Glossary of Algebraic Geometry

Glossary of algebraic geometryFrom Wikipedia, the free encyclopedia

This is a glossary of algebraic geometry.

See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For thenumber-theoretic applications, see glossary of arithmetic and Diophantine geometry.

For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed basescheme S and a morphism an S-morphism.

Contents :!$@ · A · B · C · D · E · F · G · H · I · J · K · L · M · N · O · P · Q · R · S · T · U · V · W · XYZ · See also · References

!$@

A generic point.

F(n), F(D)1. If X is a projective scheme with Serre's twisting sheaf and if F is an -module, then

2. If D is a Cartier divisor and F is an -module (X arbitrary), then If D is a Weil divisorand F is reflexive, then one replaces F(D) by its reflexive hull (and call the result still F(D).)

|D|The complete linear system of a Weil divisor D on a normal complete variety X over an algebraically closed field k;that is, . There is a bijection between the set of k-rational points of |D| and the set of

effective Weil divisors on X that are linearly equivalent to D.[1] The same definition is used if D is a Cartier divisoron a complete variety over k.

[X/G]The quotient stack of, say, an algebraic space X by the action of a group scheme G.

The GIT quotient of a scheme X by the action of a group scheme G.

Ln

An ambiguous notation. It usually means an n-th tensor power of L but can also mean the self-intersection number ofL. If , the structure sheaf on X, then it means thfe direct sum of n copies of .

The tautological line bundle. It is the dual of Serre's twisting sheaf .

Serre's twisting sheaf. It is the dual of the tautological line bundle . It is also called the hyperplane bundle.

1. If D is an effective Cartier divisor on X, then it is the inverse of the ideal sheaf of D.2. Most of the times, is the image of D under the natural group homomorphism from the group of Cartierdivisors to the Picard group of X, the group of isomorphism classes of line bundles on X.3. In general, is the sheaf corresponding to a Weil divisor D (on a normal scheme). It need not be locallyfree, only reflexive.

4. If D is a ℚ-divisor, then is of the integral part of D.

Glossary of algebraic geometry - Wikipedia, the free encyclopedia https://en.wikipedia.org/w/index.php?title=Glossary_of_algebraic_geom...

1 af 19 29-10-2015 15:23

1. is the sheaf of Kähler differentials on X.2. is the p-th exterior power of .

1. If p is 1, this is the sheaf of logarithmic Kähler differentials on X along D (roughly differential forms with simplepoles along a divisor D.)2. is the p-th exterior power of .

P(V)Unfortunately, the notation is ambiguous. Its traditional meaning is the projectivization of a finite-dimensionalk-vector space V; i.e.,

(the Proj of the ring of polynomial functions k[V]) and its k-points correspond to lines in V. In contrast, Hartshorneand EGA write P(V) for the Proj of the symmetric algebra of V.

Q-factorialA normal variety is -factorial if every -Weil divisor is -Cartier.

Spec(R)The set of all prime ideals in a ring R with Zariski topology; it is called the prime spectrum of R.

Specan(R)The set of all valuations for a ring R with a certain weak topology; it is called the Berkovich spectrum of R.

A

abelian1. An abelian variety is a complete group variety.2. An abelian scheme is a (flat) family of abelian varieties.

adjunction formula1. If D is an effective Cartier divisor on an algebraic variety X, both admitting dualizing sheaves , then theadjunction formula says:

.

2. If, in addition, X and D are smooth, then the formula is equivalent to saying:

where are canonical divisors on D and X.

affine1. Affine space is roughly a vector space where one has forgotten which point is the origin2. An affine variety is a variety in affine space3. An affine scheme is a scheme that is the prime spectrum of some commutative ring.4. A morphism is called affine if the preimage of any open affine subset is again affine. In more fancy terms, affinemorphisms are defined by the global Spec construction for sheaves of OX-Algebras, defined by analogy with the

spectrum of a ring. Important affine morphisms are vector bundles, and finite morphisms.

algebraic geometryAlgebraic geometry is a branch of mathematics that studies solutions toalgebraic equations.

algebraic geometry over the field with one elementOne goal is to prove the Riemann hypothesis. See also the field withone element and Peña, Javier López; Lorscheid, Oliver (2009-08-31)."Mapping F_1-land:An overview of geometries over the field with one

Algebraic geometry occupied a centralplace in the mathematics of the lastcentury. The deepest results of Abel,Riemann, Weierstrass, many of the mostimportant papers of Klein and Poincarebelong to this domain. At the end of thelast and the beginning of the present


2 af 19 29-10-2015 15:23

From the preface to I.R. Shafarevich,Basic Algebraic Geometry.

element". arXiv:0909.0069. as well as [2][3] .

algebraic groupAn algebraic group is an algebraic variety that is also a group in such away the group operations are morphisms of varieties.

algebraic schemeA separated scheme of finite type over a field. For example, analgebraic variety is a reduced irreducible algebraic scheme.

algebraic setAn algebraic set over a field k is a reduced separated scheme of finitetype over . An irreducible algebraic set is called an algebraicvariety.

algebraic spaceAn algebraic space is a quotient of a scheme by the étale equivalencerelation.

algebraic varietyAn algebraic variety over a field k is an integral separated scheme of finite type over . Note, not assuming k isalgebraically closed causes some pathology; for example, is not a variety since the coordinate ring is not anintegral domain.

algebraic vector bundleA locally free sheaf of a finite rank.

ampleA line bundle on a projective variety is ample if some tensor power of it is very ample.

Arakelov geometry

Algebraic geometry over the compactification of Spec of the ring of rational integers ℤ.. See Arakelov geometry.[4]

arithmetic genusThe arithmetic genus of a projective variety X of dimension r is .

artinian0-dimensional and Noetherian. The definition applies both to a scheme and a ring.

B

Behrend's trace formulaBehrend's trace formula generalizes Grothendieck's trace formula; both formulas compute the trace of the Frobeniuson l-adic cohomology.

big

A big line bundle L on X of dimension n is a line bundle such that .

birational morphismA birational morphism between schemes is a morphism that becomes an isomorphism after restricted to some opendense subset.

blueprintA kind of semirings approach to algebraic geometry, mainly motivated by the application to the algebraic geometry

over the field with one element.[5]

century the attitude towards algebraicgeometry changed abruptly. ... The styleof thinking that was fully developed inalgebraic geometry at that time was toofar removed from the set-theoretical andaxiomatic spirit, which then determinedthe development of mathematics. ...Around the middle of the presentcentury algebraic geometry hadundergone to a large extent such areshaping process. As a result, it canagain lay claim to the position it onceoccupied in mathematics.


3 af 19 29-10-2015 15:23

C

Calabi–Yau1. The Calabi–Yau metric is a Kähler metric whose Ricci curvature is zero.

canonical1. The canonical sheaf on a normal variety X of dimension n is where i is the inclusion of the smoothlocus U and is the sheaf of differential forms on U of degree n. If the base field has characteristic zero instead ofnormality, then one replaces i by a resolution of singularities.2. The canonical class on a normal variety X is the divisor class such that .3. The canonical divisor is a representative of the canonical class denoted by the same symbol (and notwell-defined.)4. The canonical ring of a normal variety X is the section ring of the canonical sheaf.

canonical model1. The canonical model is the Proj of a canonical ring (assuming the ring is finitely generated.)

Cartier1. An effective Cartier divisor D on a scheme X over S is a closed subscheme of X that is flat over S and whose idealsheaf is invertible (locally free of rank one).

Castelnuovo–Mumford regularityThe Castelnuovo–Mumford regularity of a coherent sheaf F on a projective space over a scheme S is thesmallest integer r such that

for all i > 0.

catenaryA scheme is catenary, if all chains between two irreducible closed subschemes have the same length. Examplesinclude virtually everything, e.g. varieties over a field, and it is hard to construct examples that are not catenary.

central fiber1. A special fiber.

Chow groupThe k-th Chow group of a smooth variety X is the free abelian group generated by closed subvarieties ofdimension k (group of k-cycles) modulo rational equivalences.

classifying stackAn analog of a classifying space for torsors in algebraic geometry; see classifying stack.

closedClosed subschemes of a scheme X are defined to be those occurring in the following construction. Let J be a quasi-coherent sheaf of -ideals. The support of the quotient sheaf is a closed subset Z of X and is

a scheme called the closed subscheme defined by the quasi-coherent sheaf of ideals J.[6] The reason the definitionof closed subschemes relies on such a construction is that, unlike open subsets, a closed subset of a scheme does nothave a unique structure as a subscheme.

Cohen–MacaulayA scheme is called Cohen-Macaulay if all local rings are Cohen-Macaulay. For example, regular schemes, and Spec

k[x,y]/(xy) are Cohen–Macaulay, but is not.

coherent sheafA coherent sheaf on a Noetherian scheme X is a quasi-coherent sheaf that is finitely generated as OX-module.

conicAn algebraic curve of degree two.

connected


4 af 19 29-10-2015 15:23

The scheme is connected as a topological space. Since the connected components refine the irreducible componentsany irreducible scheme is connected but not vice versa. An affine scheme Spec(R) is connected iff the ring Rpossesses no idempotents other than 0 and 1; such a ring is also called a connected ring. Examples of connectedschemes include affine space, projective space, and an example of a scheme that is not connected is Spec(k[x]×k[x])

compactificationSee for example Nagata's compactification theorem.

Cox ringA generalization of a homogeneous coordinate ring. See Cox ring.

crepantA crepant morphism between normal varieties is a morphism such that .

curveAn algebraic variety of dimension one.

D

deformationLet be a morphism of schemes and X an S-scheme. Then a deformation X' of X is an S'-scheme together witha pullback square in which X is the pullback of X' (typically X' is assumed to be flat).

degeneration1. A scheme X is said to degenerate to a scheme (called the limit of X) if there is a scheme withgeneric fiber X and special fiber .2. A flat degeneration is a degeneration such that that is flat; e.g., toric degeneration.

dimensionThe dimension, by definition the maximal length of a chain of irreducible closed subschemes, is a global property. Itcan be seen locally if a scheme is irreducible. It depends only on the topology, not on the structure sheaf. See alsoGlobal dimension. Examples: equidimensional schemes in dimension 0: Artinian schemes, 1: algebraic curves, 2:algebraic surfaces.

degree

1. The degree of a line bundle L on a complete variety is an integer d such that .

2. If x is a cycle on a complete variety over a field k, then its degree is .2. For the degree of a finite morphism, see morphism of varieties#Degree of a finite morphism.

derived algebraic geometryAn approach to algebraic geometry using (commutative) ring spectra instead of commutative rings; see derivedalgebraic geometry.

divisorialA divisorial sheaf on a normal variety is a reflexive sheaf of the form OX(D) for some Weil divisor D.

dominantA morphism f : X → Y is called dominant, if the image f(X) is dense. A morphism of affine schemes Spec A → Spec Bis dense if and only if the kernel of the corresponding map B → A is contained in the nilradical of B.

E

Éléments de géométrie algébrique

The EGA was an incomplete attempt to lay a foundation of algebraic geometry based on the notion of scheme, ageneralization of an algebraic variety. Séminaire de géométrie algébrique picks up where the EGA left off. Today it isone of the standard references in algebraic geometry.


5 af 19 29-10-2015 15:23

The title page of Éléments

de géométrie algébrique,

one of the standard

references in algebraic

geometry.

elliptic curveAn elliptic curve is a smooth projective curve of genus one.

essentially of finite typeLocalization of a finite type scheme.

étaleA morphism f : Y → X is étale if it is flat and unramified. There areseveral other equivalent definitions. In the case of smooth varieties and over an algebraically closed field, étale morphisms are preciselythose inducing an isomorphism of tangent spaces ,which coincides with the usual notion of étale map in differentialgeometry.

Étale morphisms form a very important class of morphisms; they areused to build the so-called étale topology and consequently the étalecohomology, which is nowadays one of the cornerstones of algebraicgeometry.

Euler sequenceThe exact sequence of sheaves:

where Pn is the projective space over a field and the last nonzero term is the tangent sheaf, is called the Eulersequence.

equivariant intersection theorySee Chapter II of http://www.math.ubc.ca/~behrend/cet.pdf

F

F-regular

Related to Frobenius morphism.[7]

FanoA Fano variety is a smooth projective variety X whose anticanonical sheaf is ample.

fiberGiven between schemes, the fiber of f over y is, as a set, the pre-image ; ithas the natural structure of a scheme over the residue field of y as the fiber product , where has thenatural structure of a scheme over Y as Spec of the residue field of y.

fiber product1. Another term for the "pullback" in the category theory.2. A stack given for : an object over B is a triple (x, y, ψ), x in F(B), y in H(B), ψan isomorphism in G(B); an arrow from (x, y, ψ) to (x', y', ψ') is a pair of morphisms

such that . The resulting square with obvious projections does notcommute; rather, it commutes up to natural isomorphism; i.e., it 2-commutes.

finalOne of Grothendieck's fundamental ideas is to emphasize relative notions, i.e. conditions on morphisms rather thanconditions on schemes themselves. The category of schemes has a final object, the spectrum of the ring of integers;so that any scheme is over , and in a unique way.

finiteThe morphism f : Y → X is finite if may be covered by affine open sets such that each isaffine — say of the form — and furthermore is finitely generated as a -module. See finite morphism.


6 af 19 29-10-2015 15:23

Finite morphisms are quasi-finite, but not all morphisms having finite fibers are quasi-finite, and morphisms of finitetype are usually not quasi-finite.

finite type (locally)The morphism f : Y → X is locally of finite type if may be covered by affine open sets such that eachinverse image is covered by affine open sets where each is finitely generated as a -algebra.The morphism f : Y → X is of finite type if may be covered by affine open sets such that each inverseimage is covered by finitely many affine open sets where each is finitely generated as a -algebra.

finite fibersThe morphism f : Y → X has finite fibers if the fiber over each point is a finite set. A morphism is quasi-finite if it is of finite type and has finite fibers.

finite presentationIf y is a point of Y, then the morphism f is of finite presentation at y (or finitely presented at y) if there is an open

affine subset U of f(y) and an open affine neighbourhood V of y such that f(V) ⊆ U and is a finitely presentedalgebra over . The morphism f is locally of finite presentation if it is finitely presented at all points of Y. If X

is locally Noetherian, then f is locally of finite presentation if, and only if, it is locally of finite type.[8] The morphismf : Y → X is of finite presentation (or Y is finitely presented over X) if it is locally of finite presentation, quasi-compact, and quasi-separated. If X is locally Noetherian, then f is of finite presentation if, and only if, it is of finite

type.[9]

flag varietyThe flag variety parametrizes a flag of vector spaces.

flatA morphism is flat if it gives rise to a flat map on stalks. When viewing a morphism f : Y → X as a family ofschemes parametrized by the points of , the geometric meaning of flatness could roughly be described by sayingthat the fibers do not vary too wildly.

formalSee formal scheme.

G

Gabriel–Rosenberg reconstruction theoremThe Gabriel–Rosenberg reconstruction theorem states a scheme X can be recovered from the category of quasi-

coherent sheaves on X.[10] The theorem is a starting point for noncommutative algebraic geometry since, taking thetheorem as an axiom, defining a noncommutative scheme amounts to defining the category of quasi-coherent sheaveson it. See also http://mathoverflow.net/questions/16257/how-to-unify-various-reconstruction-theorems-gabriel-rosenberg-tannaka-balmers

G-bundleA principal G-bundle.

generic pointA dense point.

genusSee #arithmetic genus, #geometric genus.

genus formulaThe genus formula for a nodal curve in the projective plane says the genus of the curve is given as

where d is the degree of the curve and δ is the number of nodes (which is zero if the curve is smooth).

geometric genus


7 af 19 29-10-2015 15:23

The geometric genus of a smooth projective variety X of dimension n is

(where the equality is Serre's duality theorem.)

geometric pointThe prime spectrum of an algebraically closed field.

geometric propertyA property of a scheme X over a field k is "geometric" if it holds for for any field extension

.

geometric quotientThe geometric quotient of a scheme X with the action of a group scheme G is a good quotient such that the fibers areorbits.

GIT quotientThe GIT quotient is when and when .

good quotientThe good quotient of a scheme X with the action of a group scheme G is an invariant morphism such that

Gorenstein1. (Note the terminology is inconsistent with the usage in commutative algebra). An algebraic variety is Gorenstein ifit is normal and the canonical sheaf of it is invertible (i.e., Cartier). Some authors require it also to be Cohen–Macaulay, but that is not standard.

2. A normal variety is ℚ-Gorenstein if the canonical divisor on it is ℚ-Cartier (and need not be Cohen–Macaulay).

Grauert–Riemenschneider vanishing theoremThe Grauert–Riemenschneider vanishing theorem extends the Kodaira vanishing theorem to higher direct imagesheaves; see also http://arxiv.org/abs/1404.1827

Grothendieck's vanishing theoremGrothendieck's vanishing theorem concerns local cohomology.

group schemeA group scheme is a scheme whose sets of points have the structures of a group.

group varietyAn old term for a "smooth" algebraic group.

H

Hilbert polynomialThe Hilbert polynomial of a projective scheme X over a field is the Euler characteristic .

Hodge bundleThe Hodge bundle on the moduli space of curves (of fixed genus) is roughly a vector bundle whose fiber over a curveC is the vector space .

hyperplane bundleAnother term for Serre's twisting sheaf . It is the dual of the tautological line bundle (whence the term).

I

imageIf f : Y → X is any morphism of schemes, the scheme-theoretic image of f is the unique closed subscheme i : Z → X


8 af 19 29-10-2015 15:23

which satisfies the following universal property:f factors through i,1.

if j : Z′ → X is any closed subscheme of X such that f factors through j, then i also factors through j.[11][12]2. This notion is distinct from that of the usual set-theoretic image of f, f(Y). For example, the underlying space of Zalways contains (but is not necessarily equal to) the Zariski closure of f(Y) in X, so if Y is any open (and not closed)subscheme of X and f is the inclusion map, then Z is different from f(Y). When Y is reduced, then Z is the Zariskiclosure of f(Y) endowed with the structure of reduced closed subscheme. But in general, unless f is quasi-compact, theconstruction of Z is not local on X.

immersionImmersions f : Y → X are maps that factor through isomorphisms with subschemes. Specifically, an open immersionfactors through an isomorphism with an open subscheme and a closed immersion factors through an isomorphism

with a closed subscheme.[13] Equivalently, f is a closed immersion if, and only if, it induces a homeomorphism fromthe underlying topological space of Y to a closed subset of the underlying topological space of X, and if the morphism

is surjective.[14] A composition of immersions is again an immersion.[15] Some authors, such asHartshorne in his book Algebraic Geometry and Q. Liu in his book Algebraic Geometry and Arithmetic Curves,define immersions as the composite of an open immersion followed by a closed immersion. These immersions areimmersions in the sense above, but the converse is false. Furthermore, under this definition, the composite of twoimmersions is not necessarily an immersion. However, the two definitions are equivalent when f is quasi-

compact.[16] Note that an open immersion is completely described by its image in the sense of topological spaces,while a closed immersion is not: and may be homeomorphic but not isomorphic. This happens,for example, if I is the radical of J but J is not a radical ideal. When specifying a closed subset of a scheme withoutmentioning the scheme structure, usually the so-called reduced scheme structure is meant, that is, the schemestructure corresponding to the unique radical ideal consisting of all functions vanishing on that closed subset.

ind-schemeAn ind-scheme is an inductive limit of closed immersions of schemes.

invertible sheafA locally free sheaf of a rank one. Equivalently, it is a torsor for the multiplicative group (i.e., line bundle).

integralA scheme that is both reduced and irreducible is called integral. For locally Noetherian schemes, to be integral isequivalent to being a connected scheme that is covered by the spectra of integral domains. (Strictly speaking, this isnot a local property, because the disjoint union of two integral schemes is not integral. However, for irreducibleschemes, it is a local property.) For example, the scheme Spec k[t]/f, f irreducible polynomial is integral, while SpecA×B. (A, B ≠ 0) is not.

irreducibleA scheme X is said to be irreducible when (as a topological space) it is not the union of two closed subsets except ifone is equal to X. Using the correspondence of prime ideals and points in an affine scheme, this means X isirreducible iff X is connected and the rings Ai all have exactly one minimal prime ideal. (Rings possessing exactly

one minimal prime ideal are therefore also called irreducible.) Any noetherian scheme can be written uniquely as theunion of finitely many maximal irreducible non-empty closed subsets, called its irreducible components. Affine space

and projective space are irreducible, while Spec k[x,y]/(xy) = is not.

J

Jacobian varietyThe Jacobian variety of a projective curve X is the degree zero part of the Picard variety .

K


9 af 19 29-10-2015 15:23

Kempf vanishing theoremThe Kempf vanishing theorem concerns the vanishing of higher cohomology of a flag variety.

Kodaira dimension1. The Kodaira dimension (also called the Iitaka dimension) of a semi-ample line bundle L is the dimension of Projof the section ring of L.2. The Kodaira dimension of a normal variety X is the Kodaira dimension of its canonical sheaf.

Kodaira vanishing theoremSee the Kodaira vanishing theorem.

L

Lelong numberSee Lelong number.

level structuresee http://math.stanford.edu/~conrad/248BPage/handouts/level.pdf

linearlizationAnother term for the structure of an equivariant sheaf/vector bundle.

localMost important properties of schemes are local in nature, i.e. a scheme X has a certain property P if and only if forany cover of X by open subschemes Xi, i.e. X= Xi, every Xi has the property P. It is usually the case that is enough to

check one cover, not all possible ones. One also says that a certain property is Zariski-local, if one needs todistinguish between the Zariski topology and other possible topologies, like the étale topology. Consider a scheme Xand a cover by affine open subschemes Spec Ai. Using the dictionary between (commutative) rings and affine

schemes local properties are thus properties of the rings Ai. A property P is local in the above sense, iff the

corresponding property of rings is stable under localization. For example, we can speak of locally Noetherianschemes, namely those which are covered by the spectra of Noetherian rings. The fact that localizations of aNoetherian ring are still noetherian then means that the property of a scheme of being locally Noetherian is local inthe above sense (whence the name). Another example: if a ring is reduced (i.e., has no non-zero nilpotent elements),then so are its localizations. An example for a non-local property is separatedness (see below for the definition). Anyaffine scheme is separated, therefore any scheme is locally separated. However, the affine pieces may glue togetherpathologically to yield a non-separated scheme. The following is a (non-exhaustive) list of local properties of rings,which are applied to schemes. Let X = Spec Ai be a covering of a scheme by open affine subschemes. For

definiteness, let k denote a field in the following. Most of the examples also work with the integers Z as a base,though, or even more general bases. Connected, irreducible, reduced, integral, normal, regular, Cohen-Macaulay,locally noetherian, dimension, catenary,

local complete intersectionThe local rings are complete intersection rings. See also: regular embedding.

local uniformizationThe local uniformization is a method of constructing a weaker form of resolution of singularities by means ofvaluation rings.

locally factorialThe local rings are unique factorization domains.

locally of finite typeThe morphism f : Y → X is locally of finite type if may be covered by affine open sets such that eachinverse image is covered by affine open sets where each is finitely generated as a -algebra.

locally NoetherianThe Ai are Noetherian rings. If in addition a finite number of such affine spectra covers X, the scheme is called

noetherian. While it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse is


10 af 19 29-10-2015 15:23

Kollár, János, Chapter 1, "Book onModuli of Surfaces".

false. For example, most schemes in finite-dimensional algebraic geometry are locally Noetherian, but is not.

logarithmic geometry

log structureSee log structure. The notion is due to Fontaine-Illusie and Kato.

loop groupSee loop group (the linked article does not discuss a loop group in algebraic geometry; for now see also ind-scheme).

M

moduliSee for example moduli space.

Mori's minimal model programThe minimal model program is a research program aiming to dobirational classification of algebraic varieties of dimension greater than 2.

morphism1. A morphism of algebraic varieties is given locally by polynomials.2. A morphism of schemes is a morphism of locally ringed spaces.3. A morphism of stacks (over, say, the category ofS-schemes) is a functor such that where arestructures maps to the base category.

multiconeThe multicone of line bundles are the Spec of the sectionring of .

N

nefSee nef line bundle.

nonsingularAn archaic term for "smooth" as in a smooth variety.

normalAn integral scheme is called normal, if the local rings are integrally closed domains. For example, all regular schemesare normal, while singular curves are not.

normal1. If X is a closed subscheme with ideal sheaf I, then the normal sheaf to X is . It is locally free

and called the normal bundle if X is embedded into the ambient scheme as a regular embedding.2. The normal cone to X is ; it is isomorphic to the normal bundle to X if X is regularly embeddedinto the ambient scheme.

normal crossingsSee normal crossings.

O

openA morphism f : Y → X of schemes is called open (closed), if the underlying map of topological spaces is open (closed,respectively), i.e. if open subschemes of Y are mapped to open subschemes of X (and similarly for closed). Forexample, finitely presented flat morphisms are open and proper maps are closed.

While much of the early work onmoduli, especially since [Mum65], putthe emphasis on the construction offine or coarse moduli spaces, recentlythe emphasis shifted towards the studyof the families of varieties, that istowards moduli functors and modulistacks. The main task is to understandwhat kind of objects form “nice”families. Once a good concept of “nicefamilies” is established, the existenceof a coarse moduli space should benearly automatic. The coarse modulispace is not the fundamental objectany longer, rather it is only aconvenient way to keep track ofcertain information that is only latentin the moduli functor or moduli stack.


11 af 19 29-10-2015 15:23

An open subscheme of a scheme X is an open subset U with structure sheaf .[14]. {{{content}}}

orbifold

Nowadays an orbifold is often defined as a Deligne–Mumford stack over the category of differentiable manifolds.[17]

P

p-divisible groupSee p-divisible group (roughly an analog of torsion points of an abelian variety).

Picard groupThe Picard group of X is the group of the isomorphism classes of line bundles on X, the multiplication being thetensor product.

Plücker embeddingThe Plücker embedding is the closed embedding of the Grassmannian variety into a projective space.

plurigenusThe n-th plurigenus of a smooth projective variety is . See also Hodge number.

Poincaré residue mapSee Poincaré residue.

pointA scheme is a locally ringed space, so a fortiori a topological space, but the meanings of point of are threefold:

a point of the underlying topological space;1. a -valued point of is a morphism from to , for any scheme ;2. a geometric point, where is defined over (is equipped with a morphism to) , where is a field, is amorphism from to where is an algebraic closure of .

3.

Geometric points are what in the most classical cases, for example algebraic varieties that are complex manifolds,would be the ordinary-sense points. The points of the underlying space include analogues of the generic points (inthe sense of Zariski, not that of André Weil), which specialise to ordinary-sense points. The -valued points arethought of, via Yoneda's lemma, as a way of identifying with the representable functor it sets up. Historicallythere was a process by which projective geometry added more points (e.g. complex points, line at infinity) to simplifythe geometry by refining the basic objects. The -valued points were a massive further step. As part of thepredominating Grothendieck approach, there are three corresponding notions of fiber of a morphism: the first beingthe simple inverse image of a point. The other two are formed by creating fiber products of two morphisms. Forexample, a geometric fiber of a morphism is thought of as

.This makes the extension from affine schemes, where it is just the tensor product of R-algebras, to all schemes of thefiber product operation a significant (if technically anodyne) result.

polarizationan embedding into a projective space

ProjSee Proj construction.

projective1. A projective variety is a closed subvariety of a projective space.2. A projective scheme over a scheme S is an S-scheme that factors through some projective space as aclosed subscheme.{{{content}}}

projectively bundleIf E is a locally free sheaf on a scheme X, the projective bundle P(E) of E is the global Proj of the symmetric algebraof the dual of E:


12 af 19 29-10-2015 15:23

Note this definition is standard nowadays (e.g., Fulton's Intersection theory) but differs from EGA and Hartshorne(they don't take a dual).

projectively normalA projective variety is projectively normal with respect to some fixed embedding into a projective space if thehomogeneous coordinate ring is an integrally closed domain.

properA morphism is proper if it is separated, universally closed (i.e. such that fiber products with it are closed maps), andof finite type. Projective morphisms are proper; but the converse is not in general true. See also complete variety. Adeep property of proper morphisms is the existence of a Stein factorization, namely the existence of an intermediatescheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism.

property PLet P be a property of a scheme that is stable under base change (finite-type, proper, smooth, étale, etc.). Then arepresentable morphism is said to have propert P if, for any with B a scheme, the base change

has property P.

pure dimensionA scheme has pure dimension d if each irreducible component has dimension d.

Q

quasi-coherentA quasi-coherent sheaf on a Noetheiran scheme X is a sheaf of OX-modules that is locally given by modules.

quasi-compactA morphism f : Y → X is called quasi-compact, if for some (equivalently: every) open affine cover of X by some Ui =

Spec Bi, the preimages f−1(Ui) are quasi-compact.

quasi-finiteThe morphism f : Y → X has finite fibers if the fiber over each point is a finite set. A morphism is quasi-finite if it is of finite type and has finite fibers.

quasi-projectiveA quasi-projective variety is a locally closed subvariety of a projective space.

quasi-separatedA morphism f : Y → X is called quasi-separated or (Y is quasi-separated over X) if the diagonal morphism

Y → Y ×XY is quasi-compact. A scheme Y is called quasi-separated if Y is quasi-separated over Spec(Z).[18]

Quot schemeA Quot scheme parametrizes quotients of locally free sheaves on a projective scheme.

quotient stackUsually denoted by [X/G], a quotient stack generalizes a quotient of a scheme or variety.

R

rationalA variety is rational if it is birational to a projective space; the notion applies in particular to curves and surfaces; seerational curve and rational surface.

rational functionAn element in the function field where the limit runs over all coordinates rings of open subsets U of

an (irreducible) algebraic variety X. See also function field (scheme theory).

rational normal curveA rational normal curve is the image of


13 af 19 29-10-2015 15:23

.If d = 3, it is also called the twisted cubic.

rational singularitiesA variety X over a field of characteristic zero has rational singularities if there is a resolution of singularities

such that and .

reducedThe local rings are reduced rings. Equivalently, none of its rings of sections (U any open subset of X) has any

nonzero nilpotent element. Any variety is reduced (by definition) while Spec k[x]/(x2) is not.

reflexive sheafA coherent sheaf is reflexive if the canonical map to the second dual is an isomorphism.

regularA regular scheme is a scheme where the local rings are regular local rings. For example, smooth varieties over a field

are regular, while Spec k[x,y]/(x2+x3-y2)= is not.

regular embeddingA closed immersion is a regular embedding if each point of X has an affine neighborhood in Y so that theideal of X there is generated by a regular sequence. If i is a regular embedding, then the conormal sheaf of i, that is,

when is the ideal sheaf of X, is locally free.

regular functionA morphism from an algebraic variety to the affine line.

representable morphismA morphism of stacks such that, for any morphism from a scheme B, the base change is ascheme. (Some authors impose algebraic space conditions.)

resolution of singularitiesA resolution of singularities of a scheme X is a proper birational morphism such that Z is smooth.

Riemann–Hurwitz formulaGiven a finite separable morphism between smooth projective curves, if is tamely ramified (no wildramification); for example, over a field of characteristic zero, then the Riemann–Hurwitz formula relates the degreeof π, the genuses of X, Y and the ramification indices:

.

Nowadays, the formula is viewed as a consequence of the more general formula (which is valid even if π is not tame):

where means a linear equivalence and is the divisor of the relative cotangent sheaf

(called the different).

Riemann–Roch formula1. If L is a line bundle of degree d on a smooth projective curve of genus g, then the Riemann–Roch formulacomputes the Euler characteristic of L:

.For example, the formula implies the degree of the canonical divisor K is 2g - 2.

2. The general version is due to Grothendieck and called the Grothendieck–Riemann–Roch formula. It says: if is a proper morphism with smooth X, S and if E is a vector bundle on X, then as equality in the Chow

group

where , means a Chern character and a Todd class of the tangent bundle of a space, and,


14 af 19 29-10-2015 15:23

[1] (http://www.landsburg.com/grothendieck/mclarty1.pdf)

over the complex numbers, is an integration along fibers. For example, if the base S is a point, X is a smooth curveof genus g and E is a line bundle L, then the left-hand side reduces to the Euler characteristic while the right-hand sideis

rigidEvery infinitesimal deformation is trivial. For example, the projective space is rigid since (andusing the Kodaira–Spencer map).

rigidifyA heuristic term, roughly equivalent to "killing automorphisms". For example, one might say "we introduce levelstructures to rigidify the geometric situation."

S

scheme

A scheme is a locally ringed space that is locally a prime spectrum of acommutative ring.

Schubert1. A Schubert cell is a B-orbit on the Grassmannian where B isthe standard Borel; i.e., the group of upper triangular matrices.2. A Schubert variety is the closure of a Schubert cell.

section ringThe section ring or the ring of sections of a line bundle L on a scheme Xis the graded ring .

Serre's conditions SnSee Serre's conditions on normality. See also http://mathoverflow.net/questions/22228/what-is-serres-condition-s-n-for-sheaves

Serre dualitySee #dualizing sheaf

semi-ampleA semi-ample line bundle is a line bundle such that some tensor power ofit is generated by global sections of the power.

separatedA separated morphism is a morphism such that the fiber product of with itself along has its diagonal as a closedsubscheme — in other words, the diagonal map is a closed immersion.

As a consequence, a scheme is separated when the diagonal of within the scheme product of with itself is aclosed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated ifthe unique morphism is separated.

Notice that a topological space Y is Hausdorff iff the diagonal embedding

is closed. In algebraic geometry, the above formulation is used because a scheme which is a Hausdorff space isnecessarily empty or zero-dimensional. The difference between the topological and algebro-geometric context comesfrom the topological structure of the fiber product (in the category of schemes) , which is different fromthe product of topological spaces.

Any affine scheme Spec A is separated, because the diagonal corresponds to the surjective map of rings (hence is aclosed immersion of schemes):

On Grothendieck’s own view thereshould be almost no history ofschemes, but only a history of theresistance to them: ... There is noserious historical question of howGrothendieck found his definition ofschemes. It was in the air. Serre haswell said that no one invented schemes(conversation 1995). The question is,what made Grothendieck believe heshould use this definition to simplifyan 80 page paper by Serre into some1000 pages of Éléments de géométriealgébrique?


15 af 19 29-10-2015 15:23

.

sheaf generated by global sectionsA sheaf with a set of global sections that span the stalk of the sheaf at every point. See Sheaf generated by globalsections.

smooth1.

Main article: smooth morphism

The higher-dimensional analog of étale morphisms are smooth morphisms. There are many different characterisations ofsmoothness. The following are equivalent definitions of smoothness of the morphism f : Y → X:

1) for any y ∈ Y, there are open affine neighborhoods V and U of y, x=f(y), respectively, such that the restriction of f toV factors as an étale morphism followed by the projection of affine n-space over U.2) f is flat, locally of finite presentation, and for every geometric point of Y (a morphism from the spectrum of analgebraically closed field to Y), the geometric fiber is a smooth n-dimensional varietyover in the sense of classical algebraic geometry.

2. A smooth scheme over a perfect field k is a scheme X that is of locally of finite type and regular over k.3. A smooth scheme over a field k is a scheme X that is geometrically smooth: is smooth.

spherical varietyA spherical variety is a normal G-variety (G connected reductive) with an open dense orbit by a Borel subgroup of G.

stable1. A stable curve is a curve with some "mild" singularity, used to construct a good-behaving moduli space of curves.2. A stable vector bundle is used to construct the moduli space of vector bundles.

stackA stack parametrizes sets of points together with automorphisms.

subschemeA subscheme, without qualifier, of X is a closed subscheme of an open subscheme of X.

surfaceAn algebraic variety of dimension two.

symmetric varietyAn analog of a symmetric space. See symmetric variety.

T

tangent spaceSee Zariski tangent space.

tautological line bundleThe tautological line bundle of a projective scheme X is the dual of Serre's twisting sheaf ; that is, .

theoremSee Zariski's main theorem, theorem on formal functions, cohomology base change theorem, Category:Theorems inalgebraic geometry.

torus embeddingAn old term for a toric variety


16 af 19 29-10-2015 15:23

toric varietyA toric variety is a normal variety with the action of a torus such that the torus has an open dense orbit.

tropical geometryA kind of a piecewise-linear algebraic geometry. See tropical geometry.

torusA torus is a product of finitely many multiplicative groups .

U

universal1. If a moduli functor F is represented by some scheme or algebraic space M, then a universal object is an element ofF(M) that corresponds to the identity morphism M → M (which is an M-point of M). If the values of F areisomorphism classes of curves with extra structure, say, then a universal object is called a universal curve. Atautological bundle would be another example of a universal object.2. Let be the moduli of smooth projective curves of genus g and that of smooth projective curves ofgenus g with single marked points. In literature, the forgetful map

is often called a universal curve.

universallyA morphism has some property universally if all base changes of the morphism have this property. Examples includeuniversally catenary, universally injective.

unramifiedFor a point in , consider the corresponding morphism of local rings

.Let be the maximal ideal of , and let

be the ideal generated by the image of in . The morphism is unramified (resp. G-unramified) if it is locallyof finite type (resp. locally of finite presentation) and if for all in , is the maximal ideal of and the inducedmap

is a finite separable field extension.[19] This is the geometric version (and generalization) of an unramified fieldextension in algebraic number theory.

V

varietya synonym with "algebraic variety".

very ampleA line bundle L on a variety X is very ample if X can be embedded into a projective space so that L is the restrictionof Serre's twisting sheaf O(1) on the projective space.

W

weakly normala scheme is weakly normal if any finite birational morphism to it is an isomorphism.

Weil divisorAnother but more standard term for a "codimension-one cycle"; see divisor.

Weil reciprocitySee Weil reciprocity.


17 af 19 29-10-2015 15:23

Z

Zariski–Riemann spaceA Zariski–Riemann space is a locally ringed space whose points are valuation rings.

Notes

Proof: Let D be a Weil divisor on X. If D' ~ D, then there is a nonzero rational function f on X such that D + (f) = D'and then f is a section of OX(D) if D' is effective. The opposite direction is similar. □

1.

Deitmar, Anton (2006-05-16). "Remarks on zeta functions and K-theory over F1". arXiv:math/0605429.2. Flores, Jaret (2015-03-08). "Homological Algebra for Commutative Monoids". arXiv:1503.02309.3. Durov, Nikolai (2007-04-16). "New Approach to Arakelov Geometry". arXiv:0704.2030.4. Lorscheid, Oliver (2013-01-01). "A blueprinted view on $\mathbb F_1$-geometry". arXiv:1301.0083.5. Grothendieck & Dieudonné 1960, 4.1.2 and 4.1.36. Smith, Karen E.; Zhang, Wenliang (2014-09-03). "Frobenius Splitting in Commutative Algebra". arXiv:1409.1169.7. Grothendieck & Dieudonné 1964, §1.48. Grothendieck & Dieudonné 1964, §1.69. Brandenburg, Martin (2014-10-07). "Tensor categorical foundations of algebraic geometry". arXiv:1410.1716.10. Hartshorne 1977, Exercise II.3.11(d)11. The Stacks Project (http://www.math.columbia.edu/algebraic_geometry/stacks-git/morphisms.pdf), Chapter 21, §4.12. Grothendieck & Dieudonné 1960, 4.2.113. Hartshorne 1977, §II.314. Grothendieck & Dieudonné 1960, 4.2.515. Q. Liu, Algebraic Geometry and Arithmetic Curves, exercise 2.316. Harada, Megumi; Krepski, Derek (2013-02-02). "Global quotients among toric Deligne-Mumford stacks".arXiv:1302.0385.

17.

Grothendieck & Dieudonné 1964, 1.2.118. The notion G-unramified is what is called "unramified" in EGA, but we follow Raynaud's definition of "unramified",so that closed immersions are unramified. See Tag 02G4 in the Stacks Project (http://stacks.math.columbia.edu/tag/02G4) for more details.

19.

References

Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series ofModern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveysin Mathematics] 2, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas".Publications Mathématiques de l'IHÉS 4. doi:10.1007/bf02684778. MR 0217083.Grothendieck, Alexandre; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentairede quelques classes de morphismes". Publications Mathématiques de l'IHÉS 8. doi:10.1007/bf02699291.MR 0217084.Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique desfaisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS 11. doi:10.1007/bf02684274.MR 0217085.Grothendieck, Alexandre; Dieudonné, Jean (1963). "Éléments de géométrie algébrique: III. Étude cohomologique desfaisceaux cohérents, Seconde partie". Publications Mathématiques de l'IHÉS 17. doi:10.1007/bf02684890.MR 0163911.Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémaset des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS 20.doi:10.1007/bf02684747. MR 0173675.Grothendieck, Alexandre; Dieudonné, Jean (1965). "Éléments de géométrie algébrique: IV. Étude locale des schémaset des morphismes de schémas, Seconde partie". Publications Mathématiques de l'IHÉS 24. doi:10.1007/bf02684322.


18 af 19 29-10-2015 15:23

MR 0199181.Grothendieck, Alexandre; Dieudonné, Jean (1966). "Éléments de géométrie algébrique: IV. Étude locale des schémaset des morphismes de schémas, Troisième partie". Publications Mathématiques de l'IHÉS 28.doi:10.1007/bf02684343. MR 0217086.Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémaset des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS 32.doi:10.1007/bf02732123. MR 0238860.Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics 52, New York: Springer-Verlag,ISBN 978-0-387-90244-9, MR 0463157Kollár, János, "Book on Moduli of Surfaces" available at his website [2] (https://web.math.princeton.edu/~kollar/)Martin's Olsson's course notes written by Anton, http://math.berkeley.edu/~anton/written/Stacks/Stacks.pdfA book (http://www.math.uzh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1) worked out by manyauthors.

See also

Glossary of arithmetic and Diophantine geometryGlossary of classical algebraic geometryGlossary of differential geometry and topologyGlossary of Riemannian and metric geometryList of complex and algebraic surfacesList of surfacesList of curves

Retrieved from "https://en.wikipedia.org/w/index.php?title=Glossary_of_algebraic_geometry&oldid=687650152"

Categories: Glossaries of mathematics Algebraic geometry Scheme theory

This page was last modified on 26 October 2015, at 22:09.Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By usingthis site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the WikimediaFoundation, Inc., a non-profit organization.


19 af 19 29-10-2015 15:23

Documents

Glossary of Algebraic Geometry