/Λ thentF = tR/Λ.

If f : F → G is a morphism of functors of Artin rings then the induced maptF → tG is called the differential of f and it is denoted by d f . It is straightforwardto check that if F and G satisfy H0 and Hε then d f is k-linear.

Every functor of Artin rings F can be extended to a functor

F : AΛ −→ (sets)

by letting, for every R in ob(AΛ):

F(R) = lim←− F(R/mn+1R )

and for every ϕ : R→ S:F(ϕ) : F(R)→ F(S)

be the map induced by the maps F(R/mnR)→ F(S/mn

S), n ≥ 1.An element u ∈ F(R) is called a formal element of F . By definition u can be

represented as a system of elements un ∈ F(R/mn+1R )n≥0 such that for every n ≥ 1

the mapF(R/mn+1

R )→ F(R/mnR)

induced by the projection R/mn+1R → R/mn

R sends

un −→ un−1 (2.7)

u is also called a formal deformation of u0. If f : F → G is a morphism of functorsof Artin rings then it can be extended in an obvious way to a morphism of functorsf : F → G.

Lemma 2.2.2. Let R be in ob(AΛ). There is a 1–1 correspondence between F(R)and the set of morphisms of functors

h R/Λ −→ F (2.8)


Proof. To a formal element u ∈ F(R) there is associated a morphism of functors(2.8) in the following way. Each un ∈ F(R/mn+1

R ) defines a morphism of func-tors h(R/mn+1)/Λ → F . The compatibility conditions (2.7) imply that the followingdiagram commutes:

h(R/mn)/Λ → h(R/mn+1)/Λ

↓

F

for every n. Since for each A in ob(AΛ)

h(R/mn)/Λ(A)→ h(R/mn+1)/Λ(A)

is a bijection for all n 0 we may define

h R/Λ(A)→ F(A)

as

limn→∞[h(R/mn+1)/Λ(A)→ F(A)]

Conversely, each morphism (2.8) defines a formal element u ∈ F(R), where un ∈F(R/mn+1

R ) is the image of the canonical projection R→ R/mn+1R via the map

h R/Λ(R/mn+1R )→ F(R/mn+1

R )

Definition 2.2.3. If R is in ob(AΛ) and u ∈ F(R), we call (R, u) a formal couplefor F. The differential tR/Λ→ tF of the morphism h R/Λ→ F defined by u is calledthe characteristic map of u (or of the formal couple (R, u)) and is denoted by du.If (R, u) is such that the induced morphism (2.8) is an isomorphism, then F is prorep-resentable, and we also say that F is prorepresented by the formal couple (R, u). Inthis case u is called a universal formal element for F, and (R, u) is a universalformal couple.

An ordinary couple (R, u) with R in AΛ defines a special case of formal couple,in which (Rn, un) = (Rn+1, un+1) for all n 0. A universal formal couple sel-dom exists; we will therefore need to introduce some weaker properties of a formalcouple. They are based on the following definition.

Definition 2.2.4. Let f : F → G be a morphism of functors of Artin rings. f iscalled smooth if for every surjection µ : B → A in AΛ the natural map:

F(B)→ F(A)×G(A) G(B) (2.9)


induced by the diagram:F(B) → G(B)↓ ↓

F(A) → G(A)

is surjective. The functor F is called smooth if the morphism from F to the constantfunctor

G(A) = one element all A ∈ ob(AΛ)

is smooth; equivalently, if

F(µ) : F(B)→ F(A)

is surjective for every surjection µ : B → A in AΛ.

Note that, since every surjection in AΛ factors as a finite sequence of small ex-tensions (i.e. surjections with one-dimensional kernel), for f (or F) to be smooth itis necessary and sufficient that the defining condition is satisfied for small extensionsin AΛ.

The next proposition states some properties of the notion of smoothness of amorphism of functors of Artin rings.*

Proposition 2.2.5. (i) Let f : R→ S be a homomorphism in AΛ. Then f is formallysmooth if and only if the morphism of functors h f : hS/Λ→ h R/Λ induced by fis smooth.

(ii) If f : F → G is smooth then f is surjective, i.e. F(B)→ G(B) is surjective forevery B in ob(AΛ). In particular the differential d f : tF → tG is surjective.

(iii) If f : F → G is smooth, then F is smooth if and only if G is smooth.(iv) If f : F → G is smooth then the induced morphism of functors f : F → G is

surjective, i.e. F(R)→ G(R) is surjective for every R in ob(AΛ).(v) If F → G and G → H are smooth morphisms of functors then the composition

F → H is smooth.(vi) If F → G and H → G are morphisms of functors and F → G is smooth, then

F ×G H → H is smooth.(vii) If F and G are smooth functors then F × G is smooth.

Proof. (i) Let B → A be a surjection in AΛ and let

A ← S↑ ↑ fB ← R

be a commutative diagram of homomorphisms of Λ-algebras. The formal smooth-ness of f is equivalent to the existence, for each such diagram, of a morphism S→ Bsuch that the resulting diagram

A ← S↑ ↑ fB ← R

∗ Here all functors are assumed to satisfy Ho.


is commutative. This is another way to express the condition that the map

hS/Λ(B) −→ hS/Λ(A)×h R/Λ(A) h R/Λ(B)

is surjective, i.e. that h f is smooth.(ii) follows immediately from the definition.(iii) Let µ : B → A be a surjection in AΛ, and consider the diagram:

F(B) → F(A)×G(A) G(B) → G(B)↓ ↓ G(µ)

F(A) −→f (A)

G(A)

Suppose G is smooth and let ξ ∈ F(A). By the smoothness of G there exists η ∈G(B) such that G(µ)(η) = f (A)(ξ). By the smoothness of f there is ζ ∈ F(B)which is mapped to (ξ, η) ∈ F(A)×G(A) G(B). It follows that F(µ)(ζ ) = ξ and Fis smooth.

The converse is proved similarly using the surjectivity of f (A).(iv) Let v = vn ∈ G(R). Since f is smooth the map

F(R/m2R)→ F(k)×G(k) G(R/m2

R) = G(R/m2R)

is surjective. Therefore there exists w1 ∈ F(R/m2R) such that f (R/m2

R)(w1) = v1.Let’s assume that for some n ≥ 1 there exist wi ∈ F(R/mi+1

R ), i = 1, . . . , n suchthat:

f (R/mi+1R )(wi ) = vi

and wi −→ wi−1 under F(R/mi+1R ) −→ F(R/mi

R).The surjectivity of the map:

F(R/mn+1R ) −→ F(R/mn

R)×G(R/mn) G(R/mn+1R )

implies that there exists wn+1 ∈ F(R/mn+1R ) whose image is (wn, vn+1). By induc-

tion we conclude that there exists w ∈ F(R) whose image under f is v .The proofs of (v), (vi) and (vii) are straightforward. We now introduce the notions of “versality” and “semiuniversality”, which are

slightly weaker than universality.

Definition 2.2.6. Let F be a functor of Artin rings. A formal element u ∈ F(R),for some R in ob(AΛ), is called versal if the morphism h R/Λ → F defined byu is smooth; u is called semiuniversal if it is versal and moreover, the differentialtR/Λ→ tF is bijective.We will correspondingly speak of a versal formal couple (respectively a semiuniver-sal formal couple).

It is clear from the definitions that:

u universal ⇒ u semiuniversal ⇒ u versal


but none of the inverse implications is true (see Examples 2.6.8). We can also de-scribe these properties as follows.

Assume given µ : (B, ξB) → (A, ξA), a surjection of couples for F (i.e. µ issurjective), and a homomorphism ϕ : R → A such that ϕ(u) = ξA. Then u is versalif for all such data there is a lifting ψ : R → B of ϕ (i.e. µψ = ϕ) such thatψ(u) = ξB :

Bψ

R ↓ µϕ

A

ξB

u ↓

ξA

u is universal if, moreover, the lifting ψ is unique. u is semiuniversal if it is versaland, moreover, the lifting ψ is unique when µ : k[ε] → k.

Let (R, u) and (S, v) be two formal couples for F . A morphism of formal couples

f : (R, u)→ (S, v)

is a morphism f : R → S in A such that F( f )(u) = v . We will call f an isomor-phism of formal couples if in addition f : R→ S is an isomorphism.

It is obvious that with this definition the formal couples for F and theirmorphisms form a category containing the category IF of couples for F as a fullsubcategory.

Proposition 2.2.7. Let F be a functor of Artin rings. Then:

(i) If (R, u) and (S, v) are universal formal couples for F there exists a unique iso-morphism of formal couples (R, u) ∼= (S, v).

(ii) If (R, u) and (S, v) are semiuniversal formal couples for F there exists an iso-morphism of formal couples (R, u) ∼= (S, v), which is not necessarily uniquebut the induced isomorphism tS ∼= tR is uniquely determined.

(iii) If (R, u) is a semiuniversal formal couple for F and (S, v) is versal, then thereis an isomorphism, not necessarily unique:

ϕ : R[[X1, . . . , Xr ]] → S

for some r ≥ 0, such that F(ϕ j)(u) = v , where j : R ⊂ R[[X1, . . . , Xr ]] is theinclusion.

Proof. (i) By the universality of (R, u) for every n ≥ 1 there exists a unique fn ∈h R/Λ(S/mn+1

S ) such that fn → vn ∈ F(S/mn+1S ) under the isomorphism associated

to u. In this way we obtain

f = lim← fn : (R, u)→ (S, v)

which is uniquely determined. Analogously, we can construct a uniquely determinedg : (S, v)→ (R, u). By universality the compositions


g f : (R, u)→ (R, u)

andf g : (S, v)→ (S, v)

are the identity.

(ii) Using versality, and proceeding as above we can construct morphisms offormal couples

f : (R, u)→ (S, v)

and

g : (S, v)→ (R, u)

We obtain a commutative diagram:

tR/Λ

↑↓ tS/Λ −→ tF

where the vertical arrows are the differentials of f and g, and the other arrows are thecharacteristic maps du and d v . From this diagram we deduce that d f = (du)−1d vand dg = (d v)−1du are uniquely determined. Since

d(g f ) = (dg)(d f ) = identity of tS/Λ

andd( f g) = (d f )(dg) = identity of tR/Λ

it follows that f and g are bijections inverse of each other.

(iii) By the versality of (R, u) we can find a morphism of formal couples

f : (R, u)→ (S, v)

We obtain a commutative diagram:

tR/Λ

↑ tS/Λ −→ tF

where du : tR/Λ→ tF is bijective because (R, u) is semiuniversal, and d v : tS/Λ→tF is surjective because (S, v) is versal. Hence d f : tS/Λ → tR/Λ is surjective. Thismeans that f induces an inclusion

t∨R/Λ ⊂ t∨S/Λ

Let x1, . . . , xr ∈ S be elements which induce a basis of

t∨S/R = mS/[m2S + f (m R)]


and define:ϕ : R[[X1, . . . , Xr ]] → S

by ϕ(Xi ) = xi , i = 1, . . . , r . Let j : R ⊂ R[[X1, . . . , Xr ]] be the inclusion. Letting

w = F( j)(u) ∈ F(R[[X1, . . . , Xr ]])we have a commutative diagram of formal couples:

(R[[X1, . . . , Xr ]], w)ϕ−→ (S, v)

↑ j f

(R, u)

such that F(ϕ j)(u) = v . Since

ϕ1 : R[[X1, . . . , Xr ]]/M2 → S/m2S

is an isomorphism (here M ⊂ R[[X1, . . . , Xr ]] denotes the maximal ideal) ϕ issurjective. Let

ψ1 : S/m2S → R[[X1, . . . , Xr ]]/M2

be the inverse of ϕ1. We have

F(ψ1)(v1) = w1

hence, by the versality of (S, v), it is possible to find a lifting of ψ1:

ψ : S→ R[[X1, . . . , Xr ]]such that F(ψ)(v) = w. Since by construction

d(ψϕ) = dψ dϕ = identity of tR[[X ]]

it follows that ψϕ is an automorphism of R[[X1, . . . , Xr ]]. In particular, we deducethat ϕ is injective.

The following is useful:

Proposition 2.2.8. If a functor of Artin rings F : A → (sets) satisfying H0 has asemiuniversal formal element and tF = (0) then F = hk, the constant functor.

Proof. The assumptions imply that there is a smooth morphism hk → F . Since asmooth morphism of functors satisfying H0 is surjective, the conclusion follows.

∗ ∗ ∗ ∗ ∗ ∗For the rest of this section we will only consider functors of Artin rings satisfying

conditions H0 and Hε .


Definition 2.2.9. Let F be a functor of Artin rings. Suppose that υ(F) is a k-vectorspace such that for every A in ob(AΛ) and for every ξ ∈ F(A) there is a k-linearmap

ξυ : ExΛ(A, k)→ υ(F)

with the following property:

ker(ξυ) consists of the isomorphism classes of extensions ( A, ϕ) such that

ξ ∈ Im[F( A)→ F(A)]

Then υ(F) is called an obstruction space for the functor F. If F has (0) as an ob-struction space then it is called unobstructed.

If F is prorepresented by the formal couple (R, u), then o(R/Λ) is an obstructionspace for F , as it follows from the functorial characterization of o(R/Λ) given in§ 2.1. Conversely, if υ(F) is an obstruction space for F , and F is prorepresented bythe formal couple (R, u), then it follows immediately from the definition that υ(F)is a relative obstruction space for R/Λ.

One can show that under relatively mild conditions a functor F has an obstructionspace. We will not discuss this matter here, and we refer the interested reader to [54].

The following are some basic properties of obstruction spaces.

Proposition 2.2.10. (i) F has (0) as an obstruction space if and only if it is smooth.(ii) Let f : F → G be a smooth morphism of functors of Artin rings. If υ(G) is an

obstruction space for G then it is also an obstruction space for F.

Proof. (i) is obvious.

(ii) Let A be in ob(AΛ) and ξ ∈ F(A), and let

ξυ := f (A)(ξ)υ : ExΛ(A, k)→ υ(G)

the map defined by f (A)(ξ) ∈ G(A). If A → A = A/(t) defines an element ofker(ξυ) then, since υ(G) is an obstruction space for G, there is η ∈ G( A) such thatη → f (A)(ξ) under the map G( A)→ G(A). From the smoothness of f it followsthat the map

F( A)→ F(A)×G(A) G( A)

is surjective, hence there is ξ ∈ F( A) which maps to the formal couple (ξ, η). Itfollows that ξ → ξ under F( A)→ F(A).

Conversely, if A is such that there exists ξ ∈ F( A) such that

F( A) → F(A)

ξ → ξ


then from the diagram:F( A) → F(A)

↓ f ( A) ↓ f (A)

G( A) → G(A)

we see that f ( A)(ξ ) → f (A)(ξ) hence f (A)(ξ) ∈ Im[G( A)→ G(A)]. Thereforethe extension A → A defines an element of ker[ f (A)(ξ)υ ]. This proves that ξυsatisfies the conditions of Definition 2.2.9. Corollary 2.2.11. Let F : A→ (sets) be a functor of Artin rings, and suppose that(R, u) is a versal formal couple for F. If F has a finite-dimensional obstructionspace υ(F) then

dimk(tR) ≥ dim(R) ≥ dimk(tR)− dimk[υ(F)]Proof. From the definition of versal formal couple and from Proposition 2.2.10 wededuce that υ(F) is an obstruction space for h R , hence for R. The conclusion followsfrom 2.1.6.

Corollary 2.2.11 is very important in deformation theory since it gives basic geo-metrical information about a deformation problem. If F is induced by a k-pointedscheme (M, m), i.e. it is of the form (2.5), then 2.2.11 gives information about thedimension of M at m. We refer the reader to Proposition 2.3.6 for a result aboutobstruction spaces which is often used in applications.

NOTES

1. For an interesting discussion on the geometric motivations behind the consideration offunctors of Artin rings we refer the reader to [72], part C.1.

2. Corollary 2.2.11 is due to Mumford (unpublished, see [189], p. 126). A proof is outlinedin [157], p. 153. The proof given here is based on such outline, and has appeared already in[163], Corollary (B2.4).

3. Let R, P be in A, ϕ : P → R a homomorphism and F : h R → h P be the correspond-ing morphism between the prorepresentable functors

h R, h P : A→ (sets)

Then F(A) : h R(A)→ h P (A) is injective for all A in ob(A) if and only if ϕ is surjective, i.e.if and only if R = P/J for an ideal J ⊂ P . (Proof: left to the reader.)

2.3 The theorem of Schlessinger

In this section we will prove a well-known theorem of Schlessinger which givesnecessary and sufficient conditions, easy to verify in practice, for the existence ofa (semi)universal element for a functor of Artin rings. Before stating the theoremwe want to make some introductory remarks, which can be useful in what follows.Let’s fix Λ ∈ ob(A) throughout this section. We start with a characterization ofprorepresentable functors:

2.3 The theorem of Schlessinger 55

Proposition 2.3.1. Let F : AΛ → (sets) be a functor of Artin rings satisfying con-dition H0. Then F is prorepresentable if and only if it is left exact and has finite-dimensional tangent space, i.e. it has properties H and H f .

Proof. The “only if” implication is obvious (see also § 2.2). So let’s assume that Fsatisfies H and H f . Then, by Proposition E.9, the category IF of couples for F iscofiltered and

F = lim→ (X,ξ)h X

Let (Ai , ξi ) ∈ ob(IF ) and consider all the subrings of Ai images of morphisms(A j , ξ j )→ (Ai , ξi ). By the descending chain condition there is ( Ai , ξi ) such that

Ai =⋂

Im[A j → Ai ]

By construction the couples ( Ai , ξi ) form a full subcategory of IF in which all mapsare surjective. Moreover, the corresponding category of representable functors h Ai /Λis clearly cofinal in (IF ), and therefore

F = lim→ h Ai /Λ

Therefore replacing IF by this subcategory we can assume that all homomorphismsare surjective, and therefore we have:

F =⋃

h Ai /Λ

Moreover, since H f holds, F(k[ε]) = h Ai (k[ε]) when i 0. We can discard allthose Ai for which this is not true and we get again a full subcategory. Therefore

F = lim→ h Ai /Λ

and F(k[ε]) = h Ai /Λ(k[ε]) for all i . Then for each i we can find a surjection

Λ[[X1, . . . , Xr ]] → Ai

with r = dim(F(k[ε])), and these surjections are compatible, i.e. the diagrams

Λ[[X1, . . . , Xr ]] → Ai

↓A j

are commutative. Define B1 = Λ[[X1, . . . , Xr ]]/(X)2 = Ai/m2i for all i . Then fix

ν ≥ 2, and set Ai,ν = Ai/mν+1i . All Ai,ν’s are quotients of Λ[[X1, . . . , Xr ]]/(X)ν+1

and form a projective system:

· · · → A j,ν → Ai,ν → · · ·


LetBν = lim← Ai,ν = Ai,ν i 0

Then by construction· · · Bν+1 → Bν → · · ·

form a projective system andF = lim→ hBν/Λ

ThenB := lim← Bν

is in ob(AΛ) and prorepresents F . Unfortunately this characterization of prorepresentable functors is not very useful

in practice because given homomorphisms

A′ A′′

A

in AΛ we have

Spec(A′ ×A A′′) = Spec(A′)⋃

Spec(A)

Spec(A′′)

and this is not easy to visualize. That’s why left exactness of a functor of Artin ringsF is hard to check. On the other hand, if at least one of the above homomorphisms issurjective then Spec(A′ ×A A′′) is easier to describe. The theorem of Schlessinger re-duces the prorepresentability to the verification of the condition of left exactness onlyin cases when at least one of the above maps is surjective. An analogous condition isgiven for the existence of a semiuniversal element. The result is the following.

Theorem 2.3.2 (Schlessinger [155]). Let F : AΛ → (sets) be a functor of Artinrings satisfying condition H0. Let A′ → A and A′′ → A be homomorphisms in AΛ

and letα : F(A′ ×A A′′) −→ F(A′)×F(A) F(A′′) (2.10)

be the natural map. Then:

(i) F has a semiuniversal formal element if and only if it satisfies the following con-ditions:H) if A′′ → A is a small extension then the map (2.10) is surjective.Hε) If A = k and A′′ = k[ε] then the map (2.10) is bijective.H f ) dimk(tF ) <∞

(ii) F has a universal element if and only if it also satisfies the following additionalcondition:H) the natural map

F(A′ ×A A′) −→ F(A′)×F(A) F(A′)

is bijective for every small extension A′ → A in AΛ.


Remark 2.3.3. It will be helpful to explain the meaning of the conditions of thetheorem before giving the proof. Consider a small extension in AΛ:

0→ (t)→ A′ µ−→ A→ 0

Assume that F ∼= h R/Λ is prorepresentable. Then two Λ-homomorphisms f, g :R → A′ have the same image in HomΛ−alg(R, A) if and only if there exists a Λ-derivation d : R→ k (which is uniquely determined) such that

g(r) = f (r)+ d(r)t

equivalently if and only if g and f differ by an element of

DerΛ(R, k) = tR/Λ

Therefore the fibres of

F(µ) : HomΛ−alg(R, A′)→ HomΛ−alg(R, A)

that are nonempty are principal homogeneous spaces under the above action oftR/Λ = tF .

Assume now that F is just a functor of Artin rings having properties H0 and Hε ,so that it has tangent space tF . We can define an action of tF on F(A′) by means ofthe composition

τ : tF × F(A′) α−1−→ F(k[ε] ×k A′) F(b)−→ F(A′)

where α−1 exists by property Hε and F(b) is induced by the morphism

b : k[ε] ×k A′ −→ A′

(x + yε, a′) −→ a′ + yt

The action τ maps the fibres of F(µ) into themselves. Indeed the isomorphism

γ : k[ε] ×k A′ −→ A′ ×A A′

(x + yε, a′) −→ (a′ + yt, a′)

induces a map

β : tF × F(A′) α−1−→ F(k[ε] ×k A′) F(γ )−→ F(A′ ×A A′) −→ F(A′)×F(A) F(A′)

which coincides with

tF × F(A′) −→ F(A′)×F(A) F(A′)

(v, ξ) −→ (τ (v, ξ), ξ)


In case F is prorepresentable we have just given another description of the action oftF on the fibres of F(µ) introduced before. In general the map β is neither injective(i.e. in general the action τ is not free on the fibres of F(µ)) nor surjective (i.e. τ isnot transitive on the fibres of F(µ)). This depends on the properties of the map

α′ : F(A′ ×A A′) −→ F(A′)×F(A) F(A′)

If F is left exact then α′ is bijective, hence β is bijective, and the action of tF onthe fibres of F(µ) is free and transitive, as expected, since F is prorepresentable byProposition 2.3.1.

Conversely, what this analysis shows is that for τ to be free and transitive on thefibres of F(µ) we only need α′ bijective, i.e. the condition H , weaker than H. Honly guarantees the transitivity of such action: the failure from prorepresentability istherefore related to the existence of fixed points of this action. In applications thisis usually due to the existence of automorphisms of geometric objects associated toan element ξ ∈ F(A) which don’t lift to automorphisms of objects associated to anelement ξ ′ ∈ F(µ)−1(ξ) (see § 2.6).

Since Theorem 2.3.2 guarantees that F has a semiuniversal element if it satisfiesH0, Hε and H , we can summarize this discussion in the following statement.

Proposition 2.3.4. Let F be a functor of Artin rings satisfying conditions H0 and Hε

and let0→ (t)→ A′ → A→ 0

be a small extension in A. Then there is an action τ of tF on F(A′) such that:

(a) if F has a semiuniversal formal element then tF acts transitively on the non-empty fibres of F(A′)→ F(A);

(b) if F is prorepresentable then tF acts freely and transitively on the non-emptyfibres of F(A′)→ F(A).

The following definition introduces a convenient terminology.

Definition 2.3.5. Let f : F → G be a morphism of functors of Artin rings havinga semiuniversal formal element and obstruction spaces υ(F) and υ(G) respectively.An obstruction map for f is a linear map

o( f ) : υ(F)→ υ(G)

such that for each A in ob(A) and for each ξ ∈ F(A) the diagram:

Exk(A, k) ξυ ηυ

υ(F)o( f )−→ υ(G)

(2.11)

is commutative, where η denotes f (A)(ξ). If f has an injective obstruction map wesay that F is less obstructed than G.


With this terminology we can state a result often used in concrete situations:

Proposition 2.3.6. Let f : F → G be a morphism of functors of Artin rings havinga semiuniversal formal element and obstruction spaces υ(F) and υ(G) respectively.Consider the following conditions:

(a) d f is surjective(b) F is less obstructed than G.

If they are both satisfied then f is smooth. If condition (b) is satisfied (but not neces-sarily (a)) and G is smooth then F is smooth.

Proof. Let ϕ : A → A be a small extension and assume (a) and (b) hold. Considerthe map

F( A)→ F(A)×G(A) G( A) (2.12)

and let (ξ, η) ∈ F(A) ×G(A) G( A). Since η → η = f (A)(ξ) we have ηυ(ϕ) = 0.By (b) and the commutativity of (2.11) we also have ξυ(ϕ) = 0 and therefore thereexists ξ ∈ F( A) such that ξ → ξ . Let η′ = f ( A)(ξ ) ∈ G( A). We have

η η′

η

Since by Theorem 2.3.2 the functor G satisfies condition H , there is w ∈ tG suchthat τ(w, η′) = η. From the surjectivity of d f it follows that there is v ∈ tF

such that v → w. Now τ(v, ξ ) ∈ F( A) satisfies

τ(v, ξ ) → (ξ, η)

because the action τ is functorial (as it easily follows from its definition). This showsthat (2.12) is surjective and f is smooth.

Now assume that only (b) holds and that G is smooth. We must show that

F( A)→ F(A)

is surjective or equivalently that ξυ(ϕ) = 0. Let ξ ∈ F(A) and let η := f (A)(ξ) ∈G(A). Since G is smooth we have ηυ(ϕ) = 0. By (b) and the commutativity of (2.11)we also have ξυ(ϕ) = 0. Corollary 2.3.7. Let f : F → G be a morphism of functors of Artin rings having asemiuniversal formal element. Assume that F is smooth, G has an obstruction spaceυ(G) and d f is surjective. Then f and G are smooth.

Proof. Since F is smooth it has (0) as an obstruction space and the obvious map(0)→ υ(G) is an obstruction map. Therefore f is smooth by Proposition 2.3.6 andwe conclude by Proposition 2.2.5(iii).


Remark 2.3.8. When F and G are prorepresentable the first part of Proposition 2.3.6is a reformulation of Theorem 2.1.5(iii) and Corollary 2.3.7 is a reformulation ofCorollary 2.1.6(i). If in Corollary 2.3.7 we replace the condition “d f surjective” by“d f bijective” we cannot conclude that f is an isomorphism unless both functors areprorepresentable: if they are then f is an isomorphism by Corollary 2.1.6(ii) becausea formally etale homomorphism of k-algebras in A is an isomorphism.

Proof of Theorem 2.3.2(i) Let’s assume that F has a semiuniversal formal element (R, u). Consider a

homomorphism f : A′ → A and a small extension π : A′′ → A both in AΛ, and let

(ξ ′, ξ ′′) ∈ F(A′)×F(A) F(A′′)

withF( f )(ξ ′) = F(π)(ξ ′′) =: ξ ∈ F(A)

By the versality of (R, u) the following maps are surjective:

HomΛ−alg(R, A′) −→ F(A′)

HomΛ−alg(R, A′′) −→ HomΛ−alg(R, A)×F(A) F(A′′) (2.13)

therefore there are

g′ ∈ HomΛ−alg(R, A′), g′′ ∈ HomΛ−alg(R, A′′)

such thatF(g′)(u) = ξ ′

and g′′ → ( f g′, ξ ′′) under the map (2.13). This last condition gives:

πg′′ = f g′, F(g′′)(u) = ξ ′′

and consequently, F(πg′′)(u) = ξ . Using the morphism g′ × g′′ : R → A′ ×A A′′we obtain an element

ζ := F(g′ × g′′)(u) ∈ F(A′ ×A A′′)

which, by construction, is mapped to (ξ ′, ξ ′′) by (2.13). This proves that (R, u) sat-isfies condition H .

If A′′ = k[ε] and A = k the map (2.13) reduces to the bijection

HomΛ−alg(R, k[ε]) −→ tF (2.14)

In this case if ζ1, ζ2 ∈ F(A′ ×A k[ε]) are such that α(ζ1) = α(ζ2) = (ξ ′, ξ ′′) chooseg′ ∈ HomΛ−alg(R, A′) as before. By the versality the map

HomΛ−alg(R, A′ ×k k[ε]) −→ F(A′ ×k k[ε])×F(k[ε]) HomΛ−alg(R, k[ε])‖

F(A′ ×k k[ε])


induced by the projection A′ ×k k[ε] → k[ε] is surjective. Hence we obtain twomorphisms:

g′ × gi : R −→ A′ ×k k[ε]such that

F(g′ × gi )(u) = ζi

i = 1, 2. But then F(gi )(u) = ξ ′′, i = 1, 2, hence, by the bijectivity of (2.14),g1 = g2, i.e. ζ1 = ζ2. This proves that F satisfies condition Hε . Condition H f issatisfied because the differential tR/Λ → tF is linear and is a bijection by definitionof a semiuniversal formal couple.

Conversely, let’s assume that F satisfies conditions H , Hε and H f . We will find asemiuniversal formal couple (R, u) by constructing a projective systemRn; pn+1 : Rn+1 → Rnn≥0 of Λ-algebras in AΛ and a sequence un ∈ F(Rn)n≥0such that F(pn)(un) = un−1, n ≥ 1.

We take R0 = k and u0 ∈ F(k) its unique element. Let r = dimk(tF ),e1, . . . , er a basis of tF and letting Λ[[X ]] = Λ[[X1, . . . , Xr ]] we set

R1 = Λ[[X ]]/((X)2 + mΛΛ[[X ]])Since we have

R1 = k[ε] ×k · · · ×k k[ε] (r times)

from Hε we deduce that F(R1) = tF × · · · × tF .Let’s take u1 = (e1, . . . , er ) ∈ F(R1). Note that the map induced by u1:

kr ∼= ((X)/(X)2)∨ = HomΛ−alg(R1, k[ε]) −→ tF

is the isomorphism(λ1, . . . , λr ) −→

∑j

λ j e j

Let’s proceed by induction on n: assume that couples

(R0, u0), (R1, u1), . . . , (Rn−1, un−1)

such that

Rh = Λ[[X ]]/Jh, uh ∈ F(Rh), uh → uh−1

h = 1, . . . , n − 1 have been already constructed. In order to construct (Rn, un) weconsider the family I of all ideals J ⊂ Λ[[X ]] having the following properties:

(a) Jn−1 ⊃ J ⊃ (X)Jn−1;(b) there exists u ∈ F(Λ[[X ]]/J ) such that the map

F(Λ[[X ]]/J ) −→ F(Rn−1)

sends u → un−1.


I = ∅ because Jn−1 ∈ I. Moreover, I has a minimal element Jn . This will beproved if we show that I is closed with respect to finite intersections. Let I, J ∈ Iand K = I ∩ J . It is obvious that K satisfies condition (a). We may assume thatI + J = Jn−1 (making J larger without changing K if necessary). This implies thatthe natural homomorphism

Λ[[X ]]/K −→ Λ[[X ]]/I ×Rn−1 Λ[[X ]]/J

is an isomorphism. By H the map

α : F(Λ[[X ]]/K ) −→ F(Λ[[X ]]/I )×F(Rn−1) F(Λ[[X ]]/J )

is surjective, therefore there exists u ∈ F(Λ[[X ]]/K ) whose image in F(Rn−1) isun−1; this means that K satisfies condition (b) as well, hence it is in I.

We take Rn = Λ[[X ]]/Jn and un ∈ F(Rn) an element which is mapped to un−1.By induction we have constructed a formal couple (R, u). We now show that it is asemiuniversal formal couple for F .

As already remarked, u induces an isomorphism of tangent spaces tR/Λ∼= tF .

Therefore we only have to prove versality, namely that the map

uπ : HomΛ−alg(R, A′) −→ HomΛ−alg(R, A)×F(A) F(A′)

is surjective for every small extension π : A′ → A.Let ( f, ξ ′) ∈ HomΛ−alg(R, A)×F(A) F(A′), i.e. F( f )(u) = F(π)(ξ ′). We must

find f ′ ∈ HomΛ−alg(R, A′) such that uπ ( f ′) = ( f, ξ ′), i.e. such that

(1) π f ′ = f ; (2) F( f ′)(u) = ξ ′

Let’s consider the commutative diagram:

HomΛ−alg(R, k[ε])× HomΛ−alg(R, A′) → tF × F(A′)↓ β1 ↓ β2

HomΛ−alg(R, A′)×Hom(R,A) HomΛ−alg(R, A′) → F(A′)×F(A) F(A′)(2.15)

where the horizontal arrows are induced by u. The map β1 is a bijection because theaction of HomΛ−alg(R, k[ε]) on the fibres of

HomΛ−alg(R, A′)→ HomΛ−alg(R, A)

is free and transitive (see Remark 2.3.3). From H it follows that β2 is surjective(Remark 2.3.3 again). Therefore if f ′ ∈ HomΛ−alg(R, A′) satisfies (1) then, letting

η′ := F( f ′)(u) ∈ F(A′)

there exist v ∈ HomΛ−alg(R, k[ε]) = tF , f ′′ ∈ HomΛ−alg(R, A′) such that indiagram (2.15) we have:

(v, f ′) −→ (v, η′)↓ ↓

( f ′′, f ′) −→ (ξ ′, η′)


This means that uπ ( f ′′) = ( f, ξ ′). It follows that it suffices to find f ′ satisfyingcondition (1).

Let n > 0 be such that f factors as

R→ Rn−1fn−1−→ A

Then f ′ exists if and only if there exists ϕ which makes the following diagram com-mutative:

Rnϕ−→ A′

↓ pn ↓ πRn−1 −→ A

equivalently, if and only if the extension

f ∗n (A′, π) : 0→ ker(π) → Rn ×A A′ π ′−→ Rn → 0‖ ↓ f ′n ↓ fn

0→ ker(π) → A′ π−→ A → 0

is trivial.Suppose not. This means that π ′ induces an isomorphism of tangent spaces (see

1.1.2(ii)), i.e. that there exists an ideal I ⊂ Λ[[X ]] such that

Rn ×A A′ = Λ[[X ]]/I

By constructionJn−1 ⊃ I ⊃ (X)Jn−1

Moreover, since by H the map

F(Rn ×A A′) −→ F(Rn)×F(A) F(A′)

is surjective, there exists u ∈ F(Rn ×A A′) inducing un ∈ F(Rn), hence inducingun−1 ∈ F(Rn−1). It follows that I satisfies conditions (a) and (b) and, by the mini-mality of Jn in I, it follows that Jn ⊂ I . But this is a contradiction because from thefact that π is a surjection with nontrivial kernel it follows that I is properly containedin Jn . This proves that f ∗n (A′, π) is trivial and concludes the proof of the fact that(R, u) is semiuniversal and of part (i) of the theorem.

(ii) If F is prorepresentable then it trivially satisfies conditions H , Hε and H f ,as already remarked.

Conversely, suppose that F satisfies conditions H , H , Hε and H f . We have justproved that F has a semiuniversal formal couple (R, u). We will prove that this is auniversal formal couple by showing that for every A in AΛ the map

u(A) : HomΛ−alg(R, A) −→ F(A)

induced by u is bijective.


This is clearly true if A = k. We will proceed by induction on dimk(A). Letπ : A′ → A be a small extension in AΛ. By the inductive hypothesis

HomΛ−alg(R, A) −→ F(A)

is bijective and, by the versality, the map

uπ : HomΛ−alg(R, A′)→ HomΛ−alg(R, A)×F(A) F(A′) ∼= F(A′)

is surjective. The map β2 in diagram (2.15) is bijective by condition H , and thisimplies that uπ is bijective.

NOTES

1. Theorem 2.3.2 has been published in [155]. It had also appeared in [154]. See also[119].

2. From Theorem 2.3.2 it follows that if F has a semiuniversal element then it has atangent space which is of finite dimension, because F satisfies H0, Hε and H f . This propertywas not explicitly stated in the definition.

2.4 The local moduli functors

2.4.1 Generalities

If X is an algebraic scheme then for every A in ob(A) we let

DefX (A) = deformations of X over A/isomorphism

By the functoriality properties already observed in § 1.2 this defines a functor ofArtin rings

DefX : A→ (sets)

which is called the local moduli functor of X . If X = Spec(B0) is affine, we willoften write DefB0 instead of DefX . We can define the subfunctor

Def′X : A→ (sets)

byDef′X (A) = locally trivial deformations of X over A/isomorphism

called the locally trivial moduli functor of X .

Theorem 2.4.1. (i) For any algebraic scheme X the functors DefX and Def′X satisfyconditions H0, H , Hε of Schlessinger’s theorem. Therefore, if DefX (k[ε]) (resp.Def′X (k[ε])) is finite dimensional, then DefX (resp. Def′X ) has a semiuniversalelement.

2.4 The local moduli functors 65

(ii) There is a canonical identification of k-vector spaces

Def′X (k[ε]) = H1(X, TX ) (2.16)

In particular, if X is nonsingular then

DefX (k[ε]) = Def′X (k[ε]) = H1(X, TX )

(iii) If X is an arbitrary algebraic scheme then we have a natural identification

DefX (k[ε]) = Exk(X,OX )

and an exact sequence:

0→ H1(X, TX )τ−→ DefX (k[ε]) −→ H0(X, T 1

X )ρ−→ H2(X, TX ) (2.17)

In particular,DefB0(k[ε]) = T 1

B0

if X = Spec(B0) is affine.(iv) If X is a reduced algebraic scheme then there is an isomorphism

DefX (k[ε]) ∼= Ext1OX(Ω1

X ,OX )

and the exact sequence (2.17) is isomorphic to the local-to-global exact se-quence for Exts:

0→ H1(X, TX )→ Ext1OX(Ω1

X ,OX )

→ H0(X, Ext1OX

(Ω1X ,OX ))→ H2(X, TX )

(2.18)

Proof. Obviously DefX and Def′X satisfy condition H0. To verify the other condi-tions we assume first that X = Spec(B0) is affine.

Let’s prove that DefB0 satisfies H . Let

A′ A′′

A

be homomorphisms in A, with A′′ → A a small extension. Letting A = A′ ×A A′′we have a commutative diagram with exact rows:

0 → (ε) → A → A′ → 0‖ ↓ ↓

0 → (ε) → A′′ → A → 0(2.19)

Consider an element of

DefB0(A′)×DefB0 (A) DefB0(A′′)


which is represented by a pair of deformations f ′ : A′ → B ′ and f ′′ : A′′ → B ′′ ofB0 such that A→ B ′ ⊗A′ A and A→ B ′′ ⊗A′′ A are isomorphic deformations. As-sume that the isomorphism is given by A-isomorphisms B ′ ⊗A′ A ∼= B ∼= B ′′ ⊗A′′ A,where A→ B is a deformation. In order to check H it suffices to find a deformationf : A→ B inducing ( f ′, f ′′). Let

B = B ′ ×B B ′′

endowed with the obvious homomorphism f : A→ B. It is elementary to check thatthere are an A′-isomorphism B⊗ A A′ ∼= B ′ and an A′′-isomorphism B⊗ A A′′ ∼= B ′′.

Therefore we only need to check that f is flat. Tensoring diagram (2.19) with⊗ A B we obtain the following diagram with exact rows:

(ε)⊗ A B → B → B ′ → 0‖ ↓ ↓

0 → B0 → B ′′ → B → 0

where the second row is given by Lemma A.9. This diagram shows that

(ε)⊗ A B → B

is injective: the flatness of f follows from Lemma A.9.Let’s prove that DefB0 satisfies Hε . Assume in the above situation that A′′ = k[ε]

and A = k, and let f : A → B be a deformation such that α( f ) = ( f ′, f ′′). Thenthe diagram

B → B ⊗ A A′′ ∼= B ′′↓ ↓

B ⊗ A A′ ∼= B ′ → B0

commutes: the universal property of the fibre product implies that we have a homo-morphism γ : B → B of deformations, hence an isomorphism by Lemma A.4. Thisproves that the fibres of α contain only one element, i.e. α is bijective. ThereforeDefB0 satisfies condition Hε .

Let’s prove (i) for X arbitrary. Consider a diagram in A:

A′ A′′

A

with A′′ → A a small extension and let A = A′ ×A A′′. Consider an element

([X ′], [X ′′]) ∈ DefX (A′)×DefX (A) DefX (A′′)

Therefore we have a diagram of deformations:

X ′ X ′′ f ′ f ′′

↓ X ↓Spec(A′) ↓ Spec(A′′)

Spec(A)


where the morphisms f ′ and f ′′ induce isomorphisms of deformations

X ′ ×Spec(A′) Spec(A) ∼= X ∼= X ′′ ×Spec(A′′) Spec(A)

Consider the sheaf of A-algebras OX ′ ×OX OX ′′ on X . Then X := (|X |,OX ′ ×OXOX ′′) is a scheme over Spec( A) (by the proof of the affine case). Reducing to theaffine case one shows that X is flat over Spec( A). Therefore X is a deformation ofX over Spec( A) inducing the pair ([X ′], [X ′′]). This shows that the map

DefX ( A)→ DefX (A′)×DefX (A) DefX (A′′)

is surjective, proving H for DefX . Moreover, if the deformations X ′ and X ′′ arelocally trivial then so is X , and therefore H holds for the functor Def′X as well.

Now assume that A′′ = k[ε] and A = k. Then the previous diagram becomes

X ′ X ′′ f ′ f ′′

X

In this case any X → Spec( A) inducing the pair

([X ′], [X ′′]) ∈ DefX (A′)×DefX (A) DefX (A′′)

is such that the isomorphisms

X ×Spec( A) Spec(A′) ∼= X ′, X ×Spec( A) Spec(A′′) ∼= X ′′

induce the identity on X = X ×Spec( A) Spec(k). Therefore X fits into a commutativediagram

X

X ′ X ′′ f ′ f ′′

X

By the universal property of the fibred sum of schemes we then get a morphism ofdeformations X → X , which is necessarily an isomorphism. This proves propertyHε for DefX . The proof for Def′X is similar.

(ii) The identification (2.16) has been proved in 1.2.9. The verification that it isk-linear is elementary and will be left to the reader.

(iii) IfX ⊂ X↓ ↓

Spec(k) → Spec(k[ε])


is a first-order deformation of X then X ⊂ X is an extension of X by OX because bythe flatness of X over Spec(k[ε]) we have εOX ∼= OX (Lemma A.9). Conversely,given such an extension X ⊂ X we have an exact sequence

0→ OXj→ OX → OX → 0

OX has a natural structure of k[ε]-algebra by sending, for any open U ⊂ X , ε →j (1). It follows from Lemma A.9 that X is flat over Spec(k[ε]).

Let’s assume first that X = Spec(B0) is affine. Then

Exk(X,OX ) = H0(X, T 1B0

) = T 1B0

and the exact sequence (2.17) reduces to the isomorphism

DefB0(k[ε]) ∼= T 1B0

which holds by what we have just remarked.Let’s assume that X is general. The map τ corresponds to the inclusion

Def′X (k[ε]) ⊂ DefX (k[ε]) in view of (2.16). The map associates to a first-orderdeformation ξ of X the section of T 1

X defined by the restrictions ξ|Ui for someaffine open cover Ui . It is clear that Im(τ ) = ker().

We now define ρ. Let h ∈ H0(X, T 1X ) be represented, in a suitable affine open

cover U = Ui = Spec(Bi ) of X , by a collection of k-extensions Ei of Bi by Bi .Since h is a global section there exist isomorphisms σi j : E j |Ui∩U j

∼= Ei |Ui∩U j . Theseisomorphisms patch together to give an extension E if and only if h ∈ Im() if andonly if we can find new isomorphisms σ ′i j such that

σ ′i jσ′jk = σ ′ik (2.20)

on Ui ∩U j ∩Uk . Such isomorphisms are of the form

σ ′i j = σi jθi j

where θi j is an automorphism of the extension E j |Ui∩U j . The collection of auto-morphisms (θi j ) corresponds, via Lemma 1.2.6, to a 1-cochain (ti j ) ∈ C1(U , TX );conversely, every 1-cochain (ti j ) defines a system of isomorphisms (σ ′i j ); and thecondition (2.20) is satisfied if and only if (ti j ) is a 1-cocycle. Therefore we defineρ(h) to be the class of the 2-cocycle (ti j + t jk − tik). With this definition we clearlyhave ker(ρ) = Im(). We leave to the reader to verify that the definition of ρ doesnot depend on the choices made.

(iv) Since we have a natural identification DefX (k[ε]) = Exk(X,OX ) we con-clude by Theorem 1.1.10. The exact sequences (2.17) and (2.18) are isomorphic inview of the isomorphism

T 1X∼= Ext1

OX(Ω1

X ,OX )

given by Corollary 1.1.11, page 18.


Corollary 2.4.2. If X is a projective scheme or an affine scheme with at most isolatedsingularities then DefX has a semiuniversal formal element.

Proof. The condition implies that H1(X, TX ) and H0(X, T 1X ) are finite-dimensional

vector spaces. Therefore the conclusion follows from Theorem 2.4.1 and Sch-lessinger’s theorem.

The stronger property of being prorepresentable is not satisfied in general byDefX . We will discuss this matter in § 2.6. The case X affine will be taken up againand analysed in detail in Section 3.1.

Definition 2.4.3. If (R, u) is a semiuniversal couple for DefX then the Krull di-mension of R (i.e. the maximum of the dimensions of the irreducible componentsof Spec(R)) is called the number of moduli of X and it is denoted by µ(X).

Remark 2.4.4. Let X be a reduced scheme and let ξ : X → Spec(k[ε]) be a first-order deformation of X . Then the conormal sequence of X ⊂ X

0→ OX → Ω1X |X → Ω1

X → 0 (2.21)

is exact and defines the element of Ext1OX(Ω1

X ,OX ) which corresponds to ξ in the

identification DefX (k[ε]) = Ext1OX(Ω1

X ,OX ) of Theorem 2.4.1(iv) (see also Theo-rem 1.1.10).

Given an infinitesimal deformation ξ : X → Spec(A) of X we have a Kodaira–Spencer map κξ : tA → Ext1OX

(Ω1X ,OX ) which associates to a tangent vector θ ∈ tA

the conormal sequence 2.21 of the pullback of ξ to Spec(k[ε]) defined by θ .

Example 2.4.5. Let X be a projective curve. Then H2(X, TX ) = (0) and the exactsequence (2.17) shows that if h0(X, T 1

X ) = 0 then DefX (k[ε]) = (0) and X is notrigid. In particular, if X is reduced and has at least one nonrigid singular point then Xis not rigid. By the way, it is not known whether rigid curve singularities exist at all.

2.4.2 Obstruction spaces

The elementary analysis of obstructions to lifting infinitesimal deformations carriedout in Chapter 1 can be interpreted as the description of obstruction spaces for thecorresponding deformation functors. More precisely, we have the following:

Proposition 2.4.6. Let X be a nonsingular algebraic variety. Then H2(X, TX ) is anobstruction space for the functor DefX . If X is an arbitrary algebraic scheme thenH2(X, TX ) is an obstruction space for the functor Def′X .

Proof. The proposition is just a rephrasing of 1.2.12. Corollary 2.4.7. Let X be a nonsingular projective algebraic variety. Then

h1(X, TX ) ≥ µ(R) ≥ h1(X, TX )− h2(X, TX )

The first equality holds if and only if X is unobstructed.

Proof. It is an immediate consequence of 2.4.6 and of Corollary 2.2.11.


If X is a singular algebraic scheme the previous results give no information aboutobstructions of the functor DefX . The following proposition addresses the case of areduced local complete intersection (l.c.i.) scheme.

Proposition 2.4.8. Let X be a reduced l.c.i. algebraic scheme X, and assume char(k) =0. Then Ext2OX

(Ω1X ,OX ) is an obstruction space for the functor DefX .

Proof. Let A be in ob(A) and let

ξ :X → X↓ ↓ f

Spec(k) → Spec(A)

be a deformation of X over A. We need to define a k-linear map

oξ : Exk(A, k)→ Ext2OX(Ω1

X ,OX )

having the properties of an obstruction map according to Definition 2.2.9. Consideran element of Exk(A, k) represented by an extension

e : 0→ (t)→ A→ A→ 0

and the conormal sequence of e:

0→ (t)→ Ω A/k ⊗ A A→ ΩA/k → 0 (2.22)

which is exact by Lemma B.10. Since f is flat, pulling back (2.22) to X we obtainthe exact sequence

0→ OX → f ∗(Ω1Spec( A)|Spec(A)

)→ f ∗(Ω1Spec(A))→ 0 (2.23)

Since X is a reduced l.c.i. the morphism f satisfies the hypothesis of Theorem D.2.8.Therefore the relative cotangent sequence of f :

0→ f ∗(Ω1Spec(A))→ Ω1

X → Ω1X /Spec(A)→ 0 (2.24)

is exact. Composing (2.23) and (2.24) we obtain the 2-term extension


) → Ω1X → Ω1

X /Spec(A) → 0

which defines an element

oξ (e) ∈ Ext2OX(Ω1

X /Spec(A),OX ) = Ext2OX(Ω1

X ,OX )

This defines the map oξ . The linearity of oξ is a consequence of the linearity ofthe map Exk(A, k) → Ext1A(ΩA/k, k) associating to an extension e the conormalsequence (2.22).


Assume that there is a lifting of ξ to A, i.e. that we have a diagram:

X ⊂ X↓ f ↓ f

Spec(A) ⊂ Spec( A)

Then we have a commutative and exact diagram as follows:

0↓


) → f ∗(Ω1Spec(A)) → 0

‖ ↓ ↓0→ OX → Ω1

X |X → Ω1X → 0

↓ ↓(Ω1

X /Spec( A))|X = Ω1

X /Spec(A)

↓ ↓0 0

In this diagram the first row is the pullback of the second row and this implies thatoξ (e) = 0.

Conversely, assume that oξ (e) = 0. Then we have a commutative and exactdiagram as follows:

0↓


) → f ∗(Ω1Spec(A)) → 0

‖ ↓ ↓0→ OX → E → Ω1

X → 0↓

Ω1X /Spec(A)↓

0

for some coherent sheaf E on X . By the construction of Theorem 1.1.10 one finds asheaf of A-algebras OX and an extension of sheaves of A-algebras

0→ OX → OX → OX → 0

such that E = Ω1X |X . It remains to be shown that OX can be given a structure of

sheaf of flat A-algebras. The shortest way to see this is to use Corollary 3.1.13(ii), tobe proved in the next chapter: since X is a l.c.i. it implies that there are no obstruc-tions to lifting X locally to A. This means that the restriction of OX to every affine

open subset U ⊂ X defines a lifting of X|U to A, and therefore it is a sheaf of flatA-algebras.


Example 2.4.9. If X is a reduced l.c.i. curve then Ext1OX

(Ω1X ,OX ) is a torsion sheaf

and Ext2OX

(Ω1X ,OX ) = 0; therefore

Hi (X, Ext jOX

(Ω1X ,OX )) = (0)

for all i, j with i+ j = 2. It follows that Ext2OX(Ω1

X ,OX ) = 0 and X is unobstructedby Proposition 2.4.8.

For a discussion of obstructions in the affine case we refer the reader to Subsec-tion 3.1.2.

2.4.3 Algebraic surfaces

In this subsection we will assume char(k) = 0. We will denote by S a projectivenonsingular connected algebraic surface. Let (R, u) be a semiuniversal formal de-formation of S, and denote by

µ(S) := dim(R)

the number of moduli of S.

Proposition 2.4.10.

10χ(OS)− 2(K 2)+ h0(S, TS) ≤ µ(S) ≤ h1(S, TS) (2.25)

If h2(S, TS) = 0 then both inequalities are equalities.

Proof. A direct application of the Riemann–Roch formula gives

h1(S, TS)− h2(S, TS) = 10χ(OS)− 2(K 2)+ h0(S, TS)

By applying Corollary 2.4.7 we obtain the conclusion. The first inequality was proved by Enriques (see [52]).

Examples 2.4.11. (i) If S is a minimal ruled surface and S → C is the ruling over aprojective nonsingular curve C , then letting S(x) ∼= IP1 be the fibre of any x ∈ Cwe have

h1(S(x), TS|S(x)) = 0

as an immediate consequence of the exact sequence

0→ TS(x) → TS|S(x)→ NS(x)/S → 0‖

OS(x)

Therefore R1 p∗TS = 0 by Corollary 4.2.6, and the Leray spectral sequence impliesthat h2(S, TS) = 0. Therefore S is unobstructed and µ(S) = h1(S, TS). This com-putation for the rational ruled surfaces Fm is also done in Example B.11(iii), page288.


(ii) Assume that S is a K3-surface. Then

h2(S, TS) = h0(S,Ω1S) = h1(S,OS) = 0

Therefore S is unobstructed. Moreover,

h0(S, TS) = h2(S,Ω1S) = h1(S, ωS) = 0

and DefS is prorepresentable (Corollary 2.6.4). Formula (2.25) gives in this case

µ(S) = h1(S, TS) = 20

(iii) Let π : X → S be the blow-up of S at a point s and E = π−1(s) theexceptional curve. Then we have a standard exact sequence

0→ TXdπ−→ π∗TS → OE (−E)→ 0

(to see it one can use the exact sequence (3.50), page 172, and note that coker(dπ) =Nπ , the normal sheaf of π , which is defined on page 162). We thus see that

H2(X, TX ) ∼= H2(S, TS)

This implies, for example, that non-minimal rational or ruled surfaces and blow-upsof K3-surfaces are unobstructed.

(iv) When the Kodaira dimension of S is ≥ 1 then in general

h2(S, TS) = h0(S,Ω1 ⊗ ωS) = 0

and in fact such surfaces can be obstructed. Examples are given in Theorem 3.4.26,page 185. If we assume that h0(S, TS) = 0 the estimate for µ(S) given by Proposi-tion 2.4.10 becomes

10χ(OS)− 2(K 2) ≤ µ(S) ≤ h1(S, TS) = 10χ(OS)− 2(K 2)+ h0(S,Ω1 ⊗ ωS)

and to give an upper bound for µ(S) amounts to giving one for h0(S,Ω1 ⊗ ωS) interms of pa, K 2, q. We refer the reader to [27] for a more detailed discussion of thispoint. For examples of obstructed surfaces see also [24], [87], [96], [28].

(v) If h0(S, ωS) > 0 and q > 0 then certainly

h0(S,Ω1 ⊗ ωS) > 0

This is because there is a bilinear pairing

H0(S,Ω1S)× H0(S, ωS)→ H0(S,Ω1 ⊗ ωS)

which is non-degenerate on each factor.


For example, take S = C1 × C2 where C1 and C2 are projective nonsingularconnected curves of genera g1 ≥ 2 and g2 ≥ 2 respectively. Then h0(S, ωS) = g1g2,q = g1 + g2 and

TS = p∗1ω−1C1⊕ p∗2ω

−1C2

where pi : S→ Ci is the i-th projection, i = 1, 2. Therefore

h0(TS) = 0, h1(TS) = 3(g1 + g2)− 6, h2(TS) = g2(3g1 − 3)+ g1(3g2 − 3)

On the other hand, S is unobstructed. In fact there is a natural morphism of functors

f : DefC1 × DefC2 → DefS

which is clearly an isomorphism on tangent spaces: in fact

(DefC1 × DefC2)(k[ε]) := DefC1(k[ε])× DefC2(k[ε]) =

= H1(C1, ω−1C1

)⊕ H1(C2, ω−1C2

) = H1(S, TS)

Therefore, since DefC1 × DefC2 is smooth because both factors are (Proposition2.2.5(vii)), DefS is smooth by Corollary 2.3.7. Actually, f is an isomorphism be-cause both functors are prorepresentable (see Remark 2.3.8, page 60). This examplecan be generalized to any finite product of nonsingular projective connected curvesof genera ≥ 2.

If S is an abelian surface then

h0(S,Ω1 ⊗ ωS) = 2 = h0(S, TS)

Formula (2.25) gives2 ≤ µ(S) ≤ h1(S, TS) = 4

In fact the second equality holds because abelian surfaces are unobstructed despitethe fact that h2(S, TS) = 0. This is a property common to all abelian varieties (see[137] and [135]).

(vi) One should keep in mind that µ(S) is defined as the number of moduli ofS in a formal sense. This is because the semiuniversal formal deformation (R, u)can be non-algebraizable (see § 2.5 for the notion of algebraizability). For exam-ple µ(S) = 20 for a K3-surface, but every algebraic family of K3-surfaces withinjective Kodaira–Spencer map at every point has dimension ≤ 19 (see § 3.3). Simi-larly an abelian surface has µ(S) = 4 but every algebraic family of abelian surfaceswith injective Kodaira–Spencer map at every point has dimension ≤ 3 (see Example3.3.13). In order to give an algebraic meaning to the number of moduli one shouldtake the maximum dimension of a semiuniversal formal deformation of a pair (S, L)where L is an ample invertible sheaf on S. See § 3.3 and the appendix by Mumfordto Chapter V of [189].

2.5 Formal versus algebraic deformations 75

2.5 Formal versus algebraic deformations

We have already mentioned (see Examples 1.2.5 and 1.2.11) that infinitesimal de-formations do not explain faithfully some of the phenomena which can occur whenone considers deformations parametrized by algebraic schemes or by the spectrumof an arbitrary noetherian, or even e.f.t., local ring. In this section we will make suchstatements precise. We start with a few definitions and some terminology.

Let (S, s) be a pointed scheme. An etale neighbourhood of s in S is an etalemorphism of pointed schemes f : (T, t) → (S, s) such that the following diagramcommutes:

T

t ↓ f

Spec(k(s))s−→ S

The definition implies that k(s) ∼= k(t), i.e. f induces a trivial extension of theresidue fields at s and t ; therefore OS,s ∼= OT,t by C.7. Affine neighbourhoods of sare particular etale neighbourhoods.

Given two etale neighbourhoods (T, t) and (U, u) of s ∈ S a morphism (T, t)→(U, u) is given by a commutative diagram of pointed schemes:

(T, t) → (U, u)

(S, s)

Lemma 2.5.1. Let f : X → Y be an etale morphism and g : Y → X a section of f .Then g is etale.

Proof. Use C.2(v). Proposition 2.5.2. Let S be a scheme. The etale neighbourhoods of a given s ∈ Sform a filtered system of pointed schemes.

Proof. Given two etale neighbourhoods (S′, s′) and (S′′, s′′), they are dominated bya third, namely:

S′ ×S S′′ → S′′↓ ↓S′ → S

Now let f1, f2 : (S′′, s′′) → (S′, s′) be two morphisms between etale neighbour-hoods. Then there exists a third etale neighbourhood (S′′′, s′′′) and a morphism(S′′′, s′′′)→ (S′′, s′′) which equalizes them. In fact, consider the diagram:

S′′ ×S S′ pr ′−→ S′↓ pr ′′ ↓S′′ → S


We can shrink S′ and S′′ so that S′ is affine and S′′ is connected. Then the graphsΓ1 and Γ2 of f1 and f2 are closed, because S′ is affine, and open, because theyare images of sections of the etale morphism pr ′′, which are etale. Therefore theyare connected components of S′′×S S′. But (s′′, s′) ∈ Γ1∩Γ2 and therefore Γ1 = Γ2.It follows that f1 = f2 on S′′ = Γ1 = Γ2. Definition 2.5.3. Given a scheme S and a point s ∈ S we define the local ring of Sin s in the etale topology to be

OS,s := lim→ (S′,s′)OS′,s′

where the limit is taken for (S′, s′) varying through all the etale neighbourhoods of s.The ring OS,s is also called the henselization of OS,s . (Note that OS,s is a local ring,because it is a limit of a filtering system of local rings and local homomorphisms.)

A local ring A is called henselian if for the closed point s of S = Spec(A) onehas

A := OS,s = OS,s = A

The henselization of an e.f.t. local k-algebra is called an algebraic local ring.

Therefore the local ring in the etale topology of a point of an algebraic scheme isan algebraic local ring.

For a given scheme S and point s ∈ S there is a canonical homomorphismOS,s → OS,s which is flat and induces an isomorphism of the completions

OS,s ∼= OS,s

because every OS,s → OS′,s′ does. Moreover,

OS,s ⊂ OS,s ⊂ OS,s

because OS,s → OS,s is faithfully flat and OS,s is separated for the m-adic topology.In particular, we see that OS,s = OS,s if OS,s = OS,s , i.e. a local k-algebra in A(i.e. complete noetherian with residue field k) is henselian.

Theorem 2.5.4 (Nagata). If A is a noetherian local ring then A is noetherian.

Proof. We have A ⊂ A ⊂ A and A = A. Moreover,

A = lim→ A′

with A′ local algebras etale over A and inducing trivial residue field extension. Toprove that A is noetherian it suffices to prove that every ascending chain of finitelygenerated ideals of A

a1 ⊆ a2 ⊆ · · · ⊆ an ⊆stabilizes. The chain an A stabilizes because A is noetherian. Therefore it sufficesto prove that if a, b ⊂ A are finitely generated ideals such that a A = bA then


a = b. Since a and b are finitely generated one can find A′ ⊃ A as above and finitelygenerated ideals a′, b′ ⊂ A′ such that a = a′ A, b = b′ A. It follows that a′ A = b′ A.But since A′ is noetherian it follows that a′ = b′ and therefore a = b.

The following proposition gives a geometrical characterization of thehenselization.

Proposition 2.5.5. Let A be a local ring, S = Spec(A), s ∈ S the closed point. A ishenselian if and only if every morphism f : Z → S such that there is a point z ∈ Zwith f (z) = s, k(s) = k(z) and f etale at z, admits a section.

Proof. Assume the condition is satisfied. If A → A′ is an etale homomorphisminducing an isomorphism of the residue fields then the induced morphism f :Spec(A′)→ S admits a section, which defines an isomorphism A′ ∼= A; therefore Ais henselian. Conversely, assume A is henselian and let f : Z → S be a morphismsatisfying the stated conditions. Then f induces an isomorphism A ∼= OZ ,z becauseA is henselian. The section is the composition

S = Spec(A) ∼= Spec(OZ ,z) ⊂ Z

∗ ∗ ∗ ∗ ∗ ∗

We will need the notion of “formal deformation of an algebraic scheme”, alreadyintroduced in § 2.2 for a general functor of Artin rings. Let X be an algebraic schemeand let A be in ob(A). Then a formal deformation of X over A is just a formalelement η ∈ DefX ( A) for DefX : by definition it is a sequence ηn of infinitesimaldeformations of X

ηn :X

fn−→ Xn

↓ ↓ πn

Spec(k) → Spec( An)

where An = A/mn+1A

, such that for all n ≥ 1 ηn induces ηn−1 by pullback under the

natural inclusion Spec( An−1)→ Spec( An), i.e. we have:

ηn−1 :X

fn−1−→ Xn ⊗ AnAn−1 = Xn−1

↓ ↓ πn−1Spec(k) → Spec( An−1)

We will denote such a formal deformation by ( A, ηn) or by ( A, η). It can be alsoviewed as the morphism of formal schemes

π : XX → Specf( A)

whereXX = (X, lim← OXn ), π = lim← πn

(see [84] and [16] for the definition and main properties of formal schemes).


Note that π is a flat morphism of formal schemes, namely for every x ∈ Xthe local ring OXX ,x is flat over A = OSpecf( A),π(x). This is an almost immediateconsequence of the general version of the local criterion of flatness ([3], Expose IV,Corollaire 5.8).

The formal deformation ( A, ηn) will be called trivial (resp. locally trivial) ifeach ηn is trivial (resp. locally trivial).

A formal deformation ( A, ηn) is not to be confused with a deformation of Xover Spec( A), just as a formal scheme over Specf( A) is not in general a formal com-pletion of a scheme over Spec( A) (see Definition 2.5.10 below and the discussionfollowing it).

Let X be a projective scheme and consider a flat family of deformations of Xparametrized by an affine scheme S = Spec(B), with B in (k-algebras)

η :X

f−→ X↓ ↓ π

Spec(k)s−→ S

namely a cartesian diagram with π projective and flat. The deformation η is calledalgebraic if B is a k-algebra of finite type. If B is in ob(A∗) then η is called a localdeformation of X . If B is in ob(A) we obtain an infinitesimal deformation of X ; itis simultaneously local and algebraic. We will identify the deformation η with thecouple (S, η) or (B, η) and we will also denote it by (S, s, η) or (B, s, η).

Given such a deformation (S, s, η) let ηn be the infinitesimal deformation in-duced by pulling back η under the natural closed embedding

Spec(OS,s/mn+1)→ S

We have OS,s/mn+1 = OS,s/mn+1 and therefore it follows that (OS,s, ηn) is aformal deformation of X . It will be called the formal deformation defined by (orassociated to) η.

(S, s, η) is called formally trivial (resp. formally locally trivial) if the formaldeformation defined by η is trivial (resp. locally trivial).

Lemma 2.5.6. Let (S, s, η) be a deformation of an algebraic scheme X, f : (S, s)→(S, s) an etale neighbourhood of s in S and (S, s, η) the deformation of X obtainedby pulling back η by f . Then the formal deformations of X defined by η and by η areisomorphic.

Proof. We have a cartesian diagram of formal schemes:

XX → XX↓ ¯π ↓ π

Specf(OS,s)→ Specf(OS,s)

where ¯π and π are the formal deformations defined by η and by η respectively. Sincef is etale it induces an isomorphism OS,s ∼= OS,s and the conclusion follows.


Definition 2.5.7. A deformation (S, s, η) of X is called formally universal (resp. for-mally semiuniversal, formally versal) if the formal deformation (OS,s, ηnn≥0) as-sociated to η is universal (resp. semiuniversal, versal). An algebraic formally versaldeformation of X is also called with general moduli.

A flat family π : X → S is called formally universal (resp. formally semiuniver-sal, formally versal, with general moduli) at a k-rational point s ∈ S if

η :X (s) ⊂ X↓ ↓ π

Spec(k)s−→ S

is a formally universal, (resp. formally semiuniversal, formally versal, with generalmoduli) deformation of X (s).

The expression “general moduli” goes back to the classical geometers. Infor-mally, it means that the family parametrizes all possible “sufficiently small” deforma-tions of X (s); when the family π parametrizes varieties for which there is a modulischeme, or just a moduli stack (whatever this means for the reader since we have notintroduced these notions), π with general moduli means that the functorial morphismfrom S to the moduli scheme or stack is open at s. An expression like “consider avariety X with general moduli” is used to mean: “choose X as a general fibre in afamily with general moduli”.

The early literature on deformation theory of complex analytic manifolds (in theapproach of Kodaira and Spencer) considered only families parametrized by complexanalytic manifolds. In that context the expression “effectively parametrized” wasused to mean “with injective Kodaira–Spencer map”, and the word “complete” wasused to mean the analogue of the notion of formal versality, in the category of germsof complex analytic manifolds.

Proposition 2.5.8. Let (S, s, η) be a deformation of X. Then:

(i) If η is formally versal (resp. formally semiuniversal or formally universal) thenthe Kodaira–Spencer map

κπ,s : Ts S→ DefX (k[ε])is surjective (resp. an isomorphism).

(ii) If S is nonsingular at s and the Kodaira–Spencer map κπ,s is surjective (resp. anisomorphism) then η is formally versal (resp. formally semiuniversal) and X isunobstructed, i.e. the functor DefX is smooth.

Proof. (i) is obvious in view of the definitions of versality and semiuniversality of aformal couple applied to hOS,s

→ DefX .(ii) follows from Proposition 2.2.5(iii) applied to f : hOS,s

→ DefX . The proposition applies in particular to an algebraic deformation, giving a crite-

rion for it to have general moduli. A classical result (see [107]) states the complete-ness of a complex analytic family of compact complex manifolds if the map κπ,s is


surjective. Part (ii) of Proposition 2.5.8 is the algebraic version of this result. It turnsout to be very useful because it reduces the verification of formal versality and ofunobstructedness to the computation of the Kodaira–Spencer map.

Definition 2.5.9. A formal deformation ( A, ηn) of X is called algebraizable if thereexists an algebraic deformation (S, s, ξ) of X and an isomorphism OS,s ∼= A sendingηn to ξn for all n (i.e. ( A, ηn) is isomorphic to the formal deformation defined byξ ). The deformation (S, s, ξ) is called an algebraization of ( A, ηn).

It goes without saying that an algebraization of a formal versal (resp. semiuniver-sal, universal) deformation is formally versal (resp. formally semiuniversal, formallyuniversal). The existence of algebraizations is a highly nontrivial problem. It can beconsidered as the counterpart of the convergence step in the construction of localfamilies of deformations in the Kodaira–Spencer theory of deformations of com-pact complex manifolds. But the algebraic case presents some characteristic featureswhich make the two theories radically different in methods and in results. In par-ticular, a projective algebraic variety X defined over the field of complex numbersneed not have an algebraic formally versal deformation, even in the case when DefX

is prorepresentable and unobstructed (see Example 2.5.12 below); but, according tothe KNSK theory (see the Introduction), such a variety has a complete deformationin the complex analytic sense.

∗ ∗ ∗ ∗ ∗ ∗For the rest of this section we will need to assume some familiarity with formal

schemes: for some definitions and results not contained in [84] we will refer to [1].The following definition introduces an important notion weaker than algebraizability.

Definition 2.5.10. Let X be an algebraic scheme and let A be in ob(A). A formaldeformation ( A, η) of X is called effective if there exists a deformation

η :X → X↓ ↓ π


of X over A such that η is the formal deformation associated to η. Equivalently,letting π : XX → Specf( A) be the morphism of formal schemes associated to theformal deformation ( A, η), its effectivity means that there is a deformation η suchthat XX is the formal completion of X along X and π is the morphism of formalschemes induced by π ; in symbols:

XX ∼= Specf( A)×Spec( A) X

It is obvious that the trivial formal deformation XX → Specf( A), where

XX = X × Specf( A) = (X, lim← OX ⊗k A/mn)

is effective: it is the formal completion of the trivial deformation


X × Spec( A)→ Spec( A)

In particular, for any A in ob(A) the formal projective space over A

PrA:= (IPr , lim← OIPr

An)

is effective, being the formal completion of IPr × Spec( A) along the closed fibreIPr = IPr×Spec(k). A consequence of Grothendieck’s theorem of formal functionsis that a formal deformation of a proper algebraic scheme can be effective in at mostone way; more precisely we have:

Theorem 2.5.11. Let X be a proper algebraic scheme. If

η :X → X↓ ↓ π

Spec(k) → Spec( A)ξ :

X → Y↓ ↓ q


are two deformations of X with A in ob(A) such that the associated formal defor-mations ( A, η), ( A, ξ ) are isomorphic, then η and ξ are isomorphic deformations.

Proof. Let S = Spec( A) and S = Specf( A). For a given proper S-scheme f : Z →S we denote by Z the formal completion of Z along Z(s) = f −1(s), where s ∈ Sis the closed point, and by F the OZ -sheaf obtained as the formal completion ofa quasi-coherent OZ -sheaf F . It suffices to show that the correspondence ϕ → ϕdefines a one-to-one correspondence

HomS(X ,Y)→ HomS(X , Y)

because it will follow that X ∼= Y , i.e. ( A, η) ∼= ( A, ξ ), implies X ∼= Y .The proof goes as follows. For any proper S-scheme f : Z → S and coherent

sheaves F and G on Z the theorem of formal functions ([84], p. 277) implies thatthere is an isomorphism

Hom(F ,G)∧ ∼= lim← HomOZn(Fn,Gn) = HomZ (F, G)

Moreover, since Hom(F ,G) is a finitely generated A-module, it coincides with itsm A-adic completion ([127], Theorem 8.7, p. 60); thus we have an isomorphism

Hom(F ,G) ∼= HomZ (F , G)

It maps a homomorphism u : F → G to its completion u : F → G. It is easy tosee that u is injective (resp. surjective) if and only if u is injective (resp. surjective)([1], ch. III, Cor. 5.1.3). In particular, we have a one-to-one correspondence betweensheaves of ideals of OZ and of OZ , equivalently between closed subschemes of Z

and closed formal subschemes of Z .Now take Z = X ×S Y and view it as an S-scheme by any of the projections.

Then we have a natural identification Z = X ×S Y ([1], ch I, Prop. 10.9.7). We


deduce a one-to-one correspondence between closed subschemes of X ×S Y andof X ×S Y , in particular between graphs of S-morphisms X → Y and graphs of

morphisms X → Y of S-formal schemes. In general, not every formal deformation of a given algebraic scheme need to be

effective, just as deformations of algebraic varieties defined over the field of complexnumbers need not be algebraic. Here is a typical example.

Example 2.5.12 (K3-surfaces). For this example we will need a result to be provedin Chapter 3. Let X be a projective K 3-surface, and assume for simplicity thatPic(X) = ZZ [H ] for some divisor H . Let ( A, η) be the formal universal deforma-tion of X and XX the corresponding formal scheme over S = Specf( A). By Example2.4.11(ii) we know that

A ∼= k[[X1, . . . , X20]]where X1, . . . , X20 are indeterminates.

Claim: ( A, η) is not effective.Otherwise there is a proper smooth morphism f : X → S = Spec( A) such

that X ∼= XX , where X is the formal completion of X along X = f −1(s), s ∈S the closed point. Since X is of finite type over the integral scheme S, it has anontrivial line bundle L which we may assume to correspond to a Cartier divisorwhose support does not contain X and has a nonempty intersection with X ; thereforeL has a nontrivial restriction to the closed fibre X , say L ⊗ OX ∼= nH , n = 0.By [1], Ch. III, Cor. 5.1.6, L corresponds to a line bundle L on XX which extendsL ⊗ OX ∼= nH . But nH cannot be extended to all of XX because it extends to a19-dimensional space of first-order deformations of X (see page 150). Therefore wehave a contradiction and the claim is proved.

The following is a basic result of effectivity of formal deformations.

Theorem 2.5.13 (Grothendieck [71]). Let X be a projective scheme. Then:

(i) Let π : XX → Specf( A), A in ob(A), be a formal deformation of X. Assume thatthere is a closed embedding of formal schemes j : XX ⊂ Pr

Asuch that π = pj

where p : PrA→ Specf( A) is the projection. Then π is effective.

(ii) Assume that h2(X,OX ) = 0. Then every formal deformation of X is effective.

Proof. (i) The structure sheaf OXX and the ideal sheaf IXX are coherent sheaves ofOPr

A-modules. Repeating verbatim the proof of the classical result of Serre (see [84],

Theorem II.5.15, p. 121), one shows that there is an exact sequence

N⊕i=1

OPrA(−ni )

g−→ OPrA→ OXX → 0

for some positive integers n1, . . . , nN . We have


g ∈N⊕

i=1

H0(PrA,OPr

A(ni ))

and, by the theorem on formal functions ([84], ch. III, Theorem 11.1) we have

H0(PrA,OPr

A(ni )) = H0(IPr

A,OIPr

A(ni ))

∧

Since H0(IPrA,OIPr

A(ni )) is an A-module of finite type we have

H0(IPrA,OIPr

A(ni ))

∧ = H0(IPrA,OIPr

A(ni ))

([127], Theorem 8.7, p. 60) and therefore

H0(PrA,OPr

A(ni )) = H0(IPr

A,OIPr

A(ni ))

This implies that the homomorphism g is induced by a homomorphism

g :N⊕

i=1

OIPrA(−ni )→ OIPr

A

whose cokernel is the structure sheaf of a closed subscheme X ⊂ IPrA

having XXas formal completion along X (o). The flatness of X over Spec( A) follows fromTheorem A.5.

(ii) Let ( A, ηn) be a formal deformation of X , where:

ηn :X

fn−→ Xn

↓ ↓ πn

Spec(k) → Spec( An)

For each n ≥ 1 we have an exact sequence:

0→ mnA

mn+1A

⊗k OXexp−→ O∗Xn

ρ−→ O∗Xn−1→ 0

where ρ is the natural restriction, exp(σ ) = 1+ σ and where we have identified

mnA

mn+1A

⊗k OX =mn

A

mn+1A

⊗ AnOXn

using the flatness of πn . From the hypothesis h2(X,OX ) = 0 we deduce that thehomomorphism of Picard groups Pic(Xn) → Pic(Xn−1) is surjective. This impliesthat if we fix a very ample line bundle L on X such that H1(X, L) = 0 then for eachn ≥ 1 we can find a line bundle Ln on Xn such that Ln|Xn−1 = Ln−1. By Note 3of § 4.2, Ln is very ample and defines an embedding Xn ⊂ IP N

Anwhere N + 1 =

h0(X, L). This means that the formal scheme XX is a formal closed subscheme ofPN

A. Now we conclude by part (i). After these preliminaries we can state the following special case of Artin’s alge-

braization theorem.


Theorem 2.5.14 (Artin [12]). Let X be a projective scheme and let ( A, η) be aneffective formal versal deformation of X. Then ( A, η) is algebraizable.

The universal formal deformation ( A, η) of a K3-surface is not effective (Exam-ple 2.5.12) and in fact it is not algebraizable: this is well known if k = C becauseK3-surfaces have arbitrarily small deformations which are non-algebraic complexmanifolds.

∗ ∗ ∗ ∗ ∗ ∗To give a complete proof of Theorem 2.5.14 is beyond our goals. Nevertheless,

we want to introduce some relevant definitions and preliminary results which areneeded for its general statement (see below) and proof.

Consider a covariant functor

F : (k-algebras)→ (sets)

Suppose given R in ob(A∗) and an element u ∈ F(R). An important question iswhether it is possible to approximate u in some way by an element u ∈ F(R). Indeed,every algebraization problem can be reduced to such a question for an appropriatefunctor F . In this context the following is a natural definition of approximation:

Definition 2.5.15. Suppose given R in ob(A∗), u ∈ F(R) and u ∈ F(R). If c > 0 isan integer we say that u and u are congruent modulo mc, in symbols

u ≡ u (modulo mc)

if u and u induce the same element in F(R/mcR) = F(R/mc

R).

In order to make the problem tractable it turns out to be natural to impose thefollowing finiteness condition on the functor F .

Definition 2.5.16. A functor


is said to be locally of finite presentation if for every filtering inductive system ofk-algebras Bi the canonical map

lim→ F(Bi )→ F(lim→ Bi )

is bijective. A functor locally of finite presentation is sometimes also called limitpreserving.

This is a natural finiteness property, first introduced in [1], ch. IV, Prop. 8.14.2,which is usually satisfied by the functors arising in algebraic geometry.

An illustration of the meaning of Definition 2.5.16 is given by the followingproposition and by its corollary.


Proposition 2.5.17. Let R be a noetherian ring and B an R-algebra. Then B is an R-algebra of finite type if and only if for every filtering inductive system of R-algebrasCi the canonical map

lim→[HomR−alg(B, Ci )

]→ HomR−alg(B, lim→ Ci )

is bijective.

Proof. Assume that the condition of the statement is satisfied, and write

B = lim→ Bi

where the Bi ⊂ B are R-subalgebras of finite type. Then the hypothesis implies thatthe identity B → B factors as B → Bi → B for some i , and this implies B = Bi .Therefore B is of finite type.

Conversely, assume that B is of finite type, and let t1, . . . , tn be a system of gen-erators of B as an R-algebra. Let Ci be a filtering inductive system of R-algebras,C = lim Ci and let

lim→[HomR−alg(B, Ci )

]→ HomR−alg(B, C) (2.26)

be the natural map. Consider two elements of lim[HomR−alg(B, Ci )

], given by two

compatible systems of homomorphisms (θi ) and (θ ′i ), defining the same homomor-phism θ : B → C . Then, letting ϕi : Ci → C and ϕ j i : Ci → C j be the canonicalhomomorphisms of the inductive system, for each s = 1, . . . , n there is an index is

such that

θ(ts) = ϕis (θis (ts)) = ϕis (θ′is(ts))

We may assume that all the is are equal, say to i0. In the same way we can find anindex j0 ≥ i0 such that

ϕ j0i0(θi0(ts)) = ϕ j0i0(θ′i0(ts))

for all s = 1, . . . , n; this means that θ j0(ts) = θ ′j0(ts), i.e. θ j0 = θ ′j0 . Therefore themap (2.26) is injective.

Now let θ : B → C be a homomorphism, and let cs = θ(ts), s = 1, . . . , n. WriteB = R[T1, . . . , Tn]/J , and let Pkk=1,...,m be a system of generators of the ideal J .Then we have

Pk(c1, . . . , cn) = θ(Pk(t1, . . . , tn)) = 0, k = 1, . . . , m

Let i0 be an index such that there exist elements x1, . . . , xn ∈ Ci0 such that ϕi0(xs) =cs , i = 1, . . . , n. One has

ϕi0(Pk(x1, . . . , xn)) = Pk(c1, . . . , cn) = 0 k = 1, . . . , m


Then there is an index j0 ≥ i0 such that

0 = ϕ j0i0((Pk(x1, . . . , xn)) = Pk(ϕ j0i0(x1), . . . , ϕ j0i0(xn)) k = 1, . . . , m

One deduces the existence of a k-algebra homomorphism θ j0 : B → C j0 such thatθ j0(ts) = ϕ j0i0(xs), s = 1, . . . , n. This implies the existence of a homomorphismθ j = ϕ j j0θ j0 : B → C j for every j ≥ j0. It follows that θ is the inductive limit ofthe system (θ j ), and therefore the map (2.26) is also surjective.

The following is an immediate consequence.

Corollary 2.5.18. Let B be a k-algebra and

F = Homk−alg(B,−) : (k-algebras)→ (sets)

Then F is locally of finite presentation if and only if B is a k-algebra of finite type.

Therefore we see that the condition of being locally of finite presentation for arepresentable functor coincides with the condition of being represented by an algebraof finite type.

For the statement of the algebraization theorem we need some terminology whichgeneralizes notions we have already introduced.

Consider a covariant functor


Let u0 ∈ F(k). A couple (A, u) where A is in ob(A) and u ∈ F(A) is called aninfinitesimal deformation of u0 if u → u0 under the map

F(A)→ F(A/m A) = F(k)

We will denote byFu0 : A→ (sets)

the functor of infinitesimal deformations of u0, i.e.:

Fu0(A) = u ∈ F(A) : u → u0 under F(A)→ F(k)This is a functor of Artin rings and, according to the definitions given in § 2.2, wehave the notions of formal deformation of u0, and of universal (resp. semiuniversal,resp. versal) formal deformation of u0.

Definition 2.5.19. A local deformation of u0 ∈ F(k) is a couple (A, u) where u ∈F(A), A is in A∗ and u → u0 under the map

F(A)→ F(A/m A) = F(k)

An algebraic deformation of u0 is a triple (B, x, u) where B is a k-algebra of finitetype, x ∈ Spec(B) is a k-rational point and u ∈ F(B) is such that u → u0 underthe map F(B)→ F(k) induced by the composition

B → OSpec(B),x → k(x) = k


To every local deformation (A, u) of u0 one associates the formal deformation( A, un), where A is the m A-adic completion of A, and un ∈ F(A/mn+1

A ) =F( A/mn+1

A) is the image of u under the map F(A) → F(A/mn+1

A ). We call

( A, un) the formal deformation of u0 associated to (or defined by) (A, u). Simi-larly, the formal deformation of u0 defined by an algebraic deformation (B, x, u) ofu0 is the formal deformation (OSpec(B),x , un) where u → un under the map

F(B)→ F(OSpec(B),x/mn+1)

induced by the natural homomorphism

B → OSpec(B),x/mn+1 = OSpec(B),x/mn+1

A local deformation is called formally universal (resp. formally semiuniversal, resp.formally versal) if the associated formal deformation has the corresponding property.A similar definition is given for algebraic deformations.

Definition 2.5.20. A formal deformation ( A, un) of u0 ∈ F(k) is called algebraiz-able if it is associated to some algebraic deformation (B, x, u). In this case the de-formation (B, x, u) is called an algebraization of ( A, un).

A formal deformation ( A, un) of u0 is called effective if there exists u ∈ F( A)which induces un. In this case we will call ( A, u) an effective formal deformationas well.

It is clear from the definition that an algebraic deformation (B, x, u) of u0 is analgebraization of the effective formal deformation ( A, u) if and only if A = OX,x

and u ≡ u mod mn for all n ≥ 0.Note that if the formal deformation ( A, un) of u0 is associated to a local de-

formation (A, u) then it is effective: in fact it follows that un is also associated tothe deformation u ∈ F( A) which is the image of u under the map F(A) → F( A).Similarly, an algebraizable deformation is effective.

The algebrization theorem states that the converse is true for versal deformations.Namely:

Theorem 2.5.21 (algebraization theorem [12]). Let


be a functor locally of finite presentation and let u0 ∈ F(k). Then every effectiveversal formal deformation of u0 is algebraizable.

Theorem 2.5.21 is a generalization of Theorem 2.5.14 because the deformationfunctor DefX of a projective scheme X can be extended to a functor locally of finitepresentation defined on the category (k-algebras).


The main technical ingredient in the proof of the algebraization theorem is thefollowing:

Theorem 2.5.22 (approximation theorem [11]). Let


be a functor locally of finite presentation, A an e.f.t. local k-algebra and u ∈ F( A).Then for every positive integer c there is an element u ∈ F( A) such that u ≡ umodulo mc.

Outline of the proof of Theorem 2.5.21 (in a special case, when it follows almostdirectly from the approximation theorem.)

Let ( A, u) be an effective formal versal deformation of u0. We will assume thatA = A, where A is an e.f.t. local k-algebra in A∗. Suppose that we can find u ∈ F( A)whose associated formal deformation is un. Then, since

A = lim→ OS,s

where (S, s) runs through all the etale neighbourhoods of (Spec(A), m A), and sinceF is locally of finite presentation there is an etale neighbourhood (Spec(B), x) of(Spec(A), m A) and u ∈ F(B) such that u → u under F(B) → F( A). Again,because F is locally of finite presentation we may assume that Spec(B) is an affinealgebraic scheme: therefore (B, x, u) is an algebraization of ( A, u). Therefore weonly need to find u as above. By the approximation theorem there exists u ∈ F( A)such that u ≡ u mod m2. Therefore the homomorphism ψ1 : A → A/m2 sendsu → u1 = u1. By versality there is a compatible sequence of homomorphismsψn : A → A/mn+1 lifting ψ1 and such that F(ψn)(u) = un for all n. We obtain

an induced local homomorphism ψ : A → A = A such that F(ψ)(u) ≡ un modmn+1 for all n. It suffices to prove that ψ is an isomorphism. By construction ψ isthe identity mod m2: the conclusion now follows from Lemma C.5.

NOTES

1. The terminology introduced in Definition 2.5.10 is the most commonly used today butit differs from Grothendieck’s: in [71] he calls a formal deformation “algebraizable” if it iseffective. The same terminology is used in [84], p. 195.

2. Other references for Theorem 2.5.13 are [1], ch. III, Th. 5.4.5, and [3], Exp. III, Prop.7.2.

2.6 Automorphisms and prorepresentability 89

2.6 Automorphisms and prorepresentability

. . . chaque fois que . . . une variete de modules . . . ne peut exister, malgre de bonnes hypothesesde platitude, proprete, et non singularite eventuellement, la raison en est seulement l’existenced’automorphismes de la structure . . .(Grothendieck [40], p. 94.)

2.6.1 The automorphism functor

The following theorem gives a criterion on an algebraic scheme X to decide whetherDefX , resp. Def′X , has a universal formal element and not merely a semiuniversalone.

Theorem 2.6.1. Assume that X is an algebraic scheme such that DefX has a semiu-niversal element (e.g. X affine with isolated singularities or X projective). Then thefollowing conditions are equivalent:

(i) DefX is prorepresentable(ii) for each small extension A′ → A in A, and for each deformation X ′ of X over

Spec(A′), every automorphism of the deformation

X ′ ×Spec(A′) Spec(A)→ Spec(A)

is induced by an automorphism of X ′.A similar statement holds for the functor Def′X .

Proof. (i)⇒ (ii) Let X = X ′ ×Spec(A′) Spec(A) and let f : X → X ′ be the inducedmorphism; assume that θ is an automorphism of X . Letting A = A′ ×A A′, one canconstruct two deformations Z and W of X over A as we did in the proof of Theorem2.4.1 as fibred sums fitting into the two diagrams:

Z

X ′ X ′ f θ f

X

W

X ′ X ′ f f

X

Since [Z], [W] ∈ DefX ( A) have the same image ([X ′], [X ′]) under the map

DefX ( A)→ DefX (A′)×DefX (A) DefX (A′)

and since this map is bijective by (i), we have an isomorphism of deformationsρ : Z ∼= W . The isomorphism ρ induces automorphisms θ1 and θ2 of X ′ and anautomorphism ψ of X such that:

θ1 f θ = f ψ, θ2 f = f ψ

This equality implies f θ = θ−11 θ2 f :


X ′θ−1

1 θ2−→ X ′↑ f ↑ f

X θ−→ X

and therefore θ−11 θ2 induces θ .

(ii) ⇒ (i) Since DefX has a semiuniversal element, it suffices to show that itsatisfies condition H of Theorem 2.3.2. Let A′ → A be a small extension in A:letting A = A′ ×A A′ we must show that the map

α : DefX ( A)→ DefX (A′)×DefX (A) DefX (A′)

is bijective. Given deformations X ′ and X ′ of X over A′ inducing the deformation Xover A, we have the “fibred sum” deformation X over A, which fits into the diagram:

X

X ′ X ′ f f

X

and satisfies α([X ]) = ([X ′], [X ′]). Suppose that Z is another deformation of Xover A such that α([Z]) = ([X ′], [X ′]). We have isomorphisms of deformationsinduced by the two projections:

X ′ ∼= Z ×Spec( A) Spec(A′) ∼= X ′

There remains induced an automorphism θ of X as the composition:

X ∼= X ′ ×Spec(A′) Spec(A) ∼= Z ×Spec( A) Spec(A) ∼= X ′ ×Spec(A′) Spec(A) ∼= X

and θ fits into the commutative diagram:

Z

X ′ X ′ f f

X θ−→ X

By (ii) we can lift θ to an automorphism σ : X ′ ∼= X ′. Replacing the lower left mapf by σ f we obtain the commutative diagram

Z

X ′ X ′ σ f f

X

By the universal property of the fibred sum we obtain an isomorphism X ∼= Z whichis an isomorphism of deformations. Therefore [Z] = [X ] and α is bijective.


In the case of Def′X the proof is similar. When X is a projective scheme, condition (ii) of Theorem 2.6.1 can be stated in

a different way by means of the automorphism functor, which we now introduce.Assume that X is an algebraic scheme such that DefX has a semiuniversal couple

(R, u). Consider the functor of Artin rings

Autu : AR → ( sets)Autu(A) = the group of automorphisms of the deformation XA

where XA is the deformation induced by u under the morphism R → A. Then wehave the following:

Proposition 2.6.2. If X is projective then Autu has H0(X, TX ) as tangent space andis prorepresented by a complete local R-algebra S. Moreover, the deformation func-tor DefX is prorepresentable if and only if S is a formally smooth R-algebra, i.e. ifit is a power series ring over R.

Proof. Obviously, Autu satisfies condition H0 because by definition the only auto-morphism of the deformation Xk = X is the identity. Now consider a diagram inAR :

A′ A′′

A

with A′′ → A a small extension and let A = A′ ×A A′′. There is induced a diagramof deformations:

X AXA′ XA′′

XA

and therefore a natural homomorphism:

OX A→ OXA′ ×OXA

OXA′′ (2.27)

Since OXA′ = OX A⊗ A A′ it follows from Lemma A.4 that (2.27) is an isomorphism;

in particular, we obtain an induced isomorphism

O∗X A

∼= O∗XA′ ×O∗XAO∗XA′′

and therefore

H0(O∗X A) ∼= H0(O∗XA′ )×H0(O∗XA

) H0(O∗XA′′ ) (2.28)

Now note that for every A in ob(AR) the elements of Autu(A) are identified withthose elements of H0(O∗XA

) which restrict to 1 ∈ O∗X . Hence we see that (2.28)immediately implies the bijection

Autu( A) ∼= Autu(A′)×Autu(A) Autu(A′′)

Therefore the functor Autu also satisfies conditions H and Hε .


From Lemma 1.2.6 it follows that

Autu(k[ε]) ∼= H0(X, TX ) (2.29)

which has finite dimension since X is projective, and also H f is satisfied. This con-cludes the proof of the first part.

Condition (ii) of Theorem 2.6.1 can be rephrased by saying that the functor Autuis smooth over DefX .

H0(X, TX ) is usually called the space of infinitesimal automorphisms of X . As acorollary we obtain:

Corollary 2.6.3. If X is a projective scheme then the following conditions are equiv-alent:

(i) h0(X, TX ) = 0, i.e. X has no infinitesimal automorphisms;(ii) Autu ∼= DefX ;(iii) every infinitesimal deformation of X has no nontrivial automorphisms.

Proof. It is an immediate consequence of the proposition. It can be also proved di-rectly without using Proposition 2.6.2: just observe that if H0(X, TX ) = 0 then usingLemma 1.2.6 one shows by induction that every infinitesimal deformation of X hasno automorphisms.

An important application of the above proposition is the following result, whichcan be considered as the scheme-theoretic version of a classical theorem due toKodaira–Nirenberg–Spencer [106]:

Corollary 2.6.4. If X is a projective scheme such that h0(X, TX ) = 0 then DefX

is prorepresentable. If, moreover, X is nonsingular and h2(X, TX ) = 0 thenDefX is prorepresented by a formal power series ring.

Proof. From (2.29) it follows that S = R if h0(X, TX ) = 0 and in particular S is aformally smooth R-algebra. Then the first part follows from Proposition 2.6.2. Thelast assertion is a consequence of Corollary 2.4.7.

The condition h0(X, TX ) = 0 (no infinitesimal automorphisms) implies that thegroup Aut(X) is finite. We thus see that the existence of a nontrivial automorphismgroup does not prevent DefX from being prorepresentable provided there are no in-finitesimal automorphisms. On the other hand, the existence of any automorphismsis a source of difficulties when one considers local deformations (see Subsection2.6.2).

The prorepresentability criterion for DefX given by Proposition 2.6.2 is not easyto apply when h0(X, TX ) > 0. Note that the condition h0(X, TX ) = 0 is notnecessary for the prorepresentability of DefX . An example is given by X = IPr ,r ≥ 1: in this case DefX is trivially prorepresentable because X is rigid, buth0(X, TX ) = (r + 1)2 − 1 > 0. For another example see the following proposi-tion.

Corollary 2.6.4 can be generalized in a straightforward way to conclude thatany functor of Artin rings F classifying isomorphism classes of deformations of


a scheme with some additional structure or of any other algebro-geometric object (a morphism, etc.) is prorepresentable provided F has a semiuniversal element and has no infinitesimal automorphisms. This remark will be applied in the proof ofthe following:

Proposition 2.6.5. Let X be a projective irreducible and nonsingular curve of genus1. Then DefX is prorepresentable.

Proof. Fix a closed point p ∈ X . For each A in ob(A) we define a deformation ofthe pointed curve (X, p) to be a pair (ξ, σ ) where

ξ :X → X↓ ↓ π

Spec(k) → Spec(A)

is an infinitesimal deformation of X over A and σ : Spec(A) → X is a section ofπ such that Im(σ ) = p. We have an obvious definition of isomorphism of twodeformations of (X, p) over A, and we define a functor of Artin rings

Def(X,p)→ (sets)

byDef(X,p)(A) = isom. classes of deformations of (X, p) over A

We have a morphism of functors:

φ : Def(X,p)→ DefX

induced by the correspondence(ξ, σ ) → ξ

which forgets the section σ . The proposition is a consequence of the following twofacts:

(a) φ is an isomorphism of functors.(b) Def(X,p) is prorepresentable.

To prove (a) let A ∈ ob(A) and consider an infinitesimal deformation ξ of X overA. The point p defines a morphism Spec(k) → X making the following diagramcommutative:

Xp ↓ π

Spec(k) → Spec(A)

By the smoothness of X over Spec(A) there is an extension of p to a section σ :Spec(A)→ X of π ; this proves that Def(X,p)(A)→ DefX (A) is surjective. Now let(ξ, σ ) and (η, τ ) be two deformations with section of X over A, where

η :X → Y↓ ↓ q

Spec(k) → Spec(A)


and suppose that there is an isomorphism of deformations

X

X ψ−→ Y

Spec(A)

Then ψσ, τ : Spec(A)→ Y are two sections of q. We now use the fact that Y has astructure of group scheme over Spec(A) with identity τ (in outline this can be seenas follows: X is a group scheme with identity p and the group structure is given by amultiplication morphism µ : X × X → X ; the group operation on Y is defined bya morphism µA : Y ×A Y → Y which extends µ and which exists because we havea commutative diagram:

X × Xµ−→ X ⊂ Y

∩ ↓Y ×A Y → Spec(A)

and Y is smooth over Spec(A)). Replacing ψ by by ψ ′ = (ψσ)−1ψ we obtainan isomorphism of deformations ψ ′ such that ψ ′σ = τ and therefore ψ ′ definesan isomorphism of (ξ, σ ) with (η, τ ); this proves that Def(X,p)(A) → DefX (A) isinjective as well, and (a) is proved.

In particular, it follows that Def(X,p) has a semiuniversal element because DefX

does. Now observe that the vector space of automorphisms of the trivial deformationof (X, p) can be identified with the vector subspace of H0(X, TX ) = H0(X,OX )consisting of the derivations D : OX → OX vanishing at p, and this is equal toH0(X,OX (−p)) = (0). Now the remark following Corollary 2.6.4 applies to con-clude that Def(X,p) is prorepresentable, i.e. (b) holds.

The following corollary computes in particular the number of moduli of projec-tive nonsingular curves.

Corollary 2.6.6. If X is a projective nonsingular connected curve of genus g thenDefX is prorepresentable. More precisely, DefX = h R where

R =⎧⎨⎩

k if g = 0k[[X ]] if g = 1k[[X1, . . . , X3g−3]] if g ≥ 2

.

Proof. X is unobstructed by Proposition 2.4.6 and

h1(X, TX ) = 0 if g = 0

1 if g = 13g − 3 if g ≥ 2

So it remains to be shown that DefX is prorepresentable. In the case g = 0 this isbecause IP1 is rigid; in the case g ≥ 2, since deg(TX ) = 2 − 2g < 0 we haveH0(X, TX ) = 0 and therefore 2.6.4 applies. If g = 1 we use 2.6.5.


Recalling Theorems 2.5.13 and 2.5.14 we deduce the following:

Theorem 2.6.7. Let X be a projective nonsingular connected curve. Then X has analgebraic formally universal deformation.

Examples 2.6.8. (i) ([154]) Let C = Spec(B), where B = k[x, y]/(xy), be areducible affine plane conic. Then DefC is not prorepresentable although C has asemiuniversal deformation by Corollary 2.4.2. In fact, consider the deformation of Cover k[ε] given by xy + ε = 0 and its automorphism:

x → x + xεy → y

This automorphism does not extend to an automorphism of xy+t = 0 over k[t]/(t3);if it did it would be of the form

x → x + x t + at2

y → y + bt2

for some a, b ∈ k[x, y]. But this implies that bx + ay = −1 in k[x, y], which isimpossible. From Theorem 2.6.1 we deduce that DefC is not prorepresentable.

This holds more generally for the union of the coordinate axes in An , n ≥ 2 (see[154]).

(ii) The condition of Corollary 2.6.4 is not satisfied by the rational ruled surfacesFm , m ≥ 0 (see Example B.11(iii)). Since h1(TF0) = 0 = h1(TF1) we find that F0and F1 are rigid; in particular, DefF0 and DefF1 are prorepresentable. On the otherhand, DefFm is unobstructed when m ≥ 2 (since h2(TFm ) = 0) and has a semiuni-versal element but it is not prorepresentable. To see it we can argue as follows. Wecan identify Fm with the hypersurface m of IP2 × IP1 of equation

xmv − ymu = 0

where (u, v, w; x, y) are bihomogeneous coordinates in IP2×IP1 (for the elementaryproof of this fact see [5], p. 55). For simplicity let’s consider the case m = 2. Thelinear pencil

V ⊂ IP2 × IP1 × A1

of equation:x2v − y2u − t xyw = 0

defines a flat family V → A1 such that V(0) = 2 and V(t) ∼= 0 for all t = 0. Infact, we have an isomorphism V\V(0)→ V(1)× A1\0 over A1\0 sending

(x, y; u, v, w; t) → (x, y; u, v, tw; t);on the other hand, 0 ∼= V(1) by the isomorphism

(x, y; u, v, w) → (x, y;−x2uw − xyw2, xyu2 + y2vw, x2u2 + 2xyuw + y2w2)


(V is essentially the family considered in Example 1.2.11(iii) for m = 2). The pull-back

Vε = V (x2v − y2u − εxyw) ⊂ IP2 × IP1 × Spec(k[ε])has the automorphism defined by sending w → w + εu and leaving all the othercoordinates unchanged. We leave the reader to check that this automorphism doesnot extend to the pullback of V over Spec(k[t]/(t3)). From Theorem 2.6.1 we deducethat DefF2 is not prorepresentable.

2.6.2 Isotriviality

The notions of triviality and of formal triviality of an algebraic deformation are re-lated in a quite subtle way, as shown by Example 1.2.11(ii). This example is a specialcase of an important phenomenon, called isotriviality, considered for the first time inthe literature in [165].

Definition 2.6.9. Let π : X → S be a flat family of schemes. Then:

(i) π is called isotrivial if there is an etale cover (i.e. a finite surjective etale mor-phism) f : S′ → S such that the family πS′ : S′ ×S X → S′ is trivial. IfS′ ×S X ∼= X × S′ we say that π is isotrivial with fibre X.

(ii) If s ∈ S is a k-rational point then π is called locally isotrivial at s if thereis an etale neighbourhood f : (S′, s′) → (S, s) such that the pullback πS′ :S′ ×S X → S′ of π is trivial. π is called locally isotrivial if it is locally isotrivialat every k-rational point of S.

Every trivial family is isotrivial. A rational ruled surface π : Fm → IP1 withm ≥ 2 is locally isotrivial because locally around each point of IP1 it is the productfamily with fibre IP1; on the other hand, π is not isotrivial because it is not trivialand the identity IP1 → IP1 is the only connected etale cover of IP1. In particular,the trivial family IP1 × IP1 → IP1 cannot be obtained by pulling back π by anetale cover of IP1. Therefore the notions of isotriviality and of local isotriviality aredifferent.

If π : X → S is isotrivial then all the fibres over the k-rational points areisomorphic. The next proposition considers the opposite implication.

Proposition 2.6.10. Let π : X → S be a flat family of algebraic schemes, and lets ∈ S be a closed point. If π is locally isotrivial at s then the formal deformation ofX (s) associated to π is trivial. If the morphism π is projective then the converse isalso true.

Proof. The first part follows immediately from Lemma 2.5.6. Conversely, let X =X (s) and A = OS,s , and assume that π is projective and that the formal deformation( A, η) of X associated to the family π is trivial. Let ( A, η) be the deformation ofX over Spec( A) induced by π under the morphism Spec( A) → S. By Theorem2.5.11 this deformation is uniquely determined by the formal deformation ( A, η)and therefore it is trivial.


Extend DefX to a functor F locally of finite presentation defined on (k-algebras).We may assume that S = Spec(B) is affine. We have a commutative diagram offunctors defined on the category (k-algebras):

h A hB hB

F

The lower arrows are induced by the two algebraic deformations π : X → S andp : X × S → S of X (by Yoneda’s lemma), while the upper arrows are the sameand correspond to the natural homomorphism B → A. There remains induced amorphism of functors

h A → hB ×F hB

which corresponds to an effective formal element for the functor hB ×F hB . SinceF is locally of finite presentation and B is of finite type the functor hB ×F hB islocally of finite presentation as well (a fibred product of functors locally of finitepresentation is again locally of finite presentation: see [12], p. 33) so that we canapply the approximation theorem 2.5.22. Therefore we can find an etale neighbour-hood Spec(B ′) → Spec(B) such that both deformations π and p pull back to thesame deformation over Spec(B ′); in particular, π pulls back to a trivial deformationover Spec(B ′).

The inverse implication of Proposition 2.6.10 is false in general: families of non-singular affine schemes are formally trivial but not isotrivial in general, as shown byExample 1.2.5, page 26. The following is immediate.

Corollary 2.6.11. If π : X → S is a smooth projective family of algebraic schemeswhich is locally isotrivial at a k-rational point s ∈ S then the Kodaira–Spencer map

κπ,s : Ts S→ H1(X (s), TX (s))

is 0.

Example 2.6.12. The converse of Corollary 2.6.11 is false. For example, consider asmooth projective family π : X → Spec(k[t]) such that the induced deformationsover k[t]/(tn) are nontrivial for n 0; in particular, π is not locally isotrivial atthe point t = 0. Let q : Spec(k[u]) → Spec(k[t]) be the morphism defined byq : k[t] → k[u] sending t → u2. Then q has zero differential at 0 and therefore thepulled back family

X ×Spec(k[t]) Spec(k[u])→ Spec(k[u])has a vanishing Kodaira–Spencer map at 0. But the restriction of this family tok[u]/(un) is nontrivial for n 0 and therefore the family is not locally isotrivialat s by Proposition 2.6.10.


The existence of nontrivial isotrivial deformations of a scheme X is closely re-lated to the existence of nontrivial automorphisms of X . Before investigating thisfact we give some examples.

Examples 2.6.13. (i) Let X be a quasi-projective scheme such that there is a finitenontrivial subgroup G ⊂ Aut(X). Let S′ be a quasi-projective scheme on which Gacts freely and let S := S′/G be the quotient. Then G acts on X× S′ componentwiseand the action is easily seen to be free. The quotient X := (X × S′)/G exists and isan algebraic scheme (see [166] or [3], exp. V). Since the projection X × S′ → S′ isG-equivariant it induces a morphism

π : X → S

and we have a commutative diagram:

X × S′ → X↓ ↓ πS′ → S

(2.30)

where the horizontal arrows are etale morphisms and all the fibres of π over theclosed points are isomorphic to X . Moreover, (2.30) is a cartesian diagram (there isan S′-morphism X × S′ → S′ ×S X which is easily seen to be an isomorphism) andπ is flat (use A.5).Claim: The family π is not trivial.In fact, since (2.30) is cartesian, if π were trivial the action of G on X × S′ would betrivial on the first factor, a contradiction.

It follows that π is an isotrivial nontrivial family.

(ii) Let G be a nontrivial finite group scheme and Z a quasi-projective schemeon which G acts freely. Then the quotient scheme Z/G exists (see [166] or [3], exp.V). Let π : Z → Z/G be the canonical morphism. Then π is an etale cover; inparticular, it is flat. Moreover, we have a commutative diagram

Z ×Z/G Z → Z↓ π ′ ↓ πZ → Z/G

and since the action is free we have an isomorphism G × Z → Z ×Z/G Z inducedby the map (g, z) → (z, gz). Therefore π ′ is the trivial family and π is isotrivial. IfZ is integral then π has no sections and it follows that π is not trivial. The morphismπ is called a principal G-bundle.

(iii) Perhaps the simplest examples of isotrivial nontrivial families are those de-scribed in Example 1.2.11(ii), page 31. We leave to the reader to verify that they areisotrivial.

We need the following well-known lemma ([165], n. 1.5).


Lemma 2.6.14. Let f : Z → Y be an etale cover of algebraic schemes. Then thereis an etale cover ϕ : P → Z such that the composition

f ϕ : P → Z → Y

is a principal G-bundle with respect to a finite group G. In particular, f is isotrivialand, if Z is integral and deg( f ) > 1, it is nontrivial.

Proof. Let n = deg( f ). In the n-fold fibre product Z ×Y Z ×Y · · · ×Y Z considerthe set P of points (z1, . . . , zn) such that zi = z j for all i = j . Then P is a unionof connected components of Z ×Y Z ×Y · · · ×Y Z which is stable under the naturalaction of the symmetric group Sn . The natural morphism φ : P → Y is an etalecover of degree n! and therefore it induces an isomorphism P/Sn ∼= Y . Therefore φis a principal Sn-bundle. Moreover, the first projection ϕ : P → Z is etale of degree(n − 1)! and satisfies f ϕ = φ.

In order to prove the last assertion recall that by Example 2.6.13(ii) φ is isotrivial.More precisely, we have a commutative diagram:

P ×Y P ∼= Sn × P → P↓ ↓ φP → Y

The left vertical morphism is the projection and, because of the factorization f ϕ =φ, it factors as

Sn × P → P ×Y Z → P

It follows that P ×Y Z → P is trivial as well and this implies that f is isotrivial. IfZ is integral and deg( f ) > 1 then f cannot be trivial because it has no sections.

Theorem 2.6.15. The following conditions are equivalent on a quasi-projectivescheme X:

(a) There exists a nontrivial isotrivial algebraic family with fibre X.(b) Aut(X) contains a nontrivial finite subgroup.

Proof. (b)⇒ (a) has been proved in Example 2.6.13(i).(a)⇒ (b). Let π : X → S be a family as in (a). By hypothesis there is an etale

cover φ : P → S such that πP : P ×S X → P is trivial with fibre X ; let’s identifyX × P = P ×S X and let ψ : X × P → X be the projection:

X × Pψ−→ X

↓ ↓ π

Pφ−→ S

For each p ∈ P denote by ψp : X → X the morphism defined by

ψp(x) = ψ(x, p)


By Lemma 2.6.14 we may assume that φ is a principal G-bundle with respect tosome finite group G. We define an action of G on X × P by the following rule:

g(x, p) = (ψ−1gp ψ(x, p), gp)

(X × P)/G exists and, since the action is clearly free and transitive on the fibres ofψ , there is an induced morphism (X× P)/G → X which is an etale cover of degreeone, i.e. it is an isomorphism. Fix p ∈ P and define

G × X → X

by gx = ψ−1gp ψ(x, p). Then one checks immediately that this is an action of G on X .

Since π is nontrivial it is quite clear that this action cannot be trivial for all p ∈ P .Therefore for some p ∈ P the above action defines a homomorphism G → Aut(X)whose image is = 1.

If a scheme X has an isotrivial local deformation η which is nontrivial then thelocal moduli functor

DefX : A∗ → (sets)

considered in the Introduction cannot be representable, i.e. the local deformationυ considered there cannot exist. Assume by contradiction that υ exists; then, sinceη is nontrivial it must be pulled back from υ by a nonconstant morphism g :Spec(A)→ Spec(O). On the other hand, since η is isotrivial its pullback to Spec( A)is trivial and is therefore obtained by pulling back υ in two different ways: by theconstant morphism and by the composition

Spec( A)ϕ−→ Spec(A)

g−→ O

which is nonconstant because ϕ is faithfully flat hence surjective; this contradicts theuniversality of υ.

These remarks explain why we cannot expect to be able to construct familiesrepresenting functors defined on A∗ or on (k-algebras) or on (schemes), which clas-sify isomorphism classes of schemes having nontrivial isotrivial deformations; andthe existence of such deformations is closely related to the existence of nontrivialautomorphisms of such schemes, as we saw in Theorem 2.6.15.

This discussion suggests that while the consideration of isomorphism classes ofdeformations is not a drawback when one is studying infinitesimal deformations, itbecomes inadequate for the classification of algebraic deformations and for globalmoduli problems. In other words, because of the presence of nontrivial automor-phisms we cannot in general expect to find a scheme structure on the set M ofisomorphism classes of objects we want to classify in such a way that it reflectsfaithfully the functorial properties of families. For example, in the case of projectivenonsingular curves of genus 0 one should have M = Spec(k) and the only candidatefor being the universal family is IP1 → Spec(k) because there is only one isomor-phism class of such curves; but the families Fm → IP1 (Example B.11(iii)) cannotbe pulled back from it, thus a universal family cannot exist in this case.


That’s why it would be more natural, instead of taking isomorphism classes ofdeformations, to consider all families together and analyse them and their isomor-phisms. This will result in a more general structure, called a stack, which contains allthe information about families and deformations of the objects of the set M we wantto classify. We refer to [43], [115] and [185] for foundational material about stacks,and to [18], [46], [53], [65] for expository treatments.

3

Examples of deformation functors

This chapter is devoted to the study of the most important functors of Artin ringswhich arise when one considers deformations of various algebro-geometric objects.We will verify Schlessinger’s conditions and we will describe first-order deforma-tions, i.e. the tangent spaces of these functors, and obstruction spaces. We will focuson several examples and applications. It will emerge from the treatment that there is apattern common to almost all examples: the tangent space and the obstruction spaceof a given functor will be respectively isomorphic to Hi and to Hi+1 of a certainsheaf which depends on the functor. It will be i = 0 if the deformation problem hasno automorphisms, while i = 1 if there are automorphisms; in this case the H0 willclassify the infinitesimal automorphisms.

3.1 Affine schemes

In this section we study the deformation functor of an affine scheme. We alreadyknow that such a functor verifies Schlessinger’s conditions H0, H , Hε and we com-puted its tangent space (Proposition 2.4.1). In particular, we proved that it has a semi-universal element if the scheme has isolated singularities (Corollary 2.4.2). Here wewill analyse this case in more detail. We will start by recalling the description of thetangent space.

3.1.1 First-order deformations

Let B0 be a k-algebra, and let X0 = Spec(B0). We continue the study of infinitesimaldeformations of X0, equivalently of B0, started in Section 1.2 in the nonsingular case.Let’s recall the following fact (see 2.4.1(iii)).

Proposition 3.1.1. There is a natural isomorphism

DefB0(k[ε]) ∼= T 1B0

104 3 Examples of deformation functors

where the class of trivial deformations corresponds to 0 ∈ T 1B0

. If B0 is reduced then

DefB0(k[ε]) ∼= Ext1k(ΩB0/k, B0)

Proof. A first-order deformation of B0 consists of a flat k[ε]-algebra B, plus ak-isomorphism B ⊗k[ε] k ∼= B0. This set of data determines a k-extension:

0 → B0j−→ B → B0 → 0

‖εB

(3.1)

obtained after tensoring the exact sequence

0→ (ε)→ k[ε] → k→ 0

by⊗k[ε]B. Isomorphic deformations give rise to isomorphic extensions. Conversely,given a k-extension (3.1), B has a structure of k[ε]-algebra given by

ε → j (1)

B is k[ε]-flat by Lemma A.9. If B0 is reduced then

T 1B0= Ext1k(ΩB0/k, B0)

by Corollary 1.1.11. The following is an immediate consequence of 3.1.1, 2.2.8 and Schlessinger’s

theorem (2.3.2).

Corollary 3.1.2. If dimk(T 1B0

) <∞ then DefB0 has a semiuniversal formal element.

This happens in particular if X0 = Spec(B0) has isolated singularities. If T 1B0= 0

then B0 is rigid.

We will give some indications for the practical computation of T 1B0

when B0 ise.f.t..

Let B0 = P/J , where P is a smooth k-algebra of the form

P = ∆−1k[X1, . . . , Xd ]

for some multiplicative system ∆ ⊂ k[X1, . . . , Xd ], and J ⊂ P is an ideal.Consider the exact sequence

0→ Hom(ΩB0/k, B0)→ Hom(ΩP/k⊗B0, B0)δ∨−→ Hom(J/J 2, B0) −→ T 1

B0→ 0

The moduleHom(ΩP/k ⊗ B0, B0) = Derk(P, B0)

3.1 Affine schemes 105

consists of all derivations D of the form

D(p) =d∑

j=1

b j∂p

∂X j

for given b j ∈ B0, and

δ∨(D)( f ) = D( f ) =d∑

j=1

b j∂ f

∂X jf ∈ J

Assume that J = ( f1, . . . , fn) and let

0→ Rι−→ Pn j−→ J → 0

be the corresponding presentation. We have the exact sequence:

0→ Hom(J/J 2, B0)j∨−→ Hom(Bn

0 , B0)ι∨−→ Hom(R, B0)

where j∨ identifies Hom(J/J 2, B0) with the submodule of Hom(Bn0 , B0) consisting

of those homomorphisms which are 0 on R. Identifying Hom(Bn0 , B0) = Bn

0 , therebyviewing its elements as column vectors, we see that the condition for

q =⎛⎝ q1

...qn

⎞⎠ ∈ Bn0

to be in Hom(J/J 2, B0) is that t q · r = 0 for each r ∈ R (where we are viewing Ras consisting of column vectors as well). j∨ associates to a homomorphism

ϕ : J/J 2 → B0∑b j f j → ∑

b jϕ( f j )

the column vector ⎛⎝ ϕ( f1)...

ϕ( fn)

⎞⎠Therefore Im(δ∨) ⊂ Bn

0 is generated by the column vectors corresponding toδ∨( ∂

∂X1), . . ., δ∨( ∂

∂Xd), i.e. by the classes mod J of:

⎛⎜⎝∂ f1∂X1...

∂ fn∂X1

⎞⎟⎠ , · · · ,⎛⎜⎝

∂ f1∂Xd...

∂ fn∂Xd

⎞⎟⎠


If J = ( f1, . . . , fn) is generated by a regular sequence then ι∨ = 0, equivalently j∨is an isomorphism, and it follows that

T 1B0∼= Pn⎛⎜⎜⎝

⎛⎜⎜⎝∂ f1∂X1...

∂ fn∂X1

⎞⎟⎟⎠ , · · · ,

⎛⎜⎜⎝∂ f1∂Xd...

∂ fn∂Xd

⎞⎟⎟⎠⎞⎟⎟⎠⊗P B0

In particular, if B0 = P/( f ) then

T 1B0∼= P(

f, ∂ f∂X1

, · · · , ∂ f∂Xd

)It follows from this description that the hypersurface V ( f ) is rigid if and only ifit is nonsingular. A similar remark holds for complete intersections. In particular,recalling Definition D.1, we can state the following, for future reference:

Proposition 3.1.3. An e.f.t. complete intersection ring B0 such that Spec(B0) is sin-gular is not rigid.

Example 3.1.4. Let P be the local ring of a nonsingular algebraic surface X at ak-rational point p, m = (x, y) its maximal ideal, and B0 = P/( f ) the local ring ofa curve C ⊂ X at p. Let’s compute T 1

B0in some cases.

(a) Node (ordinary double point). By definition B0 ∼= k[[X, Y ]]/(X2 + Y 2). Thenf = x2 + y2+ higher-order terms, and

T 1B0∼= B0

(x, y)∼= k

if char(k) = 2.(b) Ordinary cusp. In this case B0 ∼= k[[X, Y ]]/(X2 + Y 3). Then f = x2 + y3+

higher-order terms, and

T 1B0∼= B0

(y2, x)∼= k2

if char(k) = 2, 3.(c) Tacnode. We have in this case B0 ∼= k[[X, Y ]]/(Y (Y + X2)) and

T 1B0∼= B0

(x2 + 2y, xy)∼= k3

if char(k) = 2.

Conversely, we have the following result:

Proposition 3.1.5. Assume char(k) = 0. Let P be the local ring of a nonsingularalgebraic surface X at a k-rational point p, m = (x, y) its maximal ideal, andB0 = P/( f ) the local ring of a curve C ⊂ X at p; let t = dimkT 1

B0. Then


(a) t = 0 if and only if B0 is regular (a DVR).(b) t = 1 if and only if B0 is the local ring of a node.(c) t = 2 if and only if B0 is the local ring of an ordinary cusp.(d) t = 3 if and only if B0 is the local ring of a tacnode.

Proof. The “if” implication follows from the above computations. We have

t = dimk P/( f, fx , fy)

f ∈ m3P immediately implies t ≥ 4; then f ∈ m2

P and, after suitable choice ofgenerators of m P we may suppose f = y2 + xn+ higher-order terms, n ≥ 2 orf = y(y + xn)+ higher-order terms, n ≥ 2. Now the conclusion follows easily. Example 3.1.6. (The affine cone over IP1 × IP2). Let

P = k[X0, X1, X2, Y0, Y1, Y2]J = (X1Y2 − X2Y1, X2Y0 − X0Y2, X0Y1 − X1Y0)

Then B0 = P/J is the coordinate ring of the affine cone over the Segre embeddingIP1 × IP2 ⊂ IP5. We have the following presentation:

0→ P2 A−→ P3 → J → 0

where

A =( X0 Y0

X1 Y1X2 Y2

)

A direct computation shows that Hom(J/J 2, B0) is generated by the following col-umn vectors:

Y1 Y2 0 X1 X2 0−Y0 0 Y2 −X0 0 X2

0 −Y0 −Y1 0 −X0 −X1

Since these vectors are, up to permutation,

δ∨(

∂

∂X0

), δ∨

(∂

∂X1

), δ∨

(∂

∂X2

), δ∨

(∂

∂Y0

), δ∨

(∂

∂Y1

), δ∨

(∂

∂Y2

)we see that T 1

B0= 0. This implies that B0 is rigid (see Corollary 3.1.2).

More generally, one can prove that the coordinate ring of the affine cone over theSegre embedding IPn × IPm ⊂ IP(n+1)(m+1)−1 is rigid whenever n + m ≥ 3. Thishas been computed for the first time in [67] in the case n = m ≥ 2; the general caseis in [157] (see Corollary 3.1.20 below).

Example 3.1.7. Let P = k[X1, X2, X3](X), J = (X2 X3, X1 X3, X1 X2). Then B0 =P/J is the local ring at the origin of the union of the coordinate axes in A3. We havethe presentation


P3 A−→ P3 → J → 0

where

A =( X1 X1 0−X2 0 X2

0 −X3 −X3

)and the columns of A generate R.

Hom(J/J 2, B0) is generated by the following column vectors mod J :

X2 X3 0 0 0 00 0 X1 X3 0 00 0 0 0 X1 X2

and Im(δ∨) is generated by the column vectors mod J :

0 X3 X2X3 0 X1X2 X1 0

It follows at once that

T 1B0= Hom(J/J 2, B0)/Im(δ∨) ∼= k3

because there are three generators of Hom(J/J 2, B0) which are linearly independentmodulo the generators of Im(δ∨), and all other elements of Hom(J/J 2, B0) are inIm(δ∨).

In a similar vein one can consider, for any d ≥ 3

B0 = k[X1, . . . , Xd ](X)/J

whereJ = (. . . , X1 X2 · Xi · Xd , . . .)i=1,...,d

Then B0 is the local ring at the origin of the union of the coordinate axes in Ad . Onecomputes easily, along the same lines of the case d = 3, that

T 1B0∼= kd(d−2)

Example 3.1.8. Let P be the local ring of a nonsingular algebraic variety V of di-mension n ≥ 2 at a k-rational point p and let B0 = P/( f ) be the local ring of ahypersurface X ⊂ V at p. We call p a node for X if

B0 ∼= k[[X1, . . . , Xn]]/(X21 + · · · + X2

n)

equivalently, if we can choose generators x1, . . . , xn of the maximal ideal m P so thatf =∑ x2

i + higher-order terms. It can be proved immediately that if p is a node forX and char(k) = 2 then T 1

B0∼= k. In particular, T 1

k[ε] ∼= k.


3.1.2 The second cotangent module and obstructions

We will now describe an obstruction space for the functor DefB0 .Assume that B0 = P/J for a smooth k-algebra P , and an ideal J ⊂ P . Consider

a presentation:

η : 0→ Rι−→ F

j−→ J → 0 (3.2)

where F is a finitely generated free P-module. Let λ :∧2 F → F be defined by:

λ(x ∧ y) = ( j x)y − ( j y)x

and Rtr = Im(λ). Obviously Rtr ⊂ R and Rtr ⊂ J F .If J = ( f1, . . . , fn) then F = Pn and R is the module of relations among

f1, . . . , fn . Rtr is called the module of trivial (or Koszul) relations; it is generatedby the relations of the form

(0, . . . , − f j , . . . , fi , . . . , 0)

i j

It follows thatR/Rtr = H1(K•( f1, . . . , fn))

the first homology module of the Koszul complex associated to f1, . . . , fn .

Lemma 3.1.9. The P-module R/Rtr is annihilated by J and therefore it is a B0-module in a natural way.

Proof. Let x ∈ R, a ∈ J . Let y ∈ F be such that j (y) = a. Then

ax = j (y)x = j (y)x − j (x)y = λ(y ∧ x) ∈ Rtr Since Rtr ⊂ J F the presentation (3.2) induces an exact sequence of B0-modules:

R/Rtr ι−→ F ⊗P B0j−→ J/J 2 → 0 (3.3)

Definition 3.1.10. The second cotangent module of B0 is the B0-module T 2B0

definedby the exact sequence:

HomB0(J/J 2, B0)→ HomB0(F ⊗P B0, B0)→ HomB0(R/Rtr , B0)→ T 2B0→ 0

induced by (3.3). Obviously T 2B0

is a B0-module of finite type.

Lemma 3.1.11. For every e.f.t. k-algebra B0 the B0-module T 2B0

is independent ofthe presentation (3.2).


Proof. Assume that F ∼= Pn and that j : Pn → J is defined by the system ofgenerators f1, . . . , fn of J . Let

0→ S→ Pm → J → 0

be another presentation of J , defined by the system of generators g1, . . . , gm . Wemay assume that m ≥ n and that gk = fk , k = 1, . . . , n. Let

gs =∑

k

bks fk s = n + 1, . . . , m

for some bks ∈ P . Denote by α : Pn → Pm the map

α(a1, . . . , an) = (a1, . . . , an, 0, . . . , 0)

and by β : Pm → Pn the map

β(a1, . . . , am) =⎛⎝a1 +

m∑s=n+1

b1sas, . . . , an +m∑

s=n+1

bnsas

⎞⎠Evidently, α(R) ⊂ S and α(Rtr ) ⊂ Str . It is easy to verify that β(S) ⊂ R andβ(Str ) ⊂ Rtr . It follows that α and β induce homomorphisms

β : Hom(R/Rtr , B0)→ Hom(S/Str , B0)α : Hom(S/Str , B0)→ Hom(R/Rtr , B0)

whence homomorphisms

β : Hom(R/Rtr , B0)/Hom(Pn, B0)→ Hom(S/Str , B0)/Hom(Pm, B0)α : Hom(S/Str , B0)/Hom(Pm, B0)→ Hom(R/Rtr , B0)/Hom(Pn, B0)

Sinceαβ = identity of Hom(R/Rtr , B0)

it follows that

αβ = identity of Hom(R/Rtr , B0)/Hom(Pn, B0)

We now prove that

βα = identity of Hom(S/Str , B0)/Hom(Pm, B0) (3.4)

Let g ∈ Hom(S/Str , B0) be induced by the homomorphism G : S → B0. Then, if(a1, . . . , am) ∈ S and (a1, . . . , am) ∈ S/Str is its class, we have:

(βα)(g)(a1, . . . , am) = G(α(β(a1, . . . , am))) =

G(a1 +∑ms=n+1 b1sas, . . . , an +∑m

s=n+1 bnsas, 0, . . . , 0) =

G(a1, . . . , am)+ G(∑m

s=n+1 b1sas, . . . ,∑m

s=n+1 bnsas,−an+1, . . . ,−am) = (∗)


Now note that⎛⎝ m∑s=n+1

b1s ps, . . . ,

m∑s=n+1

bns ps,−pn+1, . . . ,−pm

⎞⎠ ∈ S

for every (p1, . . . , pm) ∈ Pm . Therefore letting

τ(p1, . . . , pm) = G

⎛⎝ m∑s=n+1

b1s ps, . . . ,

m∑s=n+1

bns ps,−pn+1, . . . ,−pm

⎞⎠we define a homomorphism τ : Pm → B0. It follows that

(∗) = g(a1, . . . , am)+ τ(a1, . . . , am)

Hence(βα)(g)− g ∈ Im[Hom(Pm, B0)→ Hom(S/Str , B0)]

or equivalently (3.4) holds. From the definition it easily follows that T 2

B0localizes. Namely, for every multi-

plicative subset ∆ ⊂ P we have:

∆−1T 2B0= T 2

∆−1 B0

It follows that for any scheme X we can define in an obvious way the second cotan-gent sheaf which we will denote by T 2

X . It satisfies

T 2X,x = T 2

OX,x

Proposition 3.1.12. Assume that B0 = P/J for a smooth k-algebra P. Then T 2B0

isan obstruction space for the functor DefB0 .

Proof. Let A be an object of A and let

ξ :B → B0↑ ↑A → k

be an infinitesimal deformation of B0 over A.We must associate to ξ a k-linear map

ξv : Exk(A, k)→ T 2B0

satisfying the conditions of Definition 2.2.9. Let B0 = P/J for a smooth k-algebraP and an ideal J = ( f1, . . . , fn) ⊂ P . We have an exact sequence:

0→ R→ Pn f−→ J → 0


Then, by the smoothness, in particular flatness, of P we have

B = (P ⊗k A)/(F1, . . . , Fn)

where f j = Fj (mod m A), j = 1, . . . , n, by Theorem A.10. The flatness of B overA implies that for every r = (r1, . . . , rn) ∈ R there exist R1, . . . , Rn ∈ P⊗k A suchthat r j = R j (mod. m A), j = 1, . . . , n, and

∑j R j Fj = 0, again by Theorem A.10.

Let [γ ] ∈ Exk(A, k) be represented by an extension

γ : 0→ (t)→ Aφ−→ A→ 0

Choose F1, . . . , Fn, R1, . . . , Rn ∈ P ⊗k A liftings of F1, . . . , Fn, R1, . . . , Rn re-spectively; then ∑

j

R j F j ∈ ker[P ⊗k A→ P ⊗k A] ∼= P

It is easy to check that a different choice of R1, . . . , Rn or of F1, . . . , Fn modifies∑j R j F j by an element of J or by one of the form

∑j q j r j , where q j ∈ P , respec-

tively. Therefore sendingr →

∑j

R j F j (3.5)

defines an element of

coker[Hom(Bn0 , B0)→ Hom(R, B0)]

Moreover, since if r = r i j = (0, . . . , f j , . . . ,− fi , . . . , 0) we can take

(R1, . . . , Rn) = (0, . . . , Fj , . . . ,−Fi , . . . , 0)

and we get∑

j R j F j = 0, it follows that (3.5) is zero on Hom(Rtr , B0). Therefore

the n-tuple of liftings (F) = (F1, . . . , Fn) defines an element ξv(γ ) of

coker[Hom(Bn0 , B0)→ Hom(R/Rtr , B0)] = T 2

B0

Let’s prove that the map γ → ξv(γ ) is k-linear.Let [ζ ] ∈ Exk(A, k) be another element defined by the extension:

ζ : 0→ (t)→ A′ → A→ 0

and let (F ′) = (F ′1, . . . , F ′n), F ′j ∈ P ⊗k A′ be the corresponding lifting, whichdefines ξv(ζ ). Then ξv(γ )+ ξv(ζ ) is defined by

r →∑

j

R j F j +∑

j

R′j F ′j

where R′1, . . . , R′n ∈ P ⊗k A′ are liftings of R1, . . . , Rn . Consider the diagram:


0→ k⊕ k → A ×A A′ → A → 0↓ δ ↓ σ ‖

γ + ζ : 0→ k → C → A → 0

Then ξv (γ + ζ ) is defined by

r →∑

j

# jΦ j

where #1, . . . , #n, Φ1, . . . , Φn ∈ P ⊗k C are liftings of R1, . . . , Rn, F1, . . . , Fn .Since

P ⊗k ( A ×A A′) ∼= (P ⊗k A)×A (P ⊗k A′)

letting ρ : P ⊗k ( A ×A A′) → P ⊗k C be the homomorphism induced by σ , wemay assume that

Φ j = ρ(Fj , F ′j ); # j = ρ(R j , R′j )Then: ∑

j # jΦ j =∑ j ρ(R j , R′j )ρ(Fj , F ′j ) = ρ[∑

j (R j , R′j )(Fj , F ′j )]

= ρ(∑

j R j F j ,∑

j R′j F ′j)= δ

(∑j R j F j ,

∑j R′j F ′j

)=∑ j R j F j +∑ j R′j F ′j

This proves that ξv(γ+ζ ) = ξv (γ )+ξv(ζ ). A similar argument shows that ξv (λγ ) =λξv(γ ), λ ∈ k.

Now assume that [γ ] ∈ Exk(A, k) is such that there exists an infinitesimal de-formation

ξ :B → B0↑ ↑A → k

such thatDefB0( A) → DefB0(A)

ξ → ξ

It follows that there exist liftings Fj ∈ P ⊗k A of the Fj ’s such that every r ∈ R hasa lifting R ∈ (P ⊗k A)N such that

∑j R j F j = 0. This means that ξv(γ ) = 0.

Conversely, assume that ξv(γ ) = 0, and let F1, . . . , Fn ∈ P ⊗k A be arbitraryliftings of F1, . . . , Fn . Then there exists (h1, . . . , hn) ∈ Pn such that for every choiceof a lifting R ∈ (P ⊗k A)n of a relation r ∈ R we have:∑

j

R j F j = −t∑

j

r j h j = −∑

j

R j h j

This means that the ideal (F1 + th1, . . . , Fn + thn) ⊂ P ⊗k A defines a flat defor-mation of B0 over A lifting the deformation B = (P ⊗k A)/(F1, . . . , Fn).


Any other choice of a lifting of the deformation ξ over A is of the form

(F1 + t (h1 + k1), . . . , Fn + t (hn + kn))

where k = (k1, . . . , kn) ∈ Bn0 satisfy

∑j r j k j = 0 for every relation r ∈ R. There-

fore k ∈ Hom(J/J 2, B0). It is straightforward to verify that if k ∈ Im(δ∨) thenF + th and F + t (h + k) define isomorphic liftings of ξ over A. This means that wehave an action of T 1

B0on the set of liftings of ξ over A. By construction it follows

that this action is transitive. Corollary 3.1.13. Let X0 = Spec(B0) be an affine algebraic scheme such thatdimk(T 1

B0) < ∞, and let (R, ηn) be a semiuniversal formal deformation of X0.

Then

(i)dimk(T

1B0

) ≥ dim(R) ≥ dimk(T1B0

)− dimk(T2B0

)

The first equality holds if and only if X0 is unobstructed.(ii) If B0 is an e.f.t. local complete intersection k-algebra then it is unobstructed. In

particular, hypersurface singularities are unobstructed.

Proof. (i) follows from Corollary 2.2.11.(ii) If J is generated by a regular sequence then R = Rtr and therefore

T 2B0= (0).

Proposition 3.1.14. Assume that B0 = P/J for a smooth k-algebra P. Then:

(i) If Spec(B0) is reduced then

T 2B0∼= Ext1B0

(J/J 2, B0)

(ii) If Spec(B0) is reduced and has depth at least 2 along the locus where it is not anl.c.i. (e.g. Spec(B0) is normal of dimension ≥ 2) there is an isomorphism

T 2B0∼= Ext2(ΩB0/k, B0)

Proof. We keep the notations introduced on page 109.(i) From the exact sequence (3.3) we deduce the following commutative diagram

with exact rows and columns:

Hom(ker(ι), B0)↑

Hom(J/J 2, B0)→ Hom(F ⊗ B0, B0)j∨−→ Hom(R/Rtr , B0) → T 2 → 0

‖ ‖ ∪ ∪Hom(J/J 2, B0)→ Hom(F ⊗ B0, B0) → Hom(Im(ι), B0) → E1 → 0

where E1 = Ext1(J/J 2, B0) and T 2 = T 2B0

. Since the exact sequence

η : 0→ Rι−→ F

j−→ J → 0


from which (3.3) is obtained localizes, we see that ker(ι) is supported on the lo-cus where B0 is not an l.c.i.; in particular ker(ι) is torsion. Therefore we haveHom(ker(ι), B0) = (0) and the conclusion follows.

(ii) Consider the conormal sequence

J/J 2 δ−→ ΩP/k ⊗ B0 → ΩB0/k → 0

Since Spec(B0) is reduced, ker(δ) is supported in the locus whereSpec(B0) ⊂ Spec(P) is not a regular embedding (Proposition D.1.4), and this locuscoincides with the locus where Spec(B0) is not an l.c.i. (Proposition D.2.5). Fromthe assumption about the depth of Spec(B0) it follows that

Hom(ker(δ), B0) = Ext1(ker(δ), B0) = (0)

Using this fact and recalling that Exti (ΩP/k ⊗ B0, B0) = (0), i > 0, we obtain:

Ext2(ΩB0/k, B0) ∼= Ext1(Im(δ), B0) ∼= Ext1(J/J 2, B0)

Example 3.1.15 (Schlessinger [154]). (An obstructed affine curve.) Let s be an in-determinate and let B0 = k[s7, s8, s9, s10] ⊂ k[s] be the coordinate ring of the affinerational curve C ⊂ A4 = Spec(k[x, y, z, w]) having parametric equations:

x = s7, y = s8, z = s9, w = s10

Write P = k[x, y, z, w] and

B0 = P/I

for an ideal I ⊂ P . One can check, using, for example, a computer algebra package,that I is generated by the six 2× 2 minors of the following matrix:(

x y z w2

y z w x3

)i.e. by the following polynomials:

f1 = y2 − xz; f2 = xw − yz; f3 = z2 − yw;

f4 = x4 − w2 y; f5 = x3 y − zw2; f6 = w3 − x3z

This ideal is prime of height 3. Consider a presentation:

0→ R→ F → I → 0

where F ∼= P6 with generators, say e1, . . . , e6, so that ei → fi . To describe Rone can use the beginning of the free resolution of I given by the Eagon–Northcott


complex (see [48]). One obtains a set of generators for R given by the rows of thefollowing matrix:

R1 : z y x 0 0 0R2 : w2 0 0 y −x 0R3 : 0 0 w2 0 z yR4 : 0 −w2 0 z 0 xR5 : w z y 0 0 0R6 : x3 0 0 z −y 0R7 : 0 0 x3 0 w zR8 : 0 −x3 0 w 0 y

Here each row gives the coefficients ai of the linear combination∑

i ai ei ∈ R. Wethen have an exact sequence

0→ R→ F → I/I 2 → 0

where F = F/I F and R = R/(I F ∩ R). Reducing mod I the above relations onegets the following set of generators of R as elements of F :

r1 : s9 s8 s7 0 0 0r2 : s20 0 0 s8 −s7 0r3 : 0 0 s20 0 s9 s8

r4 : 0 −s20 0 s9 0 s7

r5 : s10 s9 s8 0 0 0r6 : s21 0 0 s9 −s8 0r7 : 0 0 s21 0 s10 s9

r8 : 0 −s21 0 s10 0 s8

(3.6)

Since B0 is reduced we have

T 2B0= Ext1B0

(I/I 2, B0) = Hom(R, B0)/Hom(F, B0)

Representing an element h ∈ Hom(R, B0) as the 8-tuple (h(r1), . . . , h(r8)) ∈ B80

we see that the submodule Hom(F, B0) is generated by the columns of (3.6). There-fore to prove that B0 is obstructed it will suffice to produce a first-order deforma-tion ξ ∈ DefB0(k[ε]) whose obstruction to lifting to k[t]/(t3) is represented by anh : R→ B0 not in the submodule generated by the columns of (3.6). We define ξ bythe ideal

( f1 +∆ f1, . . . , f6 +∆ f6) ⊂ k[ε, x, y, z, w]where

∆ f := (∆ f1, . . . , ∆ f6) = (0, 0, 0, zw,w2,−x3)

this defines a deformation because R j ·∆ f ∈ I for all j = 1, . . . , 8. More precisely:

(R1 ·∆ f, . . . , R8 ·∆ f ) = (0,−w f2,− f5, f4 − w f3, 0, w f3, f6,− f5)

Therefore we see that the obstruction to lifting ξ to second order is defined by thehomomorphism h : R→ B0 represented by


(0, 0,−t20, t19, 0, 0,−t21,−t20)

Now we can immediately check that this vector is not in Hom(F, B0) and thereforethe deformation ξ cannot be lifted: thus B0 is obstructed.

3.1.3 Comparison with deformations of the nonsingular locus

Under certain conditions it is possible to compare the deformations of an affinescheme with the deformations of the open subscheme of its nonsingular points. Inthis and the following subsections we will describe the analysis made in [156] and[157], with applications to the study of certain quotient singularities. We will need apreliminary lemma.

Lemma 3.1.16. Let X be an affine scheme, Z ⊂ X a closed subscheme andG a coherent sheaf on X. Let G∨ = Hom(G,OX ). If depthZ (OX ) ≥ 2 thendepthZ (G∨) ≥ 2 and therefore

H0(X, G∨) ∼= H0(X\Z , G∨)

Proof. Consider a presentation

0→ R→ F → G → 0

where F is a free OX -module. Then we obtain an exact sequence

0→ G∨ → F∨ → Q → 0 (3.7)

where Q ⊂ R∨. Since F∨ is free we have depthZ (F∨) = depthZ (OX ) ≥ 2 andtherefore H0

Z (F∨) = 0 = H1Z (F∨) ([75], Theorem 3.8, p. 44); it follows that

H0Z (G∨) = 0. Similarly, one proves that H0

Z (R∨) = 0, and therefore H0Z (Q) = 0.

From the sequence of local cohomology associated to (3.7) we obtain H1Z (G∨) = 0

and therefore depthZ (G∨) ≥ 2 by [75], Theorem 3.8, p. 44. The last assertion fol-lows from the exact sequence

0→ H0(X, G∨)→ H0(X\Z , G∨)→ H1X (G∨)

(see [84], p. 212). Consider an affine scheme X = Spec(B) where B = P/J for a smooth k-algebra

P . Let Z = Sing(X) be the singular locus of X and U = X\Z .Let Y = Spec(P) and consider the exact sequence

0→ TX → TY |X → NX → T 1X → 0 (3.8)

where NX = NX/Y . Since T 1X is supported on Z , by restricting to U we get the exact

sequence:

0→ TU → TY |U → NX |U → 0 (3.9)


Proposition 3.1.17. (i) If depthZ (OX ) ≥ 2 (e.g. X is normal of dimension ≥ 2) wehave an exact sequence

0→ T 1B → H1(U, TU )→ H1(U, TY |U )

(ii) If depthZ (OX ) ≥ 3 thenT 1

B∼= H1(U, TU )

Proof. (i) We have the local cohomology exact sequences (see [84], p. 212):

0→ H0(X, NX )→ H0(U, NX |U )→ H1Z (NX )

0→ H0(X, TY |X )→ H0(U, TY |U )→ H1Z (TY |X )

If depthZ (OX ) ≥ 2 then from Lemma 3.1.16 we deduce that depthZ (NX ) ≥ 2 anddepthZ (TY |X ) ≥ 2. Therefore we have H1

Z (NX ) = 0 = H1Z (TY |X ) ([75], Theorem

3.8, p. 44) and

H0(X, NX ) ∼= H0(U, NX |U ), H0(X, TY |X ) ∼= H0(U, TY |U )

Comparing the exact cohomology sequences of (3.8) and (3.9) we get an exact andcommutative diagram:

H0(X, TY |X ) → H0(X, NX ) → T 1B → 0

‖ ‖ ∩H0(U, TY |U ) → H0(U, NX |U ) → H1(U, TU ) → H1(U, TY |U )

which proves (i).If depthZ (OX ) ≥ 3 then X is normal and H1

Z (TY |X ) = 0 = H2Z (TY |X ) because

TY |X is locally free; from the local cohomology exact sequence we get

H1(U, TY |U ) ∼= H1(X, TY |X ) = 0

because X is affine. Using (i) we deduce (ii). The above proposition can be applied to prove the rigidity of a large class of cones

over projective varieties. We will need the following well-known lemmas, which weinclude for the reader’s convenience.

Lemma 3.1.18. Let W ⊂ IPr be a projective nonsingular variety, CW the affinecone over W , v ∈ CW the vertex, U = CW\v and p : U → W the projection. IfG is a coherent sheaf on CW such that G |U = p∗F for some coherent F = (0) onW , then the following conditions are equivalent:

(i) depthv (G) ≥ d for some d ≥ 2;(ii) H0(CW, G) = ⊕ν∈ZZ H0(W, F(ν)) and Hk(W, F(ν)) = 0 for all 1 ≤ k ≤ d−2

and ν ∈ ZZ.


Proof. We will use the equivalence

depthv (G) ≥ d ⇔ Hkv (G) = 0, k < d

([75], Theorem 3.8, p. 44). We have an exact local cohomology sequence:

0→ H0v (G)→ H0(CW, G)→ H0(U, G|U )→ H1

v (G)→ 0

and isomorphisms:

Hk−1(U, G|U ) ∼= Hkv (G), k ≥ 2

Since G|U = p∗F with F = (0) we have depthv (G) ≥ 1, thus H0v (G) = 0. On the

other hand, since p∗G|U = p∗ p∗F = ⊕ν∈ZZ F(ν), we haveH0(U, G|U ) = ⊕ν∈ZZ H0(W, F(ν)).Now the conclusion follows. Lemma 3.1.19. Let 0 = [0, . . . , 0, 1] ∈ IPr+1, V = IPr+1\0 and let π : V → IPr

be the projection. Then

TV/IPr = π∗O(1)

Proof. It is an immediate consequence of the commutative exact diagram

0 0↑ ↑

0→ TV/IPr → TV → π∗TIPr → 0‖ ↑ ↑

0→ OV (1) → OV (1)r+2 → OV (1)r+1 → 0↑ ↑O O↑ ↑0 0

where the vertical sequences are restrictions of Euler sequences. Corollary 3.1.20. Let W ⊂ IPr be a projective nonsingular variety of dimension≥ 2. Assume that

(i) H0(IPr ,O(ν)) → H0(W,OW (ν)) surjective for all ν ∈ ZZ (W is projectivelynormal).

(ii) H1(W,OW (ν)) = 0 for all ν ∈ ZZ.(iii) H1(W, TW (ν)) = 0 for all ν ∈ ZZ.

Then the affine cone CW over W is rigid.

Proof. CW has dimension ≥ 3 and hypothesis (i) implies that it is normal ([84],p. 126). Hypotheses (i) and (ii) imply that depthv (OCW ) ≥ 3, by Lemma 3.1.18.


Therefore by 3.1.17(ii) it suffices to show that H1(U, TU ) = 0 where U = CW\v.Let p : U → W be the projection. We have

H1(U,OU ) = ⊕ν∈ZZ H1(W,OW (ν)) = 0

H1(U, p∗TW ) = ⊕ν∈ZZ H1(W, TW (ν)) = 0(3.10)

by conditions (ii) and (iii). The relative tangent sequence of p takes the followingform:

0→ OU → TU → p∗TW → 0 (3.11)

In fact it follows from Lemma 3.1.19 and from Proposition B.1(i) that we haveTπ−1(W )/W = π∗OW (1) and therefore, since U = π−1(W )\W , we have TU/W =p∗OW (1) and this is clearly equal to OU . The conclusion follows from (3.10) andfrom the cohomology sequence of (3.11). Corollary 3.1.21. (i) The affine cone over IPn × IPm in its Segre embedding is rigid

for every positive n, m such that n + m ≥ 3.(ii) The affine cone over any Veronese embedding of IPn, n ≥ 2, is rigid.(iii) If W ⊂ IPr is a projective nonsingular variety of dimension ≥ 2, such that

h1(W,OW ) = 0 = h1(W, TW ) then the affine cone over the m-th Veroneseembedding W (m) of W is rigid for every m 0.

Proof. Using 3.1.20, (i) and (ii) are easy computations and (iii) follows from Serre’svanishing theorem ([84], Th. III.5.2).

The affine cone over the quadric IP1 × IP1 ⊂ IP3 is not rigid because it is ahypersurface (see Proposition 3.1.3, page 106); it does not satisfy both conditions (ii)and (iii) of Corollary 3.1.20. The affine cone over a rational normal curve Γr ⊂ IPr ,r ≥ 2, is not rigid either (for r = 2 it is a singular quadric surface in A3; for r ≥ 3see [131], [143], [144]). For a generalization of Corollary 3.1.20 and applications see[177].

3.1.4 Quotient singularities

The analysis of the previous subsection can be applied to the study of deformationsof a class of affine singular schemes obtained as quotients of nonsingular ones by theaction of a finite group.

Let Y = Spec(P) be an affine nonsingular algebraic variety on which a finitegroup G acts. Let X = Y/G be the quotient variety, q : Y → X the projection.Assume that the action is free outside a G-invariant closed subscheme W ⊂ Y . SetV = Y\W .

Proposition 3.1.22. Assume that depthW (OY ) ≥ 2. Then

TX ∼= (q∗TY )G

where G acts on TY = Derk(P, P)∼ by D → Dg := gDg−1 for all D ∈Derk(P, P) and g ∈ G.


Proof. Consider the exact sequence of coherent sheaves on Y :

0→ TY/X → TY → q∗TX → T 1Y/X

Ω1Y/X is supported on W since q is etale outside W ; then we have TY/X = 0. Simi-

larly, T 1Y/X is supported on W so that from the above exact sequence restricted to V

we deduce an isomorphism

H0(V, TY ) ∼= H0(V, q∗TX )

Then by Lemma 3.1.16 we deduce that

H0(Y, TY ) ∼= H0(Y, q∗TX )

Note that, letting A = PG the ring of invariant elements, we have X = Spec(A) andthe above isomorphism is equivalent to an isomorphism

Derk(P, P) ∼= Derk(A, P)

Therefore it will suffice to show that

Derk(A, A) ∼= Derk(A, P)G

So let D ∈ Derk(A, P) be such that D = gDg−1 for all g ∈ G. Then for everya ∈ A we have

D(a) = g(D(g−1a)) = g(D(a))

so D(a) ∈ A and therefore Derk(A, P)G ⊂ Derk(A, A). Conversely, ifD ∈ Derk(A, A) then it defines a k-derivation of A in P which is clearly G-invariantand we also have Derk(A, A) ⊂ Derk(A, P)G . Corollary 3.1.23. Let n be the order of G. Under the assumptions of 3.1.22, ifchar(k) does not divide n then TX is a direct summand of q∗TY .

Proof. Define a homomorphism q∗TY → TX = (q∗TY )G by

D → 1

n

∑g∈G

Dg

This defines the splitting. Theorem 3.1.24. In the above situation, if the action is free outside a G-invariantclosed subscheme W of codimension ≥ 3, and char(k) does not divide the order ofG then X = Y/G is rigid.

Proof. Let Z = q(W ) where q : Y → X is the projection, V = Y\W ,U = X\Z = V/G. We have depthZ (OX ) ≥ 2 because X is normal, being the


quotient of a nonsingular variety by a finite group (for this elementary fact see e.g.[166], p. 58). Therefore

T 1X ⊂ H1(U, TU ) ∼= H1(U, (q∗TY )G) (3.12)

where the inclusion follows from 3.1.17(i) and the isomorphism is Proposition3.1.22. We also have an exact sequence

H1(Y, TY ) → H1(V, TY ) → H2W (TY )

‖H1(U, q∗TY )

where the left vector space is 0 because Y is affine and the right one is 0 becauseof the depth assumption on W . It follows that H1(U, q∗TY ) = 0 and thereforeH1(U, (q∗TY )G) = 0 as well because it is a direct summand of it by Corollary3.1.23. The conclusion now follows from (3.12).

In the theorem the hypothesis on the codimension of W cannot be removed. Infact all rational two-dimensional double points are quotient singularities and they arehypersurfaces, therefore they are not rigid (see [15]).

Example 3.1.25 (Kollar [110]). Let n ≥ 3. Consider the weighted projective space

X = IP(1, 1, 1, a3, . . . , an) = Proj(k[X0, . . . , Xn])where the Xi ’s are indeterminates with weights ai = 1 if i = 0, 1, 2 and ai ≥ 1 ifi = 3, . . . , n (see [45] for details on weighted projective spaces). Then X is locallythe quotient of an affine space by a finite cyclic group and by the choice of theweights all its singularities have codimension ≥ 3. It follows from Theorem 3.1.24that T 1

X = 0. Therefore the first-order deformations of X are classified by H1(X, TX )by the exact sequence (2.17). But the exact sequence on X :

0→ OX →⊕

i

O(ai )→ TX → 0

implies H1(X, TX ) = 0. Therefore X is rigid.

NOTES

1. The main references for this section are [154], [120], [14].

2. ([174]) Let B be an e.f.t. local k-algebra, I ⊂ B an ideal generated by a regularsequence. Prove that if B/I is rigid then B is rigid as well.

3.2 Closed subschemes

3.2.1 The local Hilbert functor

Let X ⊂ Y be a closed embedding of algebraic schemes. A cartesian diagram ofmorphisms of schemes

3.2 Closed subschemes 123

η :X → X ⊂ Y × S↓ ↓ π

Spec(k) → S

where π is flat, and it is induced by the projection from Y × S, is called a (flat)family of deformations of X in Y parametrized by S, or over S. We call S and Xrespectively the parameter scheme and the total scheme of the family. When S =Spec(A) with A in ob(A∗) (resp. in ob(A), resp. A = k[ε]) we say that η is alocal, resp. infinitesimal, resp. first-order family of deformations of X in Y . We willalso say that η is a deformation of X in Y over A. The family is called trivial ifX = X × S. X is rigid in Y if every infinitesimal deformation of X in Y is trivial.When considering a family η we speak generally of a family of closed subschemesof Y .

Let X ⊂ Y be a closed embedding of algebraic schemes. For each A in ob(A)we let

HYX (A) = deformations of X in Y over A

We can immediately verify that this defines a functor of Artin rings

HYX : A→ (sets)

called the local Hilbert functor of X in Y .

Proposition 3.2.1. Given a closed embedding of algebraic schemes X ⊂ Y then:

(i) The local Hilbert functor H YX satisfies conditions H0, Hε , H , H of Theorem 2.3.2.

(ii) There is a natural identification

HYX (k[ε]) = H0(X, NX/Y )

where NX/Y is the normal sheaf of X in Y .

Proof. (i) Obviously HYX satisfies condition H0. Let

A′ A′′

A

be homomorphisms in A, with A′′ → A a small extension. Letting A = A′ ×A A′′we have a commutative diagram with exact rows:

0 → (ε) → A → A′ → 0‖ ↓ ↓

0 → (ε) → A′′ → A → 0

Take an element of

HYX (A′)×HY

X (A) H YX (A′′)


which is represented by a pair of deformations X ′ ⊂ Y × Spec(A′) andX ′′ ⊂ Y × Spec(A′′) such that

X ′ ×Spec(A′) Spec(A) = X ′′ ×Spec(A′′) Spec(A) ⊂ Y × Spec(A)

Consider the sheaf of A-algebras OX ′ ×OX OX ′′ on X . ThenX := (|X |,OX ′ ×OX OX ′′) is a scheme over Spec( A), flat over Spec( A) (see theproof of 2.4.1). Therefore X is a deformation of X over Spec( A) inducing X ′ andX ′′. We have a commutative diagram:

X → X ′↓ ↓X ′′ → Y × Spec( A)

and the universal property of the fibred sum implies that there is a morphism

Φ : X → Y × Spec( A)

Pulling back Φ over Spec(A′′) (resp. Spec(A′)) we obtain the closed embeddingX ′′ ⊂ Y × Spec(A′′) (resp. X ′ ⊂ Y × Spec(A′)). Since Spec(A′′) ⊂ Spec( A) isa closed embedding defined by a square zero ideal, it follows that Φ is a closedembedding as well (details are left to the reader). Therefore Φ : X ⊂ Y × Spec( A)defines an element of HY

X ( A) which is mapped to (X ′,X ′′) by the map:

α : HYX (A′ ×A A′′)→ H Y

X (A′)×HYX (A) H Y

X (A′′)

It follows that α is surjective. Now let X ⊂ Y × Spec( A) be another element ofHY

X ( A) which is mapped to (X ′,X ′′). Then by the universal property of the fibredsum there is a morphism τ : X → X which, as in the proof of 2.4.1, is easily seento be an isomorphism. Moreover, the following diagram commutes:

Y × Spec( A)

X τ−→ X

Since the diagonal arrows are closed embeddings it follows that X = X as closedsubschemes of Y×Spec( A). This proves that α is actually a bijection and (i) follows.

(ii) Let I ⊂ OY be the ideal sheaf of X . A first-order deformation of X in Y , i.e.a flat family:

X → X ⊂ Y × Spec(k[ε])↓ ↓

Spec(k) → Spec(k[ε])(3.13)

is defined by a sheaf OX of flat k[ε]-algebras, with an isomorphismOX ⊗k[ε] k ∼= OX . The closed embedding X ⊂ Y × Spec(k[ε]) is determinedby a sheaf of ideals Iε ⊂ OY [ε] := OY ⊗k k[ε] such that OX = OY [ε]/Iε . Theabove data are obtained by gluing together their restrictions to an affine open cover.


On an affine open set U = Spec(P) ⊂ Y , let X ∩ U = Spec(B) where B = P/Jfor an ideal J = ( f1, . . . , fN ) ⊂ P .

Consider the exact sequence

0→ Rv−→ P N f−→ J → 0

where R is the module of relations among f1, . . . , fN . Taking HomP (−, B) weobtain the exact sequence:

0→ HomB(J/J 2, B)→ HomP (P N , B)v∨−→ HomP (R, B)

which identifies HomB(J/J 2, B) with ker(v∨). An element of ker(v∨) can be rep-resented as an N -tuple h = (h1, . . . , hN ) of elements of P which, interpreted as anelement of HomP (P N , B) by scalar product (i.e. h(p1, . . . , pN ) = ∑

j h j p j modJ ) must be zero on R. Hence∑

j

h j r j ∈ J for every (r1, . . . , rN ) ∈ R

This means that there exist ∆r1, . . . , ∆rN ∈ P such that∑j

h j r j = −∑

j

f j∆r j

or, equivalently, such that

( f + εh) t (r + ε∆r) = 0

in P ⊗k k[ε]. Therefore from Corollary A.11 it follows that f + εh generates anideal in P ⊗k k[ε] which defines a first-order deformation of Spec(B) in Spec(P)because every relation among f1, . . . , fN extends to a relation among f1+ εh1, . . . ,fN + εhN . Using the same argument backwards one sees that every first-order de-formation of Spec(B) in Spec(P) defines an element of HomB(J/J 2, B).

It follows that at the global level we have a canonical 1–1 correspondencebetween first-order deformations of X in Y and H0(X, NX/Y ). Corollary 3.2.2. Let X ⊂ Y be a closed embedding of algebraic schemes. Ifh0(X, NX/Y ) <∞, for example if X is projective, then HY

X is prorepresentable.

Proof. Follows from Proposition 3.2.1 and from Theorem 2.3.2. If X ⊂ Y is a closed embedding of projective schemes then the prorepresentabi-

lity of HYX follows directly from the existence of the Hilbert scheme HilbY because

HYX is prorepresented by the complete local ring OHilbY ,[X ] (see § 4.3).

Let A be in ob( A). A formal deformation of X in Y is a sequence

ξn :Xn ⊂ Y × Spec( An)↓

Spec( An)


of infinitesimal deformations of X in Y over An = A/mn+1A

such that ξn induces

ξn−1 by pullback under the natural inclusion Spec( An−1)→ Spec( An) for all n ≥ 1.We can describe the formal deformation ( A, ξn) of X in Y as a diagram of formalschemes

XX ⊂ Y × Specf( A)↓

Specf( A)

As in the case of the functor DefX , a formal deformation ( A, ξn) of X in Y definesan element ξ ∈ H Y

X ( A), i.e. a formal couple ( A, ξ ) for HYX , and conversely, every

such element is defined by a formal deformation of X in Y .

Considering not necessarily infinitesimal deformations of a closed subschemeone has the notion of characteristic map, as follows.

Definition 3.2.3. Let X ⊂ Y be a closed embedding of algebraic schemes, (S, s) apointed scheme and

ξ :X ⊂ X ⊂ Y × S↓ ↓ π

Spec(k)s−→ S

be a family of deformations of X in Y . By pulling back this family by morphismsSpec(k[ε]) → S with image s and applying Proposition 3.2.1 we obtain a linearmap

χξ : Ts S→ H0(X, NX/Y )

called the characteristic map of the family ξ at s.

The characteristic map is the analogous for embedded deformations of theKodaira–Spencer map for abstract ones.

Examples 3.2.4. (i) If X ⊂ IPr , r ≥ 1, is the complete intersection of r − nhypersurfaces f1, . . . , fr−n of degrees d1 ≤ d2 ≤ . . . ≤ dr−n respectively, wehave a presentation

2∧[⊕ jO(−d j )] f−→ ⊕ jO(−d j )→ IX → 0 (3.14)

Taking Hom(−,OX ) we obtain an exact sequence:

0→ NX →⊕ jOX (d j )f−→

2∧[⊕ jOX (d j )]

where NX = NX/IPr . Since each nonzero entry of the matrix defining f is one of thef j ’s, the map f is zero. Therefore

NX ∼= ⊕ jOX (d j )

Equivalently, one can remark that (3.14) induces a surjective homomorphism


⊕ jOX (−d j )→ IX/I2X

of locally free sheaves of the same rank, which must therefore be an isomorphism.In particular, if X is a hypersurface of degree d we have NX ∼= OX (d) and therefore

h0(X, NX ) =(

d + r

r

)− 1

confirming the fact that X can be deformed in IPr only inside the linear system ofhypersurfaces of degree d, which is a projective space of dimension

(d+rr

)− 1.If X is a linear subspace, i.e. d1 = · · · = dr−n = 1, then NX ∼= OX (1)⊕r−n and

thereforeh0(X, NX ) = (r − n)(n + 1)

as expected, since such linear subspaces X are parametrized by the grassmannianG(n + 1, r + 1), which is nonsingular of dimension (r − n)(n + 1).

(ii) Let X be a Cartier divisor on a connected projective scheme Y . Consider theexact sequence

0→ OY → OY (X)→ NX/Y → 0

and the cohomology sequence:

0→ H0(Y,OY )→ H0(Y,OY (X))χ−→ H0(X, NX/Y )→ H1(Y,OY ) (3.15)

Classically, the map χ was called the characteristic map of the linear system |X |.By definition, Im(χ) ∼= H0(Y,OY (X))/H0(Y,OY ) is naturally identified with thetangent space to the linear system |X |. We can verify that this is so by identifyingIm(χ) as a subvector space of first-order deformations of X in Y , as follows.

Assume that X is defined by a system of local equations fi , fi ∈ Γ (Ui ,OY ) not0-divisor, with respect to an affine cover Ui of Y . We havefi j := fi f −1

j ∈ Γ (Ui ∩ U j ,O∗Y ) for all i, j , and fi j is a Cech 1-cocycle whichdefines the line bundle OY (X). A first-order deformation of X in Y is a Cartier di-visor X ⊂ Y × Spec(k[ε]) which is determined by a system Fi = fi + εgi ,gi ∈ Γ (Ui ,OY ), such that there exist Fi j = fi j + εgi j ∈ Γ (Ui ∩U j ,O∗Y×Spec(k[ε]))(hence gi j ∈ Γ (Ui ∩ U j ,OY )) satisfying Fi = Fi j Fj for all i, j . Therefore onUi ∩U j we have:

fi + εgi = ( fi j + εgi j )( f j + εg j )

which is equivalent to the identity:

gi = fi j g j + gi j f j

This identity shows that the system gi = gi mod IX defines a section of NX/Y , asexpected. We also see that gi ∈ Im(χ) if and only if gi = fi j g j , in which caseFi j = fi j , i.e. X deforms inside the linear system |X |, as asserted.


(iii) In IP2 × IP1 with bihomogeneous coordinates (u, v, w; x, y) consider thehypersurface m , m ≥ 0, defined by the equation:

xmv − ymu = 0

One can show that m ∼= Fm , and that the structure morphism π : Fm → IP1 isinduced by the projection IP2 × IP1 → IP1. Therefore the surface m is a realiza-tion of the rational ruled surface Fm as the total scheme of a family of lines of IP2

parametrized by IP1. When m > 0 this family is nontrivial because Fm ∼= IP1× IP1

(for details see [5]).

Examples 3.2.5. The following set of examples deals with properties of projectivecurves. Proofs are straightforward and are left to the reader.

(i) If Y is a projective scheme and C ⊂ Y is a projective integral l.c.i. curve, thenthe normal sheaf NC/Y is torsion free. If C is nonsingular then NC/Y is locally free.

(ii) Consider a nonsingular curve C ⊂ IP3 and a (possibly singular) surfaceS ⊂ IP3 of degree n containing C . Prove that there is an exact sequence of locallyfree sheaves on C :

0→ a−1 ⊗ KC (−n + 4)→ NC/IP3ψ−→ OC (n)→ [OC/a](n)→ 0 (3.16)

where a ⊂ OC is the ideal sheaf generated by the restriction to C of the partialderivatives

∂F

∂X0, . . . ,

∂F

∂X3

where F = 0 is an equation of S.(Hint: Im(ψ) = OC ⊗ Im(TIP3|S → NS/IP3)).

In the case where S is nonsingular we obtain the sequence:

0→ KC (−n + 4)→ NC → OC (n)→ 0 (3.17)

Deduce from (3.16) that if a = OC (i.e. if C ∩ Sing(S) = ∅) then C is not regularlyembedded in S (yet the normal sheaf NC/S is locally free.)

(iii) Consider a nonsingular curve C ⊂ IP3 and a point p ∈ IP3\C . Then thereis an exact sequence

0→ OC (1)→ NC/IP3 → ωC (3)→ 0

which is obtained as a special case of (3.16) by taking as S the cone projecting Cfrom p. Deduce that

h1(C, NC/IP3) ≤ h1(C,OC (1))

(iv) Let C ⊂ IPr , r ≥ 4, be a nonsingular irreducible projective curve. Letp1, . . . , pk ∈ IPr , 1 ≤ k ≤ r − 3, be general points and let π : C → IPr−k bethe projection with centre the linear span 〈p1, . . . , pk〉. Prove that there is an exactsequence


0→ OC (1)k → NC/IPr → π∗Nπ(C)/IPr−k → 0 (3.18)

which splits if and only if C is contained in a hyperplane. Deduce that

h1(C, NC/IPr ) ≤ (r − 2)h1(C,OC (1)) (3.19)

In particular, H1(C, NC/IPr ) = (0) if h1(C,OC (1)) = 0. (Solution: see [163], Prop.11.2). Inequality (3.19) was already known classically as a bound on the dimensionof the Hilbert scheme ([169], s. 8). For a sharpening of (3.19) see [47].

(v) Let C ⊂ IPr be a nonsingular irreducible projective curve and letL = OC (1). Show that if the Petri map µ0(L) (see Example 3.3.9) is injective thenH1(C, NC/IPr ) = (0).

(vi) Let C ⊂ IPr be a projective irreducible nonsingular curve of degree d andgenus g. Prove that

χ(NC/IPr ) := h0(C, NC/IPr )− h1(C, NC/IPr ) = (r + 1)d + (r − 3)(1− g)

In particular, for a canonical curve of genus r + 1 ≥ 3 and degree 2r in IPr we have

χ(NC/IPr ) = h0(C, NC/IPr ) = r(r + 5)

because h1(C, NC/IPr ) = 0 since the Petri map µ0(ωC ) is injective (see (v) above).

3.2.2 Obstructions

Let X ⊂ Y be a closed embedding, A in ob(A) and let

ξ :X → X ⊂ Y × Spec(A)↓ ↓ f

Spec(k) → Spec(A)

be a deformation of X in Y over A. Let

e : 0→ k→ A→ A→ 0

be a small extension. A lifting of ξ to A is a deformation of X in Y over A:

ξ :X → X ⊂ Y × Spec( A)↓ ↓ f


whose pullback to Spec(A) is ξ .

Proposition 3.2.6. Let X ⊂ Y be a regular closed embedding (Definition D.1.1) ofalgebraic schemes with X projective. Then H1(X, NX/Y ) is an obstruction space forthe local Hilbert functor HY

X .


Proof. Given

ξ :X → X ⊂ Y × Spec(A)↓ ↓ f

Spec(k) → Spec(A)

where A is in ob(A), an infinitesimal deformation of X in Y , we will show that thereis a natural linear map

oξ/Y : Exk(A, k)→ H1(X, NX/Y )

such that, for every extension

e : 0→ εk→ A→ A→ 0

we have oξ/Y (e) = 0 if and only if ξ has a lifting to A; this will prove the proposition.Since X is regularly embedded in Y we can find an affine open cover U = Ui i∈I

of Y such that Xi := X ∩ Ui is a complete intersection in Ui for each i . LetUi = Spec(Pi ), Xi = Spec(Pi/Ii ) where Ii = ( fi1, . . . , fi N ) with fi1, . . . , fi N a regular sequence in Pi . We then have X|Ui = Spec(Pi A/Ii A) where

Ii A = (Fi1, . . . , Fi N ) ⊂ Pi A := Pi ⊗ A

and fiα = Fiα mod m A, α = 1, . . . , N . Choose arbitrarily Fi1, . . . , Fi N ∈ Pi Asuch that Fiα = Fiα mod ε. By Example A.12 Fi1, . . . , Fi N and Fi1, . . . , Fi N are regular sequences in Pi A and in Pi A respectively; in particular, letting Ii A =(Fi1, . . . , Fi N ),

Xi := Spec(Pi A/Ii A) ⊂ Ui × Spec( A)

is a lifting of X|Ui ⊂ Ui × Spec(A) to A. In order to find a lifting of X to A we mustbe able to choose the Fiα’s in such a way that

Xi |Ui j = X j |Ui j ⊂ Ui j × Spec( A) (3.20)

for each i, j ∈ I . Letting Ui j = Spec(Pi j ) and viewing the fiα’s and f jα’s aselements of Pi j via the natural maps

Pi Pj

Pi j

we haveFjα − Fiα =: εhi jα

and hi j := (hi j1, . . . , hi j N ) ∈ Γ (Ui j , NX/Y ), because NX/Y is locally free of rank

N and is trivial on each Ui . By construction, hi j ∈ Z1(U , NX/Y ). The condition

(3.20) means that we can choose the Fiα’s so that hi j = 0 all i, j . A different choice

of the Fiα’s is of the form Fiα + εhiα and hi := (hi1, . . . , hi N ) ∈ Γ (Ui , NX/Y ).Since we have


(Fjα + εh jα)− (Fiα + εhiα) = ε(hi jα + h jα − hiα) (3.21)

we see that hi j defines an element oξ/Y (e) ∈ H1(X, NX/Y ) which is zero if and

only if the Xi ’s satisfy condition (3.20) and define a lifting X of X to A. It is easy toshow that the map oξ is k-linear.

The element oξ/Y (e) ∈ H1(X, NX/Y ) is called the obstruction to lifting ξ to A;we call ξ obstructed if oξ/Y (e) = 0 for some e ∈ Exk(A, k); otherwise it is unob-structed. X is said to be unobstructed in Y if all its infinitesimal deformations in Yare unobstructed; otherwise X is said to be obstructed in Y . Examples of obstructedclosed subschemes are usually quite subtle, especially if one is interested in nonsin-gular obstructed subvarieties. In order to be able to describe them in a natural way itis necessary to know the existence of the Hilbert scheme of a projective scheme. Wewill give examples in § 4.6.

Corollary 3.2.7. Let j : X ⊂ Y be a regular closed embedding of algebraic schemeswith X projective and let (R, ξn) be the formal universal deformation of X in Y .Then

(i)h0(X, NX/Y ) ≥ dim(R) ≥ h0(X, NX/Y )− h1(X, NX/Y )

The first equality holds if and only if X is unobstructed in Y .(ii) X is rigid in Y if and only if H0(X, NX/Y ) = 0.(iii) If H1(X, NX/Y ) = (0) then X is unobstructed in Y .

The proof is left to the reader. In the case of a closed embedding X ⊂ Y whichis not regular, Proposition 3.2.6 says nothing about the obstruction space of HY

X . Werefer the reader to § 4.4 and § 4.6 for some information about the general case.

Examples 3.2.8. (i) Let C be a projective nonsingular curve contained in a nonsingu-lar surface S, and assume that C is negatively embedded in S, i.e. deg(OC (C)) < 0.Then H0(C,OC (C)) = 0 and therefore C is rigid in S.

This happens in particular when C ∼= IP1 is an exceptional curve of the first kind.Another example is when C has genus g ≥ 2, S = C × C and C is identified to thediagonal ∆ ⊂ S. In this case NC/S = TC , which has degree 2 − 2g < 0. Note thatH1(C, TC ) = (0) but C is unobstructed in S, being rigid in S. This example showsthat the sufficient condition of Corollary 3.2.7 is not necessary.

(ii) Hypersurfaces of IPr are unobstructed. In fact, if X ⊂ IPr has degree d then

h1(X, NX/IPr ) = h1(X,OX (d)) = 0

More generally, complete intersections in IPr are unobstructed (see Subsection4.6.1).

(iii) Let Q ⊂ IP3 be a quadric cone with vertex v , and L ⊂ Q a line. Then wehave an inclusion

NL/Q ⊂ NL/IP3 = OL(1)⊕OL(1)


whose cokernel is OL(2)(−v) (see Example 3.2.5(ii)). It follows that NL/Q =OL(1); in particular, it is locally trivial and H1(L , NL/Q) = 0, despite the factthat L ⊂ Q is not a regular embedding (Example 3.2.5(ii)) and L is obstructed in Q(see Note 1 of § 4.6).

(iv) Let C ⊂ IPr be a nonsingular irreducible projective curve and let L =OC (1). If the Petri map µ0(L) (see Example 3.3.9) is injective then C is unobstructedin IPr because h1(C, NC/IPr ) = 0 (Example 3.2.5(v)). In particular canonical curvesof genus g ≥ 3 are unobstructed in IPg−1 (Example 3.2.5(vi)).

3.2.3 The forgetful morphism

Let X ⊂ Y be a closed embedding of algebraic schemes. The forgetful morphism

Φ : HYX → DefX

is the morphism which associates to an infinitesimal deformation of X in Y :

ξ :X → X ⊂ Y × Spec(A)↓ ↓

Spec(k) → Spec(A)

the isomorphism class of the deformation of X :

X → X↓ ↓

Spec(k) → Spec(A)

Proposition 3.2.9. Assume that X and Y are nonsingular and that X is projective.Consider the exact sequence

0→ TX → TY |X → NX/Y → 0 (3.22)

Then:

(i) dΦ = δ : H0(X, NX/Y ) → H1(X, TX ) is the coboundary map coming from(3.22).

(ii) The coboundary map

δ1 : H1(X, NX/Y )→ H2(X, TX )

arising from the exact sequence (3.22) is an obstruction map for Φ (see Defini-tion 2.3.5).

(iii) If H 1(X, TY |X ) = 0 then Φ is smooth.(iv) If X is unobstructed in Y and δ is surjective then Φ is smooth. In particular, X

is unobstructed as an abstract variety and has rk(δ) number of moduli.


Proof. (i) Let

ξ :X ⊂ X ⊂ Y × Spec(k[ε])↓ ↓ π

Spec(k)s−→ Spec(k[ε])

be a first-order deformation of X in Y . We must show that δχ(ξ) = κ(ξ). Letχ(ξ) = h ∈ H0(X, NX/Y ). Consider an affine open cover U = Ui = Spec(Pi ) ofY , and let Xi = X ∩Ui = Spec(Pi/( fi1, . . . , fi N )). We have

Xi := X|Ui = Spec(P[ε]/( fi1 + εhi1, . . . , fi N + εhi N ))

Then (hi1, . . . , hi N ) =: hi = h|Ui∈ Γ (Ui , NX/Y ). Since Xi is affine and non-

singular the abstract deformation Xi of Xi is trivial: thus there exist isomorphismsθi : Xi × Spec(k[ε]) → Xi and κ(ξ) ∈ H1(X, TX ) is defined by the 1-cocycledi j ∈ Z1(U , TX ) corresponding to the system of automorphisms

θi j = θ−1i θ j : Xi j × Spec(k[ε])→ Xi j × Spec(k[ε])

where Xi j = X ∩ Ui j . Let’s compute δ(h). The isomorphism θi is given by anisomorphism of k[ε]-algebras

ti : Pi [ε]/( fi1 + εhi1, . . . , fi N + εhi N )→ Pi [ε]/( fi1, . . . , fi N )

which, by the smoothness of Pi , is induced by a k[ε]-automorphism

Ti : Pi [ε] → Pi [ε]of the form Ti (p + εq) = p + ε(q + di (p)) where di ∈ Derk(Pi , Pi ) = Γ (Ui , TY )is such that di (hiα) = − fiα . We have di ∈ C0(U , TY ) and δ(h) is defined by(d j − di )|Xi . Since (d j − di )|Xi = di j we conclude that δ(h) = κ(ξ).

(ii) One must show that, given an infinitesimal deformation

ξ :X ⊂ X ⊂ Y × Spec(A)↓ ↓ π

Spec(k)s−→ Spec(A)

where A is in A, we have a commutative diagram

Exk(A, k) oξ/Y oξ

H1(X, NX/Y )δ1−→ H2(X, TX )

The proof of this fact is similar to the proof of part (i) and will be omitted.(iii) The exact sequence (3.22) shows that the hypothesis implies

dΦ : H0(X, NX/Y )→ H1(X, TX ) surjective and

δ1 : H1(X, NX/Y )→ H2(X, TX ) injective

Therefore the assertion follows from Proposition 2.3.6.(iv) is left to the reader.


Remark 3.2.10. Let X ⊂ Y be a regular embedding of algebraic schemes with Xreduced and Y nonsingular, I ⊂ OY the ideal sheaf of X , and let Φ : HY

X → DefX

be the forgetful morphism. The differential

dΦ : H0(X, NX/Y ) = HomOX (I/I2,OX )→ Ext1OX(Ω1

X ,OX )

is the k-linear map which associates to σ : I/I2 → OX the pushout σ∗(S) where

S : 0→ I/I2 → Ω1Y |X → Ω1

X → 0

is the conormal sequence of X ⊂ Y . This generalizes Proposition 3.2.9. The proofconsists in considering, for a first-order deformation of X in Y

X ⊂ X ⊂ Y × Spec(k[ε])the induced diagram of conormal sequences

0→ I/I2 ⊕OX → Ω1Y |X ⊕OX → Ω1

X → 0↓ ↓ ‖

0→ OX → Ω1X |X → Ω1

X → 0

and in recognizing the second row as the pushout of the first.

Part (iii) of the proposition often gives a very effective way of proving that a givenX ⊂ Y is unobstructed as an abstract variety. If X is a curve in IPr the vanishing ofH1(X, TIPr |X ) is related to the Petri map (see Example 3.3.9).

The following examples are further applications of this principle.

Examples 3.2.11. (i) ([107]) Let X ⊂ IPr , r ≥ 3, be a nonsingular hypersurface ofdegree d ≥ 2. Then h1(NX/IPr ) = h1(OX (d)) = 0 and therefore X is unobstructedin IPr . On the other hand, from the exact sequence:

0→ TIPr (−d)→ TIPr → TIPr |X → 0

and the Euler sequence:

0→ OIPr → OIPr (1)r+1 → TIPr → 0

we deduce that:h1(TIPr |X ) = h2(TIPr (−d)) = 0 if r ≥ 4

while for r = 3 we have the exact sequence:

0← H2(TIP3(−d))∨ ← H0(OIP3(d − 4)) ← H0(OIP3(d − 5))4

‖H1(TIP3|X )∨

Therefore we see that

h1(TIPr |X ) =

1 if r = 3 and d = 4;0 otherwise


From 3.2.9 we therefore deduce that H IPr

X → DefX is smooth and X is unobstructedas an abstract variety unless r = 3 and d = 4 (this is precisely the case when X isa K3 surface). An analogous result holds more generally for complete intersections([161]).

Using (3.22) one computes easily that H2(TX ) = 0 if X is a nonsingular surfaceof degree d ≥ 5 in IP3: therefore the unobstructedness of X could not have beendeduced from 2.4.6 in this case.

One can generalize to singular hypersurfaces as follows. Consider a reducedhypersurface X ⊂ IPr , r ≥ 3, of degree d ≥ 2. Then the conormal sequence is

0→ OX (−d)→ Ω1IPr |X → Ω1

X → 0

so that H1(X, NX/IPr ) = 0 and we have the exact sequence

H0(X, NX/IPr )dΦ−→ Ext1OX

(Ω1X ,OX )→ Ext1(Ω1

IPr |X ,OX )‖ ‖

H0(X,OX (d)) H1(X, TIPr |X )

where the equality on the right is because Ω1IPr |X is locally free. Therefore as before,

we see that Φ : HYX → DefX is smooth and X is unobstructed if (r, d) = (3, 4).

(ii) The previous example can be easily generalized to nonsingular hypersurfacesof IPn × IPm , 1 ≤ n ≤ m, n + m ≥ 3. Let

IPn × IPm q−→ IPm

↓ pIPn

be the projections. Consider a nonsingular hypersurface X ⊂ IPn × IPm of bidegree(a, b), i.e. defined by an equation σ = 0 for some σ ∈ H0(O(a, b)), where

O(a, b) := p∗O(a)⊗ q∗O(b)

From the exact sequence

0→ O→ O(a, b)→ NX/IPn×IPm → 0

one deduces thatH1(NX/IPn×IPm ) = (0)

and therefore X is unobstructed in IPn× IPm . For any coherent sheaf F on IPn× IPm

we use the notationF(α, β) = F ⊗O(α, β)

Using the fact thatTIPn×IPm = p∗TIPn ⊕ q∗TIPm


and the Leray spectral sequence with respect to any one of the projections, one easilycomputes that

hi (TIPn×IPm (α, β)) = 0

when n +m ≥ 4, i = 1, 2 and (α, β) arbitrary. Moreover, when (n, m) = (1, 2) onefinds:

hi (TIP1×IP2) = 0 all i ≥ 1h2(TIP1×IP2(−a,−b)) = 0 unless (a, b) = (2, 3)

Putting all this information together and using the exact sequence

0→ TIPn×IPm (−a,−b)→ TIPn×IPm → TIPn×IPm |X → 0

one deduces thath1(TIPn×IPm |X ) = 0

unless (n, m) = (1, 2) and (a, b) = (2, 3) (this is precisely the case whenX is a K3 surface). Now as before, we conclude that the forgetful morphismH IPn×IPm

X → DefX is smooth and X is unobstructed as an abstract variety.

3.2.4 The local relative Hilbert functor

Given a projective morphism p : X → S of schemes and a k-rational point s ∈ S,consider the fibre X (s) and a closed subscheme Z ⊂ X (s). For each A in ob(A) aninfinitesimal deformation of Z in X relative to p parametrized by A is a commutativediagram:

Z ⊂ XA → X ↓ ↓ p

Spec(A)s−→ S

where the right square is cartesian, the left diagonal morphism is flat and its closedfibre is Z ; this means in particular that the morphism s has image s and thereforethat A is an OS,s-algebra. Then, letting Λ = OS,s , we can define the local relativeHilbert functor

HX /SZ : AΛ→ (sets)

by

HX /SZ (A) =

infinitesimal deformations of Z in X

relative to p parametrized by A

We have the following generalization of Corollaries 3.2.2 and 2.4.7:

Theorem 3.2.12. Let p : X → S be a projective morphism of schemes, s ∈ S ak-rational point, and Z ⊂ X (s) a closed subscheme of the fibre X (s). Denote OS,s

by Λ. Then:

(i) The local relative Hilbert functor HX /SZ : AΛ → (sets) is prorepresentable and

has tangent space H0(Z , NZ/X (s)).

3.3 Invertible sheaves 137

(ii) If Z is regularly embedded in X (s) and p is flat then H 1(Z , NZ/X (s)) is an

obstruction space for HX /SZ , and we have an exact sequence:

0→ H0(Z , NZ/X (s))→ tR → Ts S→ H1(Z , NZ/X (s)) (3.23)

where R is the local Λ-algebra prorepresenting HX /SZ .

Proof. (i) The proof of 3.2.1 can be followed almost verbatim, showing that HX /SZ

satisfies conditions H0, Hε , H , H and that H0(Z , NZ/X (s)) is its tangent space.(ii) The proof of 3.2.6 can be easily adapted to this case. The exact sequence

(3.23) follows from the above and from (2.2). Another generalization of the local Hilbert functors can be obtained with no extra

effort. Consider a projective scheme X and a formal deformation of X

π : XX → Specf(R)

where R is in ob(A) and π is a flat projective morphism of formal schemes; letZ ⊂ X be a closed subscheme. For each A in ob(A) define an infinitesimal defor-mation of Z in XX relative to π parametrized by A as a commutative diagram:

Z ⊂ XXA → XX ↓ ↓ π

Spec(A)s−→ Specf(R)

where the right square is cartesian, the left diagonal morphism is flat and its closedfibre is Z . Note that Spec(A) = Specf(A) and the morphism s is defined by a surjec-tive homomorphism R → A, so that XXA is just an ordinary scheme projective andflat over Spec(A). We can define the local relative Hilbert functor

HXX /Specf(R)Z : AR → (sets)

as above. A result analogous to 3.2.12 can be proved in this case as well with asimilar proof. Details of this straightforward generalization are left to the reader.

3.3 Invertible sheaves

3.3.1 The local Picard functors

The local Picard functors are related to the Picard schemes in the same way as the lo-cal Hilbert functors are to the Hilbert schemes, in the sense that if the Picard schemeof a given scheme exists then the local Picard functors describe its infinitesimal andlocal properties. In this section we will prove the basic properties of the local Picardfunctors. The Picard schemes will not be treated here. For a full treatment the readeris referred to Kleiman [100].


Let X be a scheme. Recall that the Picard group of X is defined to be the groupPic(X) of isomorphism classes of invertible sheaves on X ; we have the identifica-tion Pic(X) = H1(X,O∗X ) (see [84]). We denote by [L] ∈ Pic(X) the class of aninvertible sheaf L . For every A in ob(A) we will write for brevity X A instead ofX × Spec(A). For every morphism A → B in A, and for every invertible sheafL on X A we denote by L ⊗A B the invertible sheaf it induces on X B by pullback.Similarly, given λ ∈ Pic(X A) we denote by λ⊗A B ∈ Pic(X B) its image under thepullback operation. This defines a homomorphism of group

Pic(X A) = H1(X A,O∗X A)→ H1(X B,O∗X B

) = Pic(X B)

which makes A → Pic(X A) a covariant functor on A with values in the category ofabelian groups.

We fix an element λ0 ∈ Pic(X) once and for all and, for each A in ob(A), we let

Pλ0(A) := λ ∈ Pic(X A) : λ⊗A k = λ0Then Pλ0(A) is a subset of Pic(X A) whose definition is functorial in A so that wehave a functor of Artin rings:

Pλ0 : A→ (sets)

which will be called the local Picard functor of (X, λ0).Let L be an invertible sheaf on X such that λ0 = [L]. For a given A ∈ ob(A)

the elements of Pλ0(A) are the isomorphism classes of infinitesimal deformationsof L over A: an infinitesimal deformation of L over A is an invertible sheaf L onX × Spec(A) such that L = L|X×Spec(k). In the case A = k[ε] we speak of a first-order deformation of L .

The main result about this functor is the following.

Theorem 3.3.1. Let X be an algebraic scheme, λ0 ∈ Pic(X). Assume that the fol-lowing conditions are satisfied:

(a) H0(X,OX ) ∼= k(b) dimk H1(X,OX ) <∞Then Pλ0 is prorepresentable and Pλ0(k[ε]) = H1(X,OX ). Moreover, H2(X,OX )is an obstruction space for Pλ0 .

Proof. Let’s check the conditions of Theorem 2.3.2. It is clear that P = Pλ0 satisfiescondition H0. Let

A′ A′′

A

be a diagram in A with A′′ → A surjective. Consider an element

(λ′, λ′′) ∈ P(A′)×P(A) P(A′′)


and let λ = λ′ ⊗A′ A = λ′′ ⊗A′′ A. Let L ′, L ′′, L be invertible sheaves onX A′ , X A′′ , X A respectively such that [L ′] = λ′, [L ′′] = λ′′, [L] = λ. Then we havehomomorphisms L ′ → L and L ′′ → L of sheaves on |X | inducing isomorphismsL ′ ⊗A′ A ∼= L , L ′′ ⊗A′′ A ∼= L . Let B = A′ ×A A′′.

Claim: OX B∼= OX A′ ×OX A

OX A′′ .

For every open set U ⊂ |X | we have by definition:

[OX A′ ×OX AOX A′′ ](U ) = OX A′ (U )×OX A (U ) OX A′′ (U )

and by the universal property of the fibred sum we have a homomorphism

φ : OX B → OX A′ ×OX AOX A′′

which is induced by the homomorphisms OX B → OX A′ and OX B → OX A′′ comingfrom B → A′ and B → A′′ respectively. Since A′′ → A is surjective

φ ⊗B A′ : OX B ⊗B A′ → OX A′

is an isomorphism. From Lemma A.4 it follows that φ is an isomorphism.From the claim we deduce that N := L ′ ×L L ′′ is an invertible sheaf on X B and

the projections induce isomorphisms N ⊗B A′ ∼= L ′, N ⊗B A′′ ∼= L ′′. Therefore

P(B) * [N ] → (λ′, λ′′) ∈ P(A′)×P(A) P(A′′)

This shows that the map

α : P(B)→ P(A′)×P(A) P(A′′)

is surjective.Assume that M is an invertible sheaf such that α([M]) = α([N ]). This means

that there are homomorphisms M → L ′ and M → L ′′ inducing isomorphismsM⊗B A′ ∼= L ′, M⊗B A′′ ∼= L ′′. It follows that we have an automorphism θ : L → Lgiven by the composition:

L ∼= L ′ ⊗A′ A ∼= M ⊗B A ∼= L ′′ ⊗A′′ A ∼= L

which makes the following diagram commutative:

M q ′ q ′′

L ′ L ′′ u′ u′′

Lθ−→ L


By hypothesis (a) the isomorphism θ is the multiplication by a unit a ∈ A. SinceA′′ → A is surjective we can lift a to a′′ ∈ A′′ and we can change q ′′ to a′′q ′′ thusassuming that u′q ′ = u′′q ′′. It follows that we have a commutative diagram

M

L ′ L ′′

L

and this implies that M ∼= N . This shows that α is bijective. Therefore P satisfiesalso conditions Hε and H .

We have an exact sequence

0→ OXexp−→ O∗Xk[ε] → O∗X → 0

where exp( f ) = 1+ ε f . It follows that

P(k[ε]) = ker[H1(Xk[ε],O∗Xk[ε])→ H1(X,O∗X )] = H1(X,OX )

Finally, given A in ob(A) and [e] ∈ Exk(A, k) represented by an extension

0→ (t)→ A→ A→ 0

we have an exact sequence:

0→ tOX → O∗X A→ O∗X A

→ 0

which induces an exact sequence

H1(X A,O∗X A

)→ H1(X A,O∗X A)

δ−→ H2(X,OX )

Given λ ∈ P(A) it can be lifted to a λ ∈ P( A) if and only if δ(λ) = 0 ∈ H2(X,OX ).This shows that H2(X,OX ) is an obstruction space for P . Corollary 3.3.2. Let X be a projective integral scheme. Then Pλ0 is prorepresentablefor every λ0 ∈ Pic(X).

Proof. A projective integral scheme satisfies both conditions (a) and (b) of thetheorem. Remark 3.3.3. Let X be an algebraic scheme and λ0 ∈ Pic(X). Then the tangent andobstruction spaces of the functor Pλ0 , as described by Theorem 3.3.1, depend onlyon X and not on λ0. This is because, given any λ0, µ0 ∈ Pic(X), there is a canonicalisomorphism of functors Pλ0

∼= Pµ0 . We leave to the reader the easy proof of thisfact.


3.3.2 Deformations of sections, I

Let X be a projective integral scheme, and let L be an invertible sheaf on X . One candefine a homomorphism of sheaves:

m : OX → H0(X, L)∨ ⊗ L (3.24)

as follows. For every open set U ⊂ X

m(U ) : O(U )→ H0(X, L)∨ ⊗ Γ (U, L)

f → [s → f s|U ]

for every s ∈ H0(X, L). The induced maps on global sections are just given by cupproduct:

mi : Hi (X,OX )→ Hom(H0(X, L), Hi (X, L))a → [s → a ∪ s]

If L is base point free and

ϕL : X → IP := IP(H0(X, L)∨)

is the morphism defined by the sections of L , then it is easy to check that the homo-morphism (3.24) is the same as the one appearing in the pulled back Euler sequenceof IP:

0→ OX → H0(X, L)∨ ⊗ L → ϕ∗L TIP → 0

We leave this to the reader. The linear maps between cohomology groups inducedby these sheaf homomorphisms have deformation theoretic interpretations which wewill now explain.

Consider a deformation L of L over A ∈ ob(A): we have an induced restrictionmap

ρL : H0(X A,L)→ H0(X, L)

We say that a section σ ∈ H0(X, L) extends to L if σ ∈ Im(ρL).

Proposition 3.3.4. Let La be a first-order deformation of L, corresponding to anelement a ∈ H1(X,OX ). A section s ∈ H0(X, L) extends to La if and only if

a ∪ s = 0 ∈ H1(X, L)

equivalently, if and only if s ∈ ker[m1(a)] where

m1 : H1(X,OX )→ Hom(H0(X, L), H1(X, L))

is the map induced by (3.24).


Proof. Let U = Uα be an affine open covering of X such that L is represented bya system of transition functions fαβ, fαβ ∈ Γ (Uαβ,O∗X ). Then the first-order de-formation La of L can be represented, in the same covering Uα of X ×Spec(k[ε]),by transition functions:

fαβ ∈ Γ (Uαβ, O∗X×Spec(k[ε]))

such thatfαβ fβγ = fαγ (3.25)

and which restrict to the fαβ ’s modulo ε.Since O∗X×Spec(k[ε]) = O∗X + εOX we can write

fαβ = fαβ + εgαβ (3.26)

for suitable gαβ ∈ Γ (Uαβ,OX ). Identity (3.25) gives

gαβ

fαβ+ gβγ

fβγ= gαγ

fαγ

and the system (gαβ

fαβ) is a Cech 1-cocycle which defines the element a ∈ H1(X,OX ).

Let’s assume that s ∈ H0(X, L) is represented by the cocycle (sα),sα ∈ Γ (Uα,OX ), such that sα = fαβsβ on Uαβ . For s to extend to a section

s ∈ H0(X × Spec(k[ε]),La)

it is necessary and sufficient that there exist tα ∈ Γ (Uα,OX ) such that

sα + εtα = fαβ(sβ + εtβ)

on Uαβ . After replacing the fαβ ’s by the expressions (3.26) we obtain the identities

sα + εtα = ( fαβ + εgαβ)(sβ + εtβ)

which are equivalent to:gαβsβ = tα − fαβ tβ

These identities can be also written as:

gαβ

fαβsα = tα − fαβ tβ

and they mean exactly that the 1-cocycle(gαβ

fαβsα

)∈ Z1(U , L)

is a coboundary, i.e. that a ∪ s = 0.


Corollary 3.3.5. In the above situation:

(i) If a ∈ ker(m1) then all the sections of L extend to La.(ii) For every a ∈ H1(X,OX ), at least max0, h0(X, L)− h1(X, L) linearly inde-

pendent sections of L extend to La.

Proof. Immediate. Let’s fix a section s ∈ H0(X, L) and let D = div(s) ⊂ X be the divisor of s.

Consider the local Hilbert functor H XD . We have a morphism of functors:

aD : H XD → P[L]

associating to a deformation of D in X over A in ob(A), given by an effective Cartierdivisor D ⊂ X A, the element [OX A (D)] ∈ P[L](A). aD is called the Abel–Jacobimorphism of D ⊂ X . Consider the exact sequence

0→ OXs−→ L → L D → 0 (3.27)

It induces linear maps:

δ0 : H0(D, L D)→ H1(X,OX )

δ1 : H1(D, L D)→ H2(X,OX )

Noting that L D = ND/X we have:

Proposition 3.3.6. In the above situation δ0 is the differential of aD and δ1 is anobstruction map for aD.

Proof. We keep the notations of the proof of Proposition 3.3.4 and therefore weassume D defined by local equations sα = 0 where (sα) is a 0-cocycle defining swith respect to the covering U ; in particular, sα = fαβsβ . A first-order deformationD ⊂ X × Spec(k[ε]) of D is defined by local equations

sα + εtα = 0

which satisfy the cocycle conditions:

sα + εtα = ( fαβ + εgαβ)(sβ + εtβ)

These conditions can be also written as:

tα = fαβ tβ + gαβ

fαβsα (3.28)

They mean that(tα = tα mod sα)

define a section t ∈ H0(D, L D) which is the one corresponding to the first-order de-formation D of D. From the identities (3.28) we also see that δ0(t) is represented bythe 1-cocycle (

gαβ

fαβ). Since this cocycle represents aD(D) ∈ H1(X,OX ) this proves

that δ0(t) = daD (t). The proof that δ1 is an obstruction map is similar and will beleft to the reader.


Corollary 3.3.7. Under the assumptions of Proposition 3.3.6:

(i) If H1(X, L) = 0 then aD is smooth.(ii) If the natural map

H1(X, L)→ H1(D, L D)

is zero then H XD is less obstructed than P[L]. If, moreover, P[L] is smooth then

H XD is smooth.

Proof. (i) If H1(X, L) = 0 then (3.27) implies that δ0 is surjective and δ1 is injec-tive. Therefore aD is smooth by Proposition 2.3.6. The proof of (ii) is similar usingProposition 2.3.6 again. Remarks 3.3.8. (i) Assume X to be projective and nonsingular. A Cartier divisor Don X is called semiregular if the natural map

H1(X,OX (D))→ H1(D,OD(D))

is zero, i.e. if condition (ii) of the corollary is satisfied. Part (ii) of the corollary canthus be rephrased by saying that D is unobstructed in X if it is semiregular andP[L] is smooth. The smoothness of P[L] is known to be true if char(k) = 0, by ageneral theorem of Cartier: we thus recover a celebrated theorem of Severi, Kodairaand Spencer claiming the unobstructedness of semiregular divisors on a projectivenonsingular complex variety X .

These matters are covered in full detail in [130]. The original sources are [172]and [108]. For further developments and applications of the notion of semiregularitysee [21], [149] and [23].

(ii) In view of Proposition 3.3.6 the cohomology sequence of (3.27) can be inter-preted as a sequence of tangent spaces and differentials as follows:

0→ H0(X,L)〈σ 〉 → H0(D, L D)

δ0−→ H1(X,OX )

‖ ‖ ‖

0→ TD|D| → H XD (k[ε]) daD−→ P[L](k[ε])

where we used the identification of H0(X,L)〈σ 〉 with the tangent space to the linear

system |D| at D proved in Example 3.2.4(ii).

Example 3.3.9. Let X be a Gorenstein curve (e.g. a local complete intersectioncurve). Then the map m1 is dual to the map:

µ0(L) : H0(X, L)⊗ H0(X, ωX L−1)→ H0(X, ωX ) (3.29)

given by multiplication of global sections (ωX denotes the dualizing sheaf of X ).µ0(L) is called the Petri map of L . More generally, if V ⊂ H0(X, L) is a vectorspace of sections of L , one can consider the multiplication map:


µ0(V ) : V ⊗ H0(X, ωX L−1)→ H0(X, ωX )

which is called the Petri map of V . From Proposition 3.3.4 it follows thatcoker(µ0(V ))⊥ ⊂ H1(X,OX ) is the space of first-order deformations of L to whichall sections of V extend. In particular, if h0(X, L) = r + 1 and deg(L) = d thencoker(µ0(L))⊥ is the tangent space at L to the scheme W r

d (X) of linear systems ofdegree d and dimension ≥ r on X (see [9] for details).

3.3.3 Deformations of pairs (X, L)

Let X be a nonsingular projective algebraic variety and let d : OX → Ω1X be the

canonical derivation. We can define a homomorphism of sheaves of abelian groups

O∗X → Ω1X

by the rule

u → du

u

for all open sets U ⊂ X and u ∈ Γ (U,O∗X ). We have an induced group homomor-phism:

c : H1(X,O∗X )→ H1(X,Ω1X )

To simplify the notation we write c(L) instead of c([L]) for a given invertible sheafL on X . If k = C then c(L) is, up to a multiplicative constant, the Chern class of L(see [128], p. 127). Since Ω1

X is locally free we have an identification

H1(X,Ω1X ) = Ext1OX

(TX ,OX )

so that we can associate to c(L) an extension

0→ OX → EL → TX → 0 (3.30)

defined up to isomorphism, called the Atiyah extension of L . The sheaf EL is locallyfree of rank dim(X)+ 1 and

PL := E∨L ⊗OX L

is called the sheaf of (first-order) principal parts of L .Let U = Uα be an affine open covering of X such that L is represented by a

system of transition functions fαβ, fαβ ∈ Γ (Uαβ,O∗X ). Then c(L) is representedby the Cech 1-cocycle (d fαβ

fαβ

)∈ Z1(U ,Ω1

X )

The sheaf EL|Uα is isomorphic to OUα ⊕ TX |Uα . A section (aα, dα) of OUα ⊕ TX |Uα

and a section (aβ, dβ) of OUβ ⊕ TX |Uβ are identified on Uαβ if and only if dα = dβ

and aβ − aα = dα( fαβ)fαβ

.


Remark 3.3.10. We have c(L ⊗ M) = c(L) + c(M), in particular c(Ln) = nc(L)for any n ∈ ZZ . Therefore the Atiyah extension of Ln is a constant multiple of theextension (3.30). This means in particular that if n = 0 then ELn ∼= EL .

Consider, for example, X = IP := IP(V ) for some finite-dimensional k-vectorspace V . Then one can easily compute that the Euler sequence

0→ OIP → V ⊗OIP (1)→ TIP → 0

is the Atiyah extension of OIP (1). Therefore EL ∼= EO(1) = V ⊗ OIP (1) for everynontrivial line bundle L on IP .

Let A be in ob(A). An infinitesimal deformation of the pair (X, L) over A con-sists of a pair (ξ,L) (also denoted by (X ,L)), where

ξ :X → X↓ ↓

Spec(k) → Spec(A)

is an infinitesimal deformation of X over A and L is an invertible sheaf on X suchthat L = L|X . One can also say that L is a deformation of L along ξ . In the caseA = k[ε] we speak of a first order-deformation of (X, L). Two deformations (X ,L)and (X ′,L′) of (X, L) over A will be called isomorphic if there is an isomorphismof deformations f : X → X ′ and an isomorphism L→ f ∗L′.

By letting

Def(X,L)(A) =

deformations of (X, L) over A/isomorphism

we define a functor of Artin rings

Def(X,L) : A→ (sets)

called the functor of infinitesimal deformations of the pair (X, L).We will denote by [X ,L] ∈ Def(X,L)(A) the isomorphism class of a deformation

(X ,L) of (X, L) over A ∈ ob(A).

Theorem 3.3.11. Let (X, L) be a pair consisting of a nonsingular projective alge-braic variety X and an invertible sheaf L on X. Then:

(i) The functor Def(X,L) has a semiuniversal formal element.(ii) There is a canonical isomorphism

Def(X,L)(k[ε]) =1st-order deformations of (X, L)

isomorphism

∼= H1(X, EL)

and H2(X, EL) is an obstruction space for Def(X,L).


(iii) Given a first-order deformation ξ of X, there is a first-order deformation of Lalong ξ if and only if

κ(ξ) · c(L) = 0

where “·” denotes the composition:

H1(X, TX )× H1(X,Ω1X )

∪−→ H2(X, TX ⊗Ω1X )→ H2(X,OX )

of the cup product of cohomology classes ∪ with the map induced by the du-ality pairing TX ⊗ Ω1

X → OX (therefore the left-hand side is an element ofH2(X,OX )).

Proof. Let’s check the conditions of Theorem 2.3.2. It is clear that Def(X,L) satisfiescondition H0. Let

A′ A′′

A

be a diagram in A with A′′ → A surjective. Consider an element

([X ′,L′], [X ′′,L′′]) ∈ Def(X,L)(A′)×Def(X,L)(A) Def(X,L)(A′′)

Then there is an [X ,L] ∈ Def(X,L)(A) and a diagram of deformations:

X ′ X ′′ f ′ f ′′

↓ X ↓Spec(A′) ↓ Spec(A′′)

Spec(A)

where the morphisms f ′ and f ′′ induce isomorphisms of deformations:

X ′ ×Spec(A′) Spec(A) ∼= X ∼= X ′′ ×Spec(A′′) Spec(A)

Moreover, we have isomorphisms

f ′∗L′ ∼= L ∼= f ′′∗L′′

Let B = A′ ×A A′′. Then, as in the proof of Theorem 2.4.1, one sees that

X := (|X |,OX ′ ×OX OX ′′)

is a deformation of X over B inducing the pair ([X ′], [X ′′]). Define

L := L′ ×L L′′


Then L is an invertible sheaf over X which restricts on X ′ to L′ and on X ′′ to L′′.Therefore the pair (X , L) defines an element of Def(X,L)(B) such that

[X , L] → ([X ′,L′], [X ′′,L′′]) (3.31)

under the map

Def(X,L)(B)→ Def(X,L)(A′)×Def(X,L)(A) Def(X,L)(A′′)

This proves that Def(X,L) satisfies condition H of Theorem 2.3.2.In order to show that Def(X,L) satisfies condition Hε we must prove that the

element [X , L] constructed above is the unique one satisfying (3.31) if we assumethat A′′ = k[ε]. From the proof of Theorem 2.4.1 we know that [X ] ∈ DefX (B) isthe unique element inducing ([X ′], [X ′′]) ∈ DefX (A′) ×DefX (A) DefX (A′′). Usingthis fact the proof can now be completed along the lines of the proof of Theorem3.3.1. We still need to verify that Def(X,L) satisfies condition H f . This will result asa consequence of part (ii) to be proved next, because X is projective.

(ii) Let (ξ,L) be a first-order deformation of (X, L), where

ξ :X → X↓ ↓

Spec(k) → Spec(k[ε])Let U = Uα be an affine open covering such that L is given by a system of tran-sition functions fαβ ∈ Z1(U ,O∗X ) and κ(ξ) ∈ H1(X, TX ) is given by a Cech1-cocycle (dαβ) ∈ Z1(U , TX ). Let θαβ = 1 + εdαβ be the automorphism ofUαβ × Spec(k[ε]) corresponding to dαβ .

The invertible sheaf L can be given by a system of transition functions (Fαβ) ∈Z1(U ,O∗X ) which reduces to fαβ mod ε. Therefore it can be represented onUαβ × Spec(k[ε]) as

Fαβ = fαβ + εgαβ, gαβ ∈ Γ (Uαβ,OX )

and the cocycle condition translates into

Fαβθαβ(Fβγ ) = Fαγ

equivalently:Fαβ(Fβγ + εdαβ Fβγ ) = Fαγ

which means:

( fαβ + εgαβ)[ fβγ + εgβγ + εdαβ( fβγ + εgβγ )] = fαγ + εgαγ

After dividing by fαγ and equating the coefficients of ε we obtain:

gαβ

fαβ+ gβγ

fβγ− gαγ

fαγ+ dαβ fβγ

fβγ= 0 (3.32)


This identity means that the data ( gαβ

fαβ, dαβ) define an element of Z1(U , EL), and

that conversely such an element defines a first-order deformation of (X, L). Thisproves that

Def(X,L)(k[ε]) ∼= H1(X, EL)

modulo verifying that this correspondence is independent from the choice of thecovering U and of the cocycles representing (X, L); we leave this to the reader.

Consider a small extension

e : 0→ (t)→ A→ A→ 0

in A and let (ξ,L) be an infinitesimal deformation of (X, L) over A, where

ξ :X → X↓ ↓

Spec(k) → Spec(A)

Let U = Uα be an affine open cover of X ; let

θα : Uα × Spec(A)→ X|Uα

be isomorphisms so that θαβ := θ−1α θβ is an automorphism of the trivial deformation

Uαβ × Spec(A). We may assume that L is given by a system of transition functions(Fαβ), where each Fαβ is a nowhere zero function on Uαβ × Spec(A), such that


In order to see if a lifting (ξ , L) of (ξ,L) to Spec( A) exists we choose arbitrarilya collection

θαβ, Fαβwhere, for each α, β, γ :

(a) θαβ is an automorphism of the product family Uαβ × Spec( A) which restricts toθαβ on Uαβ × Spec(A).

(b) Fαβ is a nowhere zero function on Uαβ × Spec( A) which restricts to Fαβ onUαβ × Spec(A).

Such collection exists by Lemma 1.2.8. Because of (a) we have

θαβ θβγ θ−1αγ = id+ tdαβγ

where “id” here means the identity of Γ (Uαβγ ,OX ), and (dαβγ ) ∈ Z2(U , TX ) is a2-cocycle which represents the obstruction to lifting ξ over Spec( A) (see the proof ofProposition 1.2.12). Because of (b), for each α, β, γ there is gαβγ ∈ Γ (Uαβγ ,OX )such that

Fαβ θαβ(Fβγ )F−1αγ = 1+ tgαβγ


Therefore we have: (θαβ

[Fβγ θβγ (Fγ δ)F−1

βδ

])·

[Fαγ θαγ (Fγ δ)F−1αδ ]−1[Fαβ θαβ(Fβδ)F−1

αδ ][Fαβ θαβ(Fβγ )F−1αγ ]−1

= 1+ t (gβγ δ − gαγ δ + gαβδ − gαβγ )

On the other hand, the left side can be written as

θαβ θβγ (Fγ δ)[θαγ (Fγ δ)]−1 = tdαβγ ( fγ δ)

fγ δ

Therefore we see that(gαβγ , dαβγ ) ∈ Z2(U , EL) (3.33)

We are free to modify our choice by replacing the θαβ ’s and the Fαβ ’s by

Φαβ = θαβ + tdαβ, Gαβ = Fαβ + tgαβ (3.34)

for some dαβ ∈ Γ (Uαβ, TX ) and gαβ ∈ Γ (Uαβ,OX ). One checks easily that thecocycle (3.33) is replaced by the cohomologous one:

(gαβγ + gαβ − gαγ + gβγ , dαβγ + dαβ − dαγ + dβγ ) (3.35)

and it is clear that the lifting (ξ , L) exists if and only if the data (3.34) can be deter-mined so that (3.35) is zero. Therefore associating to the extension e the cohomologyclass o(ξ,L)(e) ∈ H2(X, EL) defined by the cocycle (3.33) we have defined an ob-struction map

Exk(A, k)→ H2(X, EL)

which we leave to the reader to verify to be linear. This makes H2(X, EL) an ob-struction space for the functor Def(X,L).

(iii) Observing that (dαβ fβγ

fβγ

)∈ Z2(U ,OX )

represents κ(ξ) · c(L), the identity (3.32) expresses the condition that this 2-cocycleis a coboundary, and proves (iii).

If X is a connected nonsingular projective curve then H2(X,OX ) = 0 so thatevery line bundle can be deformed along any first-order deformation of X . Moreover,H2(X, EL) = 0 for any L , thus Def(X,L) is smooth and from the exact sequence(3.30) it follows that

h1(X, EL) = 4g − 3

if X has genus g ≥ 2.For higher-dimensional varieties the situation is more complicated in general.

For example, if X is a K3-surface then the cup product

H1(X, TX )× H1(X,Ω1X )→ H2(X,OX ) ∼= k


coincides with Serre duality. Therefore

H1(X, TX )·c(L)−→ H2(X,OX )

is surjective for every nontrivial line bundle L . This means that L deforms alonga 19-dimensional subspace of H1(X, TX ), because h1(X, TX ) = 20 (see Example2.4.11(ii), page 73).

We have a natural forgetful morphism

Φ : Def(X,L)→ DefX

defined in the obvious way. The differential and the obstruction map of this mor-phism are described as follows.

Proposition 3.3.12. In the situation of Theorem 3.3.11:

(i) The differentialdΦ : Def(X,L)(k[ε])→ DefX (k[ε])

coincides with the linear map

H1(X, EL)→ H1(X, TX )

coming from the exact sequence (3.30).(ii) The map

H2(X, EL)→ H2(X, TX )

coming from the exact sequence (3.30) is an obstruction map for Φ.(iii) If H2(X,OX ) = 0 the morphism Φ is smooth.

Proof. (i) and (ii) are left to the reader. For (iii) use Proposition 2.3.6. Example 3.3.13. Assume k = C. Let A = V/Λ be an abelian variety of dimensiong, represented as the quotient of a g-dimensional complex vector space V by a latticeΛ ⊂ V . Then, letting Ω = V∨ we have

TA = V ⊗OA, Ω1A = Ω ⊗OA

Moreover,H1(A,OA) = Ω := HomC(V, C)

is the space of C-antilinear forms on V and

H2(A,OA) =2∧

Ω

(see [20], Theorem 1.4.1). Therefore

H1(A, TA) = V ⊗ H1(A,OA) = V ⊗ Ω


In particular, h1(A, TA) = g2. We also have:

H1(A,Ω1A) = Ω ⊗ H1(A,OA) = Ω ⊗ Ω

which can be identified with the space of hermitian forms on V . Let L be an ampleinvertible sheaf on X . Then

c(L) ∈ H1(A,Ω1A) = Ω ⊗ Ω

is identified with a positive definite hermitian form hL : V × V → C ([20], § 4.1). Itfollows that the map

H1(A, TA)·c(L)−→ H2(A,OA) (3.36)

of Theorem 3.3.11 is just the composition of the map

V ⊗ Ω → Ω ⊗ Ωv ⊗ → hL(v,−)⊗

(3.37)

with the canonical surjection

Ω ⊗ Ω →2∧

Ω

Since hL is positive definite the map (3.37) is an isomorphism and it follows thatthe map (3.36) is surjective. The conclusion is that L deforms along a subspace ofH1(A, TA) of dimension

g(g + 1)

2= g2 −

(g

2

)For the related case of jacobians see Example 3.4.24(iii), page 182.

3.3.4 Deformations of sections, II

Consider again a projective nonsingular variety X and an invertible sheaf L on X .Assume as above that L is given, in an affine open cover U = Uα, by transitionfunctions ( fαβ) ∈ Z1(U ,O∗X ). We can define a homomorphism of sheaves

M : EL → H0(X, L)∨ ⊗ L

in the following way. Consider a section η ∈ Γ (U, EL), where U ⊂ X is an open set;it is given by a system (aα, dα) where aα ∈ Γ (U ∩Uα,OX ), dα ∈ Γ (U ∩Uα, TX ),subject to the conditions that dβ = dα and aβ − aα = dα( fαβ)/ fαβ on U ∩Uα ∩Uβ .Then, for every s = (sα) ∈ H0(X, L) we let

M(η)(sα) = aαsα + dα(sα)


On U ∩Uα ∩Uβ we find:

fαβ M(η)(sβ) = fαβ(aβsβ + dβ(sβ)) = aβsα + fαβdβ(sβ)= sα(aα + dα( fαβ)/ fαβ)+ fαβdβ(sβ) = sαaα + dα( fαβ)sβ + fαβdβ(sβ)

= sαaα + dα( fαβsβ) = sαaα + dα(sα) = M(η)(sα)

Therefore the functions M(η)(sα) ∈ Γ (U ∩ Uα,OX ) patch together to define asection M(η)(s) ∈ Γ (U, L). This defines M . It is obvious from the definition thatwe have a commutative diagram:

m : OX → H0(X, L)∨ ⊗ L↓ ‖

M : EL → H0(X, L)∨ ⊗ L

(3.38)

namely M extends m (see (3.24), page 141). The linear map

M1 : H1(X, EL)→ Hom(H0(X, L), H1(X, L))

induced by M can be explicitly described as follows. Let η1 ∈ H1(X, EL) be repre-sented by the Cech cocycle (aαβ, dαβ) ∈ Z1(U , EL). Then

M1(η1) : H0(X, L)→ H1(X, L)

(sα) → (aαβsα + dαβsα)

The map M1 has the following deformation theoretic interpretation. Let A ∈ob(A) and let (X ,L) be an infinitesimal deformation of (X, L) over Spec(A). Thenwe say that a section s ∈ H0(X, L) extends to L if

s ∈ Im[H0(X ,L)→ H0(X, L)]Proposition 3.3.14. Let X be a projective nonsingular variety, L a line bundle onX, and (X ,L) a first-order deformation of the pair (X, L) defined by a cohomologyclass η1 ∈ H1(X, EL) according to Theorem 3.3.11(ii). Then a section s ∈ H0(X, L)extends to L if and only if s ∈ ker(M1(η1)).

Proof. With respect to an affine cover U = Uα of X we assume L representedby ( fαβ) ∈ Z1(U ,O∗X ) and η1 by (aαβ, dαβ) ∈ Z1(U , EL). Then, according tothe description given in the proof of Theorem 3.3.11, X is determined by gluingthe products Uα × Spec(k[ε]) along the open subsets Uαβ × Spec(k[ε]) by meansof the automorphisms θαβ = 1 + εdαβ , and L is determined by transition func-tions of the form Fαβ = fαβ(1 + εaαβ) satisfying the patching identities onUαβγ × Spec(k[ε]):


The condition that s = (sα) ∈ H0(X, L) extends to L is equivalent to the existenceof functions tα ∈ Γ (Uα,OX ) such that

θαβ [Fαβ(sβ + εtβ)] = sα + εtα


for all α, β. After replacing the expressions for Fαβ and θαβ we obtain the identities:

fαβ(1+ εaαβ)(sβ + εtβ)+ εdαβ( fαβsβ) = sα + εtα

By equating the coefficients of ε on both sides we obtain:

aαβsα + dαβ(sα) = tα − fαβ tβ

and this means exactly that M1(η1)(s) = 0. Corollary 3.3.15. In the above situation:

(i) If η1 ∈ ker(M1) then all the sections of L extend to the first-order deformation of(X, L) corresponding to η1.

(ii) For any η1 ∈ H1(X, EL) at least max0, h0(X, L)−h1(X, L) linearly indepen-dent sections of L extend to the first-order deformation of (X, L) correspondingto η1.

Proof. Immediate. This statement, as well as Proposition 3.3.14, show the deformation theoretic

interest of the map M1. It can be useful to have a closer picture of the relationbetween the maps m and M . Assuming that L is base point free and denoting byϕL : X → IP := IP(H0(X, L)∨) the morphism defined by the sections of L , wehave a commutative and exact diagram, which extends (3.38):

0↓

0 TX

↓ ↓0→ OX

m−→ H0(X, L)∨ ⊗ L → ϕ∗L TIP → 0↓ ‖ ↓

0→ ELM−→ H0(X, L)∨ ⊗ L → NϕL → 0

↓ ↓TX 0↓0

(3.39)

The first column is the Atiyah extension. The sheaf NϕL is the normal sheaf of themorphism ϕL . It will be considered more systematically in § 3.4, page 162. In thecase where ϕL is a closed embedding, NϕL = NϕL (X)/IP and this diagram shows that

coker(m1) ⊂ H1(X, ϕ∗TIP )

where we recall that

m1 : H1(X,OX )→ Hom(H0(X, L), H1(X, L))

is the map induced by m (see Proposition 3.3.4, page 141). Therefore, accordingto Proposition 3.2.9(iii), h2(X,OX ) = 0 and the surjectivity of m1 are sufficient


conditions for the smoothness of the forgetful morphism Φ : H IPϕL (X) → DefX . In

particular, if X is a nonsingular curve and m1 is surjective then Φ is smooth (see alsoExample 3.2.5(v)).

Example 3.3.16. Let X be a connected projective nonsingular curve of genus g.Then the map M1 is dual to

PL : H0(X, L)⊗ H0(X, ωX L−1)→ H0(X, ωX ⊗ E∨L )

More generally, for any vector subspace V ⊂ H0(X, L) we can consider the mapobtained by restricting PL :

PV : V ⊗ H0(X, ωX L−1)→ H0(X, ωX ⊗ E∨L )

It follows directly from Proposition 3.3.14 that coker(PV )⊥ ⊂ H1(X, EL) is thespace of first-order deformations of (X, L) to which all sections of V extend. PL

(resp. PV ) is called the extended Petri map of L (resp. of V ). After dualizing thecohomology diagram of (3.39) we deduce the following commutative diagram whichrelates the Petri map and the extended Petri map in the case where L is globallygenerated:

0↓

0 H0(ω2X )

↓ ↓0→ H1(NϕL )

∨ → H0(L)⊗ H0(ωX L−1)PL−→ H0(ωX ⊗ E∨L )

↓ ‖ ↓0→ H1(ϕ∗L TIP)∨ → H0(L)⊗ H0(ωX L−1)

µ0(L)−→ H0(ωX )

(3.40)

If ϕL : X ⊂ IP is an embedding then this diagram contains some significant infor-mation on the relations between H IP

X and DefX . From what has been remarked justbefore this example it follows that the injectivity of µ0(L), being equivalent to thesurjectivity of its dual m1, is a sufficient condition for the smoothness of the forget-ful morphism Φ : H IP

X → DefX . Diagram (3.40) also shows that we have an exactsequence

0→ H1(NX/IP )∨ → H1(TIP|X )∨ µ1(L)−→ H0(ω2X )

‖ ‖ker(PL) ker(µ0(L))

thus the injectivity of PL is implied by the injectivity of µ0(L) or more generally, bythe injectivity of µ1(L).

For more about this topic we refer the reader to [8].

NOTES

1. The coboundary maps δk in the cohomology sequence of the Atiyah extension (3.30)are induced by cup product with c(L), since the extension is defined by c(L). In particular,


3.3.11(iii) just says that L deforms along ξ if and only if

κ(ξ) ∈ ker[δ1 : H1(X, TX )→ H2(X,OX )],which is obvious in view of (i) and (ii).

2. The content of Theorem 3.3.11(iii) is outlined in the Appendix by Mumford to ChapterV of [189]. See also [89]. This result is related to the notion of deformation of a polarization(see [132]).

3. For a discussion of Example 3.3.13 for abelian varieties defined over a field of positivecharacteristic see Oort [137], remark on page 226.

4. The Petri map has been considered classically in [141] and it reappeared in the modernliterature for the first time in [10]. It is of great importance in the study of the Brill–Noetherloci on a curve and of their relation to moduli (see [9] for more details).

3.4 Morphisms

In this section we study deformations of a morphism between algebraic schemes. Inthe analytic case the corresponding theory has been developed by Horikawa in [85],[86], [88] and [89]. For a treatment in the analytic case we refer to [134]. Relatedwork in the algebraic case is in [114] and [148].

Definition 3.4.1. Let f : X → Y be a morphism of algebraic schemes and let Abe in ob(A) (resp. A = k[ε], resp. A in ob(A∗)). An infinitesimal (resp. first-order,resp. local) family of deformations of f parametrized by A or by S (more briefly adeformation of f over A or over S) is a cartesian diagram

X → X

↓ f ↓ F

Y → Y

↓ ↓ ψ

Spec(k)s−→ S

(3.41)

where S = Spec(A), and ψ and ψF are flat (“cartesian diagram” in this casemeans that the horizontal morphisms induce an isomorphism of the left column withthe pullback of the right column by s). If we replace S by a pointed scheme (S, s) wewill call (3.41) a family of deformations of f .

Essentially, a deformation of f consists of a morphism F between deforma-tions of X and of Y and an assigned identification with f of the restriction of Fto the closed fibre. Note that if Y = Spec(k) then a family of deformations off : X → Spec(k) is just a family of deformations of X in the sense of the defi-nition given at the beginning of § 1.2.

The notion of trivial deformation of f can be given in a obvious way, as well asthe notion of rigid morphism.

3.4 Morphisms 157

Given an infinitesimal deformation (3.41) and a small extension

e : 0→ (t)→ A→ A→ 0

a lifting of (3.41) to A is a cartesian diagram

X → X → X

↓ f ↓ F ↓ F

Y → Y → Y

↓ ↓ ψ ↓ ψ

Spec(k)s−→ Spec(A)→ Spec( A)

(i.e. the horizontal morphisms induce an isomorphism of each column with the pull-back of the column on the right by the lowest horizontal morphisms) where F andψ F are flat. Of course such a lifting defines in particular a deformation of f over A.

Definition 3.4.1 gives the most general notion of deformation of a morphism. Itcan be modified in several ways so to obtain more restricted notions each havingindependent interest and applications. We will study some of them starting from thesimplest ones. In each situation we will consider a corresponding functor of Artinrings which classifies deformations modulo an appropriate equivalence relation.

3.4.1 Deformations of a morphism leaving domain and target fixed

If in Definition 3.4.1 we impose that both X and Y are the product families, i.e. ifwe consider a cartesian diagram of the form:

X → X × S

↓ f ↓ F

Y → Y × S

↓ ↓ ψ

Spec(k)→ S

where S = Spec(A) and ψ is the projection, then we obtain the notion of deformationof f with fixed domain and target (wfdat).

Deformations of f wfdat can be interpreted as deformations of the graph of f inX × Y so that the methods introduced in § 3.2 apply. Precisely, let’s define a functorof Artin rings by setting


DefX/ f/Y (A) =

deformations of f over A wfdat

for all A in ob(A). Then we have the following:

Proposition 3.4.2. Let f : X → Y be a morphism of algebraic schemes, with Xprojective and reduced and Y nonsingular. Then:

(i) There is a natural isomorphism of functors

DefX/ f/Y ∼= H X×YΓ f

where Γ f ⊂ X×Y is the graph of f . In particular, DefX/ f/Y is prorepresentable.(ii) We have a natural isomorphism of vector spaces:

DefX/ f/Y (k[ε])→ H0(X, f ∗TY )

(iii) H1(X, f ∗TY ) is an obstruction space for the functor DefX/ f/Y .

Proof. (i) Let A be in ob(A) and let F : X × Spec(A) → Y × Spec(A) be adeformation of f . Its graph ΓF ⊂ X × Y × Spec(A) defines a deformation ofΓ f ⊂ X ×Y over A because ΓF ∼= X ×Spec(A) and the projection ΓF → Spec(A)equals the composition

ΓF ∼= X × Spec(A)→ Spec(A)

in particular, it is flat. Therefore we can define

DefX/ f/Y (A)→ H X×YΓ f

(A)

byF → (ΓF ⊂ X × Y × Spec(A))

This is an isomomorphism of functors. In fact, given a deformation

ΓA ⊂ X × Y × Spec(A)

of ΓF in X × Y the projection

X × Y × Spec(A)→ X × Spec(A)

induces a morphism ΓA → X × Spec(A) which is an isomorphism of deformations.Therefore the composition

X × Spec(A) ∼= ΓA ⊂ X × Y × Spec(A)→ Y × Spec(A)

can be identified with a deformation of f . This defines the inverse of the morphismof functors of the statement.

(ii) We have natural isomorphisms of vector spaces:

DefX/ f/Y (k[ε]) ∼= H X×YΓ f

(k[ε]) ∼= H0(Γ, NΓ/X×Y )

Since the projection p : X × Y → X is smooth and the composition pj : Γ → Xis an isomorphism, from Proposition D.2.5 it follows that j is a regular embedding.Therefore applying Proposition D.1.4 we obtain the exact sequence:

3.4 Morphisms 159

0→ IΓ /I2Γ → j∗Ω1

X×Y → Ω1Γ → 0

On the other hand, we have the exact sequence:

0→ f ∗Ω1Y → j∗Ω1

X×Y → (pj)∗Ω1X → 0

obtained by restricting to Γ the sequence

0→ q∗Ω1Y → Ω1

X×Y → p∗Ω1X → 0

(where q : X ×Y → Y is the second projection). Since (pj)∗Ω1X∼= Ω1

Γ , comparingthe two sequences we deduce that f ∗Ω1

Y∼= IΓ /I2

Γ . Therefore

H0(Γ, NΓ/X×Y ) = Hom(IΓ /I2Γ ,OΓ ) ∼= Hom( f ∗Ω1

Y ,OX ) = H0(X, f ∗TY )

and (ii) follows.Similarly, H1(Γ, NΓ/X×Y ) = H1(X, f ∗TY ) and (iii) follows as well. The notions of obstructed (resp. unobstructed) deformation, and of obstructed

(resp. unobstructed) morphism can be given in the usual way. One can give the notionof rigid morphism wfdat in an obvious way. We leave to the reader the task of provingthat, under the hypothesis of Proposition 3.4.2, H0(X, f ∗TY ) = 0 implies that f isrigid wfdat.

Let f : X → Y be a morphism of algebraic schemes with X reduced and Ynonsingular; let (3.41) be a family of deformations of f wfdat parametrized by apointed scheme (S, s). To every tangent vector t ∈ TS,s , viewed as a morphismSpec(k[ε])→ S with image s we can associate the pullback of the family F overSpec(k[ε]) by t and, using the correspondence of 3.4.2(i), we obtain an element ofH0(X, f ∗TY ) associated to s. This defines a natural linear map

TS,s → H0(X, f ∗TY )

which will be called the characteristic map of the family (3.41) wfdat.The next corollary follows immediately from the proposition. It can also be de-

duced as a consequence of Lemma 1.2.6.

Corollary 3.4.3. If X is a nonsingular projective scheme, then the space of first-order deformations of the identity X → X is H0(X, TX ).

Examples 3.4.4. (i) Let f : X → Y be a nonconstant morphism of projectivenonsingular connected curves, with g(Y ) ≥ 2. Then deg(TY ) < 0 and thereforeh0(X, f ∗TY ) = 0. Thus f is rigid as a morphism wfdat.

(ii) A morphism f from a scheme X to a projective space is given by a linearsystem on X : deformations of f wfdat can thus be interpreted as “deformations oflinear systems” in an appropriate way. We briefly discuss the case of curves. Let Xbe a projective irreducible and nonsingular curve, f : X → IPr a morphism and letL = f ∗OIPr (1), deg(L) = n. Then f is defined by a vector subspace V ⊂ H0(X, L)


of dimension r + 1 plus the choice of a basis of V . From the Euler sequence pulledback to X we have:

χ( f ∗TIPr ) = (r + 1)χ(L)− χ(OX ) = ρ(g, r, n)+ r(r + 2)

whereρ(g, r, n) := g − (r + 1)(g − n + r)

is the Brill–Noether number and

r(r + 2) = h0(TIPr ) = dim[PGL(r + 1)]Assume that r + 1 = h0(L), i.e. that f is defined by the complete linear series |L|,and consider the exact sequence

H1(OX )→ H1(L)r+1 → H1( f ∗TIPr )→ 0

obtained from the Euler sequence. It dualizes as:

0→ H1( f ∗TIPr )∨ → H0(L)⊗ H0(ωX L−1)µ0(L)−→ H0(ωX )

where µ0(L) is the Petri map (see Example 3.3.9, page 144). Therefore we see thatf is unobstructed if µ0(L) is injective. A necessary condition for this to be true isthat

(r + 1)(g − n + r) = dim[H0(L)⊗ H0(ωX L−1)] ≤ h0(ωX ) = g

i.e. that ρ(g, r, n) ≥ 0. This necessary condition is not sufficient. The simplest ex-ample is given by a nonsingular complete intersection X = Q∩S ⊂ IP3 of a quadriccone Q and of a cubic surface S. Then X is a canonical curve of genus 4 and the pro-jection from the vertex of the cone defines a complete g1

3 |L| such that ωX L−1 ∼= L .In this case ρ(4, 1, 3) = 0 but dim[ker(µ0(L))] = 1. The morphism f : X → IP1

is obstructed because h0( f ∗TIP1) = 4 but L is the unique g13 on X . Therefore the

unobstructed first-order deformations of f are only those in the three-dimensionalspace coming from the automorphisms of IP1.

(iii) If IP1 ∼= E ⊂ S is a nonsingular projective rational curve negatively embed-ded in a projective nonsingular surface S with E2 = −n < 0, n ≥ 1, then we haveh0(E, NE/S) = 0 and h0(E, TS|E ) = 3. More precisely, the exact sequence

0→ TE → TS|E → NE/S → 0

splits because Ext1OE(NE/S, TE ) = H1(E,OE (n + 2)) = 0: therefore

TS|E ∼= OE (2)⊕OE (−n)

This means that, despite the fact that E is rigid in S, the morphism f : E → Shas a three-dimensional family of deformations obtained by composing it with theautomorphisms of E .

3.4 Morphisms 161

More generally, whenever we have an embedding f : IP1 → Y with Y nonsin-gular algebraic variety, we have an inclusion

H0(IP1, TIP1) ⊂ H0(IP1, f ∗TY )

which implies h0(IP1, f ∗TY ) ≥ 3. In general, for any nonconstant morphismf : IP1 → Y the sheaf f ∗TY is locally free and splits as a direct sum of dim(Y )− 1invertible sheaves, by the structure theorem (see [136] p. 22). The study of such mor-phisms is closely related to the notions of uniruledness and rational connectedness.We refer to Debarre [42] and to Kollar [109] for a detailed treatment of these matters.

(iv) Similarly, if E ⊂ Y is an embedding of a projective nonsingular curve ofgenus 1 into a nonsingular algebraic variety Y , from the inclusion OE = TE ⊂ TY |Ewe deduce

H0(E,OE ) ⊂ H0(E, TY |E )

which implies h0(E, TY |E ) ≥ 1.

3.4.2 Deformations of a morphism leaving the target fixed

In this subsection we will follow quite faithfully the treatment given in [86].Given a morphism f : X → Y of algebraic schemes, a notion slightly more

general than the previous one is that of a deformation of f with target Y , obtainedby specializing Definition 3.4.1 to the case when Y is the product family, i.e. byconsidering a cartesian diagram of the form

X → X

↓ f ↓ F

Y → Y × S

↓ ↓ ψ

Spec(k)→ S

where S = Spec(A) with A in ob(A) (resp. in ob(A∗)) and ψ is the projection.Such a deformation can be denoted concisely by the diagram

X F−→ Y × S

S

(3.42)


The deformation (3.42) will be called locally trivial if its domain X defines a locallytrivial deformation of X . Given a deformation (3.42) and another deformation of fwith target Y parametrized by S:

X ′ F ′−→ Y × S

S

an isomorphism between them is an isomorphism of deformations of X :

X

X Φ−→ X ′

S

which makes the following diagram commutative:

X

↓ Φ Y × S

X ′

Definition 3.4.5. To a morphism f : X → Y of algebraic schemes there is associ-ated an exact sequence of coherent sheaves on X

0→ TX/YJ−→ TX

d f−→ Hom( f ∗Ω1Y ,OX )

P−→ N f → 0 (3.43)

which defines the sheaf N f called the normal sheaf of f . The morphism f is callednon-degenerate when TX/Y = 0.

The sequence (3.43) is of course related to the exact sequence (1.5) on page 16.Comparing (3.43) with (1.5) we see that N f = T 1

X/Y if X is nonsingular. If f issmooth then N f = 0. The condition that f is non-degenerate is equivalent to fbeing unramified on a dense open subset of X .

We now introduce vector spaces DX/Y and D1X/Y which will be used to describe

infinitesimal deformations of the morphism f .

Definition 3.4.6. Let f : X → Y be a morphism between algebraic schemes, withX projective. Let U = Ui i∈I be an affine open cover of X and define

3.4 Morphisms 163

DX/Y = (v, t) ∈ C0(U , f ∗TY )× Z1(U , TX ) : δv = d f (t)(d f (w), δw) : w ∈ C0(U , TX )

and

D1X/Y =

(ζ, s) ∈ C1(U , f ∗TY )× Z2(U , TX ) : δζ = d f (s)(d f (u), δu) : u ∈ C1(U , TX )

where δ is the coboundary map in Cech cohomology.

Lemma 3.4.7. In the situation of Definition 3.4.6:

(i) DX/Y and D1X/Y don’t depend on the choice of the affine cover U of X.

(ii) We have the following exact sequences:

(a) H0(X, TX )→ H0(X, f ∗TY )→ DX/Y → H1(X, TX )→ H1(X, f ∗TY )

(b) 0→ H1(X, TX/Y )→ DX/Y → H0(X, N f )→ H2(X, TX/Y )

(c) H1(X, TX )→ H1(X, f ∗TY )→ D1X/Y → H2(X, TX )→ H2(X, f ∗TY )

(d) 0→ H2(X, TX/Y )→ D1X/Y → H1(X, N f )→ H3(X, TX/Y )

(3.44)

(iii) If f is non-degenerate then

DX/Y ∼= H0(X, N f )

D1X/Y∼= H1(X, N f )

(iv) If f is smooth thenDX/Y ∼= H1(X, TX/Y )

D1X/Y∼= H2(X, TX/Y )

Proof. It is clear that (i i) ⇒ (i). Moreover, since f non-degenerate impliesTX/Y = 0, (iii) follows from the exact sequences (3.44)(b) and (3.44)(d). Similarly,f smooth implies N f = 0 and (iv) follows again from (3.44)(b) and (3.44)(d).

Therefore it suffices to prove (ii). In the first sequence the homomorphism

DX/Y → H1(X, TX )

is given by sending (v, t) → t ; similarly, we have

H0(X, f ∗TY )→ DX/Y

which sends u → (u, 0). The proof of exactness is left to the reader.In the second exact sequence the map

H1(X, TX/Y )→ DX/Y


is given byZ1(U , TX/Y ) * ψ → (0, Jψ) ∈ DX/Y

The mapDX/Y → H0(X, N f )

sends (v, t) → Pv . Every element of H0(X, N f ) is represented by somev ∈ C0(U , f ∗TY ) such that δv = d f (t) for some t ∈ C1(U , TX ). We haved f (δt) = δδt = 0 so that δt can be viewed as an element of Z2(U , TX/Y ). Thisdefines the map

H0(X, N f )→ H2(X, TX/Y )

The proof of exactness is left to the reader.The other two sequences are defined and their exactness is checked similarly. Let f : X → Y be a morphism between algebraic schemes. Define functors of

Artin rings:Def f/Y , Def′f/Y : A→ (sets)

by

Def f/Y (A) =

isomorphism classes ofdeformations of f over A with fixed target

Def′f/Y (A) =

isomorphism classes of locally trivial

deformations of f over A with fixed target

for all A in ob(A). Obviously, Def f/Y = Def′f/Y when X is nonsingular. We willconsider the locally trivial case. The main general result about Def′f/Y is the follow-ing:

Theorem 3.4.8. Let f : X → Y be a morphism of algebraic schemes with X pro-jective. Then Def′f/Y has a formal semiuniversal deformation. Its tangent space is

DX/Y and D1X/Y is an obstruction space (see Definition 3.4.6).

Proof. Let’s check the conditions of Theorem 2.3.2. Def′f/Y trivially satisfies condi-tion H0. Consider a diagram in A:

A′ A′′

A

with A′′ → A a small extension and let A = A′ ×A A′′. Let

( f A′ , f A′′) ∈ Def′f/Y (A′)×Def′f/Y (A) Def′f/Y (A′′)

wheref A′ : X ′ → Y × Spec(A′), f A′′ : X ′′ → Y × Spec(A′′)

Since Def′X satisfies H there is a deformation X of X over A such that

X ′ ∼= X ×Spec( A) Spec(A′), X ′′ ∼= X ×Spec( A) Spec(A′′)

3.4 Morphisms 165

and we have a commutative diagram:

X ′′f A′′−→ Y × Spec(A′′)

↓X ′ → X ↓ f A′

Y × Spec(A′) → Y × Spec( A)

By the universal property of the fibred sum we obtain a morphism

f A : X → Y × Spec( A)

which pulls back over Spec(A′) and Spec(A′′) to f A′ and f A′′ respectively. Thereforef A → ( f A′ , f A′′) under the map

α : Def′f/Y ( A)→ Def′f/Y (A′)×Def′f/Y (A) Def′f/Y (A′′)

and this proves H .Let A = k and A′′ = k[ε] and suppose that f A : X → Y × Spec( A) and

f : X → Y×Spec( A) are elements of Def′f/Y ( A) mapped to ( f A′ , f A′′) by α. By the

universal property of X we have a morphism X → X which, being an isomorphismmod ε, is an isomorphism. Moreover, we have a commutative diagram

X → X ↓

Y × Spec( A)

and therefore f A and f define the same element of Def′f/Y ( A); this implies that α isbijective in this case and Hε holds.

In order to describe the tangent space to Def′f , consider a first-order locally trivialdeformation

X → X

↓ f ↓ F

Y → Y × Spec(k[ε])

↓ ↓ ψ

Spec(k)→ Spec(k[ε])Then the deformation

X → X

↓ ↓ ψF

Spec(k)→ Spec(k[ε])

(3.45)


is locally trivial. Choose an affine open cover U = Ui i∈I of X and, for each indexi , let


be an isomorphism of deformations. The composition

Fi := Fθi : Ui × Spec(k[ε])→ Y × Spec(k[ε])is a deformation wfdat of fi := f|Ui and therefore it corresponds to an elementvi ∈ Γ (Ui , f ∗i TY ) = Γ (Ui , f ∗TY ) by Proposition 3.4.2. Therefore we get an ele-ment v = vi ∈ C0(U , f ∗TY ). Restricting to Ui j we have:

Fi |Ui j θ−1i θ j = Fj |Ui j (3.46)

and, by Lemma 1.2.6, θ−1i θ j corresponds to a section ti j ∈ Γ (Ui j , TX ); the collec-

tion ti j is an element t ∈ Z1(U , TX ) which defines the Kodaira–Spencer class ofthe deformation (3.45). The identity (3.46) means that v j − vi = d(ti j ) or, equiva-lently, that the pair (v, t) satisfies δv = d f (t). The pair (v, t) is defined up to a choiceof the trivializations θi or, equivalently, up to an element of the form (d f (w), δw),w ∈ C0(U , TX ). Similarly, if we replace F by an isomorphic deformation of the formF ′ = Fσ , where

σ : X × Spec(k[ε])→ X × Spec(k[ε])is an automorphism of the trivial deformation of X , we obtain the same element ofDX/Y .

Conversely, suppose given an element of DX/Y , represented by a pair(v, t) = (vi , ti j ) ∈ C0(U , f ∗TY ) × Z1(U , TX ) such that δv = d f (t). The classt ∈ H1(X, TX ) defines a first-order deformation of X

X → X

↓ ↓

Spec(k)→ Spec(k[ε])which is locally trivial. The 1-cocycle t = ti j defines local trivializations


and each vi defines a deformation wfdat Fi of fi and by composition we get a mor-phism:

X|Ui

θ−1i−→ Ui × Spec(k[ε]) Fi−→ Y × Spec(k[ε])

By construction Fi |Ui j θ−1i θ j = Fj |Ui j and therefore

Fi |Ui j θ−1i = Fj |Ui j θ

−1j

3.4 Morphisms 167

This means that the morphisms Fiθ−1i patch together and define a morphism

F : X → Y × Spec(k[ε]). This obviously gives rise to a first-order deformation off . It is a straightforward task to verify that the correspondences F → v and v → Fare the inverse of each other. Therefore Def′f/Y (k[ε]) ∼= DX/Y and in particular H f

holds.Now we want to prove the assertion about obstructions to lifting deformations.

We have to show that for every locally trivial infinitesimal deformation

X F−→ Y × S

S

of f with target Y over S = Spec(A) there is a natural map

oF/Y : Exk(A, k)→ D1X/Y

such that, for a given extension

e : 0→ k→ A→ A→ 0

we have oF/Y (e) = 0 if and only if F has a lifting to A which is a locally trivialdeformation of f with target Y . Choose U = Ui i∈I an affine open cover of X andtrivializations

θi : Ui × Spec(A)→ X|Ui

Since H1(Ui , f ∗TY ) = 0 for each i ∈ I the morphism

Fi := Fθi : Ui × Spec(A)→ Y × Spec(A)

has a lifting as a deformation of fi wfdat:

Ui × Spec(A)Fi−→ Y × Spec(A)⋂ ⋂

Ui × Spec( A)Fi−→ Y × Spec( A)

If we restrict to Ui j we have the identity:

Fi |Ui j θi j = Fj |Ui j

where we have written

θi j := θ−1i θ j : Ui j × Spec(A)→ Ui j × Spec(A)


Letθi j : Ui j × Spec( A)→ Ui j × Spec( A)

be an automorphism which restricts to θi j on Ui j×Spec(A). Then Fi |Ui j θi j and Fj |Ui j

are both liftings of Fj |Ui j . Therefore, since Γ (Ui j , f ∗TY ) acts faithfully and transi-tively on the set of such liftings (Proposition 3.4.2 (iii)), there is aζi j ∈ Γ (Ui j , f ∗TY ) which carries Fj |Ui j into Fi |Ui j θi j ; set ζ = ζi j ∈ C1(U , f ∗TY ).Now let

θi jk = θi j θ jk θ−1ik : Ui jk × Spec( A)→ Ui jk × Spec( A)

Since θi jk restricts to the identity on Ui jk ×Spec(A), by Lemma 1.2.6 it correspondsto an si jk ∈ Γ (Ui jk, TX ); by construction s := si jk is a 2-cocycle, i.e. it is anelement of Z2(U , TX ), and the pair (ζ, s) satisfies

δ(ζ ) = d f (s)

Define oF/Y (e) = (ζ, s) ∈ D1X/Y . This definition is well posed because a different

choice of the θi j ’s will replace (ζ, s) by (ζ+d f (u), s+δu) for some u ∈ C1(U , TX ).If oF/Y (e) = 0 then (ζ, s) = (d f (u), δu) for some u ∈ C1(U , TX ). The condi-

tion s = δu means that for the deformation

ξ :X → X↓ ↓

Spec(k)→ Spec(A)

we have oξ (e) = s = 0 ∈ H2(X, TX ) so that ξ has a lifting to Spec( A). Such alifting

ξ :X → X → X↓ ↓ ↓

Spec(k)→ Spec(A)→ Spec( A)

is defined by an appropriate choice of the automorphisms θi j such that θi j θ jk = θik .The condition ζ = d f (u) means that the liftings Fi and Fj can be chosen so that

Fi |Ui j θi j = Fj |Ui j

and this means that they patch together to define a lifting F of F . Conversely, if sucha lifting exists then one shows in the same way that oF/Y (e) = 0.

oF/Y (e) is called the obstruction to lifting F to A as a deformation with tar-get Y . The notions of obstructed/unobstructed deformation, obstructed/unobstructedmorphism with fixed target can be given as usual. One can give the notion of rigidmorphism with fixed target in an obvious way. We have the following:

Corollary 3.4.9. Under the hypotheses of Theorem 3.4.8 we have:

(i) If DX/Y = 0 then f is rigid as a morphism with fixed target. If D1X/Y = 0 then f

is unobstructed as a morphism with fixed target.

3.4 Morphisms 169

(ii) If f is non-degenerate then in the statement of 3.4.8 we can replace DX/Y andD1

X/Y by H0(X, N f ) and H1(X, N f ) respectively.

(iii) If f is smooth then in the statement of 3.4.8 we can replace DX/Y and D1X/Y by

H1(X, TX/Y ) and H2(X, TX/Y ) respectively.

Proof. The proof is immediate in view of Lemma 3.4.7. Remarks 3.4.10. (i) If f is a closed embedding of projective nonsingular algebraicschemes then N f = NX/Y and Theorem 3.4.8 asserts that first-order deformations,resp. obstructions, of f with target Y coincide with first-order deformations, resp.obstructions, of X in Y . This is because, more generally, every infinitesimal defor-mation of f with target Y is an infinitesimal deformation of X in Y : a proof is givenin Note 3.

(ii) If f : X → Y is a morphism between algebraic varieties with X projective,and if (3.42) is a family of locally trivial deformations of f with target Y , then, usingTheorem 3.4.8, we can define a linear map

TS,s → DX/Y

which associates to a tangent vector t : Spec(k[ε]) → S at s the element of DX/Y

corresponding to the first-order deformation obtained by pulling back F by t . Thismap is called the characteristic map of the family F .

Given a morphism f : X → Y between algebraic varieties with X projective wehave a natural morphism of functors

Φ f : Def′f/Y → Def′X

called the forgetful morphism which associates to a locally trivial deformation (3.42)over S = Spec(A), A in ob(A), the family of deformations of X obtained by forget-ting the morphism F . This morphism of functors generalizes the analogous forgetfulmorphism defined for the local Hilbert functor in Subsection 3.2.3. We have the fol-lowing generalization of Proposition 3.2.9.

Proposition 3.4.11. Let f : X → Y be a morphism between algebraic varieties withX projective, and let

Φ f : Def′f/Y → Def′Xbe the forgetful morphism. Then

(i)dΦ f : DX/Y → H1(X, TX )

is the map occurring in the exact sequence (3.44)(a).(ii) The map

D1X/Y → H2(X, TX )

occurring in the exact sequence (3.44)(c) is an obstruction map for Φ f .


(iii) Assume thatH1(X, f ∗TY ) = 0

Then Φ f is smooth.

Proof. The proofs of (i) and (ii) are straightforward.(iii) The hypothesis and the exact sequences (3.44)(a) and (3.44)(c) imply that

DX/Y → H1(X, TX ) is surjective and

D1X/Y → H2(X, TX ) is injective

Now, using parts (i) and (ii), the conclusion follows from Proposition 2.3.6. Corollary 3.4.12. (i) Let f : X → Y be a non-degenerate morphism of algebraic

schemes with X projective. Then

dΦ f : H0(X, N f )→ H1(X, TX )

ando(Φ f ) : H1(X, N f )→ H2(X, TX )

are the coboundary maps coming from the exact sequence

0→ TX → f ∗TY → N f → 0

(ii) Let f : X → Y be a smooth morphism of projective nonsingular algebraicschemes. Then

dΦ f : H1(X, TX/Y )→ H1(X, TX )

ando(Φ f ) : H2(X, TX/Y )→ H2(X, TX )

are the maps induced by the natural inclusion of sheaves TX/Y ⊂ TX .

Proof. The corollary is just a special case of the proposition. Examples 3.4.13. (i) Let m ≥ 0 be an integer, Fm = IP(Fm) where F is the locallyfree rank two sheaf on IP1:

Fm = OIP1(m)⊕OIP1

and let π : Fm → IP1 be the projection. Then we have

H1(Fm, π∗TIP1) = 0

by an easy calculation using the Leray spectral sequence. Therefore, since π issmooth, we can apply Proposition 3.4.11 to conclude that Φπ is smooth. Moreover,since Fm is unobstructed as an abstract variety because

h2(Fm, TFm ) = 0

(see (B.13)), it follows from Proposition 2.2.5(iii) that π is unobstructed.

3.4 Morphisms 171

We can actually be more precise because we have an exact sequence of locallyfree sheaves on IP1:

0→ OIP1 → Fm ⊗ F∨m → π∗TFm/IP1 → 0

which can be deduced easily from the exact sequence (4.28). Since

Fm ⊗ F∨m ∼= O⊕2 ⊕O(m)⊕O(−m)

using the Leray spectral sequence we deduce that h1(Fm, TFm/IP1) = m − 1 form ≥ 1. Therefore, recalling (B.12), we see that the map

H1(Fm, TFm/IP1)→ H1(Fm, TFm )

is not only surjective but it is actually an isomorphism.

(ii) Let f : X → Y be a smooth family of projective curves of genus ≥ 2 withX a projective nonsingular surface and Y a projective nonsingular connected curve.Assume that f is non-isotrivial (see Definition 2.6.9). Then H1(X, TX/Y ) = 0 andtherefore, by (i) and (iii) of Corollary 3.4.9, f is rigid as a morphism with fixedtarget. This theorem is due to Parshin [139] in char 0. For an exposition we refer thereader to Szpiro [178], where the theorem, and its generalization due to Arakelov[6], is proved without the restriction char(k) = 0.

(iii) Let f : X → Y be an etale morphism of projective nonsingular schemes ofdimension n. Then f is non-degenerate and d f : TX → f ∗TY is an isomorphism;therefore N f = 0 and f is rigid as a morphism with fixed target. If we only assumef to be non-degenerate, but not necessarily etale, then d f degenerates on a divisorR ⊂ X (which is the divisor Dn−1(d f ) of Example 4.2.8), called the ramificationdivisor of f . N f is supported on R, and in general f is not rigid as a morphism withfixed target. For the case when X and Y are curves see Subsection 3.4.3.

(iv) Let Y be a projective nonsingular variety, γ ⊂ Y a nonsingular closed subva-riety of pure codimension r ≥ 2 and π : X → Y the blow-up of Y with centre γ . LetE = π−1(γ ) ⊂ X be the exceptional divisor. Then E ∼= IP(Nγ /Y ) is a projectivebundle over γ : let q : E → γ be the structure morphism. Then NE/X = OE (E)and it is well known that the restriction of NE/X to each fibre IP of q is OIP (−1).Therefore by the Leray spectral sequence of q we immediately deduce that

hi (E, NE/X ) = 0 (3.47)

for all i . We have TX/Y = 0 because TX/Y is a subsheaf of the locally free TX and issupported on E ; therefore π is non-degenerate. Since π∗OX = OY and Riπ∗OX = 0for i ≥ 1, from the Leray spectral sequence we deduce that

Hi (X, π∗TY ) = Hi (Y, (π∗π∗OX )⊗ TY ) = Hi (Y, TY ), i ≥ 0 (3.48)


We have an exact and commutative diagram of locally free sheaves on E :

0 0↓ ↓

0→ TE/γ → TE → q∗Tγ → 0‖ ↓ ↓

0→ TE/γ → TX |E → q∗TY → Nπ → 0↓ ↓ ‖

0 → NE/X → q∗Nγ /Y → Nπ → 0↓ ↓0 0

(3.49)

In particular, we see that we have an exact sequence of locally free sheaves on E :

0→ NE/X → q∗Nγ /Y → Nπ → 0 (3.50)

The verification of these facts is straightforward and it is left to the reader.

For example, let π : X = Bl[1,0,0] IP2 → IP2 be the blow-up of IP2 with centrethe point [1, 0, 0]. From the exact sequence (3.50) we deduce that Nπ = OE (1).Therefore

h0(X, Nπ ) = 2, hi (X, Nπ ) = 0, i ≥ 1

In particular, π is unobstructed. Moreover,

h0(X, TX ) = h0(IP2, TIP2 ⊗ I[1,0,0]) = 6

as can be easily checked using the Euler sequence. Therefore from the exact sequence(3.43) we see that h1(X, TX ) = 0, i.e. X is rigid.

3.4.3 Morphisms from a nonsingular curve with fixed target

Theorem 3.4.8 applies in particular to a morphism

ϕ : C → Y

where C and Y are projective and nonsingular, C is a curve, and ϕ is not constant oneach component of C . Consider the exact sequence

0→ TCdϕ−→ ϕ∗TY → Nϕ → 0

The vanishing divisor (see Example 4.2.8 page 200)

Z := D0(dϕ)

of dϕ is called the ramification divisor of ϕ; the index of ramification of ϕ at p ∈ Cis the coefficient of p in Z . ϕ is unramified if and only if Z = 0. The homomorphismdϕ extends to a homomorphism

3.4 Morphisms 173

TC (Z)→ ϕ∗TY

whose cokernel we denote by Nϕ ; it is locally free. We have

Nϕ = Nϕ/Hϕ

where Hϕ ⊂ Nϕ is the torsion subsheaf; it is supported on Z . The following com-mutative and exact diagram summarizes the situation:

0↓Hϕ

↓0→ TC

dϕ−→ ϕ∗TY → Nϕ → 0↓ ‖ ↓

0→ TC (Z) → ϕ∗TY → Nϕ → 0↓0

(3.51)

We obtain:χ(Nϕ) = χ(ϕ∗TY )+ 3g − 3 (3.52)

Example 3.4.14. Assume that C is connected of genus g, that Y is a projective con-nected nonsingular curve of genus γ , and that ϕ has degree d; then

Nϕ = 0, Nϕ = Hϕ = OZ

where O(Z) = ϕ∗(TY )⊗ KC , so that

χ(Nϕ) = h0(Nϕ) = deg(Z) = 2[g − 1+ (1− γ )d]and ϕ is unobstructed because h1(Nϕ) = 0. This corresponds to the fact that thedeformations of ϕ leaving Y fixed are obtained by varying the branch points of ϕ.

Note that ϕ is rigid as a morphism with fixed target if g ≥ 2 and Z = 0, i.e. if itis unramified.

∗ ∗ ∗ ∗ ∗ ∗Assume now that

ϕ : C → S

is a nonconstant morphism from an irreducible projective nonsingular curve C ofgenus g to a projective nonsingular surface S and that ϕ is birational onto its image;let Γ = ϕ(C) ⊂ S. Then we have a commutative and exact diagram:

0↓

0→ TCdϕ−→ ϕ∗TS → Nϕ → 0

↓ ‖ ↓ j0→ ϕ∗TΓ → ϕ∗TS → ϕ∗NΓ/S


Since ϕ∗NΓ/S is invertible the homomorphism j factors through Nϕ and the abovediagram gives rise to the following:

0↓

0→ TC (Z) −→ ϕ∗TS → Nϕ → 0↓ ‖ ↓

0→ ϕ∗TΓ → ϕ∗TS → ϕ∗N ′Γ/S → 0

where Z is the ramification divisor of ϕ and N ′Γ/S = ker[NΓ/S → T 1Γ ] is the equi-

singular normal sheaf (see also § 4.7). This diagram implies the following isomor-phisms:

TC (Z) ∼= ϕ∗TΓ , Nϕ∼= ϕ∗N ′Γ/S, Hϕ

∼= coker[TC → ϕ∗(TΓ )] =: Nϕ

where we have denoted by ϕ : C → Γ the morphism induced by ϕ. In particular,ϕ∗TΓ and ϕ∗N ′Γ/S are invertible and

ϕ∗[TC (Z)] ∼= TΓ ⊗ ϕ∗OC , ϕ∗ Nϕ∼= N ′Γ/S ⊗ ϕ∗OC

On Γ we have a natural exact sequence:

0→ OΓ → ϕ∗OC → t→ 0

where t is a torsion sheaf supported on the singular locus of Γ . Since NΓ/S is inver-tible the homomorphism

NΓ/S → NΓ/S ⊗ ϕ∗OC

is injective and it follows that we have an exact sequence

0→ N ′Γ/S → ϕ∗ Nϕ → N ′Γ/S ⊗ t → 0‖

N ′Γ/S ⊗ ϕ∗OC

(3.53)

This sequence implies in particular:

h0(N ′Γ/S) ≤ h0(Nϕ) ≤ h0(Nϕ)

h1(N ′Γ/S) ≥ h1(Nϕ) = h1(Nϕ)

Lemma 3.4.15. If the singularities of Γ are nodes and ordinary cusps thenN ′Γ/S ⊗ t = 0; equivalently,

N ′Γ/S∼= ϕ∗ Nϕ

In particular, if Γ has only nodes as singularities then

N ′Γ/S∼= ϕ∗Nϕ

3.4 Morphisms 175

Proof. The exact sequence (3.53) can be embedded in the following exact and com-mutative diagram:

0 0↓ ↓

0→ N ′Γ/S → ϕ∗ Nϕ → N ′Γ/S ⊗ t → 0↓ ↓ a ↓ d

0→ NΓ/S → NΓ/S ⊗ ϕ∗OC → NΓ/S ⊗ t → 0↓ ↓ ↓ c

0→ T 1Γ

b−→ T 1Γ ⊗ ϕ∗OC → T 1

Γ ⊗ t → 0↓ ↓ ↓0 0 0

The arrow a is injective because it is a nonzero homomorphism of torsion free rankone sheaves. Because of the assumptions made on the singularities, at each singularpoint p ∈ Γ we have tp = ϕ∗OC,p/OΓ,p ∼= k. Therefore the arrow c is an isomor-phism because NΓ/S⊗ t ∼= t ∼= T 1

Γ ⊗ t. Thus d = 0. The arrow b is injective becauseat each singular point p ∈ Γ we have

(T 1Γ ⊗ ϕ∗OC )p = ϕ∗ϕ∗(T 1

Γ,p)∼=

k2 if p is a nodek3 if p is a cusp

(proved by easy local computation) while

T 1Γ,p∼=

k if p is a nodek2 if p is a cusp

(recall Example 3.1.4). The conclusion now follows from the “Snake Lemma”. If Γhas only nodes then Nϕ = Nϕ and we deduce that N ′Γ/S

∼= ϕ∗Nϕ . It is possible to show that conversely, if N ′Γ/S ⊗ t = 0 then the singularities of Γ

are nodes and ordinary cusps (see [68]).If S = IP2 then, letting L = ϕ∗O(1), d = deg(L), from the Euler sequence

restricted to C :0→ OC → L⊕3 → ϕ∗TIP2 → 0

we deduce χ(ϕ∗TIP2) = 3d + 2− 2g and from (3.52)

χ(Nϕ) = 3d + g − 1 (3.54)

By Corollary 3.4.9 the unobstructedness of ϕ is related to the vanishing of H1(C, Nϕ).From (3.51) we see that

deg(Nϕ) = c1(ϕ∗TIP2)− deg(TC ) = 3d + 2g − 2

and thath1(Nϕ) = h1(Nϕ)

Butdeg(Nϕ) = c1(Nϕ)− deg(Z)

and therefore Nϕ is a nonspecial line bundle whenever deg(Z) < 3d. We can there-fore state the following result:


Proposition 3.4.16. Let ϕ : C → IP2 be a morphism from an irreducible projectivenonsingular curve C of genus g, birational onto its image. Let d = deg(ϕ∗O(1)),and let Z be the ramification divisor of ϕ. Then

h0(C, Nϕ) ≥ 3d + g − 1

If deg(Z) < 3d then ϕ is unobstructed and the above inequality is an equality.In particular, if ϕ(C) is a plane curve having nodes and cusps as its only singu-

larities and the number κ of cusps satisfies κ < 3d then

h0(C, Nϕ) = 3d + g − 1, h1(C, Nϕ) = 0

3.4.4 Deformations of a closed embedding

The deformation theory of morphisms is more subtle if we want to allow both thedomain and the target to deform nontrivially. In this subsection we will address thiscase, considering only the simplest situation of a closed embedding.

Let j : X ⊂ Y be a closed embedding of algebraic schemes. If

X J−→ Y

S

where S = Spec(A), A in ob(A), is an infinitesimal deformation of j then J is aclosed embedding (see Note 3 at the end of of this section). It is obvious that aninfinitesimal deformation of j with fixed target is nothing but a deformation of X inY . Given another infinitesimal deformation of j :

X ′ J ′−→ Y ′

S

over the same S = Spec(A), an isomorphism between them is a pair of isomorphismsof deformations:

α : X → X ′, β : Y → Y ′

which make the diagram

X J−→ Y

↓ α ↓ β

X ′ J ′−→ Y ′

commutative.

3.4 Morphisms 177

We define

Def j (A) =

isomorphism classes ofdeformations of j over A

Def′j (A) =

isomorphism classes of locallytrivial deformations of j over A

for each A in ob(A). These are the functor of infinitesimal deformations of j and oflocally trivial infinitesimal deformations of j respectively.

The locally trivial infinitesimal deformations of a closed embedding are studiedby means of a sheaf which we now introduce.

Let’s now assume that Y is nonsingular and let IX ⊂ OY be the ideal sheaf of X .Let

TY 〈X〉 ⊂ TY

be the inverse image of TX ⊂ TY |X under the natural restriction homomorphismTY → TY |X . Then TY 〈X〉 is called the sheaf of germs of tangent vectors to Y whichare tangent to X . We clearly have an inclusion IX TY ⊂ TY 〈X〉 such that

TX = TY 〈X〉/IX TY

and an exact sequence

0→ TY 〈X〉 → TY → N ′X/Y → 0 (3.55)

where N ′X/Y ⊂ NX/Y is the equisingular normal sheaf of X in Y (introduced inProposition 1.1.9, page 16). From the definition it follows that, for every open setU ⊂ Y , Γ (U, TY 〈X〉) consists of those k-derivations D ∈ Γ (U, TY ) such thatD(g) ∈ Γ (U, IX ) for every g ∈ Γ (U, IX ). We also have the following exact com-mutative diagram:

0 0↓ ↓

IX TY = IX TY

↓ ↓

0→ TY 〈X〉 → TY → N ′X/Y → 0

↓ ↓ ‖

0→ TX → TY |X → N ′X/Y → 0↓ ↓0 0

(3.56)

Of course, N ′X/Y is replaced by NX/Y in the case where X is nonsingular.We will describe locally trivial infinitesimal deformations of a closed embedding

by means of the sheaf TY 〈X〉.


Proposition 3.4.17. Let j : X ⊂ Y be a closed embedding of projective algebraicschemes with Y nonsingular. Then Def′j has a formal semiuniversal deformation. Its

tangent space is H1(Y, TY 〈X〉) and H2(Y, TY 〈X〉) is an obstruction space.

Proof. The proof that Def′j satisfies Schlessinger’s conditions H0, H and Hε is sim-ilar to the proof given in Theorem 3.4.8 and will be left to the reader. Since Y isprojective the existence of a semiuniversal formal deformation will follow if we willprove the assertion about the tangent space of Def′j , because H1(Y, TY 〈X〉) is finitedimensional.

Let U = Ui i∈I be an affine open cover of Y and

V = Vi = X ∩Ui i∈I

the induced affine open cover of X . Every locally trivial first-order deformation of jis obtained by gluing the trivial deformations

Vi ⊂ Vi × Spec(k[ε])⋂ ⋂Ui ⊂ Ui × Spec(k[ε])

along Vi j × Spec(k[ε]) and Ui j × Spec(k[ε]). It is therefore necessary to describethe automorphisms of the trivial deformations

Vi j ⊂ Vi j × Spec(k[ε])⋂ ⋂Ui j ⊂ Ui j × Spec(k[ε])

Every such automorphism Ai j consists of a pair (θi j ,Θi j ) where

θi j : Vi j × Spec(k[ε])→ Vi j × Spec(k[ε])and

Θi j : Ui j × Spec(k[ε])→ Ui j × Spec(k[ε])are automorphisms of deformations such that the following diagram commutes:

Vi j × Spec(k[ε]) θi j−→ Vi j × Spec(k[ε])⋂ ⋂Ui j × Spec(k[ε]) Θi j−→ Ui j × Spec(k[ε])

equivalently, such that θi j = Θi j |Vi j . According to Lemma 1.2.6, Θi j and θi j cor-respond to sections Di j ∈ Γ (Ui j , TY ) and di j ∈ Γ (Vi j , TX ) respectively such thatDi j → di j when restricted to X . It follows that Di j ∈ Γ (Ui j , TY 〈X〉) and that togive Ai j is the same as to give Di j . This said, the proof of the statement about tan-gent and obstruction spaces of Def′j now proceeds in a straightforward way along thelines of the analogous proofs of 1.2.9 and of 1.2.12. We omit the details.

Also in this case we have the notions of obstructed (resp. unobstructed) deforma-tion, obstructed (resp. unobstructed) embedding, and of rigid embedding. It follows

3.4 Morphisms 179

from Proposition 3.4.17 that a closed embedding j : X ⊂ Y of projective nonsingu-lar varieties is rigid if and only if

H1(Y, TY 〈X〉) = 0

LetX J−→ Y

S

be a locally trivial deformation of j : X ⊂ Y parametrized by a pointed scheme(S, s). Then we can define a characteristic map

χJ : TS,s → H1(Y, TY 〈X〉)by associating to a tangent vector t : Spec(k[ε]) → S at s the element ofH1(Y, TY 〈X〉) corresponding to the first-order deformation of f obtained by pullingback J by t .

Remark 3.4.18. Let j : X ⊂ Y be a closed embedding of projective schemes withY nonsingular. If

H0(Y, TY 〈X〉) = 0

then Def′j is prorepresentable. In fact, letting (R, u) be the formal semiuniversaldeformation of Def′j , one can generalize Theorem 2.6.1 and its corollaries by intro-ducing an automorphism functor

Autu : AR → (sets)

in an obvious way and proving that it is prorepresentable with tangent spaceH0(Y, TY 〈X〉). This is called the space of infinitesimal automorphisms of j . Thedetails of this straightforward generalization are left to the reader.

Example 3.4.19. Let Y be a projective scheme, p ∈ Y a closed point andj : p ⊂ Y . Then

TY (〈p〉) = IpTY

where Ip ⊂ OY is the ideal sheaf of p. In this case Def′j is the functor of locallytrivial deformations of the pointed scheme (Y, p). If, for example, Y is a projectivenonsingular connected curve of genus g then TY (〈p〉) = TY (−p) and we get:

h1(Y, TY (−p)) = 0 if g = 0

1 if g = 13g − 2 if g ≥ 2

while h2(Y, TY (−p)) = 0.Of course, we can generalize by considering a set of m distinct closed points

p1, . . . , pm of Y , and the inclusion j : p1, . . . , pm → Y . Then Def′j is thefunctor of locally trivial deformations of the m-pointed scheme (Y ; p1, . . . , pm).


3.4.5 Stability and costability

Whenever we have a locally trivial infinitesimal deformation

X → Y

Spec(A)

of a closed embedding j : X ⊂ Y of projective schemes we also have a deformationof X and a deformation of Y (both locally trivial):

ξ :X → X↓ ↓

Spec(k)→ Spec(A)η :

Y → Y↓ ↓

Spec(k)→ Spec(A)

This means that we have two forgetful morphisms of functors:

Def′jΦY−→ Def′Y

↓ ΦX

Def′X

The differentials and obstruction maps of these morphisms are described as follows.

Proposition 3.4.20. If j : X → Y is a closed embedding of projective schemes withY nonsingular then

(i)dΦY : H1(Y, TY 〈X〉)→ H1(Y, TY )

ando(ΦY ) : H2(Y, TY 〈X〉)→ H2(Y, TY )

are the maps induced in cohomology by the inclusion TY 〈X〉 ⊂ TY .(ii)

dΦX : H1(Y, TY 〈X〉)→ H1(X, TX )

ando(ΦX ) : H2(Y, TY 〈X〉)→ H2(X, TX )

are the maps induced in cohomology by the restriction TY 〈X〉 → TX .

Proof. It is a straightforward consequence of the above analysis. Details are left tothe reader. Remark 3.4.21. Let j : X ⊂ Y be a closed embedding of projective nonsingularschemes. Then there is a natural morphism of functors

HYX → Def j

whose differential is easily seen to be the coboundary map

3.4 Morphisms 181

δ : H0(X, NX/Y )→ H1(Y, TY 〈X〉)determined by the exact sequence (3.55). It follows from the proposition that

ker(dΦY ) = Im(δ)

as expected, because deformations of X in Y are precisely those deformations of theembedding X ⊂ Y which induce the trivial deformation of Y .

The proposition implies also that

ker(dΦX ) = Im[H1(Y, IX TY )β−→ H1(Y, TY 〈X〉)]

where β is the map induced by the inclusion IX TY ⊂ TY 〈X〉. This kernel consistsof the first-order deformations of j which induce the trivial deformation of X . If, inparticular, H0(X, TX ) = 0 then

ker(dΦX ) = H1(Y, IX TY )

Definition 3.4.22. If j : X ⊂ Y is a closed embedding of projective schemes then Xis called stable in Y if the morphism of functors

ΦY : Def′j → Def′Y

is smooth. X is called costable in Y if the morphism of functors

ΦX : Def′j → Def′X

is smooth.

The notion of stability was introduced and studied in [104] for a compact com-plex submanifold of a complex manifold. As stated in [104], stability means that “nolocal deformation of Y makes X disappear”. Our definition of stability implies thatevery infinitesimal locally trivial deformation of Y is induced by a locally trivial de-formation of j . Costability implies that every infinitesimal locally trivial deformationof X is induced by a locally trivial deformation of j . The notion of costability hasbeen introduced in [88].

Proposition 3.4.23. Let j : X ⊂ Y be a closed embedding of projective schemes, Ynonsingular.

(i) If H1(X, N ′X/Y ) = (0) then X is stable in Y .

(ii) If H2(Y, IX TY ) = 0 then X is costable in Y .(iii) If X is nonsingular and is both stable and costable in Y then X is obstructed if

and only if Y is obstructed (as abstract varieties).

Proof. (i) From the exact sequence (3.55) it follows that dΦY is surjective and that

H2(Y, TY 〈X〉)→ H2(Y, TY )


is injective; but by Proposition 3.4.20(i) this last condition means that Def′f is lessobstructed than DefY and the conclusion follows from Proposition 2.3.6.

(ii) The proof is similar using the exact cohomology sequence of the first columnof diagram (3.56), Proposition 3.4.20 and Proposition 2.3.6.

(iii) Since the morphisms of functors

DefXΦX←− Def j

ΦY−→ DefY

are both smooth we deduce that any one of the functors DefX , Def j , DefY is smoothif and only if the others are. Examples 3.4.24. (i) (Kodaira [104], Th. 5) Let Y be a projective nonsingular vari-ety, γ ⊂ Y a nonsingular closed subvariety and π : X → Y the blow-up of Y withcentre γ . Let E = π−1(γ ) ⊂ X be the exceptional divisor; then

hi (E, NE/X ) = 0

for all i (see (3.47)). From Proposition 3.4.23 we obtain that E is a stable subvarietyof X . This is remarkable because γ has not been required to be stable in Y .

(ii) Let X be a projective nonsingular algebraic surface and Z ⊂ X an irreduciblenonsingular rational curve with self-intersection ν = Z2. Then Z is stable in X ifν ≥ −1 because H1(Z , NZ/X ) = 0 in this case. On the other hand, if ν ≤ −2 thenin general Z is not stable in X . An example is provided by the negative section Ein the rational ruled surface Fm , for m ≥ 2. In fact, E2 = −m and we have seenin Example 1.2.2(ii) that there is a family f : W → A1 of deformations of Fm forwhich [E] does not extend to the other fibres W(t), t = 0, since they are isomorphicto Fn for some 0 ≤ n < m. This implies that E is not stable.

(iii) Assume k = C. Let C be a projective irreducible nonsingular curve of genusg ≥ 3, and let α : C → JC be the Abel–Jacobi embedding of C into its jacobianvariety. Then:

• H1(JC,OJC ) ∼= H1(C,OC ) ([20], Lemma 11.3.1)• TJC = H1(C,OC )⊗OJC .

Thus the restriction TJC → TJC|C induces isomorphisms

H0(JC, TJC ) ∼= H0(C, TJC|C ) ∼= H1(C,OC )

H1(JC, TJC ) ∼= H1(C, TJC|C ) ∼= H1(C,OC )⊗ H1(C,OC )

(3.57)

In particular, in view of the second column of diagram (3.56), we have

H0(JC, IC TJC ) = 0 = H1(JC, IC TJC )

and therefore, taking cohomology of (3.56), we obtain the commutative and exactdiagram:

3.4 Morphisms 183

H0(NC/JC )→ H1(JC, TJC 〈C〉) dΦJC−→ H1(JC, TJC ) → H1(NC/JC ) → 0‖ ∩ ‖ ‖

H0(NC/JC )→ H1(C, TC )σ−→ H1(C, TJC|C ) → H1(NC/JC ) → 0

(3.58)

which implies thatH1(JC, TJC 〈C〉) ∼= H1(C, TC ) (3.59)

andH2(JC, TJC 〈C〉) ⊂ H2(JC, TJC ) (3.60)

Since (3.60) is the obstruction map of

ΦJC : Defα → DefJC

from Proposition 2.3.6 we deduce that Defα is smooth, being less obstructed thanDefJC which is smooth (see Example 2.4.11(v), page 74). On the other hand, since(3.59) is the differential of the forgetful morphism

ΦC : Defα → DefC

and Defα is smooth, we deduce by Corollary 2.3.7 that ΦC is smooth, i.e. C iscostable in JC .

Note that H0(JC, TJC 〈C〉) = 0 by the first column of diagram (3.56): thereforeα has no infinitesimal automorphisms and Defα is prorepresentable (Remark 3.4.18).It follows that ΦC is actually an isomorphism of functors (Remark 2.3.8).

We can identify Defα with DefC and the differential

dΦJC : H1(JC, TJC 〈C〉)→ H1(JC, TJC )

with the map σ in diagram (3.58). Therefore ΦJC is a closed embedding if and onlyif σ is injective. In view of the isomorphisms (3.57) σ is Serre-dual to the naturalmultiplication map:

σ∨ : H0(C, KC )⊗ H0(C, KC )→ H0(C, 2KC )

This map is surjective if and only if C is non-hyperelliptic: therefore in this caseDefC is a closed smooth subfunctor of DefJC . In the natural decomposition

H1(C,OC )⊗ H1(C,OC ) = S2 H1(OC )⊕2∧

H1(OC )

we haveIm(σ ) ⊂ S2 H1(OC )

because∧2 H1(OC ) ⊂ ker(σ∨). Therefore Im(dΦJC ) is contained in S2 H1(OC )

which is the space of first-order deformations of JC preserving the principal po-larization (compare with Example 3.3.13, page 151, and observe that in this caseΩ = H1(OC ) = V by the Hodge decomposition of H1(C, C)).

The validity of the condition “σ injective” is called the infinitesimal Torelli theo-rem: thus it holds if and only if C is non-hyperelliptic. It is not difficult to show thatα(C) is unobstructed in JC if and only if C is non-hyperelliptic (see [70], [113]).


The following result, due to Kodaira [104], gives the possibility of relating a localHilbert functor with the deformation functor of an abstract variety.

Proposition 3.4.25. Let Y be a projective nonsingular variety, γ ⊂ Y a closed non-singular subvariety of pure codimension r ≥ 2, and let π : X → Y be the blow-upof Y with centre γ . Then:

(i) There is a natural isomorphism of functors

B : H Yγ → Defπ/Y

In particular, Defπ/Y is prorepresentable.(ii) Assume that H1(Y, TY ) = 0, i.e. that Y is rigid. Then the forgetful morphism

Φπ : Defπ/Y → DefX is smooth. Therefore we have a smooth morphism offunctors BΦπ : H Y

γ → DefX . In particular, if γ is obstructed in Y then X isobstructed as an abstract variety.

Proof. (i) We defineB : HY

γ → Defπ/Y

by associating to a family of deformations

γA ⊂ Y × Spec(A)↓

Spec(A)

of γ in Y over A the blow-up

πA : X A := BlγA (Y × Spec(A))→ Y × Spec(A)

of Y × Spec(A) along γA. Note that, since OγA is A-flat and we have the exactsequence on Y × Spec(A):

0→ IγA → OY ⊗ A→ OγA → 0

the sheaf IγA is A-flat as well (Proposition A.2(VI)); moreover, IkγA

/Ik+1γA

is locallyfree over OY ⊗ A for all k ≥ 1 because γA is regularly embedded in Y × Spec(A)by Lemma D.2.3. From this it is easy to deduce that Ik

γAis A-flat for all k ≥ 1 and

thereforeX A = Proj

⊕k

IkγA

is A-flat by Proposition A.2(V). We leave to the reader to check the functoriality ofB.

The differential of B is the composition

d B : H0(γ, Nγ /Y ) ∼= H0(E, q∗Nγ /Y )→ H0(X, Nπ )

where the first map is the obvious isomorphism and the second one comes from theexact sequence (3.50); in a similar way, one describes the obstruction map of B asthe one induced by the composition

3.4 Morphisms 185

H1(γ, Nγ /Y ) ∼= H1(E, q∗Nγ /Y )→ H1(X, Nπ )

deduced from the exact sequence (3.50). These facts can be easily verified by chasingdiagram (3.49). Since Hi (E, NE/X ) = 0 for all i , we see that these maps are bothbijective, and the conclusion follows.

(ii) From (3.48) it follows that H1(X, π∗TY ) = 0. The exact sequence

0→ TX → π∗TY → Nπ → 0

and Proposition 3.4.20(i) imply that dΦπ is surjective and o(Φπ) is injective. Theconclusion is a consequence of Proposition 2.3.6. The last assertion is an obviousconsequence of the fact that the composition Φπ B : HY

γ → DefX is smooth. By applying Proposition 3.4.25 to any obstructed nonsingular curve γ ⊂ IP3

(e.g. the curve of degree 14 and genus 24 described in § 4.6) we obtain an exam-ple of obstructed projective variety of dimension 3. As an application we obtain thefollowing result, which gives examples of obstructed surfaces.

Theorem 3.4.26 (Horikawa [88]). Let γ ⊂ IP3 be an obstructed nonsingularcurve, and X the blow-up of IP3 with centre γ . If S ⊂ X is a sufficiently amplenonsingular surface then S is obstructed as an abstract variety.

Proof. By Proposition 3.4.23(iii) it is sufficient to show that S is both stable andcostable in X . We have h2(X,OX ) = 0 and, by the ampleness of S,h1(X,OX (S)) = 0 by Serre’s vanishing theorem. From the exact sequence

0→ OX → OX (S)→ NS/X → 0

we deduce that h1(S, NS/X ) = 0 and therefore S is stable in X . On the other hand,we have

H2(X, TX (−S)) = 0

by Serre’s vanishing theorem again. Therefore S is costable in X as well. We can immediately verify, using the adjunction formula, that the surfaces S

constructed in the theorem are regular and with ample canonical class. For relatedexamples see [29].

NOTES

1. The analysis of morphisms from a nonsingular curve is taken from [8]; see also [7]. ForLemma 3.4.15 see [180].

2. For the study of the functors Def f/Y and Def f under more general assumptions thanthose of Theorem 3.4.8 a more careful analysis of first-order deformations and obstructions isneeded. We refer the reader to [148] and [150] for more information about this. The deforma-tion theory of closed embeddings is studied in [134] in the analytic category.

3. LetX Φ−→ Y

S


be a commutative diagram of morphisms of algebraic schemes, with X and Y S-flat and Φprojective. Assume that Φo : X (o) → Y(o) is a closed embedding, for some k-rationalpoint o ∈ S. Then there is an open neighbourhood U ⊂ S of o such that the restrictionΦ(U ) : X (U )→ Y(U ) is a closed embedding.

Proof. Let K = coker[OY → Φ∗(OX )]. Since Φ is projective, Φ∗(OX ) is a coherent sheafand so is K. Moreover, K(o) = (0) because Φo is a closed embedding. It follows that thereis an open subset U ⊂ S containing o such that K|Y(U ) = (0). Let Z = Spec(Φ∗(OX )),h : Z → Y the induced S-morphism and

X Φ−→ Yg h

Z

the Stein factorization of Φ. Then it follows that h(U ) : Z(U ) → Y(U ) is a closed em-bedding. Moreover, since g has connected fibres and is bijective over Z(o), it follows that,modulo shrinking U if necessary, g(U ) : X (U )→ Z(U ) is an isomorphism. The conclusionfollows.

4

The Hilbert schemes and the Quot schemes

Even though this book is centred around the theme of infinitesimal and local defor-mations, in this chapter we turn our attention to global deformations. We will intro-duce the Hilbert schemes and other related objects, which are important examples ofparameter schemes for global families of deformations of algebro-geometric objects.They are used to describe and classify “extrinsic” deformations, i.e. deformations ofobjects within a given ambient space (e.g. closed subschemes of a given scheme).Their study is preliminary to the construction of “moduli schemes”. Moreover, theyprovide some of the most typical examples of constructions in algebraic geometry bythe functorial approach. We will study some of their properties and consider a fewapplications of the local theory developed so far.

4.1 Castelnuovo–Mumford regularity

In this section we introduce the notion of m-regularity, also called Castelnuovo–Mumford regularity, and we prove its main properties. They will be needed for theconstruction of the Hilbert schemes and of the Quot schemes.

Let m ∈ ZZ . A coherent sheaf F on IPr is m-regular if

Hi (F(m − i)) = (0)

for all i ≥ 1.Because of Serre’s vanishing theorem, every coherent sheaf F on IPr is m-

regular for some m ∈ ZZ .The definition of m-regularity makes sense for a coherent sheaf on any projective

scheme X endowed with a very ample line bundle O(1). For simplicity we willconsider the case X = IPr only, leaving to the reader the obvious modifications ofthe statements and of the proofs in the general case.

188 4 Hilbert and Quot schemes

Proposition 4.1.1. If F is m-regular then:

(i) The natural map

H0(F(k))⊗k H0(O(1))→ H0(F(k + 1))

is surjective for all k ≥ m.(ii) Hi (F(k)) = (0) for all i ≥ 1 and k ≥ m − i ; in particular, F is n-regular for

all n ≥ m.(iii) F(m), and therefore also F(k) for all k ≥ m, is generated by its global sections.

Proof. We prove (i) and (ii) by induction on r . If r = 0 there is nothing to prove.Assume r ≥ 1 and let H be a hyperplane not containing any point of Ass(F); itexists because Ass(F) is a finite set. Tensoring by F(k) the exact sequence:

0→ O(−H)→ O→ OH → 0

we get an exact sequence:

0→ F(k − 1)→ F(k)→ FH (k)→ 0

where FH = F ⊗OH . For each i > 0 we obtain an exact sequence

Hi (F(m − i))→ Hi (FH (m − i))→ Hi+1(F(m − i − 1))

which implies that FH is m-regular on H . It follows by induction that (i) and (ii) aretrue for FH .

Let’s consider the exact sequence

Hi+1(F(m − i − 1))→ Hi+1(F(m − i))→ Hi+1(FH (m − i))

If i ≥ 0 the two extremes are zero (the right one by (ii) for FH , the left one by the mregularity of F), therefore F is (m + 1)-regular. By iteration this proves (ii).

To prove (i) we consider the commutative diagram:

H0(F(k))⊗k H0(O(1)) −→u

H0(FH (k))⊗k H0(OH (1))↓ w ↓ t

H0(F(k))→ H0(F(k + 1)) −→v

H0(FH (k + 1))

The map u is surjective for k ≥ m because H1(F(k − 1)) = (0); moreover,t is surjective for k ≥ m by (i) for FH . Therefore vw is surjective. It followsthat H0(F(k + 1)) is generated by Im(w) and by H0(F(k)) for all k ≥ m. ButH0(F(k)) ⊂ Im(w) because the inclusion H0(F(k)) ⊂ H0(F(k + 1)) is multipli-cation by H . Therefore w is surjective.

Let’s prove (iii). Let h 0 be such that F(m + h) is generated by its globalsections. Then the composition

H0(F(m))⊗k H0(O(h))⊗k O→ H0(F(m + h))⊗k O→ F(m + h)

4.1 Castelnuovo–Mumford regularity 189

is surjective because from (i) it follows that the first map is; we deduce that thecomposition

H0(F(m))⊗k H0(O(h))⊗k O(−h)→ H0(F(m))⊗k O→ F(m)

is also surjective, hence the second map is surjective too. Note that if F is m-regular then the graded k[X1, . . . , Xr ]-module

Γ∗(F) :=⊕k∈ZZ

H0(F(k))

can be generated by elements of degree ≤ m. In fact, this is equivalent to the surjec-tivity of the multiplication maps

H0(F(m))⊗k H0(O(h))→ H0(F(m + h))

for h ≥ 1, and follows from part (i) of the proposition. In particular, if an ideal sheafI ⊂ OIPr is m-regular then the homogeneous ideal

I = Γ∗(I) ⊂ k[X0, . . . , Xr ]

is generated by elements of degree ≤ m.Note also that in proving 4.1.1 we have proved the following:

Proposition 4.1.2. If F is m-regular and

0→ F(−1)→ F → G → 0

is an exact sequence, then G is m-regular.

Conversely, we have the following:

Proposition 4.1.3. Let

0→ F(−1)→ F → G → 0

be an exact sequence of coherent sheaves on IPr , and assume that G is m-regular.Then:

(i) Hi (F(k)) = 0 for i ≥ 2 and k ≥ m − i

(i i) h1(F(k − 1)) ≥ h1(F(k)) for k ≥ m − 1

(i i i) H1(F(k)) = 0 for k ≥ (m − 1)+ h1(F(m − 1))

In particular, F is m + h1(F(m − 1))-regular.


Proof. (i) In the exact sequence

Hi−1(G(k))→ Hi (F(k − 1))→ Hi (F(k))→ Hi (G(k))

the first and the last group are zero for i ≥ 2 and k ≥ m − (i − 1). Therefore

Hi (F(m − i)) ∼= Hi (F(m − i + 1)) ∼= Hi (F(m − i + 2)) ∼= · · ·From Serre’s vanishing theorem we get Hi (F(m − i + h)) = 0 for all h 0 and (i)follows.

(ii) For k ≥ m − 1 we have the exact sequence

0→ H0(F(k−1))→ H0(F(k))vk−→ H0(G(k))→ H1(F(k−1))→ H1(F(k))→ 0

which implies (ii).(iii) Assume vk surjective, and consider the commutative diagram:

H0(F(k))⊗ H0(O(1))vk⊗id−→ H0(G(k))⊗ H0(O(1))

↓ ↓ wk

H0(F(k + 1))vk+1−→ H0(G(k + 1))

Since wk is surjective for k ≥ m, we have that vk+1 is surjective too. Therefore

H1(F(k − 1)) ∼= H1(F(k)) ∼= H1(F(k + 1)) ∼= · · · ∼= 0

If vk is not surjective then h1(F(k − 1)) > h1(F(k)). Therefore the functionk → h1(F(k)) is strictly decreasing for k ≥ m − 1, and this implies (iii).

The following is a useful characterization of m-regularity.

Theorem 4.1.4. A coherent sheaf F on IPr is m-regular if and only if it has a reso-lution of the form:

· · · → O(−m − 2)b2 → O(−m − 1)b1 → O(−m)b0 → F → 0 (4.1)

for some nonnegative b0, b1, b2, . . ..

Proof. Assume that F has a resolution (4.1) and let

R1 = ker[O(−m)b0 → F

]R j = ker

[O(−m − j + 1)b j−1 → O(−m − j + 2)b j−2

]j = 2, . . . , r

Rr+1 = O(−m − r)br+1

Using the short exact sequences:

0→ R1(m − i)→ O(−i)b0 → F(m − i)→ 0

0→ R j (m − i)→ O(−i − j + 1)b j−1 → R j−1(m − i)→ 0

0→ O(−i − r − 1)br+1 → O(−i − r)br → Rr (m − i)→ 0


we see that for all 1 ≤ i ≤ r we have:

Hi (F(m − i)) ∼= Hi+1(R1(m − i)) ∼= · · ·

· · · ∼= Hr (Rr−i (m − i)) ∼= Hr+1(Rr−i+1(m − i)) = (0)

and F is m-regular.Assume conversely that F is m-regular. By 4.1.1(iii) we have an exact sequence:

0→ R1 → O(−m)b0 → F → 0

with b0 = h0(F(m)), which defines R1.If R1 = 0 we are done; if R1 = 0 from the sequences:

0→ R1(m − i + 1)→ O(−i + 1)b0 → F(m − i + 1)→ 0

we deduce that

Hi (R1(m − i + 1)) ∼= Hi−1(F(m − i + 1)) 1 ≤ i ≤ r

hence R1 is (m + 1)-regular. Applying the same argument to R1 we find an exactsequence:

0→ R2 → O(−m − 1)b1 → O(−m)b0 → F → 0

with R2 (m + 2)-regular. This process can be repeated and gives a resolution asrequired.

We will now turn to the problem of finding numerical criteria of m-regularity fora coherent sheaf F on IPr .

Consider a sequence σ1, . . . , σN of N sections of OIPr (1). We will call it F-regular if the sequences of sheaf homomorphisms induced by multiplication byσ1, . . . , σN :

0→ F(−1)σ1−→ F → F1 → 0

0→ F1(−1)σ2−→ F1 → F2 → 0

etc., are exact.By choosing σi+1 not containing any point of Ass(Fi ) one shows that F-

regular sequences of any length exist. Therefore any general N -tuple (σ1, . . . , σN ) ∈H0(OIPr (1))N is an F-sequence.

Definition 4.1.5. Let F be a coherent sheaf on IPr , and (b) = (b0, b1, . . . , bN )a sequence of nonnegative integers such that N ≥ dim[Supp(F)]. Wewill call F a (b)-sheaf if there exists an F-regular sequence σ1, . . . , σN of sec-tions of OIPr (1) such that h0(Fi (−1)) ≤ bi , i = 0, . . . , N where F0 = F , andFi = F/(σ1, . . . , σi )F(−1), i ≥ 1.

Note that from the definition it follows immediately that if F is a (b)-sheaf thenF1 is a (b1, . . . , bN )-sheaf.


It is likewise clear that for every coherent sheaf F on IPr there is a sequence (b)such that F is a (b)-sheaf. Moreover, a subsheaf of a (b)-sheaf is easily seen to be a(b)-sheaf.

For example, every ideal sheaf I ⊂ OIPr is a (0)-sheaf, because OIPr is clearlya (0)-sheaf.

Lemma 4.1.6. Let0→ F(−1)→ F → G → 0

be an exact sequence of coherent sheaves on IPr . If

χ(F(k)) =r∑

i=0

ai

(k + i

i

)then

χ(G(k)) =r−1∑i=0

ai+1

(k + i

i

)The proof is left to the reader.

Proposition 4.1.7. Let F be a (b)-sheaf, let s = dim[Supp(F)] and

χ(F(k)) =s∑

i=0

ai

(k + i

i

)Then:

(i) For each k ≥ −1 we have h0(F(k)) ≤∑si=0 bi

(k+ii

).

(ii) as ≤ bs and F is also a (b0, . . . , bs−1, as)-sheaf.

Proof. (i) By induction on s. If s = 0 then a0 = h0(F) = h0(F(−1)) ≤ b0 andthe conclusion is obvious.

Assume s ≥ 1. We have an exact sequence

0→ F(−1)→ F → F1 → 0

with F1 a (b1, . . . , bN )-sheaf and dim[Supp(F1)] = s − 1. Then:

h0(F(k))− h0(F(k − 1)) ≤ h0(F1(k))

and

h0(F1(k)) ≤s−1∑i=0

bi+1

(k + i

i

)by the inductive hypothesis. Since h0(F(−1)) ≤ b0 by induction on k ≥ −1 we getthe conclusion.

(ii) By Lemma 4.1.6 and by induction on s we get as ≤ bs and F1 is a(b1, . . . , bs−1, as)-sheaf. The conclusion follows.


Definition 4.1.8. The following polynomials, defined by induction for each integerr ≥ −1:

P−1 = 0Pr (X0, . . . , Xr ) = Pr−1(X1, . . . , Xr )+∑r

i=0 Xi(Pr−1(X1,...,Xr )−1+i

i

)are called (b)-polynomials.

One immediately sees that

Pr (X0, . . . , Xt , 0, . . . , 0) = Pt (X0, . . . , Xt ) (4.2)

for each t < r .The following theorem gives a numerical criterion of m-regularity.

Theorem 4.1.9. Let F be a (b)-sheaf on IPr , with (b) = (b0, b1, . . . , bN ), and let

χ(F(k)) =r∑

i=0

ai

(k + i

i

)be its Hilbert polynomial. Let (c0, . . . , cr ) be a sequence of integers such thatci ≥ bi − ai , for i = 0, . . . , r , and m = Pr (c0, . . . , cr ). Then m ≥ 0 and F ism-regular. In particular, F is Ps−1(c0, . . . , cs−1)-regular if s = dim[Supp(F)].Proof. By induction on r . If r = 0 then m = 0 and F is n-regular for every n ∈ ZZ ,so the theorem is true in this case. Assume r ≥ 1. We have an exact sequence:

0→ F(−1)→ F → F1 → 0

with F1 a (b1, . . . , bN )-sheaf supported on IPr−1. From Lemma 4.1.6 and fromthe inductive hypothesis we deduce that n ≥ 0 and F1 is n-regular, where n =Pr−1(c1, . . . , cr ). From 4.1.3 we deduce that F is [n + h1(F(n − 1)]-regular andhi (F(n − 1)) = 0 for i ≥ 2. Therefore:

h1(F(n − 1)) = h0(F(n − 1))− χ(F(n − 1)) ≤r∑

i=0

(bi − ai )

(n − 1+ i

i

)by 4.1.7(i). It follows that F is n+∑r

i=0 ci(n−1+i

i

)-regular, by 4.1.1(ii). This proves

the first assertion.The last assertion follows from 4.1.7(ii) and from (4.2). Note that the integer m in the statement of the theorem depends on the coeffi-

cients of the Hilbert polynomial of F as well as on the integers bi . In the special casewhen F is a sheaf of ideals we can determine an m for which F is m-regular whichdepends only on the Hilbert polynomial of F , as stated in the next corollary.

Corollary 4.1.10. For each r ≥ 0 there exists a polynomial Fr (X0, . . . , Xr ) suchthat every sheaf of ideals I ⊂ OPr having the Hilbert polynomial

χ(I(k)) =r∑

i=0

ai

(k + r

i

)is m-regular, where m = Fr (a0, . . . , ar ), and m ≥ 0.


Proof. It suffices to observe that I is a (0)-sheaf. Therefore the corollary followsfrom Theorem 4.1.9, taking Fr (X0, . . . , Xr ) = Pr (−X0, . . . ,−Xr ).

NOTES

1. Corollary 4.1.10 is in general false for coherent sheaves which are not sheaves of ideals.An example from [130] is

F = OIP1(k)⊕OIP1(−k)

In fact χ(F) = 2 is independent of k but the least m such F is m-regular is |k|.2. If I is the sheaf of ideals of the closed subscheme X ⊂ IPr and I is m-regular with

m ≥ 0, then OX is (m − 1)-regular. Conversely, if OX is (m − 1)-regular and the restrictionmap

H0(IPr ,OIPr (m − 1))→ H0(X,OX (m − 1))

is surjective, then I is m-regular. This follows from the exact sequences

0→ I(k)→ OIPr (k)→ OX (k)→ 0

k ≥ m − 1.

3. The notion of m-regularity is related to that of bounded collection of sheaves, importantin moduli theory.

A collection of coherent sheaves Fj j∈J on a projective scheme X is said to be boundedif there is an algebraic scheme S and a coherent sheaf F on X × S such that for each j ∈ Jthere is a closed point s ∈ S such that Fj is isomorphic to the sheaf F(s) = F|X×s. Onealso says that the collection Fj j∈J is bounded by the sheaf F on X × S. For details we referto [98].

4. The notion of Castelnuovo–Mumford regularity has been introduced in [130]. Casteln-uovo studied the properties of m-regularity of projective curves in [26], where his upper boundfor the genus of projective curves was proved. The treatment of (b)-sheaves has been takenfrom [98].

4.2 Flatness in the projective case

This section is devoted to some properties of flat families of projective schemeswhich will be needed in this chapter. In particular, we will prove a powerful technicalresult due to Grothendieck and Mumford, the existence of flattening stratifications,which is a key ingredient in the construction of the Hilbert schemes, of the Quotschemes, and of related schemes like the Severi varieties. The treatment of stratifica-tions closely follows Lecture 8 of [130].

4.2.1 Flatness and Hilbert polynomials

The following result gives the name to the “Hilbert scheme”.

Proposition 4.2.1. (i) Let S be a scheme, F a coherent sheaf on IPr × S and p :IPr × S → S the projection. Then F is flat over S if and only if p∗F(h) islocally free on S for all h 0.

4.2 Flatness in the projective case 195

(ii) Assume that S is connected. For each s ∈ S let

Ps(t) = χ(F(s)(t)) =∑

i

(−1)i hi (IPr (s),F(s)(t))

be the Hilbert polynomial of F(s). If F is flat over S then Ps(t) is independentof s ∈ S. Conversely, if S is integral and Ps(t) is independent of s for all s ∈ S,then F is flat over S. If S is integral and algebraic and Ps(t) is independent of sfor all closed s ∈ S, then F is flat over S.

For the proof of this proposition we refer the reader to [84], Theorem III.9.9.

Corollary 4.2.2. IfX ⊂ IPr × S↓S

is a flat family of closed subschemes of IPr with S connected, then all fibres X (s)have the same Hilbert polynomial; in particular, they have the same degree.

Proof. It follows from 4.2.1 applied to F = OX . Examples 4.2.3. (i) Let Ui = (z0, z1) ∈ IP1 : zi = 0, U = U0

∐U1 and

f : U → IP1 the natural morphism. Then f is flat surjective and quasi-finite. Thefibres of f are 0-dimensional, hence projective, but their degree is not constant. Thisis not a contradiction with Corollary 4.2.2 because the morphism f is not projective,since U is an affine variety.

(ii) In IP3 with homogeneous coordinates X = (X0, X1, X2, X3) consider thecurve

Cu = Proj(k[X ]/(X2, X3)) ∪ Proj(k[X ]/(X1, X3 − u X0))

for every u ∈ A1. If u = 0 then Cu consists of two disjoint lines, while

C0 = Proj(k[X ]/(X1 X2, X3))

is a reducible conic in the plane X3 = 0. The Hilbert polynomials are

Pu(t) = 2t + 2 u = 0P0(t) = 2t + 1

From Corollary 4.2.2 it follows that Cu cannot be the set of fibres of a flat familyof closed subschemes of IP3.

We may try to construct a morphism whose fibres are the Cu’s by considering theclosed subscheme X ⊂ IP3 × A1 defined by the ideal

J = (X2, X3)∩ (X1, X3 − u X0) = (X1 X2, X1 X3, X2(X3 − u X0), X3(X3 − u X0))


of k[u, X0, . . . , X3]. From [84], Prop. III.9.7, it follows that X is flat over A1. Wehave:

X (u) = Cu, u = 0

X (0) = Proj(k[X ]/(X1 X2, X1 X3, X2 X3, X23))

and X (0) = C0: indeed

X (0) = C0 ∪ Proj(k[X ]/(X1, X2, X23))

is a nonreduced scheme obtained from C0 by adjoining an embedded point in(1, 0, 0, 0). In particular, we see that X (0) and C0 have the same support. Prop.III.9.8 of [84] implies that X (0) is uniquely determined by the other fibres, i.e. byX ∩ [IP3 × (A1\0)].

Fix a scheme S and a coherent sheaf F on IPr × S. Consider a morphism g :T → S and the diagram

IPr × Th−→ IPr × S

↓ q ↓ p

Tg−→ S

where h = id × g. For every open set U ⊂ S we have homomorphisms

H j (IPr ×U,F)→ H j (IPr × g−1(U ), h∗F)→ H0(g−1(U ), R j q∗(h∗F))

and therefore a homomorphism

R j p∗F → g∗[R j q∗(h∗F)]which corresponds to a homomorphism

g∗(R j p∗F)→ R j q∗(h∗F)

In the case j = 0 we have the following asymptotic result which will be applied laterin this section:

Proposition 4.2.4. For all m 0 the homomorphism

g∗(p∗F(m))→ q∗(h∗F(m))

is an isomorphism and, if T is noetherian, R j q∗(h∗F(m)) = 0 all j ≥ 1.

Proof. We haveh∗F = Γ∗(h∗F ) := [⊕mq∗(h∗F(m))

]Since F = Γ∗(F ) we also have

h∗F = h∗[Γ∗(F )

] = [⊕m g∗(p∗F)(m)]


and therefore for all m 0

g∗(p∗F(m)) ∼= q∗(h∗F(m))

For the last assertion cover T by finitely many affine open sets and apply TheoremIII.5.2 of [84].

The homomorphism of Proposition 4.2.4 is particularly important when g :Spec(k(s))→ S is the inclusion in S of a point s ∈ S; it is denoted by

t j (s) : R j p∗(F)s ⊗ k(s)→ H j (IPr (s),F(s))

The study of these homomorphisms is carried out in [1], Ch. III2 (see also Chapter III,Section 12, of [84]). Their main properties are summarized in the following theoremand in its corollary.

Theorem 4.2.5. Let S be a scheme, F a coherent sheaf on IPr × S, flat over S, s ∈ Sand j ≥ 0 an integer. Then:

(i) If t j (s) is surjective then it is an isomorphism.(ii) If t j+1(s) is an isomorphism then R j+1 p∗(F) is free at s if and only if t j (s) is

an isomorphism.(iii) If R j p∗(F) is free at s for all j ≥ j0 + 1 then t j (s) is an isomorphism for all

j ≥ j0.

Proof. (i) and (ii) are Theorem III.12.11 of [84]. (iii) follows from (i) and (ii) bydescending induction on j0. Corollary 4.2.6. Let X → S be a projective morphism, and let F be a coherentsheaf on X , flat over S. Then:

(i) If H j+1(X (s),F(s)) = 0 for some s ∈ S and j ≥ 0 then R j+1 p∗(F)s = (0),and

t j (s) : R j p∗(F)s ⊗ k(s)→ H j (X (s),F(s))

is an isomorphism.(ii) Let j0 be an integer such that

H j (X (s),F(s)) = 0

for all j ≥ j0 + 1 and s ∈ S (e.g. j0 = maxs∈Sdim[Supp(F(s))]). Then t j0(s)is an isomorphism for all s ∈ S.

(iii) Let j0 ≥ 0 be an integer. Then there is a nonempty open set U ⊂ S such thatt j0(s) is an isomorphism for all s ∈ U.

Proof. (i) follows immediately from 4.2.5. (ii) is a special case of (i).(iii) It is the open set U = ⋂

j≥ j0 U j , where U j = s ∈ S : R j p∗(F)s is free(apply 4.2.5(iii)).


4.2.2 Stratifications

Let S be a scheme. A stratification of S consists of a set of finitely many locallyclosed subschemes S1, . . . , Sn of S, called strata, pairwise disjoint and such that

S = S1 ∪ . . . ∪ Sn

i.e. such that we have a surjective morphism∐i

Si → S

Let F be a coherent sheaf on S and for each s ∈ S let

e(s) := dimk(s)[Fs ⊗ k(s)]Fix a point s ∈ S, let e = e(s) and let a1, . . . , ae ∈ Fs be such that their imagesin Fs ⊗ k(s) form a basis. From Nakayama’s lemma it follows that the homomor-phism fs : Oe

S,s → Fs defined by a1, . . . , ae is surjective; therefore there is anopen neighbourhood U of s to which f extends defining a surjective homomorphismf : Oe

U → F|U . With a similar argument applied to ker( fs) we may find an affineopen neighbourhood U (s) of s contained in U and an exact sequence

OdU (s)

g−→ OeU (s)

f−→ F|U (s)→ 0 (4.3)

It follows that:

(i) e(s′) ≤ e(s) for all s′ ∈ U (s): therefore s → e(s) is an upper semicontinuousfunction from S to ZZ .

(ii) Let (gi j ) be the e × d matrix with entries in H0(U (s),OS) which defines g.The ideal generated by the gi j ’s in H0(U (s),OS) defines a closed subschemeZs of U (s) with support equal to Ye ∩ U (s), where for each e ≥ 0 we have setYe = s ∈ S : e(s) = e. In particular, Ye is a locally closed subset of S.

Moreover:

(iii) If q : T → U (s) is a morphism, q∗(F) is locally free of rank e if and only if qfactors through the subscheme Zs .

Proof. q factors through Zs if and only if all the functions q∗(gi j ) are zero on T .Since the sequence

OdT

q∗(g)−→ OeT

q∗( f )−→ q∗(F)→ 0

is exact on T , this is equivalent to q∗( f ) being an isomorphism and this conditionimplies that q∗(F) is locally free of rank e. Conversely, if q∗(F) is locally free ofrank e, let G = ker[q∗( f )]. At every point t ∈ T we have an exact sequence:

0→ G ⊗ k(t)→ k(t)e → q∗(F)⊗ k(t)→ 0


Since q∗(F) ⊗ k(t) is a vector space of dimension e we have G ⊗ k(t) = (0).By Nakayama’s lemma, G = (0) in a neighbourhood of t and therefore G = (0)everywhere.

(iv) Since property (iii) characterizes the scheme Zs and does not depend on thepresentation (4.3), for any s, s′ ∈ S the schemes Zs and Zs′ coincide onU (s) ∩ U (s′); therefore the collection of schemes Zs : s ∈ S defines a lo-cally closed subscheme Ze of S supported on Ye. Evidently, Ze : e ≥ 0 is astratification of S.

(v) Because of (i), for each e the closure of Ze is contained in⋃

e′≥e Ze′ . In partic-ular, if E is the highest integer such that Z E = ∅, then Z E is closed.

(vi) By the right exactness of tensor product, the construction of the schemes Ze

commutes with base change (the proof is similar to that of (iii)). In other words,if q : T → S is a morphism then

Ze = f −1(Ze)

for all e, where Ze ⊂ T is the locally closed stratum associated to the sheaf q∗F .

We have proved the following:

Theorem 4.2.7. Let S be a scheme and F a coherent sheaf on S. There is a uniquestratification Zee≥0 of S such that if q : T → S is a morphism the sheafq∗(F) is locally free if and only if q factors through the disjoint union of the Ze’s:T →∐

e Ze → S.Moreover, the strata Z0, Z1, . . . are indexed so that for each e = 0, 1, . . . the

restriction of F to Ze is locally free of rank e.For a given e, Ze ⊂⋃e′≥e Ze′ . In particular, if E is the highest integer such that

Z E = ∅, then Z E is closed.The stratification Zee≥0 commutes with base change.

Theorem 4.2.7 describes a natural way to construct stratifications on a scheme.Zee≥0 is called the stratification defined by the sheaf F .

An alternative approach to the construction of this stratification is by the “Fittingideals” of the sheaf F . Let k ≥ 0; the k-th Fitting ideal of F is the ideal sheafFittFk ⊂ OS locally defined by the minors of order (e − k) of the matrix (gi j ) inthe presentation (4.3). The proof of Theorem 4.2.7 essentially shows that the Fittingideals are independent of the choice of the presentations (4.3). The closed subschemeof S defined by FittFk−1 is denoted by Nk(F). It follows directly from the definitionthat

Supp(Nk(F)) = s ∈ S : dimk(s)(Fs ⊗ k(s)) ≥ kand that Nk(F) commutes with base change. Therefore the stratification defined inTheorem 4.2.7 can be also described as follows:

Ze = Ne(F)\Ne+1(F) (4.4)

For details about the properties of the Fitting ideals see [48].


Example 4.2.8. Let ϕ : A → B be a homomorphism of locally free sheaves on thescheme S, of ranks a and b respectively. Applying Theorem 4.2.7 to coker(ϕ) weobtain a stratification of S with the property that Zb−e is supported on the locus

s ∈ S : rk[ϕ(s) : A(s)→ B(s)] = eThe scheme Zb−e of this stratification will be denoted by De(ϕ). Note that in partic-ular, the subscheme D0(ϕ), called the vanishing scheme of ϕ, is closed in S becauseof (v) above. It has the property that a morphism f : T → S satisfies f ∗(ϕ) = 0 ifand only if f factors through D0(ϕ).

The ideal sheaf of D0(ϕ) is locally generated by the entries of a matrix repre-senting ϕ. More intrinsically, it can be obtained as follows. Since ϕ ∈ Hom(A, B), itinduces by adjunction a homomorphism:

Hom(B, A)ϕ∨−→ OS

whose image is just the ideal sheaf of D0(ϕ).

Example 4.2.9. Let f : X → S be a finite morphism of algebraic schemes. Thenf∗OX is a coherent sheaf on S. The scheme Nk( f∗OX ) ⊂ S is supported on the setof points of S having ≥ k preimages (counting multiplicities). It is usually denotedby Nk( f ) and it is called the k-th multiple point scheme of f . The correspondingstratification is the multiple point stratification of S relative to f . There is a vastliterature on this stratification. For more about it we refer the reader to [61] and tothe literature quoted there.

4.2.3 Flattening stratifications

Definition 4.2.10. Let S be a scheme and F a coherent sheaf on IPr× S. A flatteningstratification for F is a stratification S1, . . . , Sn of S such that for every morphismg : T → S the sheaf

Fg := (1× g)∗(F)

on IPr × T is flat over T if and only if g factors through∐

Si .

Note that if such a stratification exists it is clearly unique. In the special caser = 0 we obtain again the notion of stratification defined by the sheaf F .

The following is a basic technical result.

Theorem 4.2.11. For every coherent sheaf F on IPr × S the flattening stratificationexists.

Proof. The theorem has already been proved in the case r = 0 (Theorem 4.2.7).Therefore we may assume r ≥ 1. We will proceed in several steps.

Step 1: There are finitely many locally closed subsets Y 1, . . . , Y k of S such thatfor each i = 1, . . . , k if we consider on Y i the reduced scheme structure thenF ⊗OY i×IPr is flat over Y i .


It follows immediately from a repeated use of the fact that there is a nonemptyopen subset U ⊂ S such that F|IPr×Ured is flat over Ured (see Note 7).

Step 2: Only finitely many polynomials P1, . . . , Ph occur as Hilbert polynomialsof the sheaves F(s), s ∈ S.

In fact from Corollary 4.2.2 it follows that, at most, as many Hilbert polynomialsoccur as the number of connected components of the sets Y 1, . . . , Y k .

Step 3: There is an integer N such that for every m ≥ 0 and for every s ∈ S wehave:

H j (IPr (s),F(s)(N + m)) = (0)

for j ≥ 1 and the natural map:

[p∗F(N + m)]s ⊗ k(s)→ H0(IPr (s),F(s)(N + m))

is an isomorphism, where p : IPr × S→ S is the projection.

For each i = 1, . . . , k consider the diagram

hi : IPr × Y i → IPr × S↓ pi ↓ pY i → S

(4.5)

and let ni 0 be so that R j pi∗[hi∗F(ni + m)] = (0) for all m ≥ 0 and all j ≥ 1(apply Proposition 4.2.4). Letting

N maxn1, . . . , nkwe may apply Proposition 4.2.4 to the diagrams (4.5) and to the sheaf F and weobtain isomorphisms

[p∗F(N + m)] ⊗OY i ∼= pi∗[hi∗F(N + m)]for all s ∈ Y i and for all i = 1, . . . , k. In particular, we have isomorphisms

[p∗F(N + m)] ⊗ k(s) ∼= pi∗[hi∗F(N + m)]s ⊗ k(s) (4.6)

for all s ∈ Y i and for all i = 1, . . . , k. We may also apply Corollary 4.2.6 to thesheaves hi∗F and to the projections pi for j0 = 0 to deduce that

H j (IPr (s),F(s)(N + m)) = (0)

for all s ∈ S, j ≥ 1 and m ≥ 0, and that

pi∗[hi∗F(N + m)]s ⊗ k(s) ∼= H0(IPr (s),F(s)(N + m)) (4.7)

for all s ∈ Y i and for all i = 1, . . . , k and all m ≥ 0.


Comparing (4.6) and (4.7) we obtain the conclusion.

Step 4: Let N be as in Step 3, and let g : T → S be a morphism. Then Fg is flatover T if and only if g∗[p∗F(N + m)] is locally free for all m ≥ 0.

Suppose that Fg is flat over T and let q : IPr × T → T be the projection. Since

H j (IPr (t),Fg(t)(N + m)) = H j (IPr (g(t)),F(g(t))(N + m)) = (0)

for all t ∈ T , m ≥ 0 and j ≥ 1, from Corollary 4.2.6(ii) we deduce that

q∗Fg(N + m)t ⊗ k(t)→ H0(IPr (g(t)),F(g(t))(N + m)) (4.8)

is an isomorphism for all t ∈ T . Theorem 4.2.5(ii) applied for j = −1 implies thatq∗Fg(N +m) is locally free for all m ≥ 0. For all t ∈ T the natural homomorphism

ϕ : g∗[p∗F(N + m)] → q∗Fg(N + m)

induces an isomorphism:

g∗[p∗F(N + m)]t ⊗ k(t) ∼= q∗Fg(N + m)t ⊗ k(t)

because both sides are isomorphic to H0(IPr (g(t)),F(g(t))(N + m)) (the first be-cause of Step 3, the second because of (4.8)). From the fact that q∗Fg(N + m) islocally free and from Nakayama’s lemma it follows that ϕ is an isomorphism. There-fore g∗[p∗F(N + m)] is locally free for every m ≥ 0.

Conversely, suppose that g∗[p∗F(N + m)] is locally free for all m ≥ 0. Sincefor all m 0 the natural map ϕ is an isomorphism (Prop. 4.2.4) it follows thatq∗Fg(N +m) is locally free for all m 0: Proposition 4.2.1 implies that Fg is flat.

Step 5: For every m ≥ 0 apply Theorem 4.2.7 to the sheaf p∗F(N + m) and letYm, j be the component of the corresponding stratification of S wherep∗F(N + m) becomes locally free of rank j . Then for each j = 1, . . . , h wehave the following equality of subsets of S:⋂

m≥0

Supp(Ym,Pi (N+m)) =⋂

m=0,...,r

Supp(Ym,Pi (N+m))

The inclusion ⊂ is obvious. For s ∈ S let Ps(t) be the Hilbert polynomial ofF(s). Then s ∈ ∩m≥0Supp(Ym,Pi (N+m)) if and only if

Ps(N+m) = h0(IPr (s),F(s)(N+m)) = dim[p∗F(N+m)s⊗k(s)] = Pi (N+m)

for all m ≥ 0, and this happens if and only if Ps(t) = Pi (t) as polynomials. On theother hand, s ∈ ∩m=0,...,r Supp(Ym,Pi (N+m)) if and only if Ps(N +m) = Pi (N +m)

for m = 0, . . . , r . Since both Ps(t) and Pi (t) have degree≤ r , it follows that Ps(t) =Pi (t) and therefore s ∈ ∩m≥0Supp(Ym,Pi (N+m)).


Step 6: Fix i between 1 and h. For each integer c ≥ 0 the finite intersection⋂m=0,...,c

Ym,Pi (N+m)

is a well-defined locally closed subscheme of S. Because of Step 5 the sub-schemes ⋂

m=0,...,c

Ym,Pi (N+m), c = r, r + 1, . . .

for a descending chain with fixed support; in particular, they form a descend-ing chain of closed subschemes of a fixed open set V ⊂ S, and therefore theystabilize. In other words the intersection

Zi =⋂m≥0

Supp(Ym,Pi (N+m))

is a well-defined locally closed subscheme of S. By Step 5 we have:

Supp(Zi ) = s ∈ S : Ps(t) = Pi (t)Step 7: The subschemes Z1, . . . , Zh form a stratification of S. It follows immedi-

ately from Step 4 that this is the flattening stratification for F . This concludesthe proof of Theorem 4.2.11.

NOTES

1. From the proof of Theorem 4.2.11 it follows that the strata Z1, . . . , Zh of the flatteningstratification for F are indexed by the Hilbert polynomials of the sheaves F(s), s ∈ S.

2. (The see-saw theorem) Let X be a projective scheme such that H0(X,OX ) = k, S analgebraic scheme and L an invertible sheaf on X × S. Then:

(i) The locus

S0 = s ∈ S : L|X×s ∼= OX is closed in S.

(ii) Letting p0 : X × S0 → S0 be the projection, there is an invertible sheaf M on S0 suchthat

L|X×S0∼= p∗0 M

Proof. A line bundle L on X is trivial if and only if it satisfies h0(X, L) > 0 andh0(X, L−1) > 0. In fact, nonzero sections σ ∈ H0(X, L) and τ ∈ H0(X, L−1) correspondto homomorphisms

OXσ−→ L

τ∨−→ OX

whose composition is multiplication by a constant (by the assumption H0(X,OX ) = k), sothat they are isomorphisms.


Therefore S0 = S+ ∩ S− where

S+ = s ∈ S : h0(X × s,L|X×s) > 0

S− = s ∈ S : h0(X × s,L−1|X×s) > 0

It follows from the semicontinuity theorem that both S+ and S− are closed in S, and (i)follows.

(ii) Since h0(X ×s,L|X×s) = 1 is constant in s ∈ S0, from Corollary 12.9 of [84] we

deduce that p0∗L⊗k(s) ∼= H0(X×s,L|X×s) for all s ∈ S0 and in particular, p0∗L =: Mis an invertible sheaf. We have a homomorphism of invertible sheaves on X × S0:

p∗0 M = p∗0 p0∗L→ L|X×S0

which is surjective because L|X×s is globally generated for all s ∈ S0. Then it is an isomor-phism.

3. Let X → S be a flat projective morphism of algebraic schemes, and L an invertiblesheaf on X . Assume that for some k-rational point o ∈ S the sheaf L(o) is very ample onX (o) and satisfies H1(X (o),L(o)) = 0. Then there is an open neighbourhood V ⊂ S of osuch that LV := L|X (V ) is very ample relative to V . In particular, L(s) is very ample onX (s) for every s ∈ V .

Proof. By Corollary 4.2.6, there is an open neighbourhood U ⊂ S of o such that(R1 f∗L)|U = (0) and

t0(u) : ( f∗L)u ⊗ k(u)→ H0(X (u),L(u))

is an isomorphism for all u ∈ U . We may even assume that f∗L is locally free of rankh0(X (o),L(o)) on U . From the surjectivity of the map t0(o) and from the fact that L(o) isglobally generated we deduce that the canonical homomorphism:

f ∗( f∗L)→ L

is surjective on X (o). Since f is projective it follows that there is an open W ⊂ U containingo such that

[ f ∗( f∗L)]|X (W ) → LW (4.9)

is surjective and moreover, [ f ∗( f∗L)]|X (W ) is locally free. The homomorphism (4.9) definesa W -morphism

X (W ) → IP( f ∗( f∗L))|X (W )

W(4.10)

whose restriction to X (o) is the embedding defined by the global sections of L(o). FromNote 3 of § 3.4 above it follows that there is an open subset V ⊂ W containing o such thatthe restriction of (4.10) to X (V ) is an embedding. This implies the conclusion (see [116],Theorem 1.2.17 for the proof of a more general statement without flatness assumption).

4. Let E be a locally free sheaf over IP1 × S, with S an algebraic integral scheme. Leto ∈ S be a k-rational point, and E(o) ∼= ⊕iO(ni

0) the fibre over o. Then:


(i) There is an open set U ⊂ S such that for each s ∈ U we have

E(s) ∼= ⊕iO(nis)

with

maxi nis ≤ maxi ni

o and mini nis ≥ mini ni

oMoreover, if E(o) is balanced (i.e. ni

o = n jo for all i, j ) then E(s) ∼= E(o) for all s ∈ U.

(ii) For each s ∈ S we have ∑i

nis =

∑i

ni0

Proof. ([25]) (i) By the structure theorem for locally free sheaves on IP1 (see [136]) we knowthat for each s ∈ S we have an isomorphism E(s) ∼= ⊕iO(ni

s) for some integers nis . Let

M0 = maxi ni0 and consider the sheaf E := E ⊗ p∗O(−M0− 1), where p : IP1× S→ IP1

is the projection. Since h0(E(0)) = 0, from the semicontinuity theorem it follows that thereis an open neighbourhood U of 0 such that h0(E(s)) = 0 for all s ∈ U ; but this means thatmaxi ni

s ≤ M0 for all s ∈ U , which is the first statement of the proposition. The statementabout the minimum is proved similarly after replacing E by its dual. The last assertion isobvious.

(ii) Applying (i) to det(E) we find that every point t ∈ S has an open neighbourhood Utwhere

∑i ni

s =∑

i nit for all s ∈ Ut . Since S is connected we deduce that

∑i ni

s is constant.

5. Let X → S be a flat projective morphism with S an algebraic scheme, and let o ∈ S bea k-rational point. Prove that:

(i) If X (o) is connected and X (s) is disconnected for all s = o in an open neighbourhood ofo then X (o) is nonreduced.In particular:

(ii) If X (o) is connected and reduced then X (s) is connected for all s in an open neighbour-hood of o.

(iii) If X (o) is disconnected then X (s) is disconnected for all s in an open neighbourhood ofo.

6. Let f : X → Y be a proper morphism of algebraic schemes with finite fibres. Letg : Y ′ → Y be an arbitrary morphism, X ′ = X ×Y Y ′, f ′ : X ′ → Y ′ and g′ : X ′ → X theprojections. Then for every quasi-coherent OX -module F we have a canonical isomorphism

g∗( f∗F) ∼= f ′∗(g′∗F)

Proof. Since it is proper and quasi-finite, f is finite, in particular it is affine. The conclusionfollows from [1] Ch. II, 1.5.2.

7. (Generic flatness) Let f : X → S be a morphism of finite type with S integral, and letF be a coherent sheaf on X . There is a dense open subset U ⊂ S such that the restriction ofF to f −1(U ) is flat over U.

Proof. See [1], ch. IV, Th. 6.9.1 or [48], Theorem 14.4, p. 308. Note that if f is not dominant then U = S\ f (X ) and f −1(U ) = ∅.


4.3 Hilbert schemes

4.3.1 Generalities

Consider a projective scheme Y , and a closed embedding Y ⊂ IPr . Let’s fix a nu-merical polynomial of degree ≤ r , i.e. a polynomial P(t) ∈ Q[t] of the form:

P(t) =r∑

i=0

ai

(t + r

i

)with ai ∈ ZZ for all i .

For every scheme S we let:

HilbYP(t)(S) =

flat families X ⊂ Y × S of closed subschemesof Y parametrized by S with fibres

having Hilbert polynomial P(t)

Since flatness is preserved under base change, this defines a contravariant functor

HilbYP(t) : (schemes) → (sets)

called the Hilbert functor of Y relative to P(t).In the case Y = IPr we will denote the Hilbert functor with the symbol Hilbr

P(t).

If the functor HilbYP(t) is representable, the scheme representing it will be called

the Hilbert scheme of Y relative to P(t), and will be denoted by HilbYP(t) (or Hilbr

P(t)

in case Y = IPr ). If P(t) = n a constant polynomial then HilbYP(t) is usually denoted

by Y [n].If the Hilbert scheme HilbY

P(t) exists then there is a universal element, i.e. thereis a flat family of closed subschemes of Y having Hilbert polynomial equal to P(t):

W ⊂ Y × HilbYP(t) (4.11)

parametrized by HilbYP(t) and possessing the following:

Universal property: for each scheme S and for each flat family X ⊂ Y × S of closedsubschemes of Y having Hilbert polynomial P(t) there is a unique morphismS→ HilbY

P(t), called the classifying morphism, such that

X = S ×HilbYP(t)

W ⊂ Y × S

The family (4.11) is called the universal family, and the pair (HilbYP(t),W) rep-

resents the functor HilbYP(t).

The family W is the universal element of HilbYP(t)(HilbY

P(t)), namely the elementcorresponding to the identity under the identification

Hom(HilbYP(t), HilbY

P(t)) = HilbYP(t)(HilbY

P(t))

4.3 Hilbert schemes 207

Example 4.3.1. Consider the constant polynomial P(t) = 1. Then we have a canon-ical identification Y [1] = Y and the universal family is the diagonal ∆ ⊂ Y × Y .

To prove it consider an element of Y [1](S) for some scheme S:

Γ ⊂ S × Y↓ fS

Then f is an isomorphism: in fact it is a one-to-one morphism and OS → f∗OΓ

is an isomorphism since f∗OΓ is an OS-algebra which is locally free of rank oneover OS . We therefore have the well-defined morphism g = q f −1 : S → Y whereq : S × Y → Y is the projection. The morphism

(g f, q) : Γ → Y × Y

factors through ∆ and induces a commutative diagram

Γ → ∆↓ ↓S −→

gY

such that Γ ∼= g∗∆. Therefore the family Γ is induced by ∆ via the morphism g.

Before proving the existence of the Hilbert schemes in general we will considertwo important special cases.

4.3.2 Linear systems

If X ⊂ IPr is a hypersurface of degree d it has Hilbert polynomial

h(t) =(

t + r

r

)−(

t + r − d

r

)= d

(r − 1)! tr−1 + · · ·

Conversely, if a closed subscheme Y of IPr has the Hilbert polynomial h(t) thenit is a hypersurface of degree d.

In fact, since h(t) has degree r − 1, Y has dimension r − 1, so Y = Y1 ∪ Z , withY1 a hypersurface and dim(Z) < r − 1. We have the exact sequence:

0→ IY1/IY → OY → OY1 → 0

where IY , IY1 ⊂ OIPr are the ideal sheaves of Y and Y1. We deduce that

h(t) = h1(t)+ k(t)

where h1(t) is the Hilbert polynomial of Y1 and k(t) is the Hilbert polynomial ofIY1/IY . Since this sheaf is supported on Z , we have deg(k(t)) < r −1; therefore wesee that

h1(t) = d

(r − 1)! tr−1 + · · ·


so Y1 is a hypersurface of degree d, and therefore h1(t) = h(t). It follows thatk(t) = 0, i.e. IY1 = IY ; equivalently, Y = Y1.

Therefore Hilbrh(t), if it exists, parametrizes a universal family of hypersurfaces

of degree d in IPr . To prove its existence let V := H0(IPr ,O(d)) and in IP(V ) takehomogeneous coordinates

(. . . , ci(0),...,i(r), . . .)i(0)+···+i(r)=d

The hypersurface H ⊂ IPr × IP(V ) defined by the equation∑ci(0),...,i(r)Xi(0)

0 · · · Xi(r)r = 0

projects onto IP(V ) with fibres hypersurfaces of degree d. It follows from Proposi-tion 4.2.1 that H is flat over IP(V ). Let’s denote by p : IPr × IP(V )→ IP(V ) theprojection, and let IH ⊂ OIPr×IP(V ) be the ideal sheaf of H. For all x ∈ IP(V ) wehave

1 = h0(IPr (x), IH(x)(d)) = h0(IPr (x), IH(d)(x))

and0 = hi (IPr (x), IH(x)(d)) = hi (IPr (x), IH(d)(x))

0 = hi (H(x),OH(x)(d))

for all i ≥ 1. Applying 4.2.5 and 4.2.2 we deduce that:

(a) R1 p∗IH(d) = 0;(b) p∗IH(d) is an invertible subsheaf of p∗OIPr×IP(V )(d) = V ⊗k OIP(V );(c) p∗OIPr×IP(V )(d)/p∗IH(d) = p∗OH(d) is locally free.

It follows thatp∗IH(d) = OIP(V )(−1)

the tautological invertible sheaf on IP(V ), and the natural map

p∗ p∗IH(d)→ IH(d)

is an isomorphism. Therefore

IH = [p∗OIP(V )(−1)](−d)

Let’s prove that H ⊂ IPr × IP(V ) is a universal family.Suppose that

X ⊂ IPr × S↓ fS

is a flat family of closed subschemes of IPr with Hilbert polynomial h(t), i.e. hy-persurfaces of degree d, and let IX ⊂ OIPr×S be the ideal sheaf of X . Arguing asabove we deduce that f∗IX (d) is an invertible subsheaf of V ⊗k OS with locallyfree cokernel f∗OX (d), and that


IX = [ f ∗ f∗IX (d)](−d)

We have an induced morphism g : S→ IP(V ) such that

g∗[OIP(V )(−1)] = f∗IX (d)

The subscheme S ×IP(V ) H ⊂ IPr × S is defined by the ideal sheaf

(1× g)∗IH = (1× g)∗[OIP(V )(−1)(−d)]

= f ∗[g∗OIP(V )(−1)](−d) = [ f ∗ f∗IX (d)](−d) = IXHence S ×IP(V ) H = X . The proof of the uniqueness of g having this property isleft to the reader. Therefore we see that H ⊂ IPr × IP(V ) is a universal family, andHilbr

h(t) = IP(V ).

4.3.3 Grassmannians

The classical grassmannians are special cases of Hilbert schemes, since they parame-trize linear spaces, which are the closed subschemes with Hilbert polynomials of theform

(t+n−1n−1

), n−1 being their dimension. Let’s fix a k-vector space V of dimension

N , and let 1 ≤ n ≤ N . Letting

GV,n(S) = loc. free rk n quotients of the free sheaf V∨ ⊗k OS on Swe define a contravariant functor:

GV,n : (schemes)o → (sets)

called the Grassmann functor; we will denote it simply by G when no confusion ispossible.

Theorem 4.3.2. The Grassmann functor G is represented by a scheme Gn(V ) to-gether with a locally free quotient of rank n

V∨ ⊗k OGn(V )→ Q

called the universal quotient bundle.

Proof. Given a scheme S and an open cover Ui of S, to give a locally free rank nquotient of V∨ ⊗k OS is equivalent to giving one such quotient over each open setUi so that they patch together on the intersections Ui ∩U j . Therefore G is a sheaf.

Let’s fix a basis ek of V∨ and choose a set J of n distinct indices in 1, . . . , N .We have an induced decomposition V∨ = E ′ ⊕ E ′′, with E ′ (resp. E ′′) a vectorsubspace of rank n (resp. N − n). We can define a subfunctor GJ of G letting:

GJ (S) =

locally free rk n quotients V∨ ⊗k OS → Finducing E ′ ⊗k OS → F surjective


Let S be any scheme and f : Hom(−, S)→ G a morphism of functors correspond-ing to a locally free rank n quotient

V∨ ⊗k OS → F

The fibred product SJ := Hom(−, S) ×G GJ is clearly represented by the opensubscheme of S supported on the points where the map E ′ ⊗kOS → F is surjective;this proves that GJ is an open subfunctor of G. Since clearly the SJ ’s cover S, wealso see that the family of subfunctors GJ is an open covering of G.

To prove that GJ is representable note that if

q : V∨ ⊗k OS → F

is an element of G(S) then the induced map

η : E ′ ⊗k OS → F

is surjective if and only if it is an isomorphism; in this case the composition

η−1 q : V∨ ⊗k OS → E ′ ⊗k OS

restricts to the identity on E ′ ⊗k OS , hence it is determined by the composition

E ′′ ⊗k OS → V∨ ⊗k OS → E ′ ⊗k OS

It follows that we can identify

GJ (S) = Hom(E ′′ ⊗k OS, E ′ ⊗k OS) = Hom(E ′′, E ′)⊗k OS

This proves that GJ is isomorphic to Hom(−, An(N−n)), hence it is representable.Now the theorem follows from Proposition E.10.

Gn(V ) is called the grassmannian of n-dimensional subspaces of V ; it is alsocalled the grassmannian of (n−1)-dimensional projective subspaces of IP(V ). WhenV = kN the grassmannian Gn(kN ) is denoted by G(n, N ).

When n = 1 the functor GV,1 is represented by

G1(V ) = IP(V ) = Proj(Sym(V∨)),

the (N − 1)-dimensional projective space associated to V . In this case Q =OIP(V )(1).

From Theorem 4.3.2 it follows that for all schemes S the morphismsf : S→ Gn(V ) are in 1–1 correspondence with the locally free rank n quotients

V∨ ⊗k OS → F

via f ↔ f ∗Q. This is the universal property of Gn(V ).The universal quotient bundle defines an exact sequence of locally free sheaves

on Gn(V ):0→ K → V∨ ⊗k OGn(V )→ Q → 0

called the tautological exact sequence; K is called the universal subbundle.


Let S be a scheme. Associating to every locally free quotient of rank n

V∨ ⊗k OS → F

the quotient(∧n V∨)⊗k OS → ∧nF

we define a morphism of functors GV,n → G∧n V,1, which is induced by a morphism

π : Gn(V )→ IP(∧n V )

π is called the Plucker morphism.

Proposition 4.3.3. The Plucker morphism is a closed embedding. In particular,Gn(V ) is a projective variety.

Proof. As in the proof of 4.3.2, we fix a basis of V∨ and we choose a set J ofn distinct indices in 1, . . . , N . We obtain a decomposition V∨ = E ′ ⊕ E ′′ withdim(E ′) = n, dim(E ′′) = N − n, and an induced one:

∧n V∨ = ⊕ni=0(∧n−i E ′)⊗k ∧i E ′′ = ∧n E ′ ⊕ F

where F = ⊕ni=1(∧n−i E ′)⊗k ∧i E ′′. For every scheme S let

IPJ (S) =

locally free rk 1 quotients ∧n V∨ → Ls.t. the induced ∧n E ′ → L is surjective

We obtain a subfunctor IPJ of G∧n V,1. As in the proof of 4.3.2, we see that the IPJ ’sform an open cover of G∧n V,1 by functors representable by affine spaces.

Note that for every locally free rank n quotient V∨ ⊗k OS → F the inducedhomomorphism:

E ′ ⊗k OS → Fis surjective if and only if∧n E ′ → ∧nF is. Therefore GJ = π−1(IPJ ) and it sufficesto prove that π : GJ → IPJ is a closed embedding.

We may identify GJ with (the affine space associated to) Homk(E ′, E ′′) and IPJ

with Homk(F,∧n E ′). Considering that

Homk(∧n−i E ′,∧n E ′) ∼= ∧i E ′

canonically via the perfect pairing:

∧i E ′ × ∧n−i E ′ → ∧n E ′

we have:

Homk(F,∧n E ′) = ⊕ni=1Homk((∧n−i E ′)⊗k ∧i E ′′,∧n E ′)

= ⊕ni=1Homk(∧i E ′′, Homk(∧n−i E ′,∧n E ′))

= ⊕ni=1Homk(∧i E ′′,∧i E ′)


and the map

π : Homk(E ′′, E ′)→ Homk(F,∧n E ′)

is

λ → (λ,∧2λ, . . . ,∧nλ)

This is the graph of a morphism of affine schemes, hence it is a closedembedding.

For some 1 ≤ n ≤ r , let G = G(n + 1, r + 1) be the grassmannian of n-dimensional projective subspaces of IPr . Consider the projective bundle over G:

I := IP(Q∨) = Proj(Sym(Q))

where Q is the universal quotient bundle on G. Because of the surjection Or+1G → Q

we have a closed embedding:

I ⊂ IPr × G↓ pG

note that Q∨ = p∗II(1) ⊂ p∗OIPr×G(1) = Or+1G . For every closed point v ∈ G

the fibre I(v) is the projective subspace IP(v) ⊂ IPr ; for this reason I is calledthe incidence relation. Since all fibres of p have Hilbert polynomial

(t+nn

), from

Proposition 4.2.1 it follows that p is a flat family. Suppose now that

Λ ⊂ IPr × S↓ qS

is another flat family whose fibres have Hilbert polynomial(t+n

n

). We have an inclu-

sion of sheaves on S

q∗IΛ(1) ⊂ q∗OIPr×S(1) = Or+1S

which has locally free cokernel q∗OΛ(1). By the universal property of G the aboveinclusion induces a unique morphism

g : S→ G

such that g∗(Q∨) = q∗IΛ(1). Since Λ = IP(q∗IΛ(1)) it follows that

Λ = S ×G I

namely, the family q is obtained by base change from the incidence relation via themorphism g. This proves that

G(n + 1, r + 1) = Hilbr(t+n

n )


4.3.4 Existence

Theorem 4.3.4. For every projective scheme Y ⊂ IPr and every numerical polyno-mial P(t), the Hilbert scheme HilbY

P(t) exists and is a projective scheme.

Proof. We will first prove the theorem in the case Y = IPr . From Corollary 4.1.10it follows that there is an integer m0 such that for every closed subscheme X ⊂ IPr

with Hilbert polynomial P(t) the sheaf of ideals IX is m0-regular. It suffices to take

m0 = Fr (−a0, . . . ,−ar−1, 1− ar )

It follows that for every k ≥ m0

hi (IPr , IX (k)) = 0 (4.12)

for i ≥ 1 and

h0(IPr , IX (k)) =(

k + r

r

)− P(k)

depends only on P(k). Moreover, by Note 2 of Section 4.1 we have

hi (X,OX (k)) = 0 (4.13)

all k ≥ m0 and all i ≥ 1. Let

N =(

m0 + r

r

)− P(m0)

V = H0(IPr ,OIPr (m0))

Consider the grassmannian G = G N (V ) of N -dimensional vector subspaces of V .Let V∨ ⊗k OG → Q be the universal quotient bundle, which is a locally free sheafof rank N on G, and

p : IPr × G → G

the projection. We may identify V ⊗k OG = p∗[OIPr×G(m0)]. The image of thecomposition

p∗Q∨(−m0) −→ V ⊗k OIPr×G(−m0) −→ OIPr×G

‖p∗ p∗[OIPr×G(m0)] ⊗OIPr×G(−m0)

is a sheaf of ideals: we will denote it by J.Let Z ⊂ IPr×G be the closed subscheme defined by J and denote by q : Z → G

the restriction of p to Z .Consider the flattening stratification

G1∐

G2∐

. . . ⊂ G


for OZ and let H be the stratum relative to the polynomial P(t). We will prove thatH = Hilbr

P(t) and that the universal family is the pullback of q to H :

W := H ×G Z → Z↓ π ↓ qH → G

By the choice of H , W defines a flat family of closed subschemes of IPr with Hilbertpolynomial equal to P(t).

Let’s prove that W has the universal property.Consider a flat family of closed subschemes of IPr with Hilbert polynomial P(t):

X ⊂ IPr × S↓ fS

From (4.12) and (4.13) and from Theorem 4.2.5 it follows that

R1 f∗IX (m0) = (0) = R1 f∗OX (m0)

In particular, we have an exact sequence on S:

0→ f∗IX (m0)→ f∗OIPr×S(m0) → f∗OX (m0)→ 0‖

V ⊗k OS

If we apply Theorem 4.2.5 for j = −1 we deduce that f∗IX (m0) and f∗OX (m0)are locally free and f∗IX (m0) has rank N .

From the universal property of G it follows that there exists a unique morphismg : S→ G such that

f∗IX (m0) = g∗Q∨

Claim: For all m m0 we have f∗OX (m) = g∗ p∗OZ (m).

Proof of the Claim: For all m m0 we have exact sequences:

0→ p∗J(m)→ p∗OIPr×G(m)→ q∗OZ (m)→ 0

on G and0→ f∗IX (m)→ f∗OIPr×S(m)→ f∗OX (m)→ 0

on S; since g∗ p∗OIPr×G(m) = f∗OIPr×S(m) it suffices to show that:

f∗IX (m) ∼= g∗ p∗J(m)

for all m m0. For all such m we have the equality on G:

p∗J(m) = Im[Q∨ ⊗ p∗O(m − m0)→ p∗OIPr×G(m)]induced by the surjections p∗Q∨(m − m0)→ J(m) of sheaves on IPr × G. Hencefor all m m0 we have:


g∗ p∗J(m) = g∗Im[Q∨ ⊗ p∗OIPr×G(m − m0)→ p∗OIPr×G(m)]= Im[g∗Q∨ ⊗ f∗OIPr×S(m − m0)→ f∗OIPr×S(m)]= Im[ f∗IX (m0)⊗ f∗OIPr×S(m − m0)→ f∗OIPr×S(m)] = f∗IX (m)

and this proves the Claim.

From the Claim it follows that:

(i) g factors through H .

Indeed, from 4.2.4 it follows that for all m m0:

g∗q∗OZ (m) = f∗(1× g)∗OZ (m)

Since g∗q∗OZ (m) = f∗OX (m) is locally free of rank P(m) for all such m Proposi-tion 4.2.1 implies that (1×g)∗OZ is flat over S with Hilbert polynomial P(t). Henceg factors by the definition of H .

(ii) X = S ×H W .

Indeed:

X = Proj[⊕m0

f∗OX (m)] = Proj[⊕m0

g∗q∗OZ (m)]

= Proj[⊕m0

g∗π∗OW (m)] = S ×H Proj[⊕m0

π∗OW (m)] = S ×H W

Properties (i) and (ii) imply that H = HilbrP(t) and that π is the universal family.

By construction, HilbrP(t) is a quasi-projective scheme. To prove that it is projec-

tive it suffices to show that it is proper over k. We will use the valuative criterion ofproperness. Let A be a discrete valuation k-algebra with quotient field L and residuefield K , and let

ϕ : Spec(L)→ HilbrP(t)

be any morphism. We must show that ϕ extends to a morphism

ϕ : Spec(A)→ HilbrP(t)

Pulling back the universal family by ϕ we obtain a flat family

X ⊂ IPr × Spec(L)

of closed subschemes of IPr with Hilbert polynomial P(t). Since Spec(A) is non-singular of dimension one and

Spec(L) = Spec(A)\closed pointProposition III.9.8 of [84] implies the existence of a flat family

X ′ ⊂ IPr × Spec(A)


which extends X . By the universal property of HilbrP(t) this family corresponds to a

morphism ϕ : Spec(A) → HilbrP(t) which extends ϕ. This concludes the proof of

the theorem in the case Y = IPr .

Let’s now assume that Y is an arbitrary closed subscheme of IPr . It will sufficeto show that the functor HilbY

P(t) is represented by a closed subscheme of HilbrP(t).

Applying Corollary 4.1.10 twice we can find an integer µ such that IY ⊂ OIPr

is µ-regular and such that for every closed subscheme X ⊂ IPr with Hilbert polyno-mial P(t) the ideal sheaf IX ⊂ OIPr is µ-regular. Let

V = H0(IPr ,OIPr (µ)), U = H0(IPr , IY (µ))

It follows from 4.2.5 and 4.2.6 that π∗IW (µ) is a locally free subsheaf of V⊗kOHilbwith locally free cokernel.

On HilbrP(t) consider the composition

# : U ⊗k OHilb → V ⊗k OHilb → V ⊗k OHilb/π∗IW (µ)

Let Z ⊂ HilbrP(t) be the closed subscheme defined by the condition # = 0, or

equivalently, by the condition

U ⊗k OZ ⊂ π∗IW (µ)⊗OZ (4.14)

Letting j : Z → HilbrP(t) be the inclusion, one easily sees that condition (4.14)

implies thatIY×Z ⊂ (1× j)∗IW ⊂ OIPr×Z

hence thatZ ×Hilb W ⊂ Y × Z ⊂ IPr × Z (4.15)

It is straightforward to check that Z = HilbYP(t) and that (4.15) is the universal family.

This concludes the proof of Theorem 4.3.4. For any projective scheme Y ⊂ IPr it is often convenient to consider the functor:

HilbY : (schemes)→ (sets)

defined as:HilbY (S) =

∐P(t)

HilbYP(t)(S)

This functor is represented by the disjoint union

HilbY =∐P(t)

HilbYP(t)

which is a scheme locally of finite type (but not of finite type because it has infinitelymany connected components unless dim(Y ) = 0). It is the Hilbert scheme of Y . Oneconvenient feature of HilbY is that it is independent on the projective embedding ofY , even though the indexing of its components HilbY

P(t) by Hilbert polynomials does


depend on the embedding. For this reason, when considering HilbY we will not needto specify a projective embedding of Y .

Let’s fix a projective scheme Y , and in the Hilbert scheme HilbY let’s considera k-rational point [X ] which parametrizes a closed subscheme X ⊂ Y . Denote byI ⊂ OY the ideal sheaf of X in Y . The local Hilbert functor HY

X is a subfunctor ofthe restriction to A of the Hilbert functor; since HilbY represents the Hilbert functorwe have, with the notation introduced in § 2.2:

HYX (A) = Hom(Spec(A), HilbY )[X ]

for every A in ob(A). In particular, HYX is prorepresented by the local ring OHilb,[X ].

We can therefore apply the results proved in § 2.4 to obtain information about thelocal properties of HilbY at [X ]. In particular, we have the following:

Theorem 4.3.5. (i) There is a canonical isomorphism of k-vector spaces:

T[X ]HilbY ∼= H0(X, NX/Y )

where NX/Y = HomOX (I/I2,OX ) is the normal sheaf of X in Y .(ii) If X ⊂ Y is a regular embedding then the obstruction space of OHilbY ,[X ] is a

subspace of H1(X, NX/Y ).

The simplest illustration of Theorem 4.3.5 is for Y [1] = Y . In this case 4.3.5(i)simply says that Hom(m p/m2

p, k) is the Zariski tangent space of Y at a k-rationalpoint p ∈ Y . The obstruction space is o(OY [1],[p]) = o(OY,p). Of course if p is a

singular point then it is not regularly embedded in Y , and H1(p, Np/Y ) = 0 is notan obstruction space for the local Hilbert functor.

Consider a flat family of closed subschemes of Y :

X ⊂ Y × S↓ fS

It induces a functorial morphism χ : S → HilbY (the classifying morphism of thefamily) whose differential at a k-rational point s ∈ S is a linear map

dχs : Ts S→ H0(X (s), NX (s)/Y ) (4.16)

called the characteristic map of the family f . Obviously, the surjectivity of dχs isa necessary condition for the smoothness of χ at s. We have the following moreprecise result.

Proposition 4.3.6. Let Y be a projective scheme,

X ⊂ Y × S↓ fS


a flat family of closed subschemes of Y with S algebraic, and

χ : S→ HilbY

the classifying map of the family. Then:

(i) If s is a nonsingular point of S and if the characteristic map

dχs : Ts S→ H0(X (s), NX (s)/Y )

is surjective then χ is smooth at s and X (s) is unobstructed in Y .(ii) If, moreover, h1(X (s), TY |X (s)) = 0 and X (s) is nonsingular then f has general

moduli at s and X (s) is unobstructed as an abstract variety.

Proof. (i) The smoothness of χ is a consequence of Theorem 2.1.5 and of the non-singularity of S at s. The unobstructedness of X (s) in Y , i.e. the nonsingularity ofHilbY at χ(s), follows from the smoothness of χ and the nonsingularity of S at s.

(ii) The condition h1(X (s), TY |X (s)) = 0 implies that the forgetful morphismHYX (s) → DefX (s) is smooth (Proposition 3.2.9) and therefore the Kodaira–Spencer

map of f at s is surjective, being the composition of two surjective maps. We obtainthe conclusion from Proposition 2.5.8.

The following criterion follows at once from Proposition 3.2.9:

Proposition 4.3.7. Let X ⊂ Y be a closed embedding of projective nonsingularschemes such that h1(X, TY |X ) = 0. Then the universal family

X ⊂ Y × HilbY

↓HilbY

has general moduli at the point [X ].

NOTES

1. The construction of the grassmannian given here is taken from [97].2. It is a classical result of Hartshorne that Hilbr

P(t) is connected for all r and P(t) (see[83] and [140]). For general Y this is no longer true: for example, if Y ⊂ IP3 is a nonsingularquadric then HilbY

t+1 has two connected components.

3. Let X ⊂ Y be a closed embedding of projective schemes. It can be easily verified thatfor any closed subscheme Z ⊂ X , the induced injective linear map

H0(Z , NZ/X )→ H0(Z , NZ/Y )

coincides with the differential at [Z ] of the closed embedding

HilbX ⊂ HilbY

4. If Z ⊂ Y is a closed embedding of projective schemes and P(t) a numerical polyno-mial, then one can define a functor

4.4 Quot schemes 219

Hilb(−Z)YP(t) : (schemes) → (sets)

by

Hilb(−Z)YP(t)(S) =

flat families X ⊂ Y × S with Hilbert polynomial

P(t) and such that X (s) ∩ Z = ∅, ∀s ∈ S

Then HilbY (−Z)Y

P(t) is representable by a closed subscheme

Hilb(−Z)YP(t) ⊂ HilbY

P(t)

LetHilbY\Z

P(t) := HilbYP(t)\Hilb(−Z)Y

P(t) ⊂ HilbYP(t)

be the corresponding open subscheme and define

HilbY\Z :=∐P(t)

HilbY\ZP(t)

Then this is a scheme locally of finite type which depends only on W := Y\Z and not onthe embedding W ⊂ Y . In the case P(t) = n, a constant polynomial, we denote HilbW

P(t) byW [n]. If W is affine then

HilbW =∐n

W [n]

The proofs of these facts are left to the reader. See also [81].

4.4 Quot schemes

4.4.1 Existence

We will now introduce an important class of schemes, the so-called Quot schemes,which generalize the Hilbert schemes. As special cases we will obtain the relativeHilbert schemes.

Let p : X → S be a projective morphism of algebraic schemes, and let OX (1)be a line bundle on X very ample with respect to p. Fix a coherent sheaf H on X anda numerical polynomial P(t) ∈ Q[t]. We define a functor

Quot X/SH,P(t) : (schemes/S) → (sets)

called the Quot functor of X/S relative to H and P(t), in the following way:

Quot X/SH,P(t)(Z → S) =

coherent quotients HZ → F , flat over Z , withHilbert polyn. P(t) on the fibres of X Z → Z

where X Z = Z ×S X and HZ denotes the pullback of H on X Z , as usual. WhenS = Spec(k) we write Quot X

H,P(t) instead of Quot X/Spec(k)H,P(t) .

This definition generalizes the Hilbert functors which are obtained in the caseS = Spec(k) and H = OX .


Theorem 4.4.1. The functor Quot = Quot X/SH,P(t) is represented by a projective S-

schemeQuotX/S

H,P(t)→ S

Proof. We first consider the case S = Spec(k) and X = IPr . From Theorem 4.1.9it follows that there is an integer m such that for each scheme Z and for each(ϕ : HZ → F) ∈ Quot (Z), letting N = ker(ϕ), all the sheaves N (z),H(z) =H, F(z), z ∈ Z , are m-regular. Therefore, letting pZ : IPr × Z → Z be the projec-tion, we obtain an exact sequence of locally free sheaves on Z :

0→ pZ∗N (m)→ H0(IPr ,H(m))⊗k OZ → pZ∗F(m)→ 0

Moreover, for each m′ ≥ m there is an exact sequence

H0(IPr ,O(m′ −m))⊗k pZ∗N (m)→ H0(IPr ,H(m′))⊗k OZ → pZ∗F(m′)→ 0

where the first map is given by multiplication of sections. This shows that pZ∗F(m)uniquely determines the sheaf of graded OZ [X0, . . . , Xr ]-modules

⊕k≥m pZ∗F(k),

which in turn determines F . Therefore, letting

H0(IPr ,H(m)) = V

we have an injective morphism of functors:

Quot → GV ,P(m)

given by:Quot (Z) → GV ,P(m)(Z)

(HZ → F) → [V ⊗k OZ → pZ∗F(m)]On G = G P(m)(V ) consider the tautological exact sequence

0→ K → V ⊗k OG → Q → 0

Let, moreover, p2 : IPr × G → G and p1 : IPr × G → IPr be the projections.On G we have Γ∗(H)⊗k OG , which is a sheaf of graded OG [X0, . . . , Xr ]-modules,and determines p∗1(H). Consider the subsheaf KOG[X0, . . . , Xr ] and the sheaf Fon G× IPr corresponding to the quotient Γ∗(H)⊗k OG/KOG[X0, . . . , Xr ], and letG P ⊂ G be the stratum corresponding to P of the flattening stratification of F . Thenwe claim that a morphism of schemes f : Z → G defines an element of Quot (Z)if and only if f factors through G P , and therefore Quot is represented by G P . Theproof of this fact is similar to the one given for the proof of Theorem 4.3.4 and willbe left to the reader.

Since G P is quasi-projective, to prove that it is projective amounts to proving thatit is proper over k, and this can be done using the valuative criterion of properness.Let A be a discrete valuation k-algebra with quotient field L and residue field K , andlet

ϕ : Spec(L)→ G P


be any morphism. We must show that ϕ extends to a morphism

ϕ : Spec(A)→ G P

The datum of ϕ corresponds to an element (ϕL : HL → FL) of Quot (Spec(L)). Theexistence of ϕ will be proved if there is a quotient ϕA : HA → FA on IPr ×Spec(A)which is flat over Spec(A) and which restricts to FL over IPr × Spec(L). Let i :IPr×Spec(L)→ IPr×Spec(A) be the inclusion, and take FA = i∗(FL). Obviously,FA restricts to FL . Moreover, if KL = ker(ϕL), we have R1i∗(KL) = 0 and thereforea surjection HA = i∗(HL)→ FA. We need the following:

Lemma 4.4.2. Let X be a scheme, U an open subset of X and i : U → X theinclusion. Then for every coherent sheaf F on U we have

Ass(i∗(F)) = Ass(F)

Proof. Since i∗(F)|U = F we have Ass(i∗(F)) ∩ U = Ass(F). Therefore we onlyneed to prove that Ass(i∗(F)) ⊂ U .

We may assume that X = Spec(A) and U = Spec(B) are affine. The inclu-sion i corresponds to an injective homomorphism A → B and F = M∼ foran f.g. B-module M . Let x ∈ Ass(i∗(F)) and assume that x ∈ X\U . Then theideal px ⊂ A annihilates an element mx ∈ i∗(F)x which corresponds to a sectionm ∈ Γ (V, i∗(F)) for some open neighbourhood V of x . Up to shrinking X we mayassume V = X , so that m ∈ Γ (X, i∗(F)) = Γ (U, F) = M is annihilated by theideal px B. But px B = B because x /∈ U and therefore m = 0: this is a contradiction.The lemma is proved.

From the lemma it follows that Ass(FA) = Ass(FL): therefore, using the factthat FL is flat over Spec(L) and [84], Prop. III.9.7, we deduce that FA is flat overSpec(A). This concludes the proof of the theorem in the case S = Spec(k) andX = IPr .

Assume now that S and X are arbitrary. Consider the closed embeddingj : X → IPr × S determined by OX (1). Replacing H by j∗H we can assumethat X = IPr × S. Let h, h′ 0 be such that we have an exact sequence:

OIPr×S(−h′)M ′ → OIPr×S(−h)M → H→ 0

for some M, M ′. Then for each S-scheme Z → S and for each

(HZ → F) ∈ Quot IPr×S/SH,P(t) (Z → S)

we obtain that the composition

OIPr×Z (−h)M → HZ → F → 0

is a surjection, i.e. is an element of Quot IPr×S/SO(−h)M ,P(t)

(Z → S). This proves that the

functor Quot IPr×S/SH,P(t) is a subfunctor of the functor Quot IPr×S/S

O(−h)M ,P(t), and this functor

is evidently represented by QuotIPr

O(−h)M ,P(t)× S.


Conversely, a quotient

(OIPr×Z (−h)M → F) ∈ Quot IPr×S/SOX (−h)M ,P(t)

(Z → S)

is in Quot IPr×S/SH,P(t) (Z → S) if and only if the composition

OIPr×Z (−h′)M ′ → OIPr×Z (−h)M → F

is zero. This means that the condition for an S-morphism

Z → QuotIPr×S/SO(−h)M ,P(t)

= QuotIPr /kO(−h)M ,P(t)

× S

to define an element of Quot IPr×S/SH,P(t) (Z → S) is that it factors through the closed

subscheme defined by the entries of the matrix of the homomorphism:

OIPr×Quot(−h′)M ′ → OIPr×Quot(−h)M

and this is a closed condition. This proves that Quot IPr×S/SH,P(t) is represented by a

closed subscheme of QuotIPr /kO(−h)M ,P(t)

× S. From the fact that QuotX/S

H,P(t) represents the functor Quot X/SH,P(t) it follows that

there is a universal quotient

(HQuot → F) ∈ Quot X/SH,P(t)(QuotX/S

H,P(t))

corresponding to the identity morphism under the identification

Hom(Quot, Quot) = Quot (Quot)

In the case H = OX the scheme QuotX/SOX ,P(t) is denoted by HilbX/S

P(t) and called therelative Hilbert scheme of X/S with respect to the polynomial P(t).

It will be sometimes convenient to consider the functor

Quot X/SH : (schemes/S) → (sets)

defined as:

Quot X/SH (Z → S) =

∐P(t)

Quot X/SH,P(t)(Z → S)

This functor is represented by the disjoint union

QuotX/SH =

∐P(t)

QuotX/SH,P(t)


which is a scheme locally of finite type, called the Quot scheme of X over S relativeto H; it carries a universal quotient HQuot → F .

Similarly, we will consider the relative Hilbert scheme of X over S:

HilbX/S =∐P(t)

HilbX/SP(t)

The construction of the Quot scheme commutes with base change; this is a resultwhich follows quite directly from the definition, but it is worth pointing it out:

Proposition 4.4.3 (base change property). Given a projective morphism X → S, acoherent sheaf H on X, and a morphism T → S, there is a natural identification:

QuotXT /THT

= T ×S QuotX/SH

Proof. Consider the product diagram

T × QuotX/SH → QuotX/S

H↓ ↓T → S

The universal quotient HX/S → F on QuotX/SH pulls back to a quotient

HXT /T → FT on T × QuotX/SH . We have immediately that the T -scheme

T × QuotX/SH endowed with this quotient represents the functor Quot XT /T

HT.

4.4.2 Local properties

Proposition 4.4.4. Let X → S be a projective morphism of algebraic schemes, Ha coherent sheaf on X, flat over S, and π : Q = QuotX/S

H → S the associatedQuot scheme over S. Let s ∈ S be a k-rational point and q ∈ π−1(s) = Q(s)corresponding to a coherent quotient f : H→ F with kernel K. Let

fs : H(s)→ F(s)

be the restriction of f to the fibre X (s), whose kernel is K(s) = K ⊗OX (s) (by theflatness of F). Then there is an exact sequence

0→ Hom(K(s),F(s))→ tq Qdπq−→ ts S→ Ext1OX (s)

(K(s),F(s))

and an inclusion:ker[o(π

q)] ⊂ Ext1OX (s)(K(s),F(s))

where o(π q ) : o(OQ,q) → o(OS,s) is the obstruction map of the local homomor-

phism π q : OS,s → OQ,q . In particular, π is smooth at q if

Ext1OX (s)(K(s),F(s)) = 0.


Proof. A vector in ker(dπq) corresponds to a commutative diagram:

Spec(k[ε]) → Q↓ ↓ π

Spec(k) → S

such that the upper horizontal arrow has image q. The above diagram correspondsto an exact and commutative diagram of sheaves on X (s):

0 0↓ ↓

K(s)ε −→ K(s) → 0↓ ↓

0→ εH(s) → H(s)[ε] −→ H(s) → 0↓ ↓ ↓

0→ εF(s) → F(s)ε −→ F(s) → 0↓ ↓ ↓0 0 0

where the middle row is exact by the flatness of H. Replacing the middle row byits pushout under εH(s) → εF(s) we see that this diagram is equivalent to thefollowing one:

0 0↓ ↓

K(s) = K(s) → 0↓ ↓

0→ εF(s) → P −→ H(s) → 0‖ ↓ ↓

0→ εF(s) → F(s)ε −→ F(s) → 0↓ ↓ ↓0 0 0

and therefore we deduce that ker(dπq) = Hom(K(s),F(s)).Now consider A in A and a commutative diagram

Aϕ←− OQ,q

↑ η ↑ π q

B ←−ϕ

OS,s

(4.17)

where η is a small extension in A. This diagram corresponds to an exact diagram ofsheaves on X :


0↓KA

↓γ : 0→ H(s) → H⊗k B −→ H⊗k A → 0

↓FA

↓0

(4.18)

where the row is exact by the flatness of H over S. By pushing out by the quotientfs : H(s) → F(s) and then pulling back by α : KA → H ⊗k A we obtain anelement

[α∗ fs∗(γ )] ∈ Ext1OX⊗A(KA,F(s)) = Ext1OX (s)(K(s),F(s))

By construction this element vanishes if and only if the previous diagram can beembedded in a commutative diagram with exact rows and columns:

0 0 0↓ ↓ ↓

0→ K(s) → KB → KA → 0↓ ↓ ↓

0→ H(s) → H⊗k B −→ H⊗k A → 0↓ ↓ ↓

0→ F(s) → FB → FA

↓ ↓ ↓0 0 0

The middle column of this diagram is an element of Quot X/SH (Spec(B)), which cor-

responds to a homomorphism ϕ′ : OQ,q → B making the diagram

Aϕ←− OQ,q

↑ η ϕ′ ↑ π q

B ←−ϕ

OS,s

commutative. Therefore we have associated an element of Ext1OX (s)(K(s),F(s)) to

each diagram (4.18). It is straightforward to check that this correspondence is linear.Taking η : k[ε] → k we get an inclusion

coker(dπq) ⊂ Ext1OX (s)(K(s),F(s))

Taking any small extension η in A we can apply Proposition 2.1.7 to yield the con-clusion.


Corollary 4.4.5. Under the assumptions of 4.4.4, if

Ext1OX (s)(K(s),F(s)) = 0

then π : Q → S is smooth at q of relative dimension dim[Hom(K(s),F(s))

].

When S = Spec(k) we obtain the following “absolute” version of Proposition4.4.4.

Corollary 4.4.6. If X is a projective scheme, H a coherent sheaf on X andf : H → F a coherent quotient of H with ker( f ) = K then, letting Q = QuotX

H,we have:

T[ f ]Q = Hom(K,F)

and the obstruction space of OQ,[ f ] is a subspace of Ext1(K,F).In particular, if Ext1(K,F) = 0 then Q is nonsingular of dimension

dim(Hom(K,F)

)at [ f ].

A special case of Proposition 4.4.4 is the following:

Proposition 4.4.7. Let p : X → S be a projective flat morphism of algebraicschemes, and π : HilbX /S → S the relative Hilbert scheme. For a closed points ∈ S let X = X (s) be the fibre over s and let Z ⊂ X be a closed subscheme withideal sheaf I ⊂ OX . Then there is an exact sequence:

0→ H0(Z , NZ/X )→ T[Z ]HilbX /S dπ[Z ]−→ Ts S→ Ext1OX(I,OZ )

If, moreover, Z ⊂ X is a regular embedding then the above exact sequence becomes:

0→ H0(Z , NZ/X )→ T[Z ]HilbX /S dπ[Z ]−→ Ts S→ H1(Z , NZ/X ) (4.19)

If Ext1OX(I,OZ ) = (0) (resp. H1(Z , NZ/X ) = (0) in the case where Z ⊂ X is a

regular embedding) then π is smooth at [Z ] of relative dimension h0(Z , NZ/X ).

NOTES

1. One should compare the statement of 4.4.7 with 3.2.12, since the local relative Hilbertfunctor of Z in X relative to X → S is prorepresented by the local ring O

HilbX/S[Z ]

.

2. Our presentation of the Quot schemes is an adaptation of the one given in [90]. For adescription of the sheaf of differentials of the Quot schemes see [117].

3. When p : X → S is a projective flat morphism of integral schemes the relative Hilbertscheme HilbX /S has a component isomorphic to S, parametrizing the fibres of p, and a com-ponent isomorphic to X , identified with X [1].

4.5 Flag Hilbert schemes 227

4.5 Flag Hilbert schemes

4.5.1 Existence

Fix an integer r ≥ 1 and an m-tuple of numerical polynomials

P(t) = (P1(t), . . . , Pm(t)), m ≥ 1

For every scheme S we let:

F HrP(t)(S) =

X1 ⊂ · · · ⊂ Xm ⊂ IPr × S(X1, . . . ,Xm) : S-flat closed subschemes

with Hilbert polynomials P(t)

This clearly defines a contravariant functor:

F HrP(t) : (schemes) → (sets)

called the flag Hilbert functor of IPr relative to P(t). When m = 1 the flag Hilbertfunctors are just ordinary Hilbert functors.

Theorem 4.5.1. For every r ≥ 1 and P(t) as above, the flag Hilbert functor F HrP(t)

is represented by a projective scheme FHrP(t), called the flag Hilbert scheme of IPr

relative to P(t), and by a universal family:

W1 ⊂ · · · ⊂Wm ⊂ IPr × FHrP(t)

↓FHr

P(t)

Proof. We will only prove the theorem in the case m = 2, leaving to the reader thetask of extending the proof to the general case.

By applying Corollary 4.1.10 twice we can find an integer µ such that simulta-neously for i = 1, 2 we have that for every closed subscheme Xi ⊂ IPr with Hilbertpolynomial Pi (t) the sheaf of ideals IXi is µ-regular. For every such X1 and X2 wethus have:

h0(IPr , IXi (µ)) =(µ+ r

r

)− Pi (t) =: Ni i = 1, 2

Let V = H0(IPr ,OIPr (µ)). Consider the Hilbert scheme Hi = HilbrP1(t)

with uni-versal family Vi ⊂ IPr ×Hi , i = 1, 2. On the product H1×H2 consider the pullbackof the two universal families with respect to the projections:

Vi ×Hi (H1 × H2) ⊂ IPr × H1 × H2, i = 1, 2

and denote by q : IPr × H1 × H2 → H1 × H2 the projection. Because of thechoice of µ and by Theorem 4.2.5 we have that q∗Ii (µ) is a locally free subsheaf


of V ⊗k OH1×H2 of rank Ni , with locally free cokernel, i = 1, 2. Consider thecomposition

ϕ : q∗I2(µ) ⊂ V ⊗k OH1×H2 → V ⊗k OH1×H2/q∗I1(µ)

and let F ⊂ H1 × H2 be the vanishing scheme of ϕ (Example 4.2.8). Note that wehave

q∗I2(µ)⊗OF ⊂ q∗I1(µ)⊗OF ⊂ V ⊗OF (4.20)

We now pull back to F the two universal families, i = 1, 2:

IPr × F IPr × Hi

∪ ∪Wi := Vi ×Hi F→ Vi

↓ k ↓

F → Hi

Claim: q∗Ii (µ)⊗OF = k∗IWi (µ), i = 1, 2.

We have natural homomorphisms:

βi : q∗Ii (µ)⊗OF → k∗IWi (µ)

Because of the µ-regularity and of Theorem 4.2.5 for every x ∈ F we have isomor-phisms

q∗Ii (µ)x ⊗ k(x) ∼= H0(IPr , IWi (x)(µ)) ∼= k∗IWi (µ)x ⊗ k(x)

Since the sheaf k∗IWi (µ) is locally free, from Nakayama’s lemma it follows that β1and β2 are isomorphisms, and this proves the claim.

From the claim and from (4.20) we deduce that

k∗IW2(µ) ⊂ k∗IW1(µ) (4.21)

The µ-regularity and Proposition 4.1.1(iii) imply that the natural homomorphisms

k∗k∗IWi (µ)→ IWi (µ), i = 1, 2

are surjective; because of (4.21) we deduce that IW2 ⊂ IW1 , hence W1 ⊂ W2.Therefore (W1,W2) ∈ F Hr

P(t)(F).

Claim: The pair (F, (W1,W2)) represents the functor F HrP(t).

Let S be a scheme and let (X1,X2) ∈ F HrP(t)(S). By definition X1 ⊂ X2 ⊂

IPr × S are flat over S with Hilbert polynomials P1(t) and P2(t) respectively. Letf : IPr × S→ S be the projection. We have induced classifying morphisms

g1 : S→ H1, g2 : S→ H2


which together define a morphism

g : S→ H1 × H2

Arguing as before, we see that

g∗q∗Ii (µ) ∼= f∗IXi (µ), i = 1, 2 (4.22)

The fact that X1 ⊂ X2 implies that IX2(µ) ⊂ IX1(µ), hence that

f∗IX2(µ) ⊂ f∗IX1(µ)

This, together with (4.22), in turn implies that g factors through F. Since clearly wehave

X1 = S ×F W1, X2 = S ×F W2

the claim follows, hence F = FHrP(t). Moreover, FHr

P(t) is projective because it is aclosed subscheme of H1 × H2.

From the definition it follows that the closed points of FHrP(t) are in 1–1 corre-

spondence with the m-tuples (X1, . . . , Xm) of closed subschemes of IPr such thatXi has Hilbert polynomial Pi (t) and

X1 ⊂ X2 ⊂ · · · ⊂ Xm

Such an m-tuple is called a flag of closed subschemes of IPr . We will denote by[X1, . . . , Xm] the point of FHr

P(t) parametrizing such a flag. From the proof of The-orem 4.5.1 it follows that FHr

P(t) is a closed subscheme of

m∏i=1

HilbrPi (t)

We will denote the projections by

pri : FHrP(t)→ Hilbr

Pi (t), i = 1, . . . , m

For every subset I ⊂ 1, . . . , m with cardinality µ we can consider the µ-tuple ofpolynomials PI(t) = (Pi1 , . . . , Piµ) and the flag Hilbert scheme FHr

PI(t). We have

natural projection morphisms

prI : FHrP(t)→ FHr

PI(t)

of which the pri ’s are special cases. The flag Hilbert schemes are generalizations ofthe flag varieties (fibres en drapeaux in [77], ch. 1, § 9.9), which parametrize flagsof linear subspaces of IPr .

If Z ⊂ IPr is a closed subscheme having Hilbert polynomial Q(t), and P(t) is anm-tuple of numerical polynomials as above, one can define the flag Hilbert schemeof Z relative to P(t)

F H ZP(t) : (schemes) → (sets)


by an obvious modification of the definition of F HrP(t). It is straightforward to prove

that F H ZP(t) is represented by a closed subscheme FHZ

P(t) of the scheme FHrP(t)

, where

P(t) = (P1(t), . . . , Pm(t), Q(t))

Precisely, lettingprQ : F Hr

P(t)→ Hilbr

Q(t)

be the projection, one has an identification of FHZP(t) with the scheme-theoretic fibre

pr−1Q ([Z ]) of prQ over the point [Z ] ∈ Hilbr

Q(t). It is convenient to consider thedisjoint union

FHZ =∐P(t)

FHZP(t)

which is a scheme locally of finite type, and call it the flag Hilbert scheme of Z .Another variation on the same theme is the following. Given closed subschemes

X ⊂ Z ⊂ IPr , and an m-tuple of numerical polynomials P(t), one can consider flagsof closed subschemes of Z containing X , namely m-tuples (Y1, . . . , Ym) of closedsubschemes of Z having Hilbert polynomials P(t) and such that

X ⊂ Y1 ⊂ Y2 · · · ⊂ Ym ⊂ Z

Again there is an obvious generalization of the definition of the corresponding flagHilbert functor and of the proof of its representability by a projective scheme. Thesegeneralized flag Hilbert schemes can be fruitfully used in concrete geometrical sit-uations, like, for example, in the study of families of pointed closed subschemes, orof reducible closed subschemes, of a given projective scheme.

4.5.2 Local properties

Consider a projective scheme Z and let (X1, . . . , Xm) be a flag of closed subschemesof Z ; let Ii ⊂ OZ be the ideal sheaf of Xi . We have inclusions

Im ⊂ Im−1 ⊂ · · · ⊂ I1

and surjectionsOXm → OXm−1 → · · · → OX1 → 0

Definition 4.5.2. The normal sheaf of (X1, . . . , Xm) in Z is the sheaf of germs ofcommutative diagrams of homomorphisms of OZ -modules of the following form:

Im ⊂ Im−1 ⊂ · · · ⊂ I1↓ σm ↓ σm−1 ↓ σ1

OXm → OXm−1 → · · · → OX1

It is denoted by N(X1,...,Xm )/Z .


Note that we have an obvious homomorphism of “projection”:

N(X1,...,Xm )/Z → N(Xi1 ,...,Xiµ)/Z

for each choice of a subset I = i1, . . . , iµ ⊂ 1, . . . , m. In particular, we havehomomorphisms:

N(X1,...,Xm )/Z → NXi /Z , i = 1, . . . , m

Proposition 4.5.3. Let Z be a projective scheme and let (X1, . . . , Xm) be a flag ofclosed subschemes of Z. Then:

(i) There is a natural identification:

T[X1,...,Xm ]FHZ = H0(Z , N(X1,...,Xm )/Z ) (4.23)

(ii) If Xi ⊂ Xi+1 is a regular embedding for all i = 1, . . . , m−1 and Xm is regularlyembedded in Z then the obstruction space of the local ring OFHZ ,[X1,...,Xm ] is

contained in H1(Z , N(X1,...,Xm )/Z ).

Proof. For simplicity we give the proof in the case m = 2, i.e. in the case of a flag(X, Y ) of closed subschemes of Z . The general case can be treated similarly.

(i) Let IX ⊂ OZ and IY ⊂ OZ be the ideal sheaves of X and of Y . We canrepresent a first-order deformation Y ⊂ Z × Spec(k[ε]) of Y in Z by an ideal sheafIY ⊂ OZ [ε] fitting into the commutative and exact diagram:

0 0↓ ↓IY → IY → 0↓ ↓

0→ εOZ → OZ [ε] → OZ → 0↓ ↓ ↓

0→ εOY → OY → OY → 0

The flatness of Y over Spec(k[ε]) follows from the exactness of the last row (LemmaA.9). This diagram is equivalent to the following one, deduced after pushing out thesecond row by the homomorphism εOZ → εOY :

0 0↓ ↓IY = IY → 0↓ ↓

0→ εOY → εOY ⊕OZ → OZ → 0‖ ↓ ↓

0→ εOY → OY → OY → 0


Therefore we have a convenient representation of the given first-order deformationas the middle vertical exact sequence of the previous diagram, which we rewrite:

0→ IY → εOY ⊕OZ → OY → 0

Composing the first inclusion with the first projection εOY ⊕ OZ → εOY we ob-tain the section σY ∈ H0(Y, NY/Z ) = HomOZ (IY ,OY ) corresponding to Y . Let’sassume now that we have an element (X ,Y) ∈ FHZ (k[ε]). By putting together theabove constructions for X and for Y we obtain the following commutative diagram:

0→ IY → εOY ⊕OZ → OY → 0∩ ↓ ↓

0→ IX → εOX ⊕OZ → OX → 0

where the condition X ⊂ Y corresponds to the condition OY → OX , and this isin turn equivalent to the condition that composing the inclusions on the left with thefirst projections in the middle terms we obtain a commutative diagram:

IY ⊂ IX

↓ σY ↓ σXεOY → εOX

This is the global section of N(X,Y )/Z corresponding to (X ,Y) in (4.23). Conversely,given a global section

IY ⊂ IX

↓ σ ↓ τεOY → εOX

∈ H0(Z , N(X,Y )/Z )

one finds a first-order deformation of (X, Y ) by repeating backwards the above con-struction.

(ii) Let A be in ob(A) and let

ξ : X ⊂ Y ⊂ Z × Spec(A)

be an infinitesimal deformation of the flag (X, Y ). Let

η : 0→ εk→ A→ A→ 0

define an element [e] ∈ Exk(A, k), where m Aε = 0. Since the embeddings X ⊂ Yand Y ⊂ Z are regular we can find an affine open cover U = Ui = Spec(Pi )i∈I

of Z such that Yi := Y ∩ Ui is a complete intersection in Ui and Xi := X ∩ Ui is acomplete intersection in Yi . Therefore, letting Xi = X ∩Ui , Yi = Y ∩Ui , there areliftings

Yi ⊂ Ui × Spec( A)∪ ∪Yi ⊂ Ui × Spec(A)


andXi ⊂ Yi

∪ ∪Xi ⊂ Yi

which together give local liftings of the flags (Xi ,Yi ):

Xi ⊂ Yi ⊂ Ui × Spec( A)∪ ∪ ∪Xi ⊂ Yi ⊂ Ui × Spec(A)

In order to find a lifting (X , Y) of (X ,Y) we must be able to choose the local liftings(Xi , Yi ) so that on every Ui j := Ui ∩U j we have

(Xi |Ui j , Yi |Ui j ) = (X j |Ui j , Y j |Ui j )

At the level of ideals we have:

IYi ⊂ IXi ⊂ Pi

IYi ⊂ IXi ⊂ Pi A := Pi ⊗k AIYi⊂ IXi

⊂ Pi A := Pi ⊗k A

We have a commutative and exact diagram:

IXi→ IXi → 0

↓ ↓0→ εPi → Pi A → Pi A → 0

↓ ↓ ↓0→ εPi/IXi → Pi A/IXi

→ Pi A/IXi → 0

which, after pushing out the middle row by εPi → εPi/IXi , gives the following one:

IXi = IXi↓ ↓0→ εPi/IXi → Qi A → Pi A → 0

‖ ↓ ↓0→ εPi/IXi → Pi A/IXi

→ Pi A/IXi → 0

where

Qi A = (εPi/IXi )∐εPi

Pi A

therefore the datum of Xi corresponds to the middle vertical sequence of this dia-gram:

0→ IXi → Qi A → Pi A/IXi→ 0 (4.24)


Repeating the analogous construction for Yi we obtain that Yi is determined by anexact sequence:

0→ IYi → Qi A → Pi A/IYi→ 0 (4.25)

Since Xi ⊂ Yi the exact sequences (4.24) and (4.25) fit together:

0→ IYi → Qi A → Pi A/IYi→ 0

∩ ‖ ↓0→ IXi → Qi A → Pi A/IXi

→ 0

A different choice of the lifting Xi corresponds to a different homomorphismIXi → Qi A, and they differ by an element

σXi∈ HomPi A

(IXi , εPi/IXi ) = HomPi (IXi , Pi/IXi )

Similarly, a different choice of the lifting Yi corresponds to an element

σYi∈ HomPi (IYi , Pi/IYi )

The condition that the pair (σXi, σYi

) defines another lifting of the flag (Xi ,Yi ) isthat the following diagram

IYi ⊂ IXi

↓ σYi↓ σXi

Pi/IYi → Pi/IXi

commutes. This is precisely the condition that (σXi, σYi

) ∈ Γ (Ui , N(X,Y )/Z ). There-

fore the set of liftings of (Xi ,Yi ) over Spec( A) is in 1–1 correspondence withΓ (Ui , N(X,Y )/Z ). It follows from this analysis that for all i, j ∈ I we have asection (σi j , τi j ) ∈ Γ (Ui j , N(X,Y )/Z ) such that (X j |Ui j , Y j |Ui j ) is obtained from

(Xi |Ui j , Yi |Ui j ) after modifying it by the section (σi j , τi j ). The collection of thesesections is a 1-cocycle

(σi j , τi j ) ∈ Z1(U , N(X,Y )/Z )

which defines a cohomology class o(X ,Y)([e]). It is straightforward to verify that thisclass is independent of the choices made and that o(X ,Y)([e]) = 0 if and only if alifting (X , Y) exists.

If a flag (X1, . . . , Xm) of closed subschemes of a projective scheme Z satisfiesthe conditions of Proposition 4.5.3(ii), i.e; if Xi ⊂ Xi+1 is a regular embeddingfor all i = 1, . . . , m − 1 and Xm is regularly embedded in Z , we say that the flag(X1, . . . , Xm) is regularly embedded in Z .

Remarks 4.5.4. (i) One can adapt the above proof to the case m = 1 to obtain an-other, more intrinsic, proof of Propositions 3.2.1(ii) and 3.2.6.

(ii) In the case m = 2 considered in the proof of Proposition 4.5.3, to the flagX ⊂ Y of closed subschemes of Z there is associated a diagram of normal sheaves:

4.6 Examples and applications 235

NY/Z

↓0→ NX/Y → NX/Z → NY/Z ⊗OZ OX

and we can immediately check that there is a natural identification

H0(Z , N(X,Y )/Z ) = H0(X, NX/Z )×H0(X,NY/Z⊗OZ OX ) H0(Y, NY/Z )

(iii) It is an immediate consequence of the definition that, given a flag (X1, . . . , Xm)of closed subschemes of a projective scheme Z , the normal sheaf N(X1,...,Xm )/Z is anOXm -module. This implies that in the case when we have a regularly embedded flagconsisting of 0-dimensional subschemes of Z , we have

H1(Z , N(X1,...,Xm )/Z ) = H1(Xm, N(X1,...,Xm )/Z ) = 0

and from Proposition 4.5.3(ii) we deduce that FHZ is nonsingular at [X1, . . . , Xm].For example, if we consider a projective nonsingular curve Z then it follows thatFHZ

(n1,...,nm ), which parametrizes flags of effective divisors (D1, . . . , Dm) of degreesn1 < n2 < · · · < nm , is nonsingular.

(iv) It is possibile to give a notion of flag Quot scheme generalizing the flagHilbert schemes. This seems not to have been considered in the literature yet.

NOTES

1. The flag Hilbert schemes have been considered in [102]. The proof of Theorem 4.5.1given here has appeared in [163]. More recent references about flag Hilbert schemes are [30]and [62] (where they are called “nested Hilbert schemes”) and [151].

4.6 Examples and applications

4.6.1 Complete intersections

We have already discussed some properties of the local Hilbert functor of a completeintersection X ⊂ IPr , which of course correspond to local properties of the Hilbertscheme Hilbr at [X ]. It is easy to check that, despite the fact that H1(X, NX ) = (0)in general, every complete intersection X is unobstructed in IPr .

We may assume dim(X) > 0. Let’s suppose that X ⊂ IPr , r ≥ 2, is the completeintersection of r − n hypersurfaces f1, . . . , fr−n of degrees d1 ≤ d2 ≤ . . . ≤ dr−n

respectively, n < r .Consider a basis Φ(1), . . . , Φ(m) of ⊕ j H0(IPr ,O(d j )) where

Φ(h) = (φ(h)1 , . . . , φ

(h)r−n)


h = 1, . . . , m, and the φ(h)j ∈ k[X0, . . . , Xr ]. Consider indeterminates u1, . . . , um

and the (r − n)-tuple

f+m∑

h=1

uhΦ(h) = ( f1+u1φ

(1)1 +· · ·+umφ

(m)1 , . . . , fr−n+u1φ

(1)r−n+· · ·+umφ

(m)r−n)

(4.26)of elements of the polynomial ring k[u, x] = k[u1, . . . , um, X0, . . . , Xr ].

Let K•(f+∑h uhΦ(h)) be the Koszul complex relative to (4.26) and

∆ := Supp[H1(K•(f+∑

h

uhΦ(h)))] ⊂ Am+r+1 = Spec(k[u, x])

Denoting by p : Am+r+1 → Am the projection, U := Am\p(∆) is the set of pointsu ∈ Am such that K•(f+∑h thΦ(h)) is exact; U is an open set containing the origin.

In IPr × Am consider the closed subscheme

X = Proj(

k[u, x]/( f1+u1φ(1)1 +· · ·+umφ

(m)1 , . . . , fr−n+u1φ

(1)r−n+· · ·+umφ

(m)r−n)

)the projection π : X → Am and its restriction πU : XU → U , whereXU := π−1(U ). All the fibres of πU are complete intersections of multidegree(d1, . . . , dr−n) and X (0) = X . The Hilbert polynomial of a complete intersectiondepends only on its multidegree because it can be computed using the Koszul com-plex: it follows that all the fibres of πU have the same Hilbert polynomial P(t) andtherefore πU is a flat family of deformations of X in IPr . In an obvious way the tan-gent space of U at 0 can be identified with⊕ j H0(IPr ,O(d j )), and the characteristicmap with the restriction

ϕ : ⊕ j H0(IPr ,O(d j ))→⊕ j H0(X,OX (d j ))

Since dim(X) > 0 the map ϕ is surjective, as one easily verifies using the Koszulcomplex; since, moreover, U is nonsingular at 0 from Proposition 4.3.6 it followsthat Hilbr

P(t) is smooth at [X ] and the classifying map is smooth. From this it alsofollows that complete intersections are parametrized by an open subset of Hilbr

P(t).It is interesting to observe that the closure of this open set may contain points

parametrizing nonsingular subschemes of IPr which are not complete intersections.An example of such a subscheme is given by a trigonal canonical curve C ⊂ IP4: thequadrics containing C intersect in a rational cubic surface S, so it is not a completeintersection since it has degree 8; but [C] is in the closure of the family of completeintersections of three quadrics. It is apparently unknown whether a similar phenom-enon may occur in IP3, namely whether there are nonsingular curves in IP3 whichare flat limits of complete intersections without being complete intersections. See[51] for more about this.

The Kodaira–Spencer map of the families πU has been studied in [161] in thecase of complete intersections of dimension ≥ 2: πU has general moduli exceptfor surfaces of multidegrees (4), (2, 3), (2, 2, 2) (respectively in IP3, in IP4 and inIP5), i.e. for complete intersection K3-surfaces. The special case of hypersurfaceshad already been considered in [107] (see Example 3.2.11(i), page 134).


4.6.2 An obstructed nonsingular curve in IP3

We will prove that the Hilbert scheme HilbIP3has an everywhere nonreduced com-

ponent whose general point parametrizes a nonsingular curve of degree 14 andgenus 24. It will follow that every curve parametrized by a general point of isobstructed in IP3. This example is due to Mumford ([129]).

A general element of is constructed as follows. Let F ⊂ IP3 be a nonsingularcubic surface, E, H ⊂ F respectively a line and a plane section in F . Let C ⊂ F bea general member of the linear system |4H+2E |. Using Bertini’s theorem one easilychecks that C is irreducible and nonsingular; its degree and genus are (C · H) = 14and 1

2 (C − H · C)+ 1 = 24. From the exact sequence:

0→ KC (H)→ NC → OC (3H)→ 0

we see that

h1(C, NC ) = h1(C,OC (3H)) = h0(C, KC (−3H)) = h0(C,OC (2E)) = 1

where the last equality follows easily from the exact sequence

0→ OF (−4H)→ OF (2E)→ OC (2E)→ 0

and from h0(OF (−4H)) = 0 = h1(OF (−4H)) and h0(OF (2E)) = h0(OF (E)) =1. Moreover, the linear system |C | = |4H + 2E | has dimension

dim(|C |) = 1+ dim(|C |C ) = h0(C, KC (H)) = 37

and therefore, since every curve C is contained in a unique cubic surface (because9 < 14), the dimension of the family W of all curves C we are considering is19 + 37 = 56 = 4 · 14 but they satisfy h0(C, NC ) = 56 + h1(C, NC ) = 57.We will prove that W is an open set of a component of HilbIP3

and this will implythat is everywhere nonreduced. Our assertion will be proved if we show that ourcurves C are not contained in a family whose general member is a curve D not con-tained in a cubic surface. But on every such D the line bundle OD(4) is nonspecialand therefore, by Riemann-Roch, h0(D,OD(4)) = 33, hence D is contained in apencil of quartic surfaces. Let G1, G2 be two linearly independent quartics contain-ing D: they are both irreducible because otherwise D would be either contained in aplane or in a quadric, which is not the case because there are no nonsingular curvesof degree 14 and genus 24 on such surfaces. We have G1 ∩ G2 = D ∪ q where q isa conic; since q has at most double points D has at most triple points and thereforeG1 and G2 cannot be simultaneously singular at any point of D, thus the generalquartic surface G containing D is nonsingular along D. By applying Riemann-Rochon G we obtain dim(|D|G) = 24. Therefore, since G is not a general quartic surface(because D is not a complete intersection), we see that the family of pairs (D, G) hasdimension ≤ 33 + 24 = 57 so that the family Z of curves D has dimension ≤ 56.This shows that the family W , which has dimension 56, cannot be in the closure ofZ and this proves the assertion.


It is instructive to observe that we can write the linear system |C | on a nonsingularcubic surface F as |4H + 2E | = |6H − 2(H − E)| and this means that we can finda sextic surface F6 such that F ∩ F6 = C ∪ q1 ∪ q2 where q1 and q2 are disjointconics; if [C] ∈ is general then one can show that q1, q2 and F6 can be chosen tobe nonsingular.

There is another component R of HilbIP3whose general point parametrizes a

nonsingular curve C ′ of degree 14 and genus 24 such that

C ′ ∪ E ∪ Γ = F3 ∩ F6

where E is a line and Γ is a rational normal cubic which are disjoint. We have in thiscase |C ′| = |6H − E − Γ | and

h1(C ′, NC ′) = h1(C ′,OC ′(3H)) = h0(C ′, KC ′(−3H))= h0(C ′,OC ′(2H − E − Γ )) = h0(F3,OF3(2H − E − Γ )) = 0

Thus C ′ is unobstructed.We refer the reader to [41] for another point of view about this example. This

is the first published example of an obstructed space curve. Many others have sinceappeared in the literature (see [78], [162], [79], [50], [103], [187], [22], [125], [80]).A final word on the search for pathologies of this kind is contained in [183], whereit is shown that virtually every singularity can appear as a point of HilbIP3

parame-trizing a nonsingular curve. For examples of obstructed nonsingular curves in higherdimensional projective spaces see [35], [55].

4.6.3 An obstructed (nonreduced) scheme

In IP3 consider the scheme

X = Proj(

k[X0, . . . , X3]/J)

whereJ = (X1 X2, X1 X3, X2 X3, X2

3)

X is supported on the reducible conic defined by the equations

X1 X2 = 0, X3 = 0

has an embedded point at (1, 0, 0, 0) and has Hilbert polynomial 2(t + 1) (see Ex-ample 4.2.3(ii)). As in 4.2.3(ii) we consider the flat family parametrized by A1:

X = Proj(

k[u, X ]/(X1 X2, X1 X3, X2(X3 − u X0), X3(X3 − u X0)))⊂ IP3 × A1

where k[u, X ] = k[u, X0, . . . , X3]. We have X = X (0). If u = 0 then X (u) is apair of disjoint lines. Let

g : A1 → Hilb32(t+1)


be the classifying map. If u = 0 we have

h1(X (u), NX (u)) = 0; h0(X (u), NX (u)) = 8

Therefore g(u) is a smooth point and the tangent space has dimension 8.In order to show that X is obstructed it suffices to show that

h0(X, NX ) > 8 (4.27)

because g(0) and g(u) belong to the same irreducible component of Hilb32(t+1).

Consider the surjection

f : OIP3(−2)⊕4 → IX → 0

determined by the four equations of degree 2 which define X . Elementary compu-tations, based on the fact that the generators of the ideal are monomials, lead to thefollowing resolution of OX which extends f:

0→ OIP3(−4)B−→ OIP3(−3)⊕4 A−→ OIP3(−2)⊕4 f−→ OIP3 → OX → 0 (4.28)

A and B being given by the following matrices:

B =⎛⎜⎝

X3−X3X2−X1

⎞⎟⎠ A =⎛⎜⎝

X3 X3 0 0−X2 0 X3 0

0 −X1 0 X30 0 −X1 −X2

⎞⎟⎠By taking Hom(−,OX ) we obtain the following exact sequence:

0→ NX → OX (2)⊕4t A−→ OX (3)⊕4

from which we deduce that

H0(X, NX ) = ker[H0(OX (2))⊕4t A−→ H0(X,OX (3))⊕4] (4.29)

Using resolution (4.28) it is easy to show that the restriction maps

ϕn : H0(IP3,O(n))→ H0(X,OX (n))

are surjective if n ≥ 2. This allows us to identify H0(X,OX (2)) and H0(X,OX (3))with the homogeneous parts of degree 2 and 3 respectively of k[X0, . . . , X3]/J .Hence using (4.29) we can represent H0(X, NX ) by 4-tuples of polynomials. Pre-cisely, H0(X, NX ) is, modulo J , the vector space of 4-tuples

q = (q1, q2, q3, q4)


of homogeneous polynomials of degree 2 such that A t q ∈ (J3)4. It is easy to find

all of them because J is generated by monomials. Computing, one finds that a basisof H0(X, NX ) is defined by the following column vectors:

X21 X1 X0 X2

2 X2 X0 X3 X0 0 0 0 0 0 0 0

0 0 0 0 0 X21 X1 X0 X3 X0 0 0 0 0

0 0 0 0 0 0 0 0 X22 X2 X0 X3 X0 0

0 0 0 0 0 0 0 0 0 0 0 X3 X0

In particular, we see that h0(X, NX ) = 12, and this proves (4.27).A little extra work shows that [X ] = g(0) belongs to two irreducible components

of Hilb32(t+1). We already know one of them of dimension 8: it contains g(u), u = 0,

and a general point of it parametrizes a pair of disjoint lines.The other component has dimension 11 and a general point of it parametrizes the

disjoint union Y = Q ∪ p of a conic Q and a point p. Note that

h0(Y, NY ) = h0(Q, NQ)+ h0(p, Np) = 8+ 3 = 11

and h1(Y, NY ) = 0. Hence Y is a smooth point of a component of dimension 11of Hilb3

2(t+1). Therefore it suffices to produce a flat family parametrized by an irre-

ducible curve, e.g. A1,Y ⊂ IP3 × A1

such that Y(0) = X , Y(1) = Y . Here it is:

Y = Proj(

k[v, X0, X1, X2, X3]/I)

whereI = (X1 X2, X1 X3 + vX1 X0, X2 X3 + vX2 X0, X2

3 − v2 X20)

Clearly Y(0) = X ; since

I = (X1, X2, X3 − vX0) ∩ (X3 + vX0, X1 X2)

it follows that for all v = 0, Y(v) is the disjoint union of a conic and a point. Theflatness of Y follows from [84], Prop. III.9.8.

This example shows that in general the Hilbert schemes are reducible and notequidimensional. For a description of Hilb3

3t+1, which presents several analogieswith the one given here of Hilb3

2(t+1), we refer to [142].

4.6.4 Relative grassmannians and projective bundles

Consider a coherent sheaf E on an algebraic scheme S, and let P(t) = n, where n isa positive integer, be a constant polynomial. Then QuotS/S

E,n is a projective S-schemewhich will be denoted by Quotn(E) in what follows. We will denote by

ρ : Quotn(E)→ S


the structural projective morphism and by

ρ∗E → Q

the universal quotient bundle; Q is locally free of rank n. If n = 1 then Q is aninvertible sheaf: it will be denoted by OQuot1(E)(1), or simply by O(1) if no confusionarises. The pair (Quotn(E)/S,Q) represents the functor

Quotn(E) : (schemes/S) → (sets)

defined by:

Quotn(E)( f : T → S) = locally free rk n quotients f ∗E → F on T On Quotn(E) we have a tautological exact sequence

0→ K→ ρ∗(E)→ Q→ 0

If E is locally free then K is locally free as well and it is called the universal subbun-dle.

If E is locally free we define

Gn(E) := Quotn(E∨)

and call it the grassmannian bundle of subbundles of rank n of E ; for n = 1 we have

G1(E) = IP(E)

the projective bundle associated to E , according to our definition (which differs fromthe one adopted in [84], p. 162). The tautological exact sequence on IP(E) is:

0→ K→ ρ∗(E∨)→ OIP(E)(1)→ 0

In particular, for a finite-dimensional k-vector space V we have

IP(V ⊗k OS) = IP(V )× S

and more generally,Gn(V ⊗k OS) = Gn(V )× S

Therefore, if E is locally free on S then Quotn(E) is locally the product of S by agrassmannian; in particular, the projection ρ : Quotn(E)→ S is a smooth morphism.

Proposition 4.6.1. Let E be a locally free sheaf on the algebraic scheme S, and let

0→ K→ ρ∗(E)→ Q→ 0 (4.30)

be the tautological exact sequence on Quotn(E) for some 1 ≤ n ≤ rk(E). Then thereis a natural isomorphism

Ω1Quotn(E)/S

∼= Hom(Q,K)

and thereforeTQuotn(E)/S

∼= Hom(K,Q)


Proof. Letting B = Quotn(E) consider the product B ×S B with projectionspri : B ×S B → B, i = 1, 2, and let EB×S B be the pullback of E on B ×S B.Denote by I∆ ⊂ OB×S B the ideal sheaf of the diagonal ∆ ⊂ B ×S B. The tautolog-ical exact sequence (4.30) pulls back to two exact sequences:

0→ pr∗i K→ E∨B×S B → pr∗i Q→ 0

on B ×S B whose restrictions to ∆ coincide, and ∆ is characterized by this prop-erty. This can be also expressed by saying that ∆ is the vanishing scheme of thecomposition

pr∗1K→ E∨B×S B → pr∗2QTherefore we have a surjective homomorphism:

Hom(pr∗2Q, pr∗1K)→ I∆

(see 4.2.8) which, restricted to ∆, gives a surjective homomorphism:

Hom(Q,K)→ I∆/I2∆ = Ω1

B/S

which has to be an isomorphism since both sheaves are locally free and have thesame rank. Proposition 4.6.2. Let

0→ E α−→ F β−→ G → 0

be an exact sequence of locally free sheaves on the algebraic scheme S, and n ≥ 1an integer. Then there is a closed regular embedding

Quotn(G) ⊂ Quotn(F)

and a natural identification:

NQuotn(G)/Quotn(F) = ρ∗E∨ ⊗Q⊗OQuotn(G)

where ρ : Quotn(F)→ S is the structure morphism and ρ∗F → Q is the universalquotient.

In particular, if n = 1 we have

NQuot1(G)/Quot1(F) = ρ∗E∨ ⊗O(1)⊗OQuot1(G)

Proof. Let f : T → S be a morphism. For every locally free rank n quotient

( f ∗G → H) ∈ Quotn(G)(T )

there is associated, by composition with the surjective homomorphismf ∗(β) : f ∗F → f ∗G, an element

( f ∗F → H) ∈ Quotn(F)(T )


Therefore Quotn(G) is a subfunctor of Quotn(F). Consider the diagram of homo-morphisms on Quotn(F):

ρ∗(E)

↓ ρ∗(α)

ρ∗(F)γ−→ Q → 0

↓ ρ∗(β)

ρ∗(G)↓0

Given a morphism f : T → S, an element of

Quotn(F)(T ) = HomS(T, Quotn(F))

belongs to Quotn(G)(T ) if and only if it factors through the closed subschemeD0(γρ

∗(α)) of Quotn(F). This proves that Quotn(G) is a closed subfunctor ofQuotn(F), and therefore the embedding Quotn(G) ⊂ Quotn(F) is closed. Moreprecisely, this analysis shows that Quotn(G) = D0(γρ

∗(α)). Since Quotn(G) hascodimension rk(E)n in Quotn(F) it follows that it is regularly embedded. Accordingto Example 4.2.8 we have a surjective homomorphism:

Hom(Q, ρ∗(E))→ I

where I ⊂ OQuotn(F) is the ideal sheaf of Quotn(G). By restricting to Quotn(G) weobtain a surjective homomorphism:

Hom(Q, ρ∗E)⊗OQuotn(G)→ I/I2 → 0

which is an isomorphism because both are locally free and of the same rank. Corollary 4.6.3. Let

0→ E → F → G → 0 (4.31)

be an exact sequence of locally free sheaves on the algebraic scheme S. Then thereis a closed immersion

IP(E) ⊂ IP(F)

and a natural identification:

NIP(E)/IP(F) = ρ∗G ⊗OIP(E)(1)

In particular, to every exact sequence (4.31) with E an invertible sheaf there corre-sponds a section

σ : S→ IP(E) ⊂ IP(F)

of the projective bundle IP(F)→ S whose normal bundle is G ⊗ E∨.


Proof. Only the last assertion requires a proof. It follows by observing that

OIP(F)(1)⊗OIP(E) = OIP(E)(1) = E∨

and therefore the formula for the normal bundle is a direct consequence of Proposi-tion 4.6.2. Remark 4.6.4. Let E be a locally free sheaf on an algebraic scheme S, and let

0→ Q∨ → ρ∗(E)→ K∨ → 0 (4.32)

be the dual of the tautological exact sequence (4.30) on Gn(E) = Quotn(E∨). Ten-soring with Q we obtain the exact sequence:

0→ Q∨ ⊗Q → ρ∗(E)⊗Q → K∨ ⊗Q → 0‖

TGn(E)/S

(4.33)

In the case n = 1 and S = Spec(k) we have E = V a vector space and G1(V ) =IP(V ) = IP; the dual of the tautological sequence is

0→ OIP (−1)→ V ⊗OIP → TIP (−1)→ 0 (4.34)

and the sequence (4.33) is the Euler sequence

0→ OIP → V ⊗OIP (1)→ TIP → 0

Therefore (4.33) is a generalization of the Euler sequence. Dualizing (4.34) we getan inclusion

I = IP(Ω1IP ) ⊂ IP × IP∨

where I := (x, H) : x ∈ H is the incidence relation (see Note 2 of Appendix B).From Corollary 4.6.3 we obtain

NI/IP×IP∨ = p∗1OIP (1)⊗ p∗2OIP∨(1)⊗OI

whereIP

p1←− IP × IP∨ p2−→ IP∨

are the projections.

Example 4.6.5. Let X be a projective nonsingular variety and L an invertible sheafon X . Consider the Atiyah extension

0→ OX → EL → TX → 0

(see page 145). Then ρ : IP(EL) → X is a IPr -bundle, r = dim(X), andIP(OX ) ⊂ IP(EL) is a section of ρ with normal bundle TX . In the case X =IP := IP(V ), where V is a finite-dimensional vector space, and L = O(1) theAtiyah extension coincides with the Euler sequence (Remark 3.3.10) so that we haveIP(EL) = IP × IP and

IP = IP(OIP ) ⊂ IP(EL) = IP × IP

is the diagonal embedding.


Example 4.6.6. Let V be a finite-dimensional k-vector space, X ⊂ IP := IP(V ) aprojective irreducible nonsingular variety and I ⊂ OIP its ideal sheaf. Then we haveinclusions of locally free sheaves on X :

N∨X/IP = I/I2 ⊂ Ω1IP|X ⊂ V∨ ⊗OX (−1)

which induce closed embeddings of projective bundles:

IP(N∨X/IP ) ⊂ IP(Ω1IP|X ) ⊂ X × IP∨‖

(x, H) : x ∈ HRecalling the exact sequence

0→ N∨X/IP → Ω1IP|X → Ω1

X → 0 (4.35)

we see that we have an identification:

IP(N∨X/IP ) = (x, H) : Tx X ⊂ H ⊂ X × IP∨ (4.36)

Lettingρ : IP(N∨X/IP )→ X

be the structural morphism, n = dim(X), and r = dim(IP), from the description(4.36) we deduce that for every x ∈ X , the fibre of ρ is identified with the (r−n−1)-dimensional linear system of hyperplanes which are tangent to X at x .

Let’s consider the special case when X is a rational normal curve in IP = IPr ,r ≥ 2. Denote by λ the invertible sheaf of degree 1 on X , so that OX (1) = λr . Wehave a morphism

q : IP(N∨X/IP )→ |λr−2| ∼= IPr−2

which associates to (x, H) the divisor q(H) of degree r − 2 such that

H · X = 2x + q(H)

The existence of q implies that IP(N∨X/IP ) ∼= X × IPr−2 and therefore

N∨X/IP = λs ⊕ · · · ⊕ λs︸︷︷︸r−1

for some s ∈ ZZ . From the exact sequence (4.35) one immediately computes that

(r − 1)s = (r + 2)(1− r)

i.e. s = −(r + 2). The conclusion is the following formula for the normal bundle ofa rational normal curve X ⊂ IPr :

NX/IPr ∼= λr+2 ⊕ · · · ⊕ λr+2︸︷︷︸r−1

(4.37)

For another approach to the same computation see [131]. The normal bundles ofrational curves of any degree in IP3 (resp. in IPr ) have been computed in [63] and[49] (resp. in [153]).


Example 4.6.7 (Zappa [188]). Let C be a projective nonsingular connected curveof genus 1 and let

0→ OC → E → OC → 0 (4.38)

be an exact sequence of locally free sheaves corresponding to a nonzero elementof Ext1OC

(OC ,OC ) = H1(C,OC ); in particular, (4.38) does not split. Consider theruled surface ρ : IP(E) → C and let C ′ := IP(OC ) ⊂ IP(E) be the section ofρ corresponding to the subsheaf OC ⊂ E in (4.38). By Corollary 4.6.3 we haveN := NC ′/IP(E) = OC ′ and therefore h0(C ′, N ) = 1. If C ′′ ⊂ IP(E) is a curvesuch that [C ′] and [C ′′] belong to the same component of HilbIP(E) then, since N isthe trivial sheaf, either C ′ = C ′′ or C ′′ ∩ C ′ = ∅. But both C ′ and C ′′ are sectionsof ρ and the second possibility implies that the exact sequence (4.38) splits. Thisis a contradiction, and we conclude that [C ′] is isolated in HilbIP(E), with a one-dimensional Zariski tangent space. Therefore C ′ is obstructed in IP(E).

∗ ∗ ∗ ∗ ∗ ∗Let H be a coherent sheaf on the projective scheme X , and let P(t) = n be a

constant polynomial, where n is a positive integer. Then we have two different Quotschemes associated to these data.

The first one is QuotX/XH,n , whose k-rational points are quotients H → F which

are locally free of rank n: it has just been considered.The other one is QuotX

H,n . A k-rational point of this scheme is nothing but a

quotient H → F such that F is a torsion sheaf with finite support and h0(F) = n.When n = 1 then F ∼= k(x) for some closed point x ∈ X : therefore we have anatural morphism

q : QuotXH,1 → X

(H→ F) → Supp(F)

and QuotXH,1 is a scheme over X . Let H→ F be a k-rational point of QuotX

H,1; thenker[H→ F] is called an elementary transform of H. The process of passing from Hto ker[H → F] is called an elementary transformation centred at x . This construc-tion is classical when X is a projective nonsingular curve and H is locally free. Forgeneralizations of it see [126]. QuotX

H,1 is the scheme of elementary transformationsof H.

Proposition 4.6.8. Assume that Supp(H) is connected. Then QuotXH,1 is connected.

Proof. The natural morphism

q : QuotXH,1 → X

(H→ F) → Supp(F)

has image Supp(H). Every H→ F factors as


H → H⊗Ox

↓F ∼= Ox

↓0

and therefore the fibre q−1(x) is identified with IP(H0(H ⊗ Ox ))∨ which is con-

nected. The conclusion follows.

4.6.5 Hilbert schemes of points

The Hilbert schemes parametrizing 0-dimensional subschemes of a given scheme areinexpectedly complicated and have a variety of applications. Since it is beyond ourscope to give an exhaustive overview of the subject, we will only explain some ofthe basic facts. The reader is referred to [91], [92], [133] for more details. See also[81] for another approach to Hilbert schemes of points.

Consider a projective scheme Y and, for a positive integer n, the Hilbert schemeY [n]. We have already seen in § 4.3 that Y [1] ∼= Y : we will therefore assume n ≥ 2.

Let Z ⊂ Y be a closed subscheme of length n. Then

H0(Z , NZ ) =⊕

y∈Supp(Z)

Hom(IZ y ,OZ y )

and

H1(Z , NZ ) = (0)

Moreover, from the spectral sequence of Exts (see [64], p. 265) we see that

Ext1(IZ ,OZ ) = H0(Y, Ext1(IZ ,OZ )) =⊕

y∈Supp(Z)

Ext1(IZ y ,OZ y )

It follows that the local properties of Y [n] at [Z ] are determined by the indepen-dent contributions from each of the points of Supp([Z ]). We immediately have thefollowing properties:

(a) If Z is reduced and supported at n distinct points of Y then [Z ] is a nonsingularpoint of Y [n] if and only if it is supported at nonsingular points of Y .

(b) If Y is reduced then the set of [Z ]’s with Z supported at n nonsingular pointsof Y is an open set of dimension n dim(Y ) contained in the nonsingular locus ofY [n].

Another important property is:

(c) [58] If Y is connected then Y [n] is also connected.


Proof. Let n ≥ 1 and let I ⊂ OY×Y [n] be the ideal sheaf of the universal family inY × Y [n]. Then we have a diagram of morphisms:

QuotY×Y [n]I,1

p q

Y [n+1] Y × Y [n]

where q is the natural morphism, which is surjective because Supp(I) = Y × Y [n].The morphism p is defined as follows.

Let (y, [Z ]) ∈ Y × Y [n] be a k-rational point and let γ : I → k(y, [Z ]) be

a quotient, which is a k-rational point of QuotY×Y [n]I,1 . Then ker(γ ) ⊂ OY×Y [n] is

an ideal sheaf such that ker(γ )OY×[Z ] has colength 1 in I ⊗ OY×[Z ]. Thereforeker(γ )OY×[Z ] defines a subscheme W ⊂ Y of length n + 1 containing Z and x ;we define p(γ ) = [W ]. The morphism p is clearly surjective. Since Y × Y [n]is connected by induction, we conclude that Y [n+1] is connected by Proposition4.6.8.

In general, Y [n] is singular and reducible even if Y is nonsingular connected.Notable exceptions are the cases dim(Y ) = 1, 2.

If C is a projective nonsingular curve and n ≥ 1 an integer, every closed sub-scheme D ⊂ C of length n is a Cartier divisor, therefore regularly embedded in C .It follows that C [n] is nonsingular of dimension

h0(D, ND) = h0(D,OD) = n

because H1(D, ND) = (0) is an obstruction space for OC [n],[D] (Theorem 4.3.5(ii)).Actually, C [n] is naturally isomorphic to C (n), the n-th symmetric product of C , theisomorphism being given by the cycle map

C [n] → C (n)

which maps a closed D ⊂ C to the associated Weil divisor.The case of surfaces is more subtle.

Theorem 4.6.9 (Fogarty [58]). If Y is a projective nonsingular connected surfacethen Y [n] is nonsingular connected of dimension 2n.

Proof. Let [Z ] ∈ Y [n]. We then have:

Exti (IZ ,OZ ) = (0) i ≥ 3

Moreover, from the exact sequence:

0→ IZ → OY → OZ → 0

we obtain the sequence:


0 → Hom(OZ ,OZ )→ Hom(OY ,OZ )→ Hom(IZ ,OZ )→ Ext1(OZ ,OZ )→ Ext1(OY ,OZ )→ Ext1(IZ ,OZ )→ Ext2(OZ ,OZ )→ Ext2(OY ,OZ )→ Ext2(IZ ,OZ )→ 0

Since Exti (OY ,OZ ) = Hi (Y,OZ ) = (0) for i ≥ 1 we see that

Ext2(IZ ,OZ ) = (0)

andExt1(IZ ,OZ ) ∼= Ext2(OZ ,OZ ) = Hom(OZ ,OZ ⊗ ωY )∨

= Hom(OZ ,OZ )∨ = H0(Z ,OZ )∨

Therefore

2∑i=0

(−1)i dim[Exti (IZ ,OZ )] = h0(Z , NZ )− h0(Z ,OZ ) = h0(Z , NZ )− n

Since the left-hand side is independent of Z , it follows that h0(Z , NZ ) is also inde-pendent of Z . But Y [n] is connected and has an open set which is nonsingular and ofdimension 2n: the conclusion follows.

To see that Y [n] is singular if dim(Y ) = 3 consider IP3 with homogeneous coor-dinates X0, . . . , X3 and the subscheme

Z = V (X21, X2

2, X23, X1 X2, X1 X3, X2 X3)

Then [Z ] ∈ (IP3)[4]. A computation similar to that of the example of Subsection4.6.3 shows that the Zariski tangent space of (IP3)[4] at [Z ] has dimension 18. But(IP3)[4] is connected and has a component which is nonsingular of dimension 12 atits general point: it follows that (IP3)[4] is singular. For more examples and for auseful detailed introduction to punctual Hilbert schemes we refer the reader to [91].

4.6.6 Schemes of morphisms

Let X and Y be schemes, with X projective and Y quasi-projective. For every schemeS let:

F(S) = HomS(X × S, Y × S)

This defines a contravariant functor:

F : (schemes) → (sets)

called the functor of morphisms from X to Y .For every Φ ∈ F(S) let ΓΦ ⊂ X × Y × S be its graph. Then ΓΦ

∼= X × Sis flat over S and therefore defines a flat family of closed subschemes of X × Yparametrized by S. This means that F is a subfunctor of HilbX×Y .

If G ⊂ X ×Y × S is a flat family of closed subschemes of X ×Y , proper over S,then the projection π : G → X × S is a family of morphisms into X and the locus


of points s ∈ S such that π(s) is an isomorphism is open (Note 2 of § 4.2). Thismeans that F is an open subfunctor of HilbX×Y , represented by an open subschemeof HilbX×Y , which we denote by Hom(X, Y ). It is called the scheme of morphismsfrom X to Y .

The scheme Hom(X, Y ) contains an open and closed subscheme isomorphic toY and consisting of the constant morphisms. In particular, if the only morphismsf : X → Y are the constant ones then

Hom(X, Y ) ∼= Y

Let X and Y be as above, and consider the contravariant functor

G : (schemes) → (sets)

defined as follows:

G(S) = S-isomorphisms X × S→ Y × SClearly, G is a subfunctor of F . It is easy to prove that G is represented by an opensubscheme Isom(X, Y ) of Hom(X, Y ), called the scheme of isomorphisms from X toY . When X = Y it is denoted by Aut(X) and called the scheme of automorphisms ofX . It is a group scheme. The following result follows immediately from Proposition3.4.2 and Corollary 3.4.3:

Proposition 4.6.10. Let f : X → Y be a morphism of algebraic schemes, with Xreduced and projective and Y nonsingular and quasiprojective. Then

T[ f ](Hom(X, Y )) ∼= H0(X, f ∗TY )

and the obstruction space of Hom(X, Y ) at [ f ] is contained in H1(X, f ∗TY ).If X is nonsingular then the tangent space to Aut(X) at 1X is H0(X, TX ).

Let j : X ⊂ Y be a closed embedding of projective nonsingular schemes. Thenj induces an inclusion

J : Aut(X) ⊂ Hom(X, Y )

such that J (1X ) = j and which is induced by the closed embedding

1X × j : X × X ⊂ X × Y.

It follows that J is a closed embedding. Its differential at 1X is the injective linearmap

H0(TX )→ H0(TY |X )

coming from the natural inclusion TX ⊂ TY |X . In fact, from the diagram of inclu-sions: X × X ⊂ X × Y

∪ ∪∆ ∼= Γ j

we deduce the commutative diagram:

N∆/X×X ⊂ NΓ j /X×Y

‖ ‖TX ⊂ TY |X

and we conclude according to § 4.3, Note 3.


Example 4.6.11. Consider X = IP1, Y = IPr and j : IP1 → IPr the r -th Veroneseembedding. Locally around [ j] we have a well-defined morphism

M : Hom(IP1, IPr )→ HilbIPr

sending [ j] → [ j (IP1)] with fibre M−1([ j (IP1)]) an open neighbourhood of theidentity in Aut(IP1). Consider the following diagram consisting of two exact se-quences:

0↑

0→ TIP1 → j∗TIPr → N j → 0↑

OIP1(r)r+1

↑OIP1

↑0

From the vertical sequence (the Euler sequence restricted to IP1) we get

h1( j∗TIPr ) = 0, h0( j∗TIPr ) = r(r + 2)

Since h1(TIP1) = 0 from the other sequence we obtain h1(N j ) = 0 and the exactsequence

0→ H0(TIP1) → H0( j∗TIPr )q−→ H0(N j ) → 0

‖ ‖ ‖T1IP1 Aut(IP1) T[ j]Hom(IP1, IPr ) T[ j (IP1)]HilbIPr

Since the map q can be identified with d M[ j] we see that M and Hom(IP1, IPr ) aresmooth at [ j] and HilbIPr

is smooth at [ j (IP1)]; moreover,

dim[ j](Hom(IP1, IPr ) = r(r + 2) = dim j (IP1)(HilbIPr)+ 3

For more on the schemes Hom(IP1, X) and applications to uniruledness see [42].

4.6.7 Focal loci

We assume char(k) = 0. Consider a flat family of closed subschemes of a projectivescheme Y :

⊂ Y × Bq1−→ Y

↓ q2B

(4.39)

parametrized by a scheme B. Let

f : ⊂ Y × B → Y


be the composition of the inclusion (4.39) with the projection q1. Denote by

N := N/Y×B

the normal sheaf of in Y × B, and let

(q∗2 TB)|

be the restriction to of the tangent sheaf along the fibres of q1. The global charac-teristic map of the family (4.39) is the homomorphism:

χ : (q∗2 TB)|→ N

defined by the following exact and commutative diagram:

0↓

(q∗2 TB)|χ−→ N

↓ ‖0 → T → TY×B| → N → 0

↓ d f ↓f ∗TY = f ∗TY

(4.40)

For each b ∈ B the homomorphism χ induces a homomorphism

χb : TB,b ⊗O(b)→ N(b)/Y

called the local characteristic map of the family (4.39) at the point b.Let

ϕ : B → HilbY

be the classifying morphism induced by the flat family (4.39). Then if (b) is con-nected, the linear map:

H0(χb) : TB,b → H0(N(b)/Y )

is dϕb, the characteristic map of (4.39) at the point b (see page 217).Assume that Y, B and the family (4.39) are smooth. In this case all the sheaves

in (4.40) are locally free. From a diagram-chasing it follows that

ker(χ) = ker(d f )

and therefore:

Proposition 4.6.12.

dim[ f ()] = dim()− rk[ker(χ)]


Let’s denote by V (χ) the closed subscheme of defined by the condition:

rk(χ) < minrk[(q∗2 TB)|], rk(N ) = mindim(B), codimY×B()We will call the points of V (χ) first-order foci of the family (4.39). V (χ) is thescheme of first-order foci, and the fibre of V (χ) over a point b ∈ B:

V (χ)b = V (χb) ⊂ (b)

is the scheme of first-order foci at b.If χ has maximal rank, i.e. if χ is injective or has torsion cokernel, then V (χ) is

a proper closed subscheme of . If χ does not have maximal rank then V (χ) = .The definition of first-order foci at a point b depends only on the geometry of the

family (4.39) in a neighbourhood of b. A focus y ∈ V (χ)b is a point where there isan intersection between the fibre (b) and the infinitesimally near ones. One defineshigher-order foci inductively: second-order foci are the first-order foci of the familyof first-order foci, and so on.

Focal loci have been studied classically in the case of families of linear spaces(see e.g. [158]). Recently, they have been applied to the geometry of the theta divisorof an algebraic curve in [37] and [38]. Related work is [31], [33], [39].

NOTES

1. Let Q ⊂ IP3 be a quadric cone with vertex v , and L ⊂ Q a line. Then

NL/Q = OL (1) ⊂ NL/IP3 = OL (1)⊕OL (1)

(see Example 3.2.8(iii), page 131); in particular, H0(L , NL/Q) = 2 and H1(L , NL/Q) = 0.

On the other hand, the Hilbert scheme HilbQ is one-dimensional at the point [L] since Lmoves in a one-dimensional family. It follows that L is obstructed in Q (see [44] for general-izations of this example).

2. Another example of reducible Hilbert scheme is the following, which appears in [170].Let Y = C ×C ′ where C and C ′ are projective nonsingular connected curves of genera g andg′ respectively and let

Yp′−→ C ′

↓ pC

be the projections; assume g, g′ ≥ 2. Consider an effective divisor D = x1+· · ·+xg of degreeg on C , and an effective divisor D′ = x ′1 + · · · + x ′g′ of degree g′ on C ′, both consisting ofdistinct points, and let

Γ = p−1(D)+ p′−1(D′) = C ′x1+ · · · + C ′xg

+ Cx ′1 + · · · + Cx ′g′

where C ′x = p−1(x) and Cx ′ = p′−1(x ′). Γ is a reduced divisor, has gg′ nodes and no othersingularity. If either D or D′ is nonspecial the curve Γ belongs to an irreducible componentH1 of HilbY of dimension g + g′ generically consisting of curves of the same form, obtainedby moving D and D′. When both D and D′ are special divisors the curve Γ belongs to alinear system of dimension ≥ 3 whose general member is a nonsingular curve and thereforebelongs to another irreducible component H2 of HilbY which has dimension g + g′ − 1. Theintersection H1 ∩ H2 is irreducible of dimension g + g′ − 2.


4.7 Plane curves

An important refinement of the Hilbert functors derives from the consideration of flatfamilies of closed subschemes of a projective scheme Y having prescribed singulari-ties, i.e. of families all of whose members have the same type of singularity in somespecified sense. This leads to the notion of equisingularity and to a related vast areaof research. In this section we will only concentrate on the specific case of familiesof plane curves with assigned number of nodes and cusps: we will show how to con-struct universal families of such curves, whose parameter schemes are called Severivarieties for historical reasons. This is a subject with a long history and a wealth ofimportant results, both classical and modern. Here we will limit ourselves to prov-ing a few basic results and to indicating some of their generalizations and the mainreferences in the literature. We will assume char(k) = 0 in this section.

4.7.1 Equisingular infinitesimal deformations

Let Y be a projective nonsingular variety and X ⊂ Y a closed subscheme with idealsheaf IX ⊂ OY . Recall ((1.3)) that on X we have an exact sequence of coherentsheaves:

0→ TX → TY |X → NX/Y → T 1X → 0

Recall that the sheafN ′X/Y := ker[NX/Y → T 1

X ]is called the equisingular normal sheaf of X in Y (see Proposition 1.1.9, page 16).Clearly, N ′X/Y = NX/Y if X is nonsingular. By definition, sections of the equisin-gular normal sheaf parametrize first-order deformations of X in Y which are locallytrivial, because they induce trivial deformations around every point of X .

We recall that an alternative description of the equisingular normal sheaf can begiven by means of the sheaf of germs of tangent vectors to Y which are tangent to X

TY 〈X〉 ⊂ TY

introduced in § 3.4. In fact, there is an exact sequence (see (3.55))

0→ TY 〈X〉 → TY → N ′X/Y → 0 (4.41)

From the definition it follows that, for every open set U ⊂ Y , Γ (U, TY 〈X〉) con-sists of those k-derivations D ∈ Γ (U, TY ) such that D(g) ∈ Γ (U, I) for everyg ∈ Γ (U, I).

Examples 4.7.1. (i) Assume that X is a hypersurface in Y ; then NX/Y ∼= OX (X).Locally on an affine open set U ⊂ Y we have (NX/Y )|U∩X ∼= OU∩X . Assume that Xcan be represented by an equation f (x1, . . . , xn) = 0 in local coordinates on U ; thenfrom the definition of T 1

X it follows that (N ′X/Y )|U∩X ⊂ OU∩X is the image of theideal sheaf (∂ f/∂x1, . . . , ∂ f/∂xn) ⊂ OU . We deduce that an equisingular first-order

4.7 Plane curves 255

deformation of X in Y corresponding to a local section g of N ′X/Y can be writtenlocally as

f (x)+ εg(x) = 0

where

g(x) = a1(x)∂ f

∂x1+ · · · + an(x)

∂ f

∂xn

restricts to g. Therefore if Y = IPn and X is a hypersurface of degree d we have anexact sequence

0→ OIPn → I(d)→ N ′X/IPn → 0

where I ⊂ OIPn is the ideal sheaf locally generated by the partial derivatives ofa local equation of X . In the special case of a curve X in a surface Y assume thatp ∈ X is a singular point and let f (x, y) = 0 be a local equation of X around p.If p is a node then (∂ f/∂x, ∂ f/∂y) = m p is just the maximal ideal of p; if p is anordinary cusp with principal tangent say y = 0 then (∂ f/∂x, ∂ f/∂y) = (x, y2) is anideal of colength 2.

(ii) Let X ⊂ IP2 be a (possibly reducible) plane curve of degree d of equationF(X0, X1, X2) = 0, having δ nodes p1, . . . , pδ and no other singularity. This caseis important because every nonsingular projective curve is birationally equivalent toa nodal plane curve. Denote by ∆ = p1, . . . , pδ ⊂ IP2 the 0-dimensional reducedscheme of the nodes of X and by ν : C → X the normalization map. The aboveanalysis shows that sections of H0(I∆(d)), i.e. curves of degree d which are adjointto X , cut on X sections of N ′

X/IP2 . This means that

ν∗(N ′X/IP2) = ν∗[OX (d)⊗ I∆]

= ν∗OX (d)(−p′1 − p′′1 − · · · − p′δ − p′′δ ) = ωC ⊗ ν∗O(3)

where ν−1(pi ) = p′i , p′′i , i = 1, . . . , δ, and therefore we have

h0(C, ν∗(N ′X/IP2)) = 3d + g − 1, h1(C, ν∗(N ′X/IP2)) = 0

where g is the geometric genus of X . Moreover, since

ν∗[ν∗(N ′X/IP2)] = ν∗[ν∗(OX (d)⊗ I∆]

= OX (d)⊗ ν∗OC (−p′1 − p′′1 − · · · − p′δ − p′′δ ) = OX (d)⊗ I∆ = N ′X/IP2

we have

h0(X, N ′X/IP2) = h0(C, ν∗(N ′

X/IP2)) = 3d + g − 1 = (d+22

)− δ − 1

h1(X, N ′X/IP2) = h1(C, ν∗(N ′

X/IP2)) = 0(4.42)

Finally, since

h0(I∆(d)) ≥(

d + 2

2

)− δ = 3d + g


and H0(I∆(d))/(F) ⊂ H0(N ′X/IP2), comparing with (4.42) we see that ∆ imposes

independent conditions to curves of degree d and that

H0(N ′X/IP2) = H0(I∆(d))/(F)

or, equivalently, the restriction map

H0(I∆(d))→ H0(N ′X/IP2)

is surjective. Note that N ′X/IP2 is a noninvertible subsheaf of NX/IP2 .

(iii) Another interesting case is obtained by taking an irreducible curve X ⊂ IP2

of degree d having δ nodes p1, . . . , pδ and κ ordinary cusps q1, . . . , qκ as its onlysingularities. This case is important because branch curves of generic projection onIP2 of projective nonsingular surfaces are curves of this type.

Let ν : C → X be the normalization map. Letting q j = ν−1(q j ), j = 1, . . . , κ ,we have in this case, according to the above description

ν∗(N ′X/P2) = OC (d)(−p′1 − p′′1 − · · · − p′δ − p′′δ − 3q1 − · · · − 3qκ)

= ωC ⊗ ν∗OX (3)(−q1 − · · · − qκ)

As before, one shows that ν∗[ν∗(N ′X/P2)] = N ′

X/P2 and therefore

h0(X, N ′X/IP2) = h0(C, ωC ⊗ ν∗OX (3)(−q1 − · · · − qκ)) ≥(

d + 2

2

)− δ− 2κ − 1

(4.43)

and in general we may have strict inequality and h1(X, N ′X/IP2) = 0 because the

invertible sheaf ωC ⊗ ν∗OX (3)(−q1 − · · · − qκ) can be special. But if κ < 3d thenit is certainly nonspecial and therefore in such a case we have

h0(X, N ′X/IP2) =

(d+22

)− δ − 2κ − 1 = 3d + g − 1− κ

h1(X, N ′X/IP2) = 0

4.7.2 The Severi varieties

Given an integer d > 0, consider the complete linear system |O(d)| of plane curvesof degree d. It is a flat family of closed subschemes of IP2 parametrized by theprojective space d = IP[H0(IP2,O(d))]:

|O(d)| :H ⊂ IP2 ×d

↓d


The linear system |O(d)| has a universal property with respect to families of planecurves of degree d because the pair (d , |O(d)|) represents the Hilbert functor:

Λd : (algschemes) → (sets)

given by:

Λd(S) =

flat families C ⊂ IP2 × S of planecurves of degree d parametrized by S

(see § 4.3). In this subsection we want to consider the problem of constructing auniversal family of reduced curves in IP2 having degree d, an assigned number δ ofnodes and κ of ordinary cusps and no other singularity. If such a universal familyexists it is parametrized by a scheme which we denote by Vδ,κ

d . These schemes havebeen studied classically: the foundations of their theory are given in [171] and theyare therefore called Severi schemes or Severi varieties.

If the Severi scheme Vδ,κd exists then, by the universal property, there is a functo-

rially defined morphismVδ,κ

d → d (4.44)

We start from the definition of the functor we want to represent.

Definition 4.7.2. Let d, δ, κ as above. Then

Vδ,κd : (algschemes) → (sets)

is defined as follows. For each algebraic scheme S

Vδ,κd (S) =

flat families C ⊂ IP2 × S of plane curves of deg. d formallylocally trivial at each k-rational s ∈ S whose geometric

fibres are curves with δ nodes and κ cusps as singularities

(see § 2.5 for the definition of formal local triviality). Obviously, Vδ,κd is a subfunctor

of Λd . The main result about Vδ,κd is the following:

Theorem 4.7.3. For each d, δ, κ as above, the functor Vδ,κd is represented by an

algebraic scheme Vδ,κd which is a (possibly empty) locally closed subscheme of d .

In the case κ = 0 we write Vδd instead of Vδ,0

d . The first published proof of thisresult is in [186]. We will not reproduce it in full generality here, but we will onlyconsider the case κ = 0, i.e. the case of nodal curves. This assumption allows atechnically simpler argument without changing the structure of the original proof.We need some lemmas.

Lemma 4.7.4. Let p ∈ IP2, O the local ring of IP2 at p, B0 = O/( f0) the localring of a plane curve having a node at p. Assume that for some A in ob(A) we havea deformation A → B of B0 over A such that T 1

B/A is A-flat. Then B is trivial and

T 1B/A∼= A.


Proof. By induction on dimk(A). The case A = k is trivial because

T 1B0= O/( f0, f0X , f0Y ) = O/( f0X , f0Y ) ∼= k

(see 3.1.4). In the general case consider a small extension

0→ (ε)→ A→ A′ → 0

and the induced deformation A′ → B ′. We have B = (A ⊗k O)/( f ) for some fwhich reduces to f0 modulo m A, and T 1

B/A = B/( fX , fY ). ThereforeB ′ = (A′ ⊗k O)/( f ′) where f ′ is obtained from f by reducing the coefficientsto A′, and

T 1B′/A′ = B ′/( f ′X , f ′Y ) = B/( fX , fY )⊗A A′ = T 1

B/A ⊗A A′

It follows that T 1B′/A′ is A′-flat and, by induction, we have

B ′ = (A′ ⊗k O)/( f0)

andT 1

B′/A′ = A′ ⊗k [O/( f0, f0X , f0Y )] = A′

Thus f = f0 + εg where g ∈ k. We have:

T 1B/A = (A ⊗k O)/( f0 + εg, f0X , f0Y ) = A/(εg)

where the last equality follows from the fact that f0 ∈ ( f0X , f0Y ). Since A/(εg) isA-flat if and only if g = 0 it follows that B = (A ⊗k O)/( f0) = A ⊗k (O/( f0)) isthe trivial deformation and T 1

B/A = A. Lemma 4.7.5. Let f : X → S be a flat morphism of algebraic schemes whichfactors as

Xj−→ Y ↓ q

S

where j is a regular embedding of codimension 1 and q is smooth. Then for everymorphism of algebraic schemes ϕ : S′ → S we have

T 1S′×S X/S′

∼= Φ∗T 1X/S

where Φ : S′ ×S X → X is the projection (i.e. T 1X/S commutes with base change).

Proof. Since the question is local we may reduce to a diagram of k-algebras of theform:

B = P/( f ) → B ′ = PA′/( f ′)↑ ↑A → A′

where P is a smooth A-algebra, f ∈ P is a regular element, and f ′ is the image off in PA′ = P ⊗A A′. Then we have


( f )/( f 2) ∼= Bδ−→ ΩP/A ⊗P B

f → d f ⊗ 1

and( f ′)/( f ′2) ∼= B ′ δ′−→ ΩP ′/A′ ⊗P ′ B ′

f ′ → d f ′ ⊗ 1

But sinceΩP ′/A′ ⊗P ′ B ′ = (ΩP/A ⊗P B)⊗B B ′

we have δ′ = δ ⊗B B ′ and

T 1B/A ⊗B B ′ = coker(δ∨)⊗B B ′ = coker(δ′∨) = T 1

B′/A′ Lemma 4.7.6. Let S be an algebraic scheme and C ⊂ IP2 × S a flat family of planecurves of degree d. Let s ∈ S be a k-rational point such that the fibre C(s) is a curvehaving at most nodes as singularities. Then T 1

C/S is S-flat at a point p ∈ C(s) if andonly if the family is formally locally trivial at s around p.

Proof. If p is a nonsingular point of C(s) then C → S is smooth at p, henceT 1(C/S,OC)p = 0 and the assertion is obvious.

Let’s assume that p is a node of C(s). Let A = OS,s , Aα := A/mα ,Sα = Spec(Aα) and Cα → Sα the induced infinitesimal deformation of C1 = C(s);let B = OC,p and Bα = OCα,p. By Lemma 4.7.5 we have:

T 1Cα/Sα,p = T 1

Bα/Aα∼= T 1(C/S,OC)p ⊗OC,p Bα = T 1

B/A ⊗B Bα (4.45)

Assume that T 1(C/S,OC)p = T 1B/A is A-flat. Then T 1

Bα/Aαis Aα-flat by (4.45) and

therefore Bα is the trivial deformation of B0 = B/m A B = OC(s),p , by Lemma 4.7.4.Since this is true for every α we conclude that the family C → S is formally locallytrivial at p.

Conversely, assume that C → S is formally locally trivial at p. Then Bα∼=

B0 ⊗k Aα for all α and T 1Bα/Aα

is Aα-flat by Lemma 4.7.4. But by (4.45) we have

T 1Bα/Aα

= T 1B/A ⊗B Bα = T 1

B/A ⊗A Aα

and from the local criterion of flatness we deduce that T 1B/A = T 1(C/S,OC)p is

A-flat. Proof of Theorem 4.7.3. Consider the universal family H ⊂ IP2×d of plane curvesof degree d and let Wi be the flattening stratification of T 1

H/d. Let W = Wi be

a stratum containing a k-rational point s ∈ d parametrizing a reduced curve H(s)having δ nodes and no other singularity, and let H′ ⊂ IP2×W be the induced familyof degree d curves. By Lemma 4.7.5 we have

T 1H′/W

∼= T 1H/d

⊗OH′


and therefore, by construction, T 1H′/W is flat over W . Moreover, since H′ ⊂ IP2×W

is a regular embedding of codimension 1, T 1H′/W is of the form OV for some closed

subscheme V ⊂ IP2 ×W . By applying Lemma 4.7.5 again we deduce

OV (s) = T 1H′/W ⊗ k ∼= T 1

H(s)

which is a reduced scheme of length δ supported at Sing(H(s)). This implies thatV → W is etale at the δ points of V (s). Therefore there is an open neighbourhoodU of s ∈ W such that V (U ) → U is etale of degree δ. If u ∈ U is a k-rationalpoint then H(u) is a curve such that T 1

H(u)∼= V (u), hence H(u) has δ singular points

p1, . . . , pδ and no other singularity, such that T 1Op j

∼= k. From Proposition 3.1.5 it

follows that H(u) is a δ-nodal curve. Therefore by applying Lemma 4.7.6 we seethat the family H′(U )→ U is an element of Vδ,0

d (U ).Putting together all these open sets we obtain a locally closed subset Ui ⊂ Wi

such that the induced family H′i ⊂ IP2×Ui defines an element of Vδ,0d (Ui ). Now let

Vδd =

⋃i

Ui

and let H ⊂ IP2×Vδd be the induced family. If C ⊂ IP2× S is an element of Vδ,0

d (S)for an algebraic scheme S then by the universal property of |O(d)|we obtain a uniquemorphism S→ d inducing the given family by pullback. By Lemma 4.7.6 and thedefining property of the flattening stratification this morphism factors through Vδ

d .

Thus (Vδd ,H) represents the functor Vδ,0

d . We now consider the local properties of the Severi varieties.

Proposition 4.7.7. Let C ⊂ IP2 be a reduced curve having degree d, δ nodes and κordinary cusps and no other singularity. Let [C] ∈ V = Vδ,κ

d be the point parame-trizing C. Then there is a natural identification:

T[C]V = H0(C, N ′C/IP2)

and H1(C, N ′C/IP2) is an obstruction space for OV,[C].

Proof. T[C]V is the subspace of T[C]d = H0(C,OC (d)) corresponding to lo-cally trivial first-order deformations, and these are the elements of H0(C, N ′

C/IP2)

by the very definition of N ′C/IP2 . From the proof of Proposition 3.2.6 it is ob-

vious that obstructions to deforming locally trivial deformations lie in the spaceH1(C, N ′

C/IP2). According to the classical terminology, we call Vδ,κ

d regular at a point [C] if

H1(C, N ′C/IP2) = 0; otherwise Vδ,κ

d is called superabundant at [C]. An irreducible

component W of Vδ,κd is called regular (resp. superabundant) if it is regular (resp.


superabundant) on a nonempty open subset. Vδ,κd is called regular if all its compo-

nents are regular; otherwise it is called superabundant. From 4.7.7 and from Example4.7.1(iii) it follows that if a component W of Vδ,κ

d is regular then it is generically non-singular of dimension 3d + g − 1 − κ , where g is the geometric genus of C , i.e. ofpure codimension δ + 2κ in d .

Corollary 4.7.8. If κ < 3d then Vδ,κd is regular at every point. In particular, Vδ

d isregular at every point, thus it is nonsingular of pure dimension

3d + g − 1 =(

d + 2

2

)− 1− δ

if it is nonempty.

Proof. It follows from Proposition 4.7.7 and from Example 4.7.1(iii). Remark 4.7.9. Note that the corollary does not claim that Vδ,κ

d = ∅. The nonempti-ness of the Severi varieties will be discussed in the next subsection. Corollary 4.7.8follows also from Proposition 3.4.16 recalling that, by Lemma 3.4.15, we have

Hi (C, N ′C/IP2) ∼= Hi (C, Nϕ), i = 0, 1

where ϕ : C → C is the normalization map. The description of the deformations ofC given by Proposition 3.4.16 can be considered as the “parametric” counterpart ofthe “cartesian” point of view of Corollary 4.7.8.

The Severi varieties Vδ,κd may have a complicated structure. If there are too many

cusps then in general a [C] ∈ Vδ,κd satisfies H1(C, N ′

C/IP2) = (0) (see Example

4.7.10 below) and in fact, Vδ,κd can be singular at such a [C]. To decide whether

this effectively happens has been a long-standing classical problem (see [189], ch.VIII, where this topic is discussed). The first example of a singular point of a Severivariety was given by Wahl [186]: it is a plane irreducible curve of degree 104 with3636 nodes and 900 cusps. For other examples see [122], [80] and [183].

Example 4.7.10. If κ > 3d then Vδ,κd can be superabundant. The following classical

example is due to B. Segre (see [159] and [189], p. 220). Consider plane curves ofthe following type:

C : [ f2m(x, y)]3 + [ f3m(x, y)]2 = 0

where f2m(x, y) and f3m(x, y) are general polynomials of the indicated degrees, andm > 2. Then d = deg(C) = 6m, δ = 0 and κ = 6m2 because the only singularitiesof C are the points of intersection of the curves f2m = 0 and f3m = 0 and they areeasily seen to be cusps. C is irreducible of geometric genus

g =(

6m − 1

2

)− 6m2


The dimension of the family of curves C is

R :=(

2m + 2

2

)+(

3m + 2

2

)− 1 = 1

2(13m + 2)(m + 1)

which is larger than

r = 6m(6m + 3)

2− 2κ = 6m2 + 9m

In fact, R − r = (m−12

). Therefore V0,6m2

6m is superabundant at all points [C].Let’s compute h1(C, N ′

C/IP2). By the analysis of Example 4.7.1(iii) we know

that h1(C, N ′C/IP2) equals the index of speciality ι of the linear system cut on the

normalization C of C by the curves of degree 6m passing through the cusps andtangent there to the cuspidal tangents. It is an easy computation (see [189] p. 220 fordetails) that

ι = R − r

The conclusion is that each [C] is a nonsingular point of a superbundant component

of V0,6m2

6m of dimension R.For a modern treatment of this example see [180].

4.7.3 Nonemptiness of Severi varieties

Even though we have proved precise results about the structure of the Severi vari-eties, it is not clear that nodal curves of given degree and number of nodes exist atall: the task of writing down explicitly the equation of such a curve is too concreteand precise to be within the reach of known techniques. Nevertheless Severi himselfoutlined a method to prove the existence of nodal curves. His approach is based on

the notion of “analytic branch” and consists in analysing the local structure of Vδd

along Vδ+1d . While his proof makes perfectly good sense over the field of complex

numbers, it is not straightforward to translate it into an algebraic proof. In this sub-section we will show that the Severi varieties Vδ

d of nodal curves are nonempty inthe expected range of δ using a different method which is entirely algebraic and el-ementary. It is based on the theory of multiple point schemes, which we will nowrecall.

Consider a finite unramified morphism f : X → Y of algebraic schemes and letNk( f ) ⊂ Y be the k-th multiple point scheme of f (for the definition see Example4.2.9, page 200). Define

Mk( f ) = f −1(Nk( f )) ⊂ X

(scheme theoretic inverse image). Note that, in particular, N1( f ) is supported on theimage of f and M1( f ) = X . Let

X ×Y X = ∆∐

X2


(where ∆ is the diagonal and the union is disjoint because f is unramified) and letf1 : X2 → X be the morphism induced by the first projection. Then f1 is called thefirst iteration morphism of f .

Lemma 4.7.11. Let f : X → Y be a finite unramified morphism of algebraicschemes. Then:

(i) The first iteration morphism f1 is finite and unramified.(ii) Nk−1( f1) = Mk( f ) for all k ≥ 2.

Proof. (i) f1 is finite and unramified because both properties are invariant under basechange and composition with a closed embedding.

(ii) By the base change property of the Fitting ideals we have

Mk( f ) = f −1(Nk( f )) = Nk(π1) = Nk−1( f1)

where π1 : X ×Y X → X is the first projection. Lemma 4.7.12. Let f : X → Y be a finite unramified morphism of purely dimen-sional algebraic schemes, such that dim(X) = dim(Y ) − 1 and Y is nonsingular.Then every irreducible component X2 of X2 satisfies

dim(X2) ≥ dim(Y )− 2

Proof. We have

X ×Y X = ( f × f )−1(∆Y )

where ∆Y ⊂ Y×Y is the diagonal. Since Y is nonsingular, ∆Y is regularly embeddedof codimension dim(Y ) in Y × Y . It follows that every component of X ×Y X hascodimension ≤ dim(Y ) in X × X (Lemma D.1.2). We deduce that every componentX2 of X2 satisfies

dim(X2) ≥ 2 dim(X)− dim(Y ) = dim(Y )− 2

If the morphism f satisfies some further assumptions then one can describe itsbehaviour quite precisely.

Definition 4.7.13. Let f : X → Y be a finite unramified morphism of algebraicschemes. Then f is called self-transverse of codimension 1 if X and Y are nonsin-gular and purely dimensional, dim(X) = dim(Y ) − 1, and for any closed pointy ∈ Y and for any r distinct points x1, . . . , xr ∈ f −1(y) the tangent spacesTx1 X, . . . , Txr X, viewed as subspaces of TyY , are in general position (i.e. their in-tersection has codimension r).

Lemma 4.7.14. If f : X → Y is a self-transverse codimension 1 morphism thenf1 : X2 → X is a self-transverse codimension 1 morphism.


Proof. Let (x1, x2) ∈ X2 and let y = f1(x1, x2) = f (x1) = f (x2). Then

T(x1,x2)X2 = Tx1 X ×TyY Tx2 X

Since f is self-transverse we have dim(T(x1,x2)X2) = dim(Y )−2. On the other hand,dim(X2) ≥ dim(Y )− 2, by Lemma 4.7.12. It follows that X2 is nonsingular of puredimension dim(Y )− 2.

For any x ∈ X and (x, x ′) ∈ f −11 (x) the differential

d f1(x,x ′) : T(x,x ′)X2 → Tx X

is an injection of codimension 1. For any (x, x2), . . . , (x, xs) ∈ f −11 (x) we have

d f1(x,x2)(T(x,x2)X2)⋂· · ·⋂

d f1(x,xs )(T(x,xs )X2) = Tx X⋂

Tx2 X⋂· · ·⋂

Txs X

viewed as subspaces of TyY . The self-transversality of f1 now follows from theanalogous property of f .

With this terminology we can state the following useful result.

Proposition 4.7.15. Assume that f : X → Y is a self-transverse codimension 1morphism and that Nr ( f ) = ∅ for some r ≥ 2. Then Nr ( f ) has pure codimension rin Y and Ns( f ) = ∅ for all 1 ≤ s ≤ r−1. In particular, Ns( f ) has pure codimensions for all 1 ≤ s ≤ r − 1.

Proof. By induction on r . If r = 2 then M2( f ) = N1( f1), by Lemma 4.7.11(ii), andhas pure dimension equal to dim(X2) = dim(Y ) − 2 by Lemma 4.7.14: thereforeN2( f ) is of pure codimension 2. Moreover, N1( f ) = f (X) = ∅ and has puredimension dim(X) = dim(Y )− 1.

Now assume r ≥ 3. Nr ( f ) has pure codimension r in Y if and only if Mr ( f ) haspure codimension r − 1 in X . By Lemma 4.7.11(ii) we have

Mr ( f ) = Nr−1( f1)

and by the inductive hypothesis Nr−1( f1) has pure codimension r − 1 because f1 isself-transverse of codimension 1. Again by the inductive hypothesis

Ms( f ) = Ns−1( f1) = ∅for all 1 ≤ s − 1 ≤ r − 2. This implies that Ns( f ) = ∅ for all 2 ≤ s ≤ r − 1. Thecase s = 1 is a consequence of the first part of the proof.

Next we will see how Proposition 4.7.15 can be applied to the study of the Severivarieties of nodal curves.

∗ ∗ ∗ ∗ ∗ ∗Consider the universal family H ⊂ IP2 × d of plane curves of degree d ≥ 2

and the cotangent sheaf T 1H/d

. Since H ⊂ IP2 × d is a regular embedding

of codimension 1, T 1H/d

is a quotient of OIP2×d. Therefore we can identify


T 1H/d

= OZ where Z ⊂ H is a closed subscheme; evidently, the set of closedpoints of Z is

(p, s) : p is a singular point of H(s) ⊂ IP2 ×d

Z can be described more precisely as follows. We let

F(X0, X1, X2) =∑

n0+n1+n2=d

An0,n1,n2 Xn00 Xn1

1 Xn22 = 0

be the equation of the universal curve H inside IP2 × d . Then Z is defined by thethree equations:

∂F

∂X0= 0,

∂F

∂X1= 0,

∂F

∂X2= 0

Denote byIP2 π1←− Z

π2−→ d

the projections.

Lemma 4.7.16. (i) Z is irreducible, nonsingular, rational of codimension 3 in IP2×d .

(ii) π2 maps Z birationally onto its image W ⊂ d , which is an irreducible rationaldivisor parametrizing all singular curves of degree d.

Proof. π1 is surjective with fibres linear systems of dimension dim(d)− 3 becausefor each p ∈ IP2 the fibre π−1

1 (p) is the linear system d(−2p) of all curves ofdegree d which are singular at p. This proves (i). Moreover, by an easy applicationof Bertini’s theorem one gets that a general element of d(−2p) is a curve having anode at p as its only singular point. This takes care of (ii).

Z contains an open subset Z1 whose set of closed points is

(p, s) : H(s) has only nodes as singularities and p is a node of H(s)From the proof of Lemma 4.7.16 it follows immediately that Z1 ⊂ Z is a nonempty(dense) open subset. Let

W1 := π2(Z1)

W1 is a dense open subset of W . It parametrizes all singular curves of degree d havingonly nodes as singularities; a general point of W1 parametrizes a curve of degree dhaving one node and no other singularities.

Consider the closed subset Bd := π2(Z\Z1) ⊂ d , which parametrizes allsingular non-nodal curves of degree d, and the morphism

π : Z1 → d\Bd

obtained by restricting π2.

Proposition 4.7.17. π is birational onto its image W1, finite, unramified and self-transverse of codimension 1.


Proof. If s ∈ W1 = π(Z1) is a closed point then π−1(s) is the scheme of nodesof H(s) which is finite and reduced. Therefore π is unramified. Moreover, π is therestriction over an open subset of d of a projective morphism. Therefore π is finite.The birationality onto its image follows from that of π2 (Lemma 4.7.16). In order toprove that π is self-transverse of codimension 1, let X ⊂ IP2 be a curve of degreed having δ nodes p1, . . . , pδ and no other singularity. Then (pi , [X ]) ∈ Z1, i =1, . . . , δ. The local analysis of Example 4.7.1(i) and the fact that π is unramifiedshow that we have

Im[dπ(pi ,[X ])] = H0(X, NX/IP2(−pi ))

whereNX/IP2(−pi ) = ker[NX/IP2 → T 1

X,pi]

Thereforeδ⋂

i=1

Im[dπ(pi ,[X ])] = H0(X, N ′X/IP2) = T[X ]Vδd

Since T[X ]Vδd has codimension δ by Corollary 4.7.8, this equality means that the

subspaces Im[dπ(pi ,[X ])] are hyperplanes of T[X ]d = H0(X, NX/IP2) in generalposition.

We can now prove the main result of this subsection.

Theorem 4.7.18 (Severi [173]). Let X ⊂ IP2 be a curve of degree d having δ nodesand no other singularity, i.e. such that [X ] ∈ Vδ

d . Then

∅ = Vδd ⊂ Vδ−1

d ⊂ · · · ⊂ V1d ⊂ V0

d = d

and Vsd has pure codimension 1 in Vs−1

d , for every s = 1, . . . , δ.

Proof. From the previous analysis it follows that for each s

Vsd = Ns(π)\Ns+1(π)

The existence of X implies that Vδd = ∅. The theorem is now an immediate conse-

quence of Propositions 4.7.17 and 4.7.15. Corollary 4.7.19. (i) For every d ≥ 2 and

0 ≤ δ ≤(

d

2

)the Severi variety Vδ

d is nonempty.(ii) For every d ≥ 2 and

0 ≤ δ ≤(

d − 1

2

)the Severi variety Vδ

d contains irreducible curves.


Proof. (i) A curve X consisting of d distinct lines no three of which pass throughthe same point defines a point of Vδo

d with δo =(d

2

)and the non-emptiness of Vδ

d forδ ≤ δo follows from the theorem.

(ii) A general projection in IP2 of a rational normal curve Γd ⊂ IPd is an irre-ducible curve Y of degree d having

(d−12

)nodes. The theorem implies that Y is in

the closure of Vδd for every δ ≤ (d−1

2

). The following lemma guarantees that for each

such δ we can find irreducible curves in Vδd .

Lemma 4.7.20. Let C ⊂ IP2 × S be a flat family of plane curves of degree d ≥ 2,with S an algebraic scheme, and let o ∈ S be a k-rational point. Assume that thefibre C(o) is reduced and irreducible. Then C(s) is reduced and irreducible for alls ∈ S in an open neighbourhood of o.

Proof. Denote by f : C → S the projection. Since Hi (C(s),OC(s)(m)) = 0 for allm ≥ d − 2, i ≥ 1, and for all s ∈ S, by Theorem 4.2.5

t0m(s) : f∗OC(m)s ⊗ k(s)→ H0(C(s),OC(s)(m))

is bijective and f∗OC(m)s is free for all m ≥ d − 2 and s ∈ S.The multiplication map

µs : H0(OC(s)(d − 1))⊗ H0(OC(s)(d − 1))→ H0(OC(s)(2d − 2))

is surjective for all s ∈ S. Then the multiplication map

µ : f∗OC(d − 1)⊗ f∗OC(d − 1)→ f∗OC(2d − 2)

is surjective and ker(µ) is locally free.Let R ⊂ f∗OC(d − 1) ⊗ f∗OC(d − 1) be the cone of reducible tensors. For

each s ∈ S it restricts to the cone R(s) of reducible tensors in H0(OC(s)(d − 1)) ⊗H0(OC(s)(d − 1)), by the bijectivity of the maps t0

m(s) for m = d − 1, 2d − 2. Thecondition that C(o) is reduced and irreducible is equivalent to R(o)∩ ker(µo) = 0.Therefore there is an open neighbourhood of U of o ∈ S such that R(s)∩ ker(µs) =0, i.e. such that C(s) is reduced and irreducible, for all s ∈ U . Remarks 4.7.21. (i) It is easy to see that Vδ

d is reducible in general. For example,V3

4 has two irreducible components and V96 has five irreducible components. An im-

portant classical problem, known as the “Severi problem”, has been to decide aboutthe irreducibility of the open set of Vδ

d parametrizing irreducible nodal curves. Thisproblem has been solved affirmatively in [82] and, independently, in [147] (see also[182]). A report of Harris’ proof is given in [121]. It is known that the open set ofVδ,κ

d parametrizing irreducible curves is reducible in general if κ > 0. For examplessee [160] (such examples are also reported in [189]).

(ii) Theorem 4.7.18 and some generalizations of it have been reconsidered in[179], but the proof given there is based on infinitesimal considerations which seemto need a further insight. For an interesting discussion see [60]. The analogous prob-lem for the varieties Vδ,κ

d is still open, i.e. we don’t know a characterization of the


values of d, δ, κ for which Vδ,κd = ∅. For partial results see [95]; for a classical dis-

cussion see [159]. The results on multiple point schemes used here are special casesof a general theory for which we refer the reader to [99], [101] and to the referencesquoted there.

(iii) The proof of Theorem 4.7.3 can be easily modified to prove the existence ofuniversal families of curves with nodes and cusps (generalized Severi varieties) on aprojective nonsingular surface Y . In such a proof one replaces d by HilbY and usesthe existence and the universal property of HilbY .

Such generalized Severi varieties behave in a way relatively similar to the Vδ,κd ’s

as long as Y has Kodaira dimension ≤ 0 (see [179], [112]). On surfaces of generaltype the situation changes radically. On such a surface Y the generalized Severi va-rieties can be superabundant even when κ = 0 and it is not known in which range ofδ they are not empty. A systematic study of them has started relatively recently. Werefer the reader to [34], [69], [32], [56], [57] for details.

A

Flatness

The algebraic notion of flatness, introduced for the first time in [164], is the basictechnical tool for the study of families of algebraic varieties and schemes. In thisappendix we will overview the main algebraic results needed. For the properties offlat morphisms between schemes we refer to [84]. See also § 4.2.

A module M over a ring A is A-flat (or flat over A, or simply flat) if the func-tor N → M ⊗A N from the category of A-modules into itself is exact. Since thisfunctor is always right exact, the flatness means that it takes monomorphisms intomonomorphisms. An A-algebra B is flat over A if B is flat as an A-module.

The A-module M is said to be faithfully flat if for every sequence of A-modulesN ′ → N → N ′′ the sequence

M ⊗A N ′ → M ⊗A N → M ⊗A N ′′

is exact if and only if the original sequence is exact. Obviously, if M is faithfully flatthen it is flat. In a similar way we give the notion of faithfully flat A-algebra. It isstraightforward to check that if A→ B is a local homomorphism of local rings, thena B-module of finite type is faithfully A-flat if and only if it is A-flat and nonzero.

Recall that the flatness of an A-module M is equivalent to any of the followingconditions:

(1) TorAi (M, N ) = (0) for all i > 0 and for every A-module N .

(2) TorA1 (M, N ) = (0) for every A-module N .

(3) TorA1 (M, N ) = (0) for every finitely generated A-module N .

(4) TorA1 (M, A/I ) = (0) for every ideal I ⊂ A.

(5) I ⊗A M → M is injective for every ideal I ⊂ A.(6) I ⊗A M → I M is an isomorphism for every ideal I ⊂ A.

Example A.1. Let k be a ring, u, v indeterminates and f : k[u, uv] → k[u, v] theinclusion. Then

k[u, uv](uv)

= k[u] u−→ k[u] = k[u, uv](uv)

270 A Flatness

is injective. Tensoring by ⊗k[u,uv]k[u, v] we obtain:

k[u, v](uv)

u−→ k[u, v](uv)

which is not injective. Therefore f is not flat.

We list without proof a few basic properties of flat modules:

Proposition A.2. (I) M is A-flat if and only if Mp is Ap-flat for every prime ideal p.(II) Every projective module is flat.(III) Assume M is finitely generated. Then M is flat if and only if it is projective; if

A is local then M is flat if and only if it is free.(IV) If S ⊂ A is a multiplicative subset then AS is A-flat.(V) A direct sum M = ⊕i∈I Mi is flat if and only if all Mi ’s are flat.(VI) Let

0→ M ′ → M → M ′′ → 0

be an exact sequence of A-modules with M ′′ flat. Then M is flat if and only if M ′is flat.

(VII) Base change: if M is A-flat and f : A → B is a ring homomorphism, thenM ⊗A B is B-flat.

(VIII) Transitivity: if B is a flat A-algebra and N is a flat B-module, then N is A-flat.(IX) If A is a noetherian ring and I is an ideal, the I -adic completion A is a flat

A-algebra. If I is contained in the Jacobson radical of A then A is a faithfullyflat A-algebra.

(X) If B is an A-algebra and if there exists a B-module M which is faithfully flat,then the morphism Spec(B)→ Spec(A) is surjective.

(XI) If X1, . . . , Xr are indeterminates, then A[X1, . . . , Xr ] and A[[X1, . . . , Xr ]] areA-flat.

The following result is frequently used:

Proposition A.3. If A is an artinian local ring with residue field k the following areequivalent for an A-module M:

(i) M is free(ii) M is flat(iii) TorA

1 (M, k) = (0)

Proof. (i)⇒ (i i)⇒ (i i i) are clear.(i i i)⇒ (i i). Let N be a finitely generated A-module and let

N = N0 ⊃ · · · ⊃ Nn = (0)

be a composition series for N such that

Ni/Ni+1 ∼= k

A Flatness 271

for i = 0, . . . , n− 1. Using the Tor exact sequences from hypothesis (iii) we deducethat Tor1(M, N ) = (0) and the flatness of M follows from (3).

Let’s now prove (i i) ⇒ (i). Let e j j∈J be a system of elements of M whichinduces a basis of M ⊗A k over k. The system e j defines a homomorphismf : AJ → M which induces an isomorphism k J → M ⊗A k. From the follow-ing lemma we find that f is an isomorphism, and therefore M is free. Lemma A.4. Let R be a ring, I an ideal and f : F → G a homomorphism ofR-modules with G flat. Assume that one of the following conditions is satisfied:

(a) I is nilpotent.(b) R is noetherian, I is contained in the Jacobson radical of R and F and G are

finitely generated.

If the induced homomorphism F/I F → G/I G is an isomorphism, then f is anisomorphism.

Proof. Let K = coker( f ). Tensoring the exact sequence

F → G → K → 0

with R/I we get K/I K = 0: from Nakayama’s lemma (which holds in either ofhypotheses (a) and (b)) it follows that K = 0, and therefore F is surjective. LettingH = ker( f ) we deduce an exact sequence

0→ H/I H → F/I F → G/I G → 0

using the flatness of G. By Nakayama again we deduce H = 0 and the conclusionfollows.

The following is a basic criterion of flatness.

Theorem A.5 (Local criterion of flatness). Suppose that ϕ : A→ B is a local ho-momorphism of local noetherian rings, and let k = A/m A be the residue field of A.If M is a finitely generated B-module, then the following conditions are equivalent:

(i) M is A-flat.(ii) TorA

1 (M, k) = 0.(iii) M ⊗A (A/mn

A) is flat over A/mnA for every integer n ≥ 1.

(iv) M ⊗A (A/mnA) is free over A/mn

A for every integer n ≥ 1.

Proof. (i)⇒ (i i) is obvious.(i i)⇒ (i) see [48], Th. 6.8, p. 167.(i)⇒ (i i i) is obvious.(i i i)⇒ (i) It suffices to show that for every inclusion N ′ → N of A-modules of

finite type we have an inclusion M ⊗A N ′ → M ⊗A N . For this purpose it sufficesto show that the kernel of this last map is contained in

Kn := ker[M ⊗A N ′ → M ⊗A (N ′/N ′ ∩ mnA N )]

272 A Flatness

for all n, because⋂

n Kn = (0). We have a commutative diagram with exact rows:

0→ Kn → M ⊗A N ′ → M ⊗A (N ′/N ′ ∩ mnA N ) → 0

↓ ↓M ⊗A N → M ⊗A (N/mn

A N ) → 0

The last vertical arrow coincides with the map obtained from the injection

N ′/N ′ ∩ mnA N → N/mn

A N

after tensoring over A/mnA with the A/mn

A-flat module M⊗A (A/mnA), and therefore

it is injective. The conclusion follows from the above diagram.(i i i)⇔ (iv) follows from Proposition A.3 because A/mn

A is artinian. For a more general version of the local criterion we refer to [3], exp. IV,

Theoreme 5.6. Note that A.3 is a special case of A.5.

Corollary A.6. Suppose that ϕ : A → B is a local homomorphism of localnoetherian rings, let k = A/m A be the residue field of A, M, N two finitelygenerated B-modules, and suppose that N is A-flat. Let u : M → N be aB-homomorphism. Then the following are equivalent:

(i) u is injective and coker(u) is A-flat.(ii) u ⊗ 1 : M ⊗ k → N ⊗ k is injective.

Proof. (i)⇒ (ii). Let G = coker(u). Tensoring by k the exact sequence

0→ Mu−→ N → G → 0

by k we obtain the exact sequence:

TorA1 (G, k)→ M ⊗A k

u⊗1−→ N ⊗A k → G ⊗A k → 0

Since G is A-flat we have TorA1 (G, k) = 0, and it follows that u ⊗ 1 is injective.

(ii)⇒ (i). Factor u ⊗ 1 as

M ⊗A kα−→ Im(u)⊗A k

β−→ N ⊗A k

Then α is an isomorphism and β is injective. Tensoring by k the exact sequence

0→ Im(u)→ N → G → 0 (A.1)

we obtain the exact sequence:

TorA1 (N , k)→ TorA

1 (G, k)→ Im(u)⊗A kβ−→ N ⊗A k → G ⊗A k → 0

Since N is A-flat we have TorA1 (N , k) = 0; from the injectivity of β we deduce

TorA1 (G, k) = 0 and from A.5 it follows that G is A-flat. Applying (VI) to the exact

sequence (A.1) we deduce that Im(u) is A-flat as well. Consider the exact sequence:

A Flatness 273

0→ ker(u)→ M → Im(u)→ 0

and tensor by k. We obtain the exact sequence:

0→ ker(u)⊗A k → M ⊗A kα−→ Im(u)⊗A k → 0

Since α is an isomorphism we deduce that ker(u)⊗A k = 0, and therefore ker(u) = 0by Nakayama’s lemma.

A related result is the following:

Lemma A.7. Let B be a local ring with residue field K , and let d : G → F be ahomomorphism of finitely generated B-modules, with F free. Then d is split injectiveif and only if d ⊗B K : G ⊗B K → F ⊗B K is injective. In such a case G is alsofree.

Proof. d is split injective if and only if coker(d) is free and d is injective. If this lastcondition is satisfied then clearly d ⊗B K is injective.

Conversely, assume that d ⊗B K is injective, and factor d as

G → Im(d)→ F

We see thatG ⊗B K → Im(d)⊗B K is bijective

Im(d)⊗B K → F ⊗B K is injective

From the exact sequence

0→ Im(d)→ F → coker(d)→ 0

we get

0→ Tor1(coker(d), K )→ Im(d)⊗B K → F ⊗B K

so Tor1(coker(d), K ) = (0) and this implies that coker(d) is free. From the aboveexact sequence we deduce that Im(d) is free as well, so that

0→ ker(d)→ G → Im(d)→ 0

is split exact. Recalling that G ⊗B K ∼= Im(d) ⊗B K we deduce thatker(d)⊗B K = (0), hence ker(d) = (0) by Nakayama.

For the reader’s convenience we include the proof of the following well-knownlemma:

Lemma A.8. Let (B, m) be a noetherian local integral domain, with residue field Kand quotient field L. If M is a finitely generated B-module and if

dimK (M ⊗B K ) = dimL(M ⊗B L) = r

then M is free of rank r .

274 A Flatness

Proof. Let m1, . . . , mr ∈ M be such that their images in M ⊗B K = M/m M forma basis. Then they define a homomorphism ϕ : Br → M and we have an exactsequence:

0→ N → Br ϕ−→ M → Q → 0

where N and Q are kernel and cokernel of ϕ. Since tensoring with K we get

K r ϕ−→ M/m M → Q/m Q → 0

and ϕ is surjective, we get Q/m Q = (0) and from Nakayama’s lemma it followsthat Q = (0): hence ϕ is surjective. Now we tensor the above exact sequence withL , which is flat over B (by (IV)), and we obtain the exact sequence:

0→ N ⊗B L → Lr ϕ−→ M ⊗B L → 0

Since M ⊗B L ∼= Lr and ϕ is surjective, it follows that N ⊗B L = ker(ϕ) = (0).Therefore N is a torsion module. But N ⊂ Br and therefore N = (0).

We have the following useful criterion:

Lemma A.9. Let A → A′ be a small extension in A, and let g : A → R be ahomomorphism of k-algebras. Let R0 = R ⊗A k. Then g is flat if and only if

ker(R→ R ⊗A A′) ∼= R0

and the homomorphism g′ : A′ → R ⊗A A′ induced by g is flat.

Proof. Assume that g is flat. Then since R ⊗A (ε) ∼= R ⊗A k = R0 andTorA

1 (R, A′) = 0, from the exact sequence

0→ TorA1 (R, A′)→ R ⊗A (ε)→ R→ R ⊗A A′ → 0 (A.2)

we deduce that the first condition is satisfied. The flatness of g′ is obvious.Assume conversely that the conditions of the statement are satisfied. Then the

sequence (A.2) implies that TorA1 (R, A′) = 0. If A′ = k the conclusion follows from

A.3. If not, from the exact sequence

0→ m A′ → A′ → k→ 0

one gets the exact sequence:

TorA1 (R, A′)→ TorA

1 (R, k)∂−→ R ⊗A m A′ → R′ → R ⊗A k → 0

‖ ‖ ‖0 R′ ⊗A′ m A′ R′ ⊗A′ k

From the flatness of R′ over A′ we deduce that ∂ = 0, hence TorA1 (R, k) = 0, and

we conclude by A.3. ∗ ∗ ∗ ∗ ∗ ∗

A Flatness 275

Flatness in terms of generators and relations

Let P be a noetherian k-algebra, J ⊂ P an ideal. Let A be in ob(A),PA = P ⊗k A, and J ⊂ PA an ideal such that (PA/J) ⊗A k ∼= P/J . We wantto find the conditions J has to satisfy so that PA/J is A-flat.

We have the following:

Theorem A.10. Let

%0 : Pn → P N → P → P/J → 0

be a presentation of P/J as a P-module. Then the following conditions are equiva-lent for an ideal J ⊂ PA:

(i) PA/J is A-flat and (PA/J)⊗A k ∼= P/J .(ii) There is an exact sequence

% : PnA → P N

A → PA → PA/J→ 0

such that %0 = %⊗A k (= %/m A%).(iii) There is a complex

% : PnA

ϕ−→ P NA → PA → PA/J→ 0

which is exact except possibly at P NA , such that %0 = %⊗A k.

Proof. (i i)⇒ (i). We have:

TorA1 (PA/J, k) = H1(%⊗ k) = H1(%0) = (0)

From A.3 it follows that PA/J is A-flat. Moreover, (i i) implies that(PA/J)⊗A k ∼= P/J .

(i) ⇒ (i i). Choose a PA-homomorphism p : P NA → J which makes the

following diagram commute:

p : P NA → J↓ ↓

p0 : P N → J

where p0 is the surjective homomorphism defined by the presentation %0. From theflatness of PA/J it follows that TorA

1 (PA/J, k) = (0); hence the exact sequence

0→ TorA1 (PA/J, k) → J⊗ k → PA ⊗ k → (PA/J)⊗A k → 0

‖ ‖P P/J

implies that J⊗ k = J . It follows that p ⊗A k = p0 and therefore

coker(p)⊗A k = coker(p0) = (0)

so that coker(p) = (0) by Nakayama’s lemma. Hence p is surjective.

276 A Flatness

Now consider the exact sequence

0→ ker(p)→ P NA → J→ 0

and the associated Tor sequence:

TorA1 (J, k)→ ker(p)/m Aker(p)→ P N → J → 0 (A.3)

From the flatness of PA/J and from the exact sequence

0→ J→ PA → PA/J→ 0

we have TorA1 (J, k) = TorA

2 (PA/J, k) = (0). Therefore from (A.3) we see that

ker(p)/m Aker(p) ∼= ker(p0)

Arguing as before we can find a surjective homomorphism q : PnA → ker(p) which

makes the following diagram commutative:

PnA

q−→ ker(p)↓ ↓Pn → ker(p0)

(i i)⇒ (i i i) is obvious.

(i i i) ⇒ (i) If % is not exact at P NA then we can add finitely many generators

of the kernel of P NA → PA to obtain an exact sequence

%′ : Pn′A

ϕ′−→ P NA → PA → PA/J→ 0

Then %′ ⊗A k has the form:

Pn′ ϕ′⊗k−→ P N → P → P/J → 0

SinceIm(ϕ ⊗ k) ⊂ Im(ϕ′ ⊗ k) ⊂ ker[P N → P]

we see that Im(ϕ′ ⊗ k) = ker[P N → P] and therefore %′ ⊗A k is exact. Now (i)follows from A.3. Corollary A.11. Assume that J = ( f1, . . . , fN ) ⊂ P and that

J = (F1, . . . , FN ) ⊂ PA

with f j = Fj (mod m A PA), j = 1, . . . , N. Then every relation among f1, . . . , fN

lifts to a relation among F1, . . . , FN if and only if PA/J is A-flat and(PA/J)⊗A k ∼= P/J .

A Flatness 277

Proof. The condition that the Fj ’s reduce to the f j ’s modulo m A PA implies that theexact sequence

P NA

F−→ PA → PA/J→ 0

reduces toP N f−→ P → P/J → 0 (A.4)

when tensored by ⊗Ak. Complete (A.4) to a presentation %0 of P/J . The conditionthat every relation among f1, . . . , fN lifts to a relation among F1, . . . , FN is a re-statement of condition (iii) of A.10. Therefore the conclusion follows from TheoremA.10. Example A.12. Let A be in ob(A). Suppose that f1, . . . , fN ∈ P form a regularsequence, and let F1, . . . , FN ∈ PA be any liftings of f1, . . . , fN , i.e. such thatf j = Fj (mod m PA), j = 1, . . . , N . Then J = (F1, . . . , FN ) ⊂ PA defines a flatfamily of deformations of X = Spec(P/J ), where J = ( f1, . . . , fN ).

In fact, every relation among f1, . . . , fN is a linear combination of the trivialones

ri j = (0, . . . , f j , . . . ,− fi , . . . , 0) 1 ≤ i < j ≤ N

and these can be lifted to the corresponding trivial relations

Ri j = (0, . . . , Fj , . . . ,−Fi , . . . , 0)

among F1, . . . , FN . Applying Corollary A.11 it is easy to show that F1, . . . , FN

form a regular sequence.

NOTES

1. In the proof of Theorem A.10 the condition that A is artinian has only been used inthe proof of (i) ⇒ (i i) in order to apply Nakayama’s lemma. In particular, the implications(i i) ⇒ (i), (i i i) ⇒ (i) and (i i) ⇒ (i i i) hold for any A ∈ ob(A∗). Using the local criterionof flatness it is easy to verify that the implication (i)⇒ (i i) (and therefore the equivalence ofthe three conditions) holds as well if A is in ob(A).

B

Differentials

Let A→ B be a ring homomorphism. As usual, we will denote by ΩB/A the moduleof differentials of B over A, and by dB/A : B → ΩB/A the canonical A-derivation.Recall that

ΩB/A := I/I 2

where I = ker(B ⊗A Bµ−→ B) is the natural map, and for each b ∈ B

dB/A(b) = b ⊗ 1− 1⊗ b

is called the differential of b. We have a natural isomorphism of B-modules

DerA(B, M) ∼= HomB(ΩB/A, M)

Note that the exact sequence

0→ ΩB/A → (B ⊗A B)/I 2 µ′−→ B → 0 (B.1)

where µ′ is induced by µ, is an A-extension of B. The ring

PB/A := (B ⊗A B)/I 2

is called the algebra of principal parts of B over A. The A-extension (B.1) is trivialbecause we have splittings:

λ1, λ2 : B → PB/A

defined by λ1(b) = b ⊗ 1, λ2(b) = 1⊗ b; note that dB/A = λ1 − λ2. We willconsider PB/A as a B-algebra via λ1.

The following are some fundamental properties of the modules of differentials:

Proposition B.1. (i) IfB↑A −→ A′

280 B Differentials

are ring homomorphisms, then:

ΩB/A ⊗A A′ ∼= ΩB⊗A A′/A′

(ii) If A→ B is a ring homomorphism and ∆ ⊂ B is a multiplicative system, then:

Ω∆−1 B/A∼= ∆−1ΩB/A

(iii) Let K → L be a finitely generated extension of fields. Then

dimL(ΩL/K ) ≥ trdeg(L/K )

and equality holds if and only if L is separably generated over K . In particular,ΩL/K = (0) if and only if K ⊂ L is a finite algebraic separable extension.

Proof. See [48]. We now introduce two standard exact sequences.

Theorem B.2 (Relative cotangent sequence). Given ring homomorphisms

Af−→ B

g−→ C

there is an exact sequence of C-modules:

ΩB/A ⊗B Cα−→ ΩC/A

β−→ ΩC/B → 0 (B.2)

where the maps are given by:

α(dB/A(b)⊗ c) = cdC/A(g(b)); β(dC/A(r)) = dC/B(r) b ∈ B, c ∈ C

Proof. See [48], prop. 16.2. When B → C is surjective we have ΩC/B = (0) and the next theorem describes

ker(α).

Theorem B.3 (Conormal sequence). Let

Af−→ B

g−→ C

be ring homomorphisms with g surjective, and let J = ker(g), so that C = B/J .Then:

(i) We have an exact sequence

J/J 2 δ−→ ΩB/A ⊗B Cα−→ ΩC/A → 0 (B.3)

where δ is the C-linear map defined by δ(x) = dB/A(x)⊗ 1.

B Differentials 281

(ii) There is an isomorphism

Ω(B/J 2)/A ⊗(B/J 2) C ∼= ΩB/A ⊗B C

In other words the conormal sequence (B.3) depends only on the first infinitesi-mal neighbourhood of Spec(C) in Spec(B).

(iii) The map δ is a split injection if and only if there is a map of A-algebrasC → B/J 2 splitting the projection B/J 2 → C.

Proof. (i) see e.g. [48], prop. 16.3.(ii) Comparing the exact sequence (B.3) with the analogous sequence associated

to A→ B/J 2 → C we get a commutative diagram:

J/J 2 → ΩB/A ⊗B C → ΩC/A → 0‖ ↓ ‖

J/J 2 → Ω(B/J 2)/A ⊗(B/J 2) C → ΩC/A → 0

and the vertical arrow, which is induced by B → B/J 2, must be an isomorphism.(iii) By (ii) we may assume that J 2 = 0, i.e. that 0 → J → B → C → 0

is an A-extension. Assume that δ : J → ΩB/A ⊗B C is a split injection, and letσ : ΩB/A ⊗B C → J be a splitting. Then the composition

Bd−→ ΩB/A ⊗B C

σ−→ J

is an A-derivation. It follows that 1−σ d : B → B is an A-homomorphism such that(1−σ d)(J ) = 0 and therefore it induces an A-homomorphism C → B which splitsg.

Conversely, assume that g : B → C has a section τ : C → B. Then we have aderivation

D : B → J ⊕ΩC/A

given by D(b) = (b − (τg)(b), dC/A(g(b))). One easily checks that D induces anisomorphism ΩB/A ⊗B C ∼= J ⊕ΩC/A, thus proving the assertion.

As an application we have the following:

Proposition B.4. Let K be a field and (B, m) a local K -algebra with residue fieldB/m = K ′. Then the map

δ : m/m2 → ΩB/K ⊗B K ′

in the exact sequence (B.3) relative to K → B → K ′ is injective if and only ifK ⊂ K ′ is a separable field extension.In particular, if B/m = K then

δ : m/m2 → ΩB/K ⊗B K

is an isomorphism. Therefore

dim(B) ≤ dimK (ΩB/K ⊗B K )

282 B Differentials

Proof. See [48], cor. 16.13. The last assertion follows from the conormal sequencerelative to K → B → K .

The following theorem describes the module of differentials for regular localrings.

Theorem B.5. Assume that K is a field and B is a local noetherian K -algebra withresidue field B/m = K . If ΩB/K is a free B-module of rank equal to dim(B) then Bis a regular local ring. If K is perfect (e.g. algebraically closed) and B is e.f.t. overK then the converse is also true.

Proof. Assume first that ΩB/K is free of rank equal to dim(B). Then

dimK (m/m2) = dim(B)

by B.4, so B is a regular local ring.Assume conversely that K is perfect and that B is a regular local ring, e.f.t. over

K . Then we have

dimK (ΩB/K ⊗B K ) = dimK (m/m2) = dim(B)

Let L be the quotient field of B. Then, by B.1(3), we have

ΩB/K ⊗B L = ΩL/K

anddimL(ΩL/K ) = trdeg(L/K ) = dim(B)

because L is separably algebraic over K , since K is perfect. Therefore we have

dimK (ΩB/K ⊗B K ) = dim(B) = dimLΩB/K ⊗B L

Since B is e.f.t. over K , ΩB/K is a finitely generated B-module, and from LemmaA.8 it follows that it is free of rank equal to dim(B).

In particular, we have the following:

Corollary B.6. Let k be an algebraically closed field, and let B be an integral k-algebra of finite type. Then B is a regular ring if and only if ΩB/k is a projectiveB-module of rank equal to dim(B).

Proof. Both conditions are satisfied if and only if they are satisfied after localizingat the maximal ideals of B. For every maximal ideal m ⊂ B the local ring Bm is ak-algebra e.f.t. with residue field k. By B.5, Bm is a regular local ring if and only ifΩBm/k = (ΩB/k)m is free of rank equal to dim(B). The conclusion follows. Proposition B.7. If the ring homomorphism A → B is e.f.t. then ΩB/A is a B-module of finite type.If, in particular, B = S−1 A[X1, . . . , Xn] for some multiplicative system S, thenΩB/A is a free B-module of rank n with basis dB/A(X1), . . . , dB/A(Xn).

B Differentials 283

Proof. The last assertion is elementary (see [48]). To prove the first, letB = (S−1 P)/J , where P = A[X1, . . . , Xn] and S ⊂ P is a multiplicativesystem. Then ΩB/A is a quotient of ΩS−1 P/A ⊗S−1 P B, by the conormal sequence.

Remark B.8. If A and B are only assumed to be noetherian then ΩB/A is not neces-sarily a B-module of finite type even if A is a field. An example is given by ΩQ[[X ]]/Q(see [1] ch. 0I V , n. 20.7.16).

Examples B.9. (i) Assume that B = S−1 A[X1, . . . , Xn] for some multiplicativesystem S. Then DerA(B, B) = HomB(ΩB/A, B) is a free module of rank n withbasis ∂

∂X1, . . . ,

∂

∂Xn

which is the dual of the basis

dB/A(X1), . . . , dB/A(Xn)

of ΩB/A, and where ∂∂X j: B → B is the partial A-derivation with respect to X j .

Let Y1, . . . , Yn ∈ B be such that the jacobian determinant

det( ∂Yi

∂X j

)is a unit in B. Then

dB/A(Y1), . . . , dB/A(Yn)is another basis of ΩB/A and we have:

dB/A(X j ) = ∂X j

∂Y1dB/A(Y1)+ · · · + ∂X j

∂YndB/A(Yn)

Dually:∂

∂X j= ∂Y1

∂X j

∂

∂Y1+ · · · + ∂Yn

∂X j

∂

∂Yn(B.4)

The proof of these statements is straightforward.

(ii) Let k be a field and let B = k[X, Y ]/(XY ), where X, Y are indeterminates.Then, since Ωk[X,Y ]/K ⊗ B ∼= Bd X⊕ BdY , using the conormal sequence we deducethat

ΩB/k ∼= Bd X ⊕ BdY

(Y d X ⊕ XdY )

It follows that the element Y d X = −XdY is killed by the maximal ideal (X, Y ) andtherefore it generates a torsion submodule

T := (Y d X) ∼= k ⊂ ΩB/k

284 B Differentials

The quotient is

ΩB/k

T= Bd X ⊕ BdY

(Y d X, XdY )∼= k[X ]d X ⊕ k[Y ]dY ∼= (X, Y ) ⊂ B

where the last isomorphism is given by

f (X)d X ⊕ g(Y )dY → f (X)X + g(Y )Y

Therefore we have an exact sequence:

0→ T → ΩB/k → B → k → 0

(iii) Let k be a field and let B = k[t, X, Y ]/( f ) where t, X, Y are indeterminatesand f = XY + t . Then arguing as before we see that

ΩB/k[t] ∼= Bd X ⊕ BdY

(Y d X ⊕ XdY )

The element Y d X = −XdY is not killed by any b ∈ B; therefore ΩB/k[t] is torsionfree of rank one. The homomorphism

ΩB/k[t] → B

sending f (t, X)d X ⊕ g(t, Y )dY → f (t, X)X + g(t, Y )Y is bijective onto the max-imal ideal (t, X, Y ) so that we have an exact sequence:

0→ ΩB/k[t] → B → k → 0

(iv) Let k be a field and let k[ε] := k[t]/(t2), where we have denoted by ε theclass of t mod (t2). Then the conormal sequence of k → k[t] → k[ε] is

(t2)/(t4)δ−→ Ωk[t]/k ⊗k[t] k[ε] → Ωk[ε]/k → 0

and the middle term is isomorphic to k[ε]. The map δ acts as

t2 → 2εt3 → 0

In particular, we see that δ is not injective. Therefore

Ωk[ε]/k =

kdε if char(k) = 2;k[ε]dε if char(k) = 2

and d : k[ε] → Ωk[ε]/k acts as d(α + εβ) = βdε.

(v) An obvious generalization of the above computation shows that ifA = k[t]/(tn), n ≥ 2 and char(k) = 0 or char(k) > n then

B Differentials 285

ΩA/k = A/(t n−1)

(vi) If B ∈ ob(A∗) then t∨B := m B/m2B and tB := (m B/m2

B)∨ are the (Zariski)cotangent space, respectively tangent space of B. We have m B/m2

B∼= ΩB/k ⊗B k

by Prop. B.4, and therefore

Derk(B, k) = HomB(ΩB/k, k) = Homk(ΩB/k ⊗B k, k) = (m B/m2B)∨

Moreover, there is a natural identification

Derk(B, k) = Homk−alg(B, k[ε])which we leave to the reader to verify.

If µ : Λ→ B is a homomorphism in A∗, the induced homomorphism

dµ∨ : mΛ/m2Λ→ m B/m2

B

is the codifferential of µ, while its transpose

dµ : tB → tΛ

is the differential of µ. We define the relative cotangent space of B over Λ to be

t∨B/Λ := coker(dµ∨) = m B/(m2B + mΛB)

and the relative tangent space of B over Λ as its dual:

tB/Λ = ker(dµ) = [m B/(m2B + mΛB)

]∨From the exact sequence

ΩΛ/k ⊗Λ B → ΩB/k → ΩB/Λ→ 0

tensored by k we deduce an identification t∨B/Λ = ΩB/Λ ⊗B k and therefore

tB/Λ = HomB(ΩB/Λ, k) = DerΛ(B, k) = HomΛ−alg(B, k[ε])where the Λ-algebra structure on k[ε] is defined by the composition Λ→ k→ k[ε](the last equality is straightforward to verify).

The following lemma describes a situation where the conormal sequence is exact.

Lemma B.10. Assume char(k) = 0. Let

e : 0→ (t)→ R′ → R→ 0

be a small extension in A. Then the conormal sequence

η : 0→ (t)δ−→ ΩR′/k ⊗R′ R→ ΩR/k → 0

is exact also on the left.

286 B Differentials

Proof. Assume first that e is trivial, so that R′ = R⊕k. Then the codifferential

ΩR′/k ⊗R′ k→ ΩR/k ⊗R k‖ ‖

m R′/m2R′ m R/m2

R

has a nontrivial kernel (Example 1.1.2) so that a fortiori

ΩR′/k ⊗R′ R→ ΩR/k

has a nontrivial kernel.Assume now that e is not trivial. Then, letting m = dim(tR′) = dim(tR), we can

writeR′ = P/J ′, R = P/J

where P = k[X1, . . . , Xm], a polynomial algebra, and J ′, J ⊂ (X)2 ⊂ P idealssuch that J ′ ⊂ J and J/J ′ ∼= (t). Let T ∈ J be such that

t = T + J ′

Since e is small we have (X)J ⊂ J ′ and therefore T /∈ (X)J .

Claim: We can choose T so that

∂T

∂Xi/∈ J for some i

If ∂T∂Xi∈ J then Xi

∂T∂Xi∈ J ′ so that we can replace T by

T1 := T − Xi∂T

∂Xi

If

−Xi∂2T

∂X2i

= ∂T1

∂Xi/∈ J

we are done, otherwise we replace T1 by

T2 := T1 − Xi

2

∂T1

∂Xi= T − Xi

∂T

∂Xi+ X2

i

2

∂2T

∂X2i

and we apply the same argument. After ν steps of this process we obtain

Tν := Tν−1 − Xi

ν

∂Tν−1

∂Xi= T +

ν∑s≥1

(−1)s Xsi

s!∂s T

∂Xsi

Since ∂Tν∂Xi= 0 for ν 0 we see that either ∂Tν

∂Xi/∈ J for some ν and we replace T

by the first Tν with this property, or we can replace T by a Tν which is constant withrespect to Xi . Repeating this process for every index i we will end up by replacing

B Differentials 287

T by a T having the required property or otherwise constant with respect to everyvariable, which is clearly a contradiction. The claim is proved.

From the claim we deduce that

dT =∑

i

∂T

∂Xid Xi /∈ JΩP/k (B.5)

where d = dP/k : P → ΩP/k is the universal derivation. But we have:

ΩR′/k = ΩP/k/(J ′ΩP/k + (dg′)g′∈J ′

)so that

ΩR′/k ⊗R′ R = ΩP/k/(JΩP/k + (dg′)g′∈J ′

)But since clearly dt = dg′ for all g′ ∈ J ′, (B.5) implies that

dt = 0 ∈ ΩR′/k ⊗R′ R

∗ ∗ ∗ ∗ ∗ ∗

If f : X → Y is a morphism of schemes, we denote by Ω1X/Y the sheaf of

relative differentials, or the relative cotangent sheaf, on X . It satisfies

Ω1X/Y,x = ΩOX,x/OY, f (x)

for all x ∈ X . If f : Spec(B)→ Spec(A) is a morphism of affine schemes then

Ω1Spec(B)/Spec(A) = (ΩB/A)

∼

We denote by

TX/Y := Hom(Ω1X/Y ,OX )

the sheaf of relative derivations, or the relative tangent sheaf of f .We will write Ω1

X and TX instead of Ω1X/Spec(k) and TX/Spec(k) respectively; they

are the cotangent sheaf and the tangent sheaf of X , respectively (cotangent and tan-gent bundles if locally free).

If X is algebraic and x ∈ X is closed then, by B.4:

Ω1X,x ⊗ k(x) = m X,x

m2X,x

is the cotangent space of X at x , and

Tx X := TX,x ⊗ k(x) =(m X,x

m2X,x

)∨ ∼= Derk(OX,x , k)

is the Zariski tangent space of X at x .

288 B Differentials

Let S be a scheme andX

g−→ Y

a morphism of S-schemes. The induced homomorphism of sheaves on X :

g∗Ω1Y/S → Ω1

X/S

is called the relative codifferential of g. The dual homomorphism:

TX/S → Hom(g∗Ω1Y/S,OX )

is the relative differential of g. When S = Spec(k) we have g∗Ω1Y → Ω1

X , which isthe codifferential of g, while its dual

dg : TX → Hom(g∗Ω1Y ,OX )

is the differential of g. Note that if Ω1Y/S is locally free then

Hom(g∗Ω1Y/S,OX ) = g∗Hom(Ω1

Y/S,OY ) = g∗TY/S

but in general the first and the second sheaves are different. The relative cotangentsequence is

g∗Ω1Y/S → Ω1

X/S → Ω1X/Y → 0 (B.6)

Conditions for the injectivity of the first map in this sequence are given in TheoremC.15, page 302 and Theorem D.2.8, page 310.

If X ⊂ Y is an embedding of schemes and I = IX/Y ⊂ OY is the ideal sheaf ofX in Y , then I/I2 is a sheaf of OX -modules in a natural way, called the conormalsheaf of X in Y . Its dual

NX/Y := HomOX (I/I2,OX ) = HomOY (I,OX )

is called the normal sheaf of X in Y . NX/Y (resp. I/I2) is called the normal bundle(resp. the conormal bundle) of X in Y if it is locally free. Given a closed embeddingof S-schemes i : X ⊂ Y , we have an exact sequence of sheaves on X :

I/I2 → i∗Ω1Y/S → Ω1

X/S → 0 (B.7)

where I ⊂ OY is the ideal sheaf of X in Y . (B.7) is called the relative conormalsequence. When S = Spec(k) we obtain the conormal sequence

I/I2 → i∗Ω1Y → Ω1

X → 0

Conditions for the injectivity of the first map in these sequences are given in Propo-sition D.1.4, page 306, and Theorem D.2.7, page 310.

Examples B.11. In the following examples we will describe the global vector fieldson the given schemes by exhibiting their restrictions to an affine open set. All will bedone by explicit computation.

B Differentials 289

(i) H0(TIP1) can be described explicitly as follows. Consider IP1 = U0 ∪ U1where U0 = Spec(k[ξ ]) and U1 = Spec(k[η]) with η = ξ−1 on U0 ∩U1. We have

∂

∂η= ∂ξ

∂η

∂

∂ξ= − 1

η2

∂

∂ξ= −ξ2 ∂

∂ξ

on U0 ∩U1. Let θ ∈ H0(TIP1); then

θ|U0 = g(ξ)∂

∂ξg(ξ) ∈ k[ξ ]

and

θ|U1 = h(η)∂

∂η, h(η) ∈ k[η]

On U0 ∩U1 we have

g(ξ)∂

∂ξ= h(η)

∂

∂η= −h(ξ−1)ξ2 ∂

∂ξ

and therefore g(ξ) = −h(ξ−1)ξ2. It follows that g(ξ) = a0 + a1ξ + a2ξ2 and

h(η) = −(a0η2 + a1η + a2), with a0, a1, a2 ∈ k. In particular, H0(TIP1) ∼= k3.

Moreover, Hi (TIP1) = 0 if i ≥ 1. For i ≥ 2 it is obvious. Let θ ∈ H1(TIP1)be represented by a Cech 1-cocycle defined by θ01 ∈ Γ (U0 ∩ U1, TIP1). It can bewritten as

θ01 =n∑

i=−m

aiξi

Letting θ1 =∑−1i=−m aiη

−i and θ0 = −∑ni=0 aiξ

i we obtain:

θ01 = θ1 − θ0

so (θ01) is a coboundary.(ii) We want to describe H0(TA1×IP1). Let A1 × IP1 = V0 ∪ V1 where

V0 = A1 ×U0 = Spec[z, ξ ])V1 = A1 ×U1 = Spec[z, η])

and η = ξ−1 on V0 ∩ V1 = Spec(k[z, ξ, ξ−1]). We have

∂

∂η= ∂ξ

∂η

∂

∂ξ= − 1

η2

∂

∂ξ= −ξ2 ∂

∂ξ

on V0 ∩ V1. Let θ ∈ H0(TA1×IP1); then

θ|V0 = g(z, ξ)∂

∂z+ h(z, ξ)

∂

∂ξg(z, ξ), h(z, ξ) ∈ k[z, ξ ]

θ|V1 = γ (z, η)∂

∂z+ χ(z, η)

∂

∂ηγ (z, η), χ(z, η) ∈ k[z, η]

290 B Differentials

On V0 ∩ V1 we have:g(z, ξ) = γ (z, ξ−1)

and therefore g(z, ξ) = g(z) is constant with respect to ξ . Moreover,

h(z, ξ)∂

∂ξ= χ(z, η)

∂

∂η= −χ(z, ξ−1)ξ2 ∂

∂ξ

and thereforeh(z, ξ) = −χ(z, ξ−1)ξ2

It follows that h(z, ξ) = a(z) + b(z)ξ + c(z)ξ2, with a(z), b(z), c(z) ∈ k[z]. Inconclusion, every θ ∈ H0(TA1×IP1) restricts to V0 as a vector field of the form

θ|V0 = g(z)∂

∂z+ (a(z)+ b(z)ξ + c(z)ξ2)

∂

∂ξ(B.8)

with g(z), a(z), b(z), c(z) ∈ k[z], and conversely, every such vector field is therestriction of a global section of TA1×IP1 . As in example (i) we also deduce thatHi (TA1×IP1) = 0 if i ≥ 1.In a similar way, one describes H0(T(A1\0)×IP1) by showing that the image of therestriction

H0(T(A1\0)×IP1)→ H0(T(A1\0)×U0)

consists of the vector fields of the form (B.8) with g(z), a(z), b(z), c(z) ∈ k[z, z−1].(iii) We now consider, for a given integer m ≥ 0, the rational ruled surface

Fm = IP(OIP1(m)⊕OIP1)

Let π : Fm → IP1 be the projection. Then Fm can be represented as

Fm = π−1(U ) ∪ π−1(U ′) = (U × IP1) ∪ (U ′ × IP1)

where U = Spec(k[z]), U ′ = Spec(k[z′]) and z′ = z−1 on U ∩U ′. We consider theaffine open sets

V0 = Spec(k[z, ξ ]) ⊂ U × IP1

V ′0 = Spec(k[z′, ξ ′]) ⊂ U ′ × IP1

where on V0 ∩ V ′0 = Spec(k[z, z−1, ξ ]) = Spec(k[z′, z′−1, ξ ′]) we have:

z′ = z−1, ξ ′ = zmξ

Therefore we have:∂∂z′ = −z2 ∂

∂z + mzξ ∂∂ξ

∂∂ξ ′ = z−m ∂

∂ξ

(B.9)

We will describe a typical element θ ∈ H0(TFm ) by describing its restriction to theopen sets V0 and V ′0. We have, by example (ii) above:

B Differentials 291

θ|V0 = g(z)∂

∂z+ (a(z)+ b(z)ξ + c(z)ξ2)

∂

∂ξ

with g(z), a(z), b(z), c(z) ∈ k[z] and similarly

θ|V ′0 = ρ(z′) ∂

∂z′+ (α(z′)+ β(z′)ξ ′ + γ (z′)ξ ′2) ∂

∂ξ ′

with ρ(z′), α(z′), β(z′), γ (z′) ∈ k[z′]. Imposing their equality on V0 ∩ V ′0 and using(B.9) we obtain the following conditions:

g(z) = −ρ(z−1)z2

a(z) = α(z−1)z−m

b(z) = β(z−1)+ ρ(z−1)mzc(z) = γ (z−1)zm

(B.10)

We distinguish the cases m = 0 and m > 0. If m = 0 (B.10) give:

g(z) = g0 + g1z + g2z2

a(z) = ab(z) = bc(z) = c

g0, g1, g2, a, b, c ∈ k

In the case m > 0 we have:

g(z) = g0 + g1z + g2z2

a(z) = 0b(z) = b − mz(g1 + g2z)c(z) = c0 + c1z + · · · + cm zm

g0, g1, g2, b, c0, . . . , cm ∈ k

Since the restriction H0(TFm ) → H0(TV0) is injective and we have described itsimage, we can conclude:

H0(TF0)∼= k6

H0(TFm ) ∼= km+5

In particular, Fm and Fn are not isomorphic if m = n (note that F0 ∼= IP1 × IP1 isnot isomorphic to F1 ∼= Bl(1,0,0) IP2).

Since, by the calculations of the previous example (ii)

hi (TU×IP1) = hi (TU ′×IP1) = hi (T(U∩U ′)×IP1) = 0, i ≥ 1 (B.11)

we deduce that:

H1(TFm ) = H0(T(U∩U ′)×IP1)/H0(TU×IP1)+ H0(TU ′×IP1)

An easy computation based on (B.10) shows that, for m ≥ 1, H1(TFm ) consists ofthe classes, modulo H0(TU×IP1)+ H0(TU ′×IP1), of the vector fields

(b1z + · · · + bm−1zm−1)∂

∂ξ

292 B Differentials

In particular,H1(TFm ) ∼= km−1 (B.12)

It also follows from (B.11) that

H2(TFm ) = (0) (B.13)

NOTES

1. Let X → Y be a morphism of algebraic schemes. Prove that there is an exact sequence

0→ Ω1X/Y → P1

X/Y → OX → 0 (B.14)

which globalizes (B.1). P1X/Y is called the sheaf of principal parts of X over Y , denoted by

P1X if Y = Spec(k).

Let X = IP(V ) for a finite-dimensional k-vector space V . Then the exact sequence (B.14)is the dual of the Euler sequence; in particular,

P1IP(V )

∼= OIP(V )(−1)⊗ V∨

Therefore (B.14) is a generalization of the Euler sequence to any X → Y .

2. Consider IP = IP(V ) for a finite-dimensional k-vector space V and the incidencerelation:

I = (x, H) : x ∈ H ⊂ IP × IP∨ (B.15)

Consider the twisted and dualized Euler sequence:

0→ Ω1IP(V )(1)→ OIP(V ) ⊗ V∨ → OIP(V )(1)→ 0

From its definition it follows that I = IP(Ω1IP(V )(1)) and IP × IP∨ = IP(OIP(V ) ⊗ V∨) and

the inclusion in (B.15) is induced by the first homomorphism in the above sequence.

C

Smoothness

The notion of “formal smoothness”, introduced in [1], Ch. IV § 17, is of crucialimportance in deformation theory, and therefore plays a special role in this book. Inthis appendix we introduce this concept from scratch, and we show how it is relatedto the notion of “smooth morphism” as introduced in [3] Expose II, and [84]. We willnot give a systematic treatment of the properties of smooth morphisms in algebraicgeometry: the reader is referred to the above quoted references for them. For moredetails on the approach taken in this section the reader can also consult [13] and [94].

Definition C.1. A ring homomorphism f : R → B is called formally smooth, andB is called a formally smooth R-algebra, if for every exact sequence:

0→ I → Aη−→ A′ → 0 (C.1)

where A and A′ are local artinian R-algebras, each R-algebra homomorphism B →A′ has a lifting B → A; equivalently, if the map:

HomR−alg(B, A)→ HomR−alg(B, A′) (C.2)

is surjective.f is called smooth if it is formally smooth and essentially of finite type (shortly e.f.t.).If the map (C.2) is bijective (instead of only being surjective) for all exact sequences(C.1), then f is formally etale; f is etale if it is formally etale and e.f.t..

Recall that e.f.t. means that B is a localization of an R-algebra of finite type(see e.g. [127]). It is easy to prove by induction that it suffices to check the aboveconditions only for the exact sequences (C.1) such that I 2 = (0), i.e. for extensionsof local artinian R-algebras.

Proposition C.2. (i) If B is a ring and ∆ ⊂ B is a multiplicative system, thenB → ∆−1 B is formally etale. In particular, B is a formally etale B-algebra.

(ii) The composition of formally smooth (resp. formally etale) homomorphisms isformally smooth (resp. formally etale).

294 C Smoothness

(iii) If f : R → B is formally smooth (resp. formally etale) and C is an R-algebra,then C → C ⊗R B is formally smooth (resp. formally etale).

(iv) A finitely generated field extension K ⊂ L is smooth if and only if L is separableover K .

(v) Let Rf−→ B

g−→ C be ring homomorphisms, and assume that f is formallyetale. Then g f is formally smooth (resp. formally etale) if and only if g is for-mally smooth (resp. formally etale).

Proof. (i) Given an exact sequence (C.1) and a commutative diagram

B → ∆−1 B↓ ϕ′ ↓ ϕA −→

pA′

we must find ϕ : ∆−1 B → A which makes it commutative. For every s ∈ ∆ chooseas ∈ A such that ϕ(s)−1 = p(as). Since

p(ϕ′(s)as) = ϕ(s)ϕ(s)−1 = 1A′

we have

ϕ′(s)as = 1A + is is ∈ I

for every s ∈ ∆. Therefore

ϕ′(s)as(1A − is) = 1A

Hence ϕ′(s) ∈ A is invertible. Now define ϕ(r/s) = ϕ′(r)ϕ′(s)−1.Noting that ϕ is uniquely determined by ϕ′ we get the assertion.(ii) and (iii) are straightforward.(iv) Assume first that K ⊂ L is separable. By (ii) it suffices to consider the cases

L = K (X) and L = K [X ]/( f (X)) where f is irreducible and f ′(x) = 0. The firstcase is left to the reader (see remark C.3(i)).In the second case consider an extension A = A/I of local artinian K -algebras,where I ⊂ A is an ideal with I 2 = (0). Let ϕ : K [X ]/( f (X))→ A be a homomor-phism, sending X → α. Choose arbitrarily α ∈ A such that α = α mod I . It willsuffice to find e ∈ I such that

f (α + e) = 0

We have f (α + e) = f (α)+ f ′(α)e. Since f ′(α) is a unit mod I it is also a unit inA, and therefore we can take e = − f (α)/ f ′(α).

Assume conversely that K ⊂ L is smooth. Then L = F[X ]/J where F is apurely transcendental extension of K and J is a principal ideal. We have an exactsequence of finite-dimensional L-vector spaces:

J/J 2 → ΩF[X ]/K ⊗ L → ΩL/K → 0

C Smoothness 295

where J/J 2 is one-dimensional. By the first part of the proof F is smooth over Kand by B.3(ii) the left map is injective because, by the smoothness of L over K , thesurjection F[X ]/J 2 → L splits. It follows that

dim(ΩL/K ) = dim(ΩF[X ]/K ⊗ L)− 1 = trdegK (F[X ])− 1

= trdegK (F) = trdegK (L)

From B.1(iii) it follows that K ⊂ L is separable.(v) “if” follows immediately from (ii); “only if” is left to the reader.

Remarks C.3. (i) Any polynomial algebra R[X1, X2 . . .] is trivially a formallysmooth R-algebra. From C.2(i) it follows that a localization of a polynomial R-algebra is also a formally smooth R-algebra.

More precisely, a localization P = S−1 R[X1, X2, . . .] of a polynomial algebraover a ring R satisfies the following condition, stronger than formal smoothness:

For every extension of R-algebras:

0→ I → A→ A′ → 0

where A and A′ are R-algebras and I 2 = 0 the map

HomR−alg(P, A)→ HomR−alg(P, A′)

is surjective.

Every R-algebra B is a quotient of a formally smooth R-algebra, because it is aquotient of a polynomial R-algebra. From C.2(i) it follows that every e.f.t. R-algebrais a quotient of a smooth R-algebra.

This is trivial for polynomial rings, and in the general case it can be provedadapting the proof of C.2(i) in an obvious way.

(ii) if R is in ob(A) then every formal power series ring R[[X1, X2, . . .]] is aformally smooth R-algebra, because local artinian R-algebras are complete. Moreprecisely, a formal power series ring R[[X1, X2, . . .]] satisfies the following condi-tion, stronger than formal smoothness over R:

For every extension:0→ I → A→ A′ → 0

of complete local R-algebras the map

HomR−alg(P, A)→ HomR−alg(P, A′)

is surjective.

The proof is straightforward and is left to the reader.

The following result characterizes an important class of formally smoothalgebras.

296 C Smoothness

Theorem C.4. Let k be a field and let (B, m) be a noetherian local k-algebra withresidue field K . Suppose that K is finitely generated and separable over k. Then thefollowing are equivalent:

(i) B is regular.(ii) B ∼= K [[X1, . . . , Xd ]], where d = dim(B).(iii) B is a formally smooth k-algebra.

Proof. (i)⇔ (ii) is standard (see [48], prop. 10.16 and exercise 19.1).(ii)⇒ (iii). It follows directly from the definition that B is formally smooth over k ifand only if B is. Since B is formally smooth over K (remark C.3(ii)), and since K issmooth over k by C.2(iv), the conclusion follows by transitivity.(iii) ⇒ (i). Let x1, . . . , xd be a system of generators of m. Then, since B/m2 iscomplete and K is separable over k, B/m2 contains a coefficient field ([48], Theorem7.8). Therefore there exists an isomorphism

v1 : B/m2 ∼= K [X1, . . . , Xd ]/M2 M = (X1, . . . , Xd)

Let v : B → B/m2 v1−→ K [X1, . . . , Xd ]/M2. By the formal smoothness of B andby induction we can find a lifting of v:

vn : B → K [X1, . . . , Xd ]/Mn+1

for every n ≥ 2. Consider the elements

vn(x1), . . . , vn(xd) ∈ M/Mn+1

Their classes generate M/M2, hence they generate M/Mn+1, by Nakayama. Thenwe have:

K [X1, . . . , Xd ]/Mn+1 = vn(B)+ (M/Mn+1)

= vn(B)+∑i vn(xi )[vn(B)+ (M/Mn+1)

] = vn(B)+ (M/Mn+1)2 = · · ·

= vn(B)+ (M/Mn+1)n+1 = vn(B)

hence vn is surjective. Since mn+1 ⊂ ker(vn) we have:

(B/mn+1) ≥ (K [X1, . . . , Xd ]/Mn+1) =(

d + n

d

)and this implies that dim(B) ≥ d. Since m is generated by d elements it follows thatB is regular.

For the reader’s convenience we include the proof of the following well-knownlemma:

Lemma C.5. (i) A surjective endomorphism f : A → A of a noetherian ring is anautomorphism.

C Smoothness 297

(ii) Let A be a complete noetherian local ring and ψ : A → A an endomorphisminducing an isomorphism ψ1 : A/m2

A → A/m2A. Then ψ is an automorphism.

Proof. (i) We have an ascending chain of ideals

ker( f ) ⊆ ker( f 2) ⊆ ker( f 3) ⊆ · · ·Since A is noetherian we have ker( f n) = ker( f n+1) = ker( f n+2) = · · · for somen, and it suffices to prove that ker( f n) = (0). After replacing f by f n we mayassume ker( f ) = ker( f 2). Let a ∈ ker( f ); by assumption there exists b ∈ A suchthat a = f (b). Then 0 = f (a) = f 2(b) and therefore b ∈ ker( f 2) = ker( f ), i.e.a = f (b) = 0.

(ii) Let gr(A) = A/m⊕m/m2⊕· · · be the associated graded ring. Since gr(A) isgenerated by m/m2 over A/m the endomorphism gr(ψ) : gr(A)→ gr(A) inducedby ψ is surjective. It follows that ψ is also surjective. In fact, given a ∈ A thesurjectivity of gr(ψ) implies that there are a1, a2, a3, . . . , b1, b2, b3, . . . ∈ A suchthat ai ∈ mi−1, bi ∈ mi , and

a = f (a1)+ b1, b1 = f (a2)+ b2, b2 = f (a3)+ b3, . . .

We obtain a convergent power series a = a1 + a2 + a3 + · · · such that

a − ψ(a1 + a2 + · · · + an) = bn ∈ mn+1

On the limit we therefore get a = ψ(a). The conclusion is now a consequenceof (i). Proposition C.6. Let f : R → B be a local homomorphism of noetherian localrings containing a field k isomorphic to their residue fields. Then the following con-ditions are equivalent:

(i) f is formally smooth.(ii) B is isomorphic to a formal power series ring over R.(iii) The homomorphism f : R→ B induced by f is formally smooth.

Proof. (i)⇒ (ii). Let m ⊂ B and n ⊂ R be the maximal ideals. Choose elementsx1, . . . , xd ∈ B inducing a k-basis of B/(m2+ f (n)), and let F = R[[X1, . . . , Xd ]],where X1, . . . , Xd are indeterminates. Denote by M ⊂ F the maximal ideal.

The homomorphismu : F → B

Xi → xi

induces an isomorphism

u1 : F/(M2 + nF)→ B/(m2 + f (n))

By the formal smoothness of f the composition

v1 : B → B → B/(m2 + f (n))u−1

1−→ F/(M2 + nF)

298 C Smoothness

can be lifted to an R-homomorphism

vk : B → F/Mk

for each k ≥ 2. Therefore the sequence vk defines an R-homomorphism

v : B → F

such that vu : F → F and uv : B → B induce isomorphisms(vu)1 : F/M2 → F/M2 and (uv)1 : B/m2 → B/m2 respectively. From LemmaC.5 it follows that u and v are isomorphisms.(ii)⇒ (iii) is obvious.(iii)⇒ (i) is left to the reader. Corollary C.7. Let f : R → B be a local homomorphism of noetherian local ringscontaining a field k isomorphic to their residue fields. Then the following conditionsare equivalent:

(i) f is formally etale.(ii) The homomorphism f : R→ B induced by f is an isomorphism.

Proof. Left to the reader.

Corollary C.8. Let R be in ob(A∗). The inclusion f : R→ R is formally etale.

The proof is obvious. We now restrict our attention to smooth homomorphisms,i.e. we add the condition that the homomorphism is e.f.t.. In this case the module ofdifferentials comes into play; moreover, the defining condition of Definition C.1 canbe replaced by the more general condition (i) in the following statement.

Theorem C.9. Let f : R → B be an e.f.t. ring homomorphism. Then the followingconditions are equivalent:

(i) For every extension of R-algebras:

0→ I → A→ A′ → 0 (C.3)

the mapHomR−alg(B, A)→ HomR−alg(B, A′)

is surjective.(ii) If B = P/J , where P = S−1 R[X1, . . . , Xd ], S ⊂ R[X1, . . . , Xd ] is a multi-

plicative system and J ⊂ P is an ideal, the conormal sequence

0→ J/J 2 δ−→ ΩP/R ⊗P B → ΩB/R → 0

is split exact. In particular, J/J 2 and ΩB/R are finitely generated projectiveB-modules.

C Smoothness 299

(iii) B is a smooth R-algebra.(iv) (Jacobian criterion of smoothness) If P and J are as in (ii) the map

(J/J 2)⊗B K (p)δ⊗B K (p)−→ ΩP/R ⊗P K (p) where K (p) = Bp/m Bp

is injective for every prime ideal p ⊂ B.

Proof. (i)⇒ (ii). The hypothesis implies that the extension:

0→ J/J 2 → P/J 2 → B → 0

splits. Therefore the conormal sequence is split exact by B.3(iii) and it follows thatJ/J 2 and ΩB/R are finitely generated projective because the module ΩP/R ⊗P B isfree of finite rank.

(ii)⇒ (i). Consider an exact sequence (C.3) and a homomorphism of R-algebrasf ′ : B → A′. By Remark C.3(i) there exists an R-homomorphism g : P → Amaking the following diagram commute:

P → B↓ g ↓ f ′A → A′

Since g(J ) ⊂ I , we see that g factors through P/J 2, so that we have a commutativediagram:

P/J 2 → B↓ g ↓ f ′A → A′

The hypothesis implies, via B.3(iii), that there exists h : B → P/J 2 a splitting ofP/J 2 → B. The composition f = gh : B → A gives a lifting of f ′.

(i)⇒ (iii) is obvious.

(iii)⇒ (iv). We may assume B and P local with residue field K . To prove thatδ ⊗B K is injective, it suffices to show that for every K -vector space V the mapinduced by δ:

HomK (ΩP/R ⊗P K , V ) → HomK ((J/J 2)⊗B K , V )‖ ‖

DerR(P, V ) HomB(J/J 2, V )

is surjective. Consider a homomorphism g : J/J 2 → V , and the associated pushoutdiagram (see § 1.1 for the definition):

Λ : 0 → J/J 2 → P/J 2 → B → 0↓ g ↓ ‖

g∗(Λ) : 0 → V → Q → B → 0

300 C Smoothness

We can write m Q = V ⊕m′, where m′ ⊂ Q is an ideal, because V is annihilated bym Q . Therefore the previous diagram can be embedded in the following:

P↓

Λ : 0 → J/J 2 → P/J 2 → B → 0↓ g ↓ ‖

g∗(Λ) : 0 → V → Q → B → 0‖ ↓ ↓ v

η : 0 → V → Q/m′ → K → 0

where η is an extension of local artinian R-algebras. From the smoothness of B wededuce the existence of v : B → Q/m′ lifting the projection v : B → K . Denotingby r : P → B the natural map, and by w : P → P/J 2 → Q → Q/m′ thecomposition, consider the homomorphism:

d = w − vr : P → V

It is easy to show that this is an R-derivation, which induces g.

(iv)⇒ (ii). From Nakayama’s lemma it follows that ker(δ) ⊗ Bp = (0) for allprime ideals p ⊂ B and therefore ker(δ) = (0). Moreover, since ΩP/R⊗B Bp is free

and finitely generated it follows that TorBp1 (ΩB/R ⊗B Bp, K (p)) = 0: it follows that

ΩB/R ⊗B Bp is flat, and therefore, being finitely generated, it is free. Thus ΩB/R isprojective, δ has a splitting and J/J 2 is also projective and finitely generated.

The following result follows easily from what we have seen so far.

Theorem C.10. Let B be an integral k-algebra of finite type and of dimension d.Then the following are equivalent:

(i) Bp is smooth over k for each prime ideal p ∈ Spec(B).(ii) B is a regular ring.(iii) ΩB/k is projective of rank d.(iv) B is smooth over k.

Proof. (ii)⇔ (iii) is Corollary B.6.(i)⇔ (ii) follows from Theorem C.4.(iv) ⇒ (i). for each p ∈ Spec(B), Bp is smooth over B by Proposition C.2(i);

from Proposition C.2(ii) it follows that Bp is smooth over k.(i)⇒ (iv). (i) implies that condition (iv) of Theorem C.9 is satisfied for all p ∈

Spec(B), so that B is smooth by Theorem C.9. From now on we will freely replace the defining property for smooth homo-

morphisms given in Definition C.1 by condition (i) of Theorem C.9. Here is a firstexample.

Proposition C.11. Let R be a ring, P an R-algebra and B = P/J for an idealJ ⊂ P. If B is a smooth R-algebra the conormal sequence

C Smoothness 301

0→ J/J 2 → ΩP/R ⊗P B → ΩB/R → 0

is split exact and ΩB/R is projective and finitely generated. If, moreover, P is asmooth R-algebra then J/J 2 is finitely generated and projective as well.

Proof. Since B is smooth the R-algebra extension

0→ J/J 2 → P/J 2 → B → 0

splits. Therefore the conormal sequence splits by Theorem B.3(iii) and ΩB/R is fi-nitely generated and projective by Theorem C.9. If P is smooth then ΩP/R is finitelygenerated and projective as well and so is J/J 2. Corollary C.12. Let P be an e.f.t. k-algebra and B = P/J for an ideal J ⊂ P.Assume that B is reduced. Then in the conormal sequence

J/J 2 δ−→ ΩP/k ⊗P B → ΩB/k → 0 (C.4)

ker(δ) is a torsion B-module whose support is contained in the singular locus ofSpec(B).

If J/J 2 is torsion free then δ is injective.

Proof. Since B is reduced there is a dense open subset U ⊂ Spec(B) such that Bp

is a regular local ring for all p ∈ U . From Theorem C.4 it follows that Bp is asmooth k-algebra for all such p and, by Propositions C.11 and B.1(ii), the conormalsequence (C.4) localized at p is split exact. It follows that ker(δ)p = (0) for allp ∈ U and the conclusion follows. The last assertion is an obvious consequence ofthe first part.

The next result explains the relation between smoothness and the relative cotan-gent sequence.

Theorem C.13. Let Kf−→ R

g−→ B be ring homomorphisms, with g smooth. Thenthe relative cotangent sequence:

0→ ΩR/K ⊗R Bα−→ ΩB/K → ΩB/R → 0

is split exact.

Proof. By Theorem B.2 it suffices to prove that α is a split injection; this is equivalentto showing that, for any B-module M , the induced map:

HomB(ΩB/K , M)α∨−→ HomB(ΩR/K ⊗R B, M)

‖ ‖DerK (B, M) DerK (R, M)

D′ → D′g

302 C Smoothness

is split surjective. Let D : R → M be a K -derivation and consider the commutativediagram:

B1B−→ B

↑ g ↑R

γ−→ B⊕M

where γ (r) = (g(r), D(r)), r ∈ R. By the smoothness of g we can find a homomor-phism of R-algebras ψ : B → B⊕M making the diagram

B1B−→ B

↑ g ψ ↑R −→

γB⊕M

commutative. The homomorphism ψ is necessarily of the form:

ψ(b) = (b, D′(b))

and D′ : B → M is a K -derivation such that D = D′g. This proves the surjectivityof α∨. Now take M = ΩR/K ⊗R B and D = dR/K ⊗ g : R→ ΩR/K ⊗R B and let

α′ : ΩB/K → ΩR/K ⊗R B

be the B-linear map corresponding to D′ : B → ΩR/K ⊗R B. Then α′α = 1M andthis proves that α is split injective.

Corollary C.14. Let Kf−→ R

g−→ B be ring homomorphisms, with g etale. Then

ΩR/K ⊗R B ∼= ΩB/K

andΩB/R = (0)

Proof. By the relative cotangent sequence the two assertions are equivalent. We willprove the first. Keeping the notations of the proof of C.13, the hypothesis that g isetale implies that the derivation D′ is unique and consequently α is an isomorphism.

∗ ∗ ∗ ∗ ∗ ∗A morphism ϕ : X → Y of algebraic schemes is smooth at a point x ∈ X if OX,x

is a smooth OY,ϕ(x)-algebra; ϕ is smooth if it is smooth at every point. The definitionof etale morphism is given similarly. This definition is equivalent to the definition ofsmooth (resp. etale) morphism as given in [3] and in [84]. The equivalence can beseen by means of the jacobian criterion of smoothness, proved in Theorem C.9, andusing [3], Expose II, Corollaire 5.9.

By translating into geometrical language the algebraic results proved above wededuce in particular the following.

Theorem C.15. Let S be an algebraic scheme, and ϕ : X → Y a morphism ofalgebraic S-schemes. Then:

C Smoothness 303

(i) If ϕ is smooth at x ∈ X then the relative cotangent sequence

0→ ϕ∗Ω1Y/S → Ω1

X/S → Ω1X/Y → 0

is split exact at x and Ω1X/Y is locally free at x. The rank of the free module

Ω1X/Y,x is called the relative dimension of ϕ at x.

(ii) ϕ is etale at x ∈ X if and only if it is smooth of relative dimension zero at x.In particular, Ω1

X/Y,x = 0 (i.e. ϕ is unramified at x) and therefore we have anisomorphism

ϕ∗Ω1Y/S,x

∼= Ω1X/S,x

(iii) If X is smooth over S at x and ϕ is a closed embedding with ideal sheaf I ⊂ OY

then the relative conormal sequence

O → I/I2 → ϕ∗Ω1Y/S → Ω1

X/S → 0

is exact at x and Ω1X/S is free at x; if, moreover, Y is also smooth over S at ϕ(x)

then I/I2 is free at x as well.

The exactness of the relative conormal sequence in part (iii) holds under moregeneral assumptions as well (see Theorem D.2.7).

D

Complete intersections

D.1 Regular embeddings

Definition D.1.1. An embedding of schemes j : X ⊂ Y is a regular embedding ofcodimension n at the point x ∈ X if j (x) has an affine open neighbourhood Spec(R)in Y such that the ideal of j (X) ∩ Spec(R) in R can be generated by a regularsequence of length n. If this happens at every point of X we say that j is a regularembedding of codimension n.

An open embedding is a regular embedding of codimension 0. If X and Y areboth nonsingular then X ⊂ Y is a regular embedding. The set of points of X wherean embedding j : X ⊂ Y is regular is open.

If X ⊂ Y is a regular embedding of codimension n then I/I2 and NX/Y are bothlocally free of rank n ([48], Exercise 17.12, p. 440). It follows from standard facts incommutative algebra (see [48], Exercise 17.16, p. 441) that Ik/Ik+1 is locally freeas well for every k ≥ 2.

A ring B is called a complete intersection if Spec(B) can be regularly embeddedin Spec(R) where R is a regular ring.

A scheme X is a local complete intersection (l.c.i.) if every local ring OX,x is acomplete intersection ring.

A nonsingular scheme X , i.e. a scheme all of whose local rings are regular, isan example of an l.c.i. scheme. If X ⊂ Y is a regular embedding and Y is an l.c.i.scheme, then X is an l.c.i. scheme.

Lemma D.1.2. Let f : X → Y be a morphism of schemes and let Z ⊂ Y be aregular embedding of codimension n. Then the induced embedding j : X ×Y Z ⊂ Xhas codimension ≤ n at every point and if equality holds at a point x ∈ X ×Y Z thenj is regular at x.

Proof. If IZ ⊂ OY is the ideal sheaf of Z in Y then the ideal sheaf f −1IZ ofX ×Y Z ⊂ X is locally generated at a point x by the n images of the local generatorsof IZ , f (x). The conclusion follows easily from this fact.

306 D Complete intersections

If we have a flag of embeddings of schemes X ⊂ Y ⊂ Z and IY ⊂ IX ⊂ OZ

are the ideal sheaves of X and Y , we have the exact sequence

0→ IY → IX → IX/Y → 0 (D.1)

where IX/Y ⊂ OY is the ideal sheaf of X in Y . After tensoring by⊗OZOX we obtainan exact sequence of coherent OX -modules:

IY

I2Y

⊗OXα−→ IX

I2X

→ IX/Y

I2X/Y

→ 0 (D.2)

Its dual is the sequence:

0→ NX/Y → NX/Z → NY/Z ⊗OX (D.3)

Lemma D.1.3. (i) If f : X ⊂ Y and g : Y ⊂ Z are regular embeddings of codi-mensions m and n respectively, then g f : X → Z is a regular embedding ofcodimension m + n.

(ii) If the embeddings f and g are both regular then we have exact sequences oflocally free sheaves on X:

0→ IY

I2Y

⊗OXα−→ IX

I2X

→ IX/Y

I2X/Y

→ 0 (D.4)

0→ NX/Y → NX/Z → NY/Z ⊗OX → 0 (D.5)

Proof. (i) Left to the reader.(ii) All sheaves in (D.4) are locally free because they are conormal bundles of regu-lar embeddings. Since Im(α) is a torsion free sheaf of the same rank of (IY /I2

Y ) ⊗OX , it follows that α must be injective. The sequence (D.5) is exact becauseExt1

OX(IX/Y /I2

X/Y ,OX ) = 0. Proposition D.1.4. Let j : X ⊂ Y be an embedding of algebraic schemes, with Xreduced and Y nonsingular. Consider the conormal sequence

I/I2 δ−→ Ω1Y |X → Ω1

X → 0 (D.6)

(where I ⊂ OY is the ideal sheaf of X). Then:

(i) The homomorphism δ is injective on the open set where j is a regular embedding.(ii) If X and Y are nonsingular then the dual sequence

0→ TX → TY |X → NX/Y → 0 (D.7)

is exact.

D.2 Relative complete intersection morphisms 307

Proof. (i) It suffices to show that δ is injective under the assumption that j is a regularembedding. In this case the conormal sheaf I/I2 is locally free of rank equal to thecodimension of X . The sequence (D.6) is exact at every nonsingular point x ∈ Xby Theorem C.15(iii). Since X is reduced, this happens on a dense open subset sothat ker(δ) is supported on a nowhere dense subset. But X has no embedded pointsbecause it is regularly embedded in Y : it follows that ker(δ) = 0.

(ii) Under the stated hypothesis, j is a regular embedding and Ω1X is locally free,

so we have Ext1(Ω1X ,OX ) = 0 and the exactness of (D.7) follows.

Remark D.1.5. If we don’t assume X reduced, part (i) of the proposition is false ingeneral. An example is given by the closed regular embedding of codimension 1:

Spec(k[ε]) ⊂ Spec(k[t]) = A1

(see Example B.9(iv)).

A morphism f : X → Y of schemes will be called a cover (or a covering) if it isfinite and surjective.

Recall that a morphism of schemes f : X → Y is called unramified at a pointx ∈ X if Ω1

X/Y,x = 0; f is unramified if it is unramified at every x ∈ X . After

identifying X with the diagonal ∆ ⊂ X ×Y X , we see that Ω1X/Y gets identified with

the conormal sheaf of this embedding. It follows that f is unramified at x if and onlyif ∆ ⊂ X ×Y X is an open embedding at x , and that the locus of x ∈ X such that fis unramified at x is open. Moreover, f is unramified if and only if ∆ is both openand closed in X ×Y X .

D.2 Relative complete intersection morphisms

We now introduce a natural class of morphisms which generalize smooth morphismsand behave well with respect to differentials and base change.

Definition D.2.1. A flat morphism of finite type f : X → S is called a relativecomplete intersection (r.c.i.) morphism at the point x ∈ X if there is an open neigh-bourhood U of x such that the restriction of f to U can be obtained as a composition

Uj−→ V

g−→ S

where j is a regular embedding and g is smooth. If f is an r.c.i. morphism at everypoint we call it an r.c.i. morphism, and we call X a complete intersection over S.

This definition is equivalent to Def. 19.3.6 of Ch. IV of [1]; the equivalence isproved in [19], Prop. 1.4. Note that in case S = Spec(k) the morphism f is an r.c.i.if and only if X is an l.c.i. of finite type.


If X → S is a flat morphism of finite type of nonsingular varieties then f is anr.c.i. because it factors as

X → X × S→ S

where the first morphism is the graph of f .Before discussing the main properties of this notion we need two lemmas.

Lemma D.2.2. Let A→ B be a ring homomorphism, M a B-module and f1, . . . , fn

an M-regular sequence of elements of B. Assume that for each i = 1, . . . , n themodule M/(

∑i−1j=1 f j M) is A-flat. Then, for every ring homomorphism A → A′,

letting B ′ = B ⊗A A′, M ′ = M ⊗A A′, and f ′i = fi ⊗ 1 (1 ≤ i ≤ n), the sequence

f ′1, . . . , f ′n of elements of B ′ is M ′-regular and the modules M ′/(∑i−1

j=1 f ′j M ′) areA′-flat.

Proof. Consider the exact sequence:

0→ Mf1−→ M → M/ f1 M → 0

Since M/ f1 M is A-flat, the sequence:

0→ M ⊗A A′ f1⊗1−→ M ⊗A A′ → (M/ f1 M)⊗A A′ → 0

is exact, and therefore f ′1 is not a zero-divisor for M ′. Let Mi = M/(∑i

j=1 f j M),

M ′i = M ′/(∑i

j=1 f ′j M ′); then we have M ′i = Mi ⊗A A′, Mi+1 = Mi/ fi+1 Mi ,M ′i+1 = M ′i/ f ′i+1 M ′i . Replacing M and f1 by Mi and fi+1 in the above argument,one deduces that f ′i+1 is not a zero-divisor for M ′i , thereby proving the first assertionby induction. The last assertion follows from A.2(VII). Lemma D.2.3. Let A → B be a local homomorphism of noetherian local rings, Ma B-module of finite type, flat over A, and f1, . . . , fn ∈ m B. For 1 ≤ i ≤ n let gi

be the image of fi in B ⊗A k, where k = A/m A is the residue field of A. Then thefollowing conditions are equivalent:

(i) f1, . . . , fn is an M-regular sequence, and Mi = M/(∑i

j=1 f j M) is A-flat forall 1 ≤ i ≤ n.

(ii) g1, . . . , gn is an (M ⊗A k)-regular sequence.

Proof. (i)⇒ (ii) follows from D.2.2 applied to A′ = k.(ii) ⇒ (i) Applying Corollary A.6, from the injectivity of

g1 : M ⊗A k → M ⊗A k we deduce that f1 : M → M is injective and thatM1 = M/ f1 M is A-flat. Proceeding by induction on i , assume Mi flat over A.Since gi+1 : Mi ⊗A k → Mi ⊗A k is injective from A.6, again we deduce thatfi+1 : Mi → Mi is injective and that Mi+1 is A-flat.

In the next proposition some general properties of r.c.i. morphisms are proved.

Proposition D.2.4. (i) An open embedding is an r.c.i. morphism. A smooth morphismof finite type is an r.c.i. morphism.


(ii) If f : X → S is an r.c.i. morphism and h : S′ → S is a morphism, thenthe morphism f ′ : X ×S S′ → S′ induced by f after base change is an r.c.i.morphism.

Proof. (i) is an immediate consequence of the definition and (ii) follows easily fromLemma D.2.2.

From D.2.4(ii) it follows in particular that if f : X → S is an r.c.i. morphismthen Xs is an l.c.i. for every k-rational point s ∈ S. So for example, a non-l.c.i.algebraic scheme cannot be the fibre of a flat morphism of algebraic nonsingularvarieties.

The next result gives a useful characterization of r.c.i. morphisms.

Proposition D.2.5. LetX

j−→ Y f g

S

(D.8)

be a commutative diagram of morphisms of algebraic schemes, where f is flat, g issmooth and j is an embedding. Then the following conditions are equivalent for ak-rational point x ∈ X:

(i) f is an r.c.i. morphism at x.(ii) Letting s = f (x), the fibre Xs is an l.c.i. at x.(iii) j is a regular embedding at x.

Proof. (i)⇒ (ii) follows from D.2.4(ii) and (iii)⇒ (i) is obvious.(ii)⇒ (iii) From (ii) it follows that the embedding js : Xs ⊂ Ys is regular at x .

Let I ⊂ OY be the ideal sheaf of X . Tensoring the exact sequence

0→ I → OY → OX → 0

by −⊗OS k we obtain the sequence

0→ I ⊗OS k→ OYs → OXs → 0

which is exact because f is flat. Therefore I⊗OS k is the ideal sheaf of j (Xs) in Ys .Consider a sequence f1, . . . , fn of sections of I in an open neighbourhood of j (x)which induce a basis of I j (x)/(msI j (x) + I2

j (x)) as a OY, j (x)/(msOY, j (x) + I j (x))-module. Then the images f1 ⊗ 1 = g1, . . . , fn ⊗ 1 = gn are generating sectionsof I ⊗OS k in an open neighbourhood of j (x) in Ys which form a regular sequencein j (x). From Nakayama’s lemma it follows that f1, . . . , fn generate I in an openneighbourhood of j (x) in Y . From Lemma D.2.3 it follows that f1, . . . , fn form aregular sequence in j (x) and therefore (iii) holds. Corollary D.2.6. Under the hypothesis of Proposition D.2.5, the locus of points x ∈X such that f is an r.c.i. at x is open. If f is proper then the locus of points s ∈ Ssuch that Xs is an l.c.i. is open.


Proof. The last assertion follows from the first because a proper map is closed. Thefirst assertion can be proved using characterization D.2.5(iii) of r.c.i. morphism andthe fact that the locus where an embedding is regular is open.

We conclude this section with two results about the relative conormal sequenceand cotangent sequence for r.c.i. morphisms.

Theorem D.2.7. LetX

j−→ Y

f ↓ g

S

be a commutative diagram of morphisms of algebraic schemes, with f an r.c.i., j animmersion and g smooth. Let J ⊂ OY be the ideal sheaf of j (X). If f is smooth ona dense open subset intersecting every fibre then the relative conormal sequence

0→ J /J 2 δ−→ j∗Ω1Y/S → Ω1

X/S → 0

is exact and J /J 2 is locally free.

Proof. From the equivalence (i) ⇔ (i i i) in Proposition D.2.5 it follows that j isa regular embedding and therefore J /J 2 is locally free. Moreover, the support ofker(δ) does not contain any generic point of X nor any fibre of f because it is con-tained in the locus where f is not smooth. Since f is generically smooth and j is aregular embedding, X has no embedded components except possibly for some unionof fibres. It follows that δ is injective. Theorem D.2.8. Let f : X → S be an r.c.i morphism of algebraic schemes, andassume f smooth on a dense open subset intersecting every fibre. Then the relativecotangent sequence

0→ f ∗Ω1S → Ω1

X → Ω1X/S → 0 (D.9)

is exact.

Proof. We only have to prove the injectivity of the left homomorphism and the ques-tion is local on X . Since all schemes are algebraic, locally on X we can construct thefollowing commutative diagram:

Xj−→ V

i−→ U

f ↓ ψ ↓ ϕ

Sh−→ W

where W,U, ψ, ϕ are smooth, i, h are closed embeddings and j is a regular closedembedding. From the smooth morphism ϕ we deduce the exact sequence of locallyfree sheaves on U :


0→ ϕ∗Ω1W → Ω1

U → Ω1U/W → 0

which restricts on X to the exact sequence:

0→ (h f )∗Ω1W → (i j)∗Ω1

U → j∗Ω1V/S → 0

Let J ⊂ OV and I ⊂ OU be the ideal sheaves of the embeddings j and i j respec-tively, and H ⊂ OW the ideal sheaf of the embedding h. Then we have an exact andcommutative diagram of coherent sheaves on X :

0↓

f ∗(H/H2) → I/I2 → J /J 2 → 0↓ ↓ ↓ δ j

0→ (h f )∗Ω1W → (i j)∗Ω1

U → j∗Ω1V/S → 0

↓ ↓ ↓f ∗Ω1

Sd f ∨−→ Ω1

X → Ω1X/S → 0

↓ ↓ ↓0 0 0

where the second and the third columns are the relative conormal sequences of i jand of j respectively; the first column is the pullback to X of the conormal sequenceof h; the first row is exact because ψ∗(H/H2) is the conormal sheaf of i ; the map δ j

is injective by Theorem D.2.7 and by the assumptions made on X and f . A diagramchasing shows that the codifferential d f ∨ is injective and proves the theorem.

NOTES

1. An algebraic scheme can have different embeddings in IPr , i.e. by means of noniso-morphic invertible sheaves, but with same normal sheaf. An example is given by a projectivenonsingular curve C of genus 1, and by the embeddings in IP3 given by two nonisomorphicinvertible sheaves L1and L2 of degree 4 such that L2

1 = L22. Then C is embedded as a non-

singular complete intersection of two quadrics by both sheaves, and the normal bundles areL2

1 ⊕ L21 = L2

2 ⊕ L22.

2. Let S be a scheme, and X, Y smooth over S. Prove that every closed S-embeddingX ⊂ Y is regular. In particular, every section of a smooth morphism f : Y → S is a regularembedding of codimension equal to the relative dimension of f .

3. Let f : X → S be a morphism of finite type and s ∈ S a k-rational point. Letms ⊂ OS,s be the maximal ideal and I = IX (s) the ideal sheaf of the fibre X (s) of f over s.Prove that we have a surjective homomorphism

ms

m2s⊗k OX (s) → I/I2

and an injection:NX (s)/X ⊂ TS,s ⊗k OX (s)

If f is flat then they are isomorphisms; in particular, if f is flat then NX (s)/X is free.

E

Functorial language

Let C be a category. A covariant (resp. contravariant) functor F from C to (sets) issaid to be representable if there is an object X in C such that F is isomorphic to thefunctor

Y → Hom(X, Y ) (E.1)

(resp. Y → Hom(Y, X)). We will denote by h X a functor of the form (E.1). Therepresentable functors are a full subcategory, isomorphic to C (resp. to C in thecontravariant case), of the category Funct (C, (sets)) of covariant functors (resp.Funct (C, (sets)) of contravariant functors) from C to (sets).

To fix ideas let’s consider covariant functors. In order to investigate conditionsfor the representability of a given functor F it is convenient to study functorial mor-phisms h X → F . Such morphisms turn out to be easy to describe, thanks to thefollowing elementary lemma:

Lemma E.1 (Yoneda). Let F : C → (sets) be a covariant functor. For each objectX in C there is a canonical bijection:

Hom(h X , F) ↔ F(X)Φ → Φ(X)(1X )

Let’s mention, in passing, that functorial morphisms F → h X are much harderto control. They are related to the notion of “coarse moduli scheme”.

We may consider couples of the form (X, ξ), where X is an object of C andξ ∈ F(X). Yoneda’s lemma implies that to give such a couple is equivalent to givinga morphism of functors h X → F ; if this morphism is an isomorphism then (X, ξ)is called a universal couple, and ξ a universal element, for F . The existence of auniversal couple is equivalent to the representability of F .

The couples for F are the objects of a category in which a morphism(X, ξ)→ (Y, η) between two couples is by definition a morphism f : X → Y in C

314 E Functorial language

such that F( f )(ξ) = η. We denote this category by IF . A morphism f : (X, ξ)→(Y, η) in IF corresponds to a commutative diagram of morphisms of functors:

h Xξ−→ F

↑ f ηhY

We have an obvious “forgetful functor”

IF → C

The fibres of this functor are precisely the sets F(X), which are embedded as sub-categories of IF by ξ → (X, ξ).(Recall that, given a functor G : C → D, the fibre G−1(D) of G over an object Dof D is a subcategory of C, consisting of all objects C such that G(C) = D and ofall morphisms f such that G( f ) = 1D . A set can be viewed as a category whoseobjects are its elements and the only morphisms are the identity morphisms.)

Lemma E.2. The functor F is representable if and only if the category IF has aninitial object (X, ξ). If this is the case, (X, ξ) is a universal couple for F.

The proof is immediate. Note that, since an initial object is unique up to isomor-phism, it follows that a representable functor has a unique universal couple, up toisomorphism.

∗ ∗ ∗ ∗ ∗ ∗Let I and D be two categories. Given an object A of D, the constant functor

cA : I → D is defined as cA(i) = A for each object i of I and cA( f ) = 1A for eachmorphism f in I . Note that cA is both covariant and contravariant. Every morphismα : A→ B in D induces an obvious morphism of functors cα : cA → cB . Considera covariant functor Φ : I → D. An inductive limit of Φ is an object A of D and afunctorial morphism λ : Φ → cA such that for every other morphism µ : Φ → cB

there is a morphism α : A→ B such that µ = cαλ.

Φλ−→ cA

µ ↓ cα

cB

From the definition it follows that an inductive limit of Φ, if it exists, is unique up tounique isomorphism, and is denoted by

lim→ Φ

In practice an inductive limit is an object A of D such that there is a morphismΦ(i)→ A for each i ∈ Ob(I ) with the condition that the diagram

Φ(i) → A↓ Φ( f )

Φ( j)

E Functorial language 315

is commutative for each morphism f : i → j in I ; moreover, these data must satisfya universal property.

Dually, one has the notion of projective limit of a covariant functor Φ : I → D: itis an object A of D and a morphism π : cA → Φ such that for every other morphismρ : cB → Φ there is a morphism β : B → A such that ρ = πcβ . The projectivelimit of Φ, if it exists, is denoted by

lim← Φ

The above notions can be defined without changes replacing the covariant functorΦ by a contravariant one. We will write Φi for Φ(i), for each object i of I , andsometimes

lim→ Φi (resp. lim← Φi ) instead of lim→ Φ (resp. lim← Φ)

Example E.3. Let J be a partially ordered set. We define a category Ord(J ) asfollows. The objects of Ord(J ) are the elements of J ; for any i, j ∈ J the setHomOrd(J )(i, j) consists of one element if i ≤ j and is ∅ otherwise. A covariant(resp. contravariant) functor Φ : Ord(J ) → D is called an inductive system (resp.a projective system) in D indexed by J ; in the case D =(sets), we obtain the usualnotions of inductive (projective) system and of inductive (projective) limit.If I is a set and Φ : I → D is a functor, where D is a category with arbitrarycoproducts, then

lim→ Φ =∐

i

Φi

Similarly, if D has products then

lim← Φ =∏

i

Φi

Proposition E.4. The inductive limit and projective limit exist for every functor Φ :I → (sets) from any category I .

Proof. We takelim→ Φ =

∐i

Φi/R

where R is the equivalence relation generated by pairs (x, y), x ∈ Φi and y ∈ Φ j ,such that there exists ϕ : i → j with Φ(x) = y. Similarly for the projective limit. Example E.5. Let F : C → (sets) be a covariant functor, and let IF be the categoryof couples for F . Then we have a contravariant functor

Φ : IF → Funct (C, (sets))

which sends a couple (X, ξ) to the functor h X : C → (sets), and a morphismf : (X, ξ) → (Y, η) to the functorial morphism h f : hY → h X induced by f .


By construction there is a morphism Φ → cF . This morphism makes F the induc-tive limit of the functor Φ (the proof is an easy exercise). We will write:

F = lim→ (X,ξ)h X

Definition E.6. A category I is filtered if:

(a) for every pair of objects i, j in I there exists an object k in I and morphisms:

i↓

j → k

(b) each pair of morphisms i→→ j has a coequalizer i

→→ j → k.

The category I is cofiltered if the dual category I is filtered.

Assume from now on that C is a category with products and fibred products.

Definition E.7. A covariant functor F : C → (sets) is called left exact if F(B×C) =F(B)× F(C) and F(B ×A C) = F(B)×F(A) F(C) for each diagram

C↓

B → A

in C (i.e. F commutes with finite products and finite fibred products).

Every representable functor is left exact by definition of product and fibred prod-uct.

Lemma E.8. Let I be a filtered category and Φ : I → Funct (C, (sets)) a covariantfunctor. Then, for each diagram in C:

C↓

B → A

there is a bijection:

lim→ Φi (B)×lim Φi (A) lim→ Φi (C) ∼= lim→ [Φi (B)×Φi (A) φi (C)]

The proof of this lemma is straightforward and we omit it. The following resultis a useful characterization of left-exact functors.

Proposition E.9. A covariant functor F : C → (sets) is left exact if and only if thecategory IF is cofiltered.


Proof. Assume that IF is cofiltered. Applying Lemma E.8 to the functor Φ of Ex-ample E.5, we see that the inductive limit F = lim(X,ξ) h X is left exact because eachfunctor h X is left exact.

Conversely, assume that F is left exact. Let (X, ξ), (Y, η) ∈ Ob(IF ); we mustfind

(Z , ζ ) → (X, ξ)↓

(Y, η)

Take (Z , ζ ) = (X × Y, (ξ, η)). Now consider (X, ξ)→→ (Y, η) coming from

φ,ψ : X → Y . We have

F(φ)(ξ) = F(ψ)(ξ) = η

Consider the diagram:

XΓφ−→ X × Y

↑ ↑ Γψ

K → X

where Γφ = (1X , φ) and Γψ = (1X , ψ) and K = X ×X×Y X . Since F is left exact

F(K ) = F(X)×F(X×Y ) F(X)

and there is χ ∈ F(K ) corresponding to (ξ, ξ):

ξF(Γφ)−→ (ξ, η)

↑ ↑ F(Γψ)

χ −→ ξ

Then (K , χ) is the equalizer of φ and ψ . Therefore IF is cofiltered. Let I be a category. A full subcategory J of I is cofinal if for each i ∈ Ob(I )

there is a morphism f : i → j for some j ∈ Ob(J ). It follows immediately fromthe definitions that if Φ : I → D is a covariant functor and ΦJ : J → D is itsrestriction, then

lim→ Φ = lim→ ΦJ

∗ ∗ ∗ ∗ ∗ ∗Let Z be a scheme. In this subsection we will consider contravariant functors

defined on (schemes/Z ). All we will say holds, with obvious modifications, forfunctors defined on (algschemes/Z ), the full subcategory of algebraic Z -schemes.A contravariant functor

F : (schemes/Z) → (sets)

defines on every Z -scheme S a presheaf of sets:

U → F(U )


for all open sets U ⊂ S. For this reason a functor as above is also called a presheaf.F is called a sheaf (more precisely a sheaf in the Zariski topology) if it defines asheaf on every scheme; namely, if for all Z -schemes S and for all open coveringsUi of S the following is an exact sequence of sets:

F(S)→∏

i

F(Ui )→→

∏i, j

F(Ui ∩U j )

The most important sheaves are the representable functors, i.e. functors isomorphicto one of the form:

S → HomZ (S, X)

for some Z -scheme X . Such a functor is called the functor of points of X/Z .It is very important to have conditions, easy to verify in practice, for a contravari-

ant functor F : (schemes/Z) → (sets) to be representable. Certainly a necessarycondition is that F is a sheaf. Another necessary condition is the following.

Recall that a subfunctor G of F is said to be an open (resp. closed) subfunctor iffor every scheme S and for every morphism of functors

Hom(−, S)→ F

the fibred product Hom(−, S)×F G, which is a subfunctor of Hom(−, S), is repre-sented by an open (resp. closed) subscheme of S. A family of open subfunctors Gi of F is a covering of F if for every Z -scheme S and for every morphism of functorsHom(−, S) → F the family Hom(−, S) ×F Gi of subschemes of S is an opencovering of S.

An obvious example is obtained by considering an open (resp. closed) subschemeX ′ of a Z -scheme X : correspondingly, we obtain an open (resp. closed) subfunctorHom(−, X ′) of Hom(−, X). An open cover Xi of X defines a cover of Hom(−, X)by open subfunctors.

Therefore a second obvious necessary condition for a functor F to be repre-sentable is that it can be covered by representable open subfunctors. We will nowshow that these two necessary conditions are also sufficient.

Proposition E.10. LetF : (schemes/Z) → (sets)

be a contravariant functor. Suppose that:

(a) F is a sheaf;(b) F admits a covering by representable open subfunctors Fi .

Then F is representable.

Proof. Letting Fi j = Fi ×F Fj , by (b) the projections Fi j → Fi correspond toopen embedding of schemes Xi j → Xi . Therefore the Fi ’s patch together to form arepresentable functor Hom(−, X), where X is the scheme obtained by patching theXi ’s together along the Xi j ’s. By (a), F and Hom(−, X) are isomorphic.

The following is an easy but important remark.


Lemma E.11. If F is a sheaf then F is determined by its restriction to the categoryof affine schemes.

Proof. In fact, if S is any Z -scheme we can consider an affine open cover Ui . Forany i, j we take an affine open cover Vi, j,α of Ui ∩U j ; composing the map

F(Ui )→→

∏i, j

F(Ui ∩U j )

with the inclusions F(Ui ∩U j )→∏α F(Vi, j,α) we obtain the exact sequence:

F(S)→∏

i

F(Ui )→→

∏i, j,α

F(Vi, j,α)

which shows that F(S) is determined by its values on affine schemes. This lemma implies that the functor of points of a scheme X

F = HomZ (−, X) : (schemes) → (sets)

is determined by its restriction to the category of affine schemes, or equivalently, byits covariant version:


Since the category of schemes is isomorphic to the category of functors of points,this means that we can define schemes as certain types of functors on (k-algebras).Thanks to Proposition E.10, we can say that these functors are precisely the sheavesadmitting an open cover by affine schemes, i.e. by representable functors. This pointof view is very fruitful because it gives the possibility of generalizing the notion ofscheme by considering more general functors. The notion of algebraic space is sucha generalization (see Artin [13]).

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List of Symbols

k, 1k(s), 1ob(C), 1C, 1A, 1A, 1A∗, 1AΛ, 1A∗Λ, 1

AΛ, 2E∨, 2IP(V ), 2IP(E), 2(R′, ϕ), 9R⊕I , 10A[ε], 11ExA(R, I ), 12T 1

R/A, 14

T 1R , 14

Ex(X/S, I), 15OX ⊕I, 16T 1

X/S , 16

T 1X , 16

N ′X/Y , 16Fm , 22κ , 29κ(ξ), 30κξ , 31κ f,s , 31

κX /S,s , 31oξ (e), 33o(R/Λ), 37o(R), 37o( f/Λ), 37o( f ), 38tF , 46d f , 46F , 46(R, u), 47DefX , 64Def′X , 64µ(X), 69OS,s , 76A, 76DefX , 77( A, ηn), 77( A, η), 77XX , 77(S, s, η), 78(B, s, η), 78Pr

A, 81

Autu , 91Def(X,p), 93T 2

B0, 109

T 2X , 111

H YX , 123

oξ/Y , 130

HX /SZ , 136

330 List of Symbols

HXX /Specf(R)Z , 137

X A, 138L ⊗A B, 138Pλ0 , 138m1, 141aD , 143µ0(L), 144µ0(V ), 145c(L), 145EL , 145PL , 145Def(X,L), 146M1, 153PL , 155PV , 155wfdat, 157DefX/ f/Y , 158ρ(g, r, n), 160N f , 162DX/Y , 162D1

X/Y , 162Def f/Y , 164Def′f/Y , 164oF/Y , 167Φ f , 169Nϕ , 173Hϕ , 173Def j , 177Def′j , 177TY 〈X〉, 177FittFk , 199Nk(F), 199De(ϕ), 200Nk( f ), 200HilbY

P(t), 206Hilbr

P(t), 206

HilbYP(t), 206

HilbrP(t), 206

Y [n], 206GV,n , 209G, 209Gn(V ), 209G(n, N ), 210I, 212

HilbY , 216HilbY , 216Hilb(−Z)Y

P(t), 219

Hilb(−Z)YP(t), 219

HilbY\ZP(t), 219

HilbY\Z , 219Quot X/S

H,P(t), 219

Quot XH,P(t), 219

QuotX/SH,P(t), 220

HilbX/SP(t), 222

Quot X/SH , 222

QuotX/SH , 222

HilbX/S , 223P(t), 227F Hr

P(t), 227FHr

P(t), 227[X1, . . . , Xm], 229pri , 229PI(t), 229prI, 229F H Z

P(t), 229

FHZP(t), 230

P(t), 230FHZ , 230N(X1,...,Xm )/Z , 230Quotn(E), 240Quotn(E), 241Gn(E), 241Hom(X, Y ), 250Isom(X, Y ), 250Aut(X), 250d , 256|O(d)|, 256Λd , 257Vδ,κ

d , 257

Vδ,κd , 257

Mk( f ), 262ΩB/A, 279dB/A, 279PB/A, 279k[ε], 284tB/Λ, 285

List of Symbols 331

Ω1X/Y , 287

TX/Y , 287Ω1

X , 287TX , 287Tx X , 287dg, 288NX/Y , 288P1

X/Y , 292

P1X , 292

h X , 313Funct (C, (sets)), 313(X, ξ), 313IF , 314lim→ Φ, 314

lim← Φ, 315

Index

Abel–Jacobiembedding, 182morphism, 143

abelian variety, 151action, 57algebra

etale, 293formally etale, 293formally smooth, 293less obstructed, 38obstructed/unobstructed, 37of dual numbers, 11of principal parts, 279rigid, 25, 26, 104, 122smooth, 293

algebraic local ring, 76algebraic space, 319algebraization, 80algebraization theorem, 83, 87ample

canonical class, 185surface, 185

approximation theorem, 88Arakelov, 171Artin, 84, 319Artin’s algebraization theorem, 83Atiyah extension, 145, 154, 244automorphism functor, 90

Bertini’s theorem, 237blow-up, 73, 171, 182, 184, 185bounded collection of sheaves, 194Brill–Noether number, 160

carpet, 16Cartier, 144Castelnuovo, 194Castelnuovo–Mumford regularity, 194Castelnuovo–Mumford regularity

numerical criterion, 193Cauchy sequence, 39characteristic map, 47, 126, 159, 169, 179,

217, 218, 252global/local, 252of a linear system, 127

Chern class, 145codifferential, 288

relative, 288cofinal subcategory, 317complete intersection, 126, 135, 160, 235,

305, 307K3 surface, 236nonrigid, 106

complex torus, 35cone, 131

over IP1 × IP2, 107over IPn × IPm , 107, 120over Veronese embedding of IPn , 120rigid, 107, 119, 120

congruence, 84conic, 95, 195, 240conormal

bundle, 288sequence, 134, 280, 288, 306sheaf, 288

coordinate axes, 95, 107costable subscheme, 181

334 Index

cotangentmodule

first, 14second, 109upper/lower, 19

sequence (relative), 280, 288, 303, 310sheaf

first, 16second, 111

sheaf/bundle, 287space, 285

couple, 313formal, 47, 126universal, 313

cover/covering, 307covering of a functor, 318curve, 155

affine obstructed, 115canonical, 129, 132canonical of genus 4, 160cubic, 26elliptic, 26Gorenstein, 144negatively embedded, 131non-hyperelliptic, 183nonsingular, 94, 150not rigid, 69obstructed, 185obstructed in IP3, 237of genus ≥ 3, 182of genus 0, 100of genus 1, 93on a surface in IP3, 128pointed, 93rational, 160, 182rational normal, 120unobstructed, 35

cusp, 106, 174cycle map, 248

Debarre, 161deformation

algebraic, 78, 86complete, 79effectively parametrized, 79first-order/infinitesimal, see first-

order/infinitesimal deformationformal, 46, 77, 78, 80, 84, 86formally locally trivial, 78formally trivial, 78

formally versal/universal/semiuniversal,79, 87

local, 20, 78, 86, 92, 123, 156locally trivial, 21of a morphism, 156of a morphism wfdat, 157of a morphism with fixed target, 161of a polarization, 156of a scheme, 20of a subscheme, 123product, 21trivial, 21, 123, 156with general moduli, 79

depth, 117diagonal, 131, 207, 263, 307differential

of aD , 143of a morphism of functors, 46of a morphism of schemes, 288of the forgetful morphism, 132, 134, 151,

169, 170, 180divisor

Cartier, 127ramification, 171, 172semiregular, 144

duality pairing, 147dualizing sheaf, 144

Eagon–Northcott complex, 116element

universal, 206elementary transformation, 246embedded point, 196embedding

regular, 305rigid, 178Segre, 107, 120Veronese, 120, 251

Enriques, 72equisingularity, 254etale

cover, 96homomorphism, 293morphism, 171, 302neighbourhood, 75topology, 76

Euler sequence, 35, 119, 134, 146, 160, 172,175, 244, 251, 292

generalization of, 244

Index 335

extended Petri map, 155extension of algebras, 9

pullback of, 12pushout of, 12small, 11splits/splitting, 10trivial, 10versal, 10

extension of schemes, 15trivial, 15

faithfully flat module, 269family

isotrivial, 96locally isotrivial, 96non-isotrivial, 171of closed subschemes, 123product, 21universal, 206, 227with general moduli, 79, 218

family of deformations, see deformationfiltered category, 316first iteration morphism, see morphism, first

iterationfirst-order foci, 253first-order/infinitesimal deformation

of ξ0 ∈ F(k), 44of a morphism, 156of a pair (X, L), 146of a scheme, 21of a subscheme, 123of an algebra, 24of an invertible sheaf, 138

Fitting ideal, 199, 263flag

Hilbert functor, 227Hilbert scheme, 227, 229of closed subschemes, 229Quot scheme, 235regularly embedded, 234variety, 229

flat module, 269flatness criterion

by Hilbert polynomial, 194in terms of generators and relations, 275local, 271

foci, 253Fogarty’s theorem, 248

forgetful morphism, 132, 134, 151, 155, 169,180, 183, 184

formalcouple, 126scheme, 126

formal couple, 47universal, 47versal/semiuniversal, 49

formal deformation, 46, 77, 86algebraizable, 80, 84, 88effective, 80, 84, 87locally trivial, 78of a subscheme, 125semiuniversal, 74, 86, 114, 164, 178trivial, 78universal, 86, 95versal, 84, 86

formal element, 46semiuniversal, 52, 56, 58, 59, 69, 104, 146universal, 47versal/semiuniversal, 49

formal projective space, 81formal scheme, 77formally etale homomorphism, 293formally smooth homomorphism, 293functor

automorphism, 90criterion of representability, 318flag Hilbert, 227Grassmann, 209left exact, 45, 316less obstructed, 58, 144, 183local Hilbert, 123local moduli, 64local Picard, 138local relative Hilbert, 136locally of finite presentation, 84, 97of deformations of a pair (X, L), 146of Artin rings, 44of deformations of a closed embedding,

177of infinitesimal deformations of u0, 86of morphisms, 249of points, 318prorepresentable, 44, 47, 60, 125, 136,

138, 158, 179, 184Quot, 219representable, 44, 313, 318

336 Index

functor (Continued.)smooth, 47, 150unobstructed, 53

general moduli, 79, 218generic flatness, 205graph, 82, 158, 212, 249, 308Grassmann functor, 209grassmannian, 127, 210grassmannian bundle, 241Grothendieck, 82, 88, 194

Harris, 267Hartshorne, 218henselian local ring, 76henselization, 76hermitian form, 152Hilbert functor, 206

local, 123local relative, 136, 137

Hilbert scheme, 206, 216bound on the dimension, 129connectedness, 218existence theorem, 213flag, 229nested, 235nonreduced, 237reducible, 253relative, 222, 223

Hodge decomposition, 183homogeneous space, 27

formal principal, 28principal, 27, 57

homomorphismessentially of finite type, 1, 293etale, 293formally etale, 293formally smooth, 293obstructed/unobstructed, 37of extensions, 9smooth, 293

Horikawa, 156, 185hypersurface

of IPn × IPm , 135of IPr , 134, 207unobstructed, 114, 131

incidence relation, 212, 244, 292index of ramification, 172

infinitesimal automorphisms, 92infinitesimal deformation, see first-

order/infinitesimal deformationof u0 ∈ F(k), 86

infinitesimal Torelli theorem, 183isomorphism

of deformations, 21of extensions, 9

jacobian criterion of smoothness, 299jacobian variety, 182

Kleiman, 137Kodaira, 182, 184

dimension, 73, 268Kodaira–Nirenberg–Spencer, 92Kodaira–Spencer, 80

class, 30, 166correspondence, 29map, 31, 69, 74, 79, 97, 126, 236map (vanishing), 97

Kollar, 122, 161Koszul

complex, 109, 236relations, 109

Krull dimension, 40, 69

lifting, 129limit

inductive, 314projective, 315

linein a quadric cone, 131

linear system, 127, 144, 207deformation of, 159

localcomplete intersection, 305criterion of flatness, 271, 277deformation, 92deformation of a scheme, 20Hilbert functor, 123Picard functor, 138relative Hilbert functor, 136, 137ring

algebraic, 76henselian, 76in the etale topology, 76

local-to-global exact sequence, 65

Index 337

module of differentials, 279moduli scheme, 44, 79, 187

coarse, 313moduli stack, 79morphism

Abel–Jacobi, 143classifying, 206etale, 171, 302first iteration, 263forgetful, see forgetful morphismnon-degenerate, 162, 171obstructed/unobstructed, 159of nonsingular curves, 159Plucker, 211quasi-finite, 195relative complete intersection, 307rigid, 156, 159, 168, 171self-transverse codimension 1, 263smooth, 302unramified, 162, 303, 307

multiple pointscheme, 200, 262stratification, 200

Mumford, 54, 74, 156, 194example of, 237

Nagata, 76node, 106, 108, 174non-degenerate morphism, 162normal bundle, 243, 288

of a rational normal curve, 245normal sheaf, 123, 128, 288

equisingular, 16, 174, 177, 254of (X1, . . . , Xm), 230of a morphism, 154, 162

number of moduli, 69, 72, 74, 94, 132

obstructednonreduced scheme, 238affine curve, 115curve, 185curve in IP3, 237surface, 185variety of dimension 3, 185

obstructed/unobstructedembedding, 178morphism, 159scheme, 34subscheme, 131

surface, 35obstructed/unobstructed deformation

of a morphism, 159of a morphism with fixed target, 168of a scheme, 34of a subscheme, 131of an embedding, 178

obstruction, 33, 131, 168map, 58, 223

for aD , 143for the forgetful morphism, 132, 151,

169, 170, 180space

for DefB0 , 111for Def′j , 178

for Def′X , 69for Def′f/Y , 164for DefX , 70for Def(X,L), 146for DefX/ f/Y , 158for Hom(X, Y ), 250for HY

X , 129

for HX /SZ , 137

for Pλ0 , 138for a functor, 53, 58for an algebra, 43for Quot, 226of OHilbY ,[X ], 217of OFHZ ,[X1,...,Xm ], 231

relative/absolute, 37Oort, 156

parameter scheme, 20, 123Parshin, 171Petri map, 129, 132, 134, 144, 160

extended, 155Picard functor (local), 138Picard group, 138Plucker morphism, 211pointed scheme, 21, 156, 159, 179polarization, 156presheaf, 318principal G-bundle, 98principal parts

algebra of, 279sheaf of, 145, 292

product of curves, 74projection of a curve, 128

338 Index

projective bundle, 2, 241prorepresentability

of DefX , 89, 92, 94of Def′X , 89

of HYX , 125

of Pλ0 , 140pseudo-torsor, 28pullback of an extension of algebras, 12pushout, 134

of an extension of algebras, 12

quadric, 22, 120cone, 131, 160

Quotfunctor, 219scheme, 223

flag, 235

ramification divisor, 171, 172rational connectedness, 161regular

embedding, 305sequence, 40Severi variety, 260

relativecodifferential, 288complete intersection morphism, 307conormal sequence, 288, 303, 310cotangent sequence, 280, 288, 303, 310cotangent sheaf, 287derivations, 287differential, 288differentials (sheaf of), 287dimension of a smooth morphism, 303Hilbert scheme, 222tangent sheaf, 287

ribbon, 16rigid

algebra, 25, 26, 104, 122cone, 107, 119, 120curve singularity, 69embedding, 178morphism, 156, 159, 168, 171nonsingular variety, 34product of projective spaces, 35projective space, 35scheme, 21, 121subscheme, 123, 131, 184weighted projective space, 122

schemeflag Hilbert, 227formal, 77, 126Hilbert, 206multiple point, 200, 262obstructed/unobstructed, 34of elementary transformations, 246of isomorphisms/automorphisms, 250of morphisms, 250pointed, 21, 156, 159, 179Quot, 223rigid, 21, 121Severi, 257unobstructed, 79vanishing, 200

Schlessinger, 115theorem of, 56, 64, 104

see-saw theorem, 203Segre

embedding, 107, 120example of, 261

semiregular, 144Serre, 82

duality, 151vanishing theorem, 185, 187, 190

Severi, 262problem, 267scheme/variety, 257variety

regular/superabundant, 260Severi–Kodaira–Spencer, 144sheaf, 209, 318

(b), 191m-regular, 187Castelnuovo–Mumford regular, 187conormal, 288dualizing, 144normal, 128, 288of germs of vectors tangent along a

subvariety, 177of principal parts, 145, 292

smoothfunctor, 47, 150homomorphism, 293morphism, 302morphism of functors, 47

smoothness of forgetful morphism, 155stable subscheme, 181stack, 101

Index 339

stratification, 198defined by a sheaf, 199flattening, 200multiple point, 200

subfunctor, 211, 243open/closed, 318

subschemecostable, 181obstructed/unobstructed, 131rigid, 123, 131, 184stable, 181unobstructed, 131

superabundant Severi variety, 260surface, 106, 182

abelian, 74K3, 73, 74, 82, 135, 136, 150, 236minimal ruled, 72obstructed, 185obstructed/unobstructed, 35of general type, 268rational, 73rational ruled, 22, 31, 35, 72, 95, 96, 128,

290ruled, 73

symmetric product, 248system

inductive/projective, 315Szpiro, 171

tacnode, 106tangent

space, 285tangent space

of Def′j , 178

of Def′X , 64of DefX , 64

of HX /SZ , 136

of Def′f/Y , 164of Def(X,L), 146of DefX/ f/Y , 158

of HilbY , 217of Hom(X, Y ), 250of HY

X , 123

of Pλ0 , 138of FHZ, 231of a functor, 46of Quot, 226

tautologicalexact sequence, 210, 241, 244invertible sheaf, 208

torsor, 27total scheme, 20, 123truncated cotangent complex, 19

uniruledness, 161, 251universal

family, deformation, element, see family,deformation, element

property, 206, 210quotient, 222quotient bundle, 209, 241subbundle, 210, 241

unobstructedcanonical curve, 132curve, 35functor, 53hypersurface, 114, 131local complete intersection, 114morphism, 170nonsingular projective variety, 69rational ruled surface, 35, 170scheme, 79subscheme, 131

unramified covering of curves, 31unramified morphism, 162, 303, 307upper semicontinuous function, 198

vanishing scheme, 200Veronese embedding, 120, 251

Wahl, 261weighted projective space, 122

Yoneda’s lemma, 97, 313

Zappa’s example, 246

A Series of Comprehensive Studies in Mathematics

A Selection

246. Naimark/Stern: Theory of Group Representations247. Suzuki: Group Theory I248. Suzuki: Group Theory II249. Chung: Lectures from Markov Processes to Brownian Motion250. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations251. Chow/Hale: Methods of Bifurcation Theory252. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampère Equations253. Dwork: Lectures on ρ-adic Differential Equations254. Freitag: Siegelsche Modulfunktionen255. Lang: Complex Multiplication256. Hörmander: The Analysis of Linear Partial Differential Operators I257. Hörmander: The Analysis of Linear Partial Differential Operators II258. Smoller: Shock Waves and Reaction-Diffusion Equations259. Duren: Univalent Functions260. Freidlin/Wentzell: Random Perturbations of Dynamical Systems261. Bosch/Güntzer/Remmert: Non Archimedian Analysis – A System Approach to Rigid

Analytic Geometry262. Doob: Classical Potential Theory and Its Probabilistic Counterpart263. Krasnosel’skiı/Zabreıko: Geometrical Methods of Nonlinear Analysis264. Aubin/Cellina: Differential Inclusions265. Grauert/Remmert: Coherent Analytic Sheaves266. de Rham: Differentiable Manifolds267. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. I268. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. II269. Schapira: Microdifferential Systems in the Complex Domain270. Scharlau: Quadratic and Hermitian Forms271. Ellis: Entropy, Large Deviations, and Statistical Mechanics272. Elliott: Arithmetic Functions and Integer Products273. Nikol’skiı: Treatise on the shift Operator274. Hörmander: The Analysis of Linear Partial Differential Operators III275. Hörmander: The Analysis of Linear Partial Differential Operators IV276. Liggett: Interacting Particle Systems277. Fulton/Lang: Riemann-Roch Algebra278. Barr/Wells: Toposes, Triples and Theories279. Bishop/Bridges: Constructive Analysis280. Neukirch: Class Field Theory281. Chandrasekharan: Elliptic Functions282. Lelong/Gruman: Entire Functions of Several Complex Variables283. Kodaira: Complex Manifolds and Deformation of Complex Structures284. Finn: Equilibrium Capillary Surfaces285. Burago/Zalgaller: Geometric Inequalities286. Andrianaov: Quadratic Forms and Hecke Operators287. Maskit: Kleinian Groups288. Jacod/Shiryaev: Limit Theorems for Stochastic Processes

Grundlehren der mathematischen Wissenschaften

289. Manin: Gauge Field Theory and Complex Geometry290. Conway/Sloane: Sphere Packings, Lattices and Groups291. Hahn/O’Meara: The Classical Groups and K-Theory292. Kashiwara/Schapira: Sheaves on Manifolds293. Revuz/Yor: Continuous Martingales and Brownian Motion294. Knus: Quadratic and Hermitian Forms over Rings295. Dierkes/Hildebrandt/Küster/Wohlrab: Minimal Surfaces I296. Dierkes/Hildebrandt/Küster/Wohlrab: Minimal Surfaces II297. Pastur/Figotin: Spectra of Random and Almost-Periodic Operators298. Berline/Getzler/Vergne: Heat Kernels and Dirac Operators299. Pommerenke: Boundary Behaviour of Conformal Maps300. Orlik/Terao: Arrangements of Hyperplanes301. Loday: Cyclic Homology302. Lange/Birkenhake: Complex Abelian Varieties303. DeVore/Lorentz: Constructive Approximation304. Lorentz/v. Golitschek/Makovoz: Construcitve Approximation. Advanced Problems305. Hiriart-Urruty/Lemaréchal: Convex Analysis and Minimization Algorithms I.

Fundamentals306. Hiriart-Urruty/Lemaréchal: Convex Analysis and Minimization Algorithms II.

Advanced Theory and Bundle Methods307. Schwarz: Quantum Field Theory and Topology308. Schwarz: Topology for Physicists309. Adem/Milgram: Cohomology of Finite Groups310. Giaquinta/Hildebrandt: Calculus of Variations I: The Lagrangian Formalism311. Giaquinta/Hildebrandt: Calculus of Variations II: The Hamiltonian Formalism312. Chung/Zhao: From Brownian Motion to Schrödinger’s Equation313. Malliavin: Stochastic Analysis314. Adams/Hedberg: Function spaces and Potential Theory315. Bürgisser/Clausen/Shokrollahi: Algebraic Complexity Theory316. Saff/Totik: Logarithmic Potentials with External Fields317. Rockafellar/Wets: Variational Analysis318. Kobayashi: Hyperbolic Complex Spaces319. Bridson/Haefliger: Metric Spaces of Non-Positive Curvature320. Kipnis/Landim: Scaling Limits of Interacting Particle Systems321. Grimmett: Percolation322. Neukirch: Algebraic Number Theory323. Neukirch/Schmidt/Wingberg: Cohomology of Number Fields

325. Dafermos: Hyperbolic Conservation Laws in Continuum Physics326. Waldschmidt: Diophantine Approximation on Linear Algebraic Groups327. Martinet: Perfect Lattices in Euclidean Spaces328. Van der Put/Singer: Galois Theory of Linear Differential Equations329. Korevaar: Tauberian Theory. A Century of Developments330. Mordukhovich: Variational Analysis and Generalized Differentiation I: Basic Theory331. Mordukhovich: Variational Analysis and Generalized Differentiation II: Applications

324. Liggett: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes

Derived Categories332. Kashiwara/Schapira: Categories and Sheaves. An Introduction to Ind-Objects and

334. Sernesi: Deformations of Algebraic Schemes335. Bushnell/Henniart: The Local Langlands Conjecture for GL(2)

333. Grimmett: The Random-Cluster Model