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The Hilbert Topics in

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

n Let X ⊂ Pk be a , and let I (X ) be the homogeneous ideal corresponding to X and consider Γ(X ) = Γ(X , OX ) = k[x0, .., xn]/I (X ). Definition The Hilbert function of X is defined as

hX : N → N m 7→ dimk (Γ(X )m)

where Γ(X )m is the m-the graded piece of Γ(X ).

Theorem n Let X ⊂ Pk be an embedded projective variety of dimension r. Then there exists a polynomial pX such that hX (m) = pX (m) for all sufficiently large m, and the degree of pX is equal to r. This polynomial is the Hilbert polynomial of X.

Rosa Schwarz The Hilbert scheme n+m Answer (see for example Emily Clader’s notes), pX (m) = n

Hilbert polynomial

Example: n What is pX (m) for X = Pk .

Rosa Schwarz The Hilbert scheme Hilbert polynomial

Example: n What is pX (m) for X = Pk . n+m Answer (see for example Emily Clader’s notes), pX (m) = n

Rosa Schwarz The Hilbert scheme Answer: pX (m) = d (constant polynomial).

Hilbert polynomial

Example: n Let X = {p1, ..., pd } ⊂ Pk be a finite collection of distinct points; what is pX (m)?

Rosa Schwarz The Hilbert scheme Hilbert polynomial

Example: n Let X = {p1, ..., pd } ⊂ Pk be a finite collection of distinct points; what is pX (m)? Answer: pX (m) = d (constant polynomial).

Rosa Schwarz The Hilbert scheme Other definitions: n Let X ⊂ Pk be a projective scheme. The Hilbert polynomial is 0 the unique polynomial such that p(m) = dimk H (X , OX (m)) for sufficiently large m. (Koll´ar) Or for F a coherent sheaf on X as the Euler characteristic ∞ X i i χ(X , F (m)) = (−1) dimk H (X , F (m)) i=0 (Fantechi e.a.)

Hilbert polynomial

Remarks: