The Hilbert Scheme - Universiteit...

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


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.


Hilbert polynomial

Example:What is pX (m) for X = Pn

k .

Answer (see for example Emily Clader’s notes), pX (m) =(n+m

n

)


Hilbert polynomial

Example:What is pX (m) for X = Pn

k .Answer (see for example Emily Clader’s notes), pX (m) =

(n+mn

)


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).


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).


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.)


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.


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.


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

).


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

).


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

}


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 ).


Hilbert Scheme

Theorem

The functor HilbP(PnS/S) is representable by a scheme

HilbP(PnS/S).


Hilbert Scheme

We may relate this statement to Theorem 1.1(a) in Hartshorne


“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}


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).


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.



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.


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)


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)



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


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.


Properties

Let Z → S be a morphism and X ⊂ PnS closed subscheme, then we

haveHilb(X ×S Z/Z ) ∼= Hilb(X/S)×S Z .


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.


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.


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}.


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).



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 .


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).



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).



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).



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.


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.


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.


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).


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.


http://www-personal.umich.edu/~eclader/HilbertPolynomials.pdf

http://www-personal.umich.edu/~eclader/HilbertPolynomials.pdf

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