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The Hilbert SchemeTopics in Algebraic Geometry
Rosa Schwarz
Universiteit Leiden
20 februari 2019
Rosa Schwarz The Hilbert scheme
Overview
Hilbert polynomial (and examples)
Hilbert functor
Hilbert scheme (and examples)
Properties
Applications: the existence of a Hom scheme
Rosa Schwarz The Hilbert scheme
The Hilbert polynomial
Let X ⊂ Pnk be a projective variety, and let I (X ) be the
homogeneous ideal corresponding to X and considerΓ(X ) = Γ(X ,OX ) = k[x0, .., xn]/I (X ).
Definition
The Hilbert function of X is defined as
hX : N→ Nm 7→ dimk(Γ(X )m)
where Γ(X )m is the m-the graded piece of Γ(X ).
Theorem
Let X ⊂ Pnk be an embedded projective variety of dimension r .
Then there exists a polynomial pX such that hX (m) = pX (m) forall sufficiently large m, and the degree of pX is equal to r . Thispolynomial is the Hilbert polynomial of X.
Rosa Schwarz The Hilbert scheme
Hilbert polynomial
Example:What is pX (m) for X = Pn
k .
Answer (see for example Emily Clader’s notes), pX (m) =(n+m
n
)
Rosa Schwarz The Hilbert scheme
Hilbert polynomial
Example:What is pX (m) for X = Pn
k .Answer (see for example Emily Clader’s notes), pX (m) =
(n+mn
)
Rosa Schwarz The Hilbert scheme
Hilbert polynomial
Example:Let X = {p1, ..., pd} ⊂ Pn
k be a finite collection of distinct points;what is pX (m)?
Answer: pX (m) = d (constant polynomial).
Rosa Schwarz The Hilbert scheme
Hilbert polynomial
Example:Let X = {p1, ..., pd} ⊂ Pn
k be a finite collection of distinct points;what is pX (m)?Answer: pX (m) = d (constant polynomial).
Rosa Schwarz The Hilbert scheme
Hilbert polynomial
Remarks:
The degree of a projective variety of dimension r (as inBezout’s theorem) is r ! times the leading coefficient of pX (m).
Other definitions:Let X ⊂ Pn
k be a projective scheme. The Hilbert polynomial isthe unique polynomial such that p(m) = dimk H
0(X ,OX (m))for sufficiently large m. (Kollar)Or for F a coherent sheaf on X as the Euler characteristic
χ(X ,F (m)) =∞∑i=0
(−1)i dimk Hi (X ,F (m))
(Fantechi e.a.)
Rosa Schwarz The Hilbert scheme
Hilbert polynomial
Remarks:
The degree of a projective variety of dimension r (as inBezout’s theorem) is r ! times the leading coefficient of pX (m).
Other definitions:Let X ⊂ Pn
k be a projective scheme. The Hilbert polynomial isthe unique polynomial such that p(m) = dimk H
0(X ,OX (m))for sufficiently large m. (Kollar)Or for F a coherent sheaf on X as the Euler characteristic
χ(X ,F (m)) =∞∑i=0
(−1)i dimk Hi (X ,F (m))
(Fantechi e.a.)
Rosa Schwarz The Hilbert scheme
Hilbert polynomial
Example:Let νn : P1
k → Pnk be the n-th Veronese embedding:
(x : y) 7→ (xn : xn−1y : ... : xyn−1 : yn)
to all monomials of total degree n in variables x and y . LetX = νn(P1), what is pX (m)?
Answer (see for example Emily Clader’s notes), pX (m) = nm + 1.
Rosa Schwarz The Hilbert scheme
Hilbert polynomial
Example:Let νn : P1
k → Pnk be the n-th Veronese embedding:
(x : y) 7→ (xn : xn−1y : ... : xyn−1 : yn)
to all monomials of total degree n in variables x and y . LetX = νn(P1), what is pX (m)?Answer (see for example Emily Clader’s notes), pX (m) = nm + 1.
Rosa Schwarz The Hilbert scheme
Hilbert polynomial
Example:Let A = k[x0, ..., xn]d and let f ∈ A be a homogeneous polynomialof degree d . Then X = V (f ) ⊂ Pn
k is a degree-d hypersurface;what is pX (m)?
Answer: pX (m) =(m+n
n
)−(m+n−d
n
).
Rosa Schwarz The Hilbert scheme
Hilbert polynomial
Example:Let A = k[x0, ..., xn]d and let f ∈ A be a homogeneous polynomialof degree d . Then X = V (f ) ⊂ Pn
k is a degree-d hypersurface;what is pX (m)?Answer: pX (m) =
(m+nn
)−(m+n−d
n
).
Rosa Schwarz The Hilbert scheme
Hilbert functor
Hartshorne works over S = Spec(k).
Definition
Let Y ⊂ PnS be a closed subscheme with Hilbert polynomial P.
Define the Hilbert functor as the functor
HilbP(PnS/S) : Schop
S → Set
T 7→{
subsch Z ⊂ PnS ×S T flat over T
whose fibers have Hilbert poly P
}
Rosa Schwarz The Hilbert scheme
Hilbert functor
Without specifying a Hilbert polynomial we have
Hilb(PnS/S) : Schop
S → Set
T 7→ {subschemes Z ⊂ PnS ×S T flat over T}
and if T is connected then
Hilb(PnS/S)(T ) =
⊔P
HilbP(PnS/S)(T ).
Rosa Schwarz The Hilbert scheme
Hilbert Scheme
Theorem
The functor HilbP(PnS/S) is representable by a scheme
HilbP(PnS/S).
Rosa Schwarz The Hilbert scheme
Hilbert Scheme
We may relate this statement to Theorem 1.1(a) in Hartshorne
Rosa Schwarz The Hilbert scheme
“Proof”
Due to Grothendieck
Definition
Let S be a scheme, E a vector bundle on S and r ∈ Z≥0. TheGrassmannian functor is
Grass(r ,E ) : SchopS → Set
T 7→ {Subvector bundles of rank r of E ×S T}
Rosa Schwarz The Hilbert scheme
Properties
Let X ⊂ PnS be a closed subscheme over S . The theorem implies
the existence of a scheme HilbP(X/S). There is a natural injection
Hilb(X/S)→ Hilb(PnS/S).
and (as in 1.8 step 4 Kollar) we can then represent Hilb(X/S) bya subscheme of Hilb(Pn
S/S).
Rosa Schwarz The Hilbert scheme
Hilbert scheme - example
Consider the constant polynomial 1. Then what is Hilb1(X/S)?(And what is Hilb0(X/S)?)
Answers: Hilb0(X/S) ∼= S and Hilb1(X/S) ∼= X .Reference: Fantechi, ea ..., Fundamental Algebraic Geometry,Grothendieck’s FGA explained, AMS, 2005, chapter 7.3 Examplesof Hilbert Schemes.
Rosa Schwarz The Hilbert scheme
Hilbert scheme - example
Consider the constant polynomial 1. Then what is Hilb1(X/S)?(And what is Hilb0(X/S)?)Answers: Hilb0(X/S) ∼= S and Hilb1(X/S) ∼= X .Reference: Fantechi, ea ..., Fundamental Algebraic Geometry,Grothendieck’s FGA explained, AMS, 2005, chapter 7.3 Examplesof Hilbert Schemes.
Rosa Schwarz The Hilbert scheme
Properties
Let X ⊂ PnS be a closed subscheme over S . The scheme
HilbP(X/S) is projective over S and Hilb(X/S) is a countabledisjoint union of the projective schemes HilbP(X/S).
Hartshorne: if S is connected, then HilbP(PnS/S) is connected.
(Reference: Robin Hartshorne, Connectedness of the HilbertScheme)
Rosa Schwarz The Hilbert scheme
Properties
Let X ⊂ PnS be a closed subscheme over S . The scheme
HilbP(X/S) is projective over S and Hilb(X/S) is a countabledisjoint union of the projective schemes HilbP(X/S).Hartshorne: if S is connected, then HilbP(Pn
S/S) is connected.(Reference: Robin Hartshorne, Connectedness of the HilbertScheme)
Rosa Schwarz The Hilbert scheme
Hilbert scheme - example
Example of a nice Hilbert scheme (see Hartshorne exercise 1):Curves in P2
kof degree d are parametrized by a Hilbert scheme
that is a(d+2
2
)− 1-dimensional projective space.
(Fantechi e.a., number (4) in section 5.1.5)For p(t) =
(n+tn
)−(n−d+t
n
)have
Hilbp(t)(Pn) ∼= P(n+dd )−1
Rosa Schwarz The Hilbert scheme
Hilbert scheme - example
Example of a nice Hilbert scheme (see Hartshorne exercise 1):Curves in P2
kof degree d are parametrized by a Hilbert scheme
that is a(d+2
2
)− 1-dimensional projective space.
(Fantechi e.a., number (4) in section 5.1.5)For p(t) =
(n+tn
)−(n−d+t
n
)have
Hilbp(t)(Pn) ∼= P(n+dd )−1
Rosa Schwarz The Hilbert scheme
Properties
Hilberts schemes can be nice sometimes, but generally horrible:Murphy’s law (Vakil, Mumford) Arbitrarily bad singularities occurin Hilbert schemes.Reference: Vakil, Murphy’s law in algebraic geometry, badlybehaved deformation spaces.
Rosa Schwarz The Hilbert scheme
Properties
Let Z → S be a morphism and X ⊂ PnS closed subscheme, then we
haveHilb(X ×S Z/Z ) ∼= Hilb(X/S)×S Z .
Rosa Schwarz The Hilbert scheme
Hilbert Scheme - example
Let C be a smooth curve over a field k. Consider the Hilbertscheme Hilbm(C ) for m ∈ Z>0.
Then the Hilbert scheme Hilbm(C ) is the collection of degree msubschemes of dimension zero. This is the set of collections of m(unordered!) points, counted with multiplicities. So C × . . .× C ,m times, quotiented by the symmetric group Sm. Again, seeFantechi, ea ..., Fundamental Algebraic Geometry, Grothendieck’sFGA explained, chapter 7.3 Examples of Hilbert Schemes.
Rosa Schwarz The Hilbert scheme
Hilbert Scheme - example
Let C be a smooth curve over a field k. Consider the Hilbertscheme Hilbm(C ) for m ∈ Z>0.Then the Hilbert scheme Hilbm(C ) is the collection of degree msubschemes of dimension zero. This is the set of collections of m(unordered!) points, counted with multiplicities. So C × . . .× C ,m times, quotiented by the symmetric group Sm. Again, seeFantechi, ea ..., Fundamental Algebraic Geometry, Grothendieck’sFGA explained, chapter 7.3 Examples of Hilbert Schemes.
Rosa Schwarz The Hilbert scheme
Applications - Hom scheme
Many representability results rely on the existence of the Hilbertscheme. For example: the Hom scheme.
Definition
Let X/S and Y /S be schemes. Define the functorHomS(X ,Y ) : Schop
S → Set by
HomS(X ,Y )(T ) = {T −morphisms : X ×S T → Y ×S T}.
Rosa Schwarz The Hilbert scheme
Application - Hom scheme
Theorem
Let X/S and Y /S be projective schemes over S . Assume that Xis flat over S . Then HomS(X ,Y ) is represented by an opensubscheme
HomS(X ,Y ) ⊂ Hilb(X ×S Y /S).
Rosa Schwarz The Hilbert scheme
Application - Hom scheme
Firstly note that there is a morphism of functors
γ : HomS(X ,Y )→ Hilb(X ×S Y /S)
given by associating the graph to a map,i.e. for an S-scheme T ,given f : X ×S T → Y ×S T , we consider the image Γf of (id, f ) inX ×S Y ×S T :
X ×S T(id,f )→ X ×S T ×T Y ×S T
∼→ X ×S Y ×S T .
Then
Graph map is a closed immersion so Γf ⊂ X ×S Y ×S T is aclosed subscheme.
X is flat over S and so X ×S T is flat over T, and soΓf∼= X ×S T is flat over T .
Rosa Schwarz The Hilbert scheme
Application - Hom scheme
Firstly note that there is a morphism of functors
γ : HomS(X ,Y )→ Hilb(X ×S Y /S)
given by associating the graph to a map,i.e. for an S-scheme T ,given f : X ×S T → Y ×S T , we consider the image Γf of (id, f ) inX ×S Y ×S T :
X ×S T(id,f )→ X ×S T ×T Y ×S T
∼→ X ×S Y ×S T .
Then
Graph map is a closed immersion so Γf ⊂ X ×S Y ×S T is aclosed subscheme.
X is flat over S and so X ×S T is flat over T, and soΓf∼= X ×S T is flat over T .
Rosa Schwarz The Hilbert scheme
Application - Hom Scheme
Note that closed subschemes Z ⊂ X ×S Y ×S T , flat over T ,correspond to a graph Γf iff the projection π : Z → X ×S T is anisomorphism.Therefore we can consider HomS(X ,Y ) as subfunctor ofHilb(X ×s Y /S).
Rosa Schwarz The Hilbert scheme
Application - Hom scheme
Now we want to show that HomS(X ,Y ) is an open subfunctor ofHilb(X ×s Y /S). That means, for all S-schemes T and mapsT → Hilb(X ×S Y /S) the fiber product
T ×Hilb Hom T
HomS(X ,Y ) Hilb(X ×S Y /S)γ
is represented by an open subscheme of T .
Then using the isomorphism Hilb(X ×S Y /S)→ Hilb(X ×S Y /S)on the RHS we get an open subscheme of Hilb(X ×S Y /S)representing HomS(X ,Y ) (as pullback of an isomorphism is anisomorphism).
Rosa Schwarz The Hilbert scheme
Application - Hom scheme
Now we want to show that HomS(X ,Y ) is an open subfunctor ofHilb(X ×s Y /S). That means, for all S-schemes T and mapsT → Hilb(X ×S Y /S) the fiber product
T ×Hilb Hom T
HomS(X ,Y ) Hilb(X ×S Y /S)γ
is represented by an open subscheme of T .Then using the isomorphism Hilb(X ×S Y /S)→ Hilb(X ×S Y /S)on the RHS we get an open subscheme of Hilb(X ×S Y /S)representing HomS(X ,Y ) (as pullback of an isomorphism is anisomorphism).
Rosa Schwarz The Hilbert scheme
Application - Hom scheme
Let T → Hilb(X ×S Y /S) be defined by Z ∈ Hilb(X ×S Y /S)(T ),then the fiber product T ×Hilb Hom is given at T ′ → T by pairs(
t : T ′ → T , f : X ×S T ′ → Y ×S T ′ | t∗Z = γ(f )).
Hence by the condition that the image in Hilb(X ×S Y /S) is agraph. Then we want to show that this is an open condition.
Rosa Schwarz The Hilbert scheme
Lemma
Suppose X ,Y are proper schemes over a locally Noetherian basescheme S , with X flat over S , and a morphism f : X → Y over S .Then the locus of points s ∈ S such that fs : Xs → Ys is anisomorphism is an open subset U of S , and f is an isomorphism onthe preimage of U.
Lemma
Let 0 ∈ T be the spectrum of a local ring. Let U/T be flat andproper and V /T arbitrary. Let p : U → V be a morphism over T .If p0 : U0 → V0 is a closed immersion (resp. an isomorphism), thenp is a closed immersion (resp. an isomorphism).
One of these lemma’s finishes the proof.
Rosa Schwarz The Hilbert scheme
Application - Isom scheme
Now we can also show that for X ,Y flat projective schemes over Sthe functor
IsomS(X ,Y ) : SchopS → Set
T 7→ {T − isomorphisms : X ×S T → Y ×S T}.
is representable.David’s exercise: Prove that the Isom scheme is a torsor under theAut scheme.
Rosa Schwarz The Hilbert scheme
Application - Cartier Divisors
Let f : X → S be flat, then D ⊂ X is an effective Cartier divisor iffor every x ∈ X there is an fx ∈ OX ,x which is not a zero divisorsuch that D = Spec(OX ,x/(fx)) in a neighborhood of x .Let X/S be flat. Consider the functor
CDiv(X/S) : SchopS → Set
CDiv(X/S)(T ) = {relative effective Cartier divisors V ⊂ X ×S T } .
Theorem
(Theorem 1.13.1 Kollar) Let X be a scheme, flat and projectiveover S . Then CDiv(X/S) is representable by an open subschemeCDiv(X/S) ⊂ Hilb(X/S).
Rosa Schwarz The Hilbert scheme
References
Robin Hartshorne, Deformation Theory, Springer, 2010,chapters 1 and 24.
Janos Kollar, Rational Curves on Algebraic Varieties, Springer(corrected second printing 1999), chapter I.1.
Emily Clader, Hilbert polynomials and the degree of aprojective variety, notes available on http://www-personal.
umich.edu/~eclader/HilbertPolynomials.pdf.
Fantechi, ea ..., Fundamental Algebraic Geometry,Grothendieck’s FGA explained, AMS, 2005, chapter 7.3Examples of Hilbert Schemes.
Brian Osserman, A Pithy Look at the Quot, Hilbert and HomSchemes.
Rosa Schwarz The Hilbert scheme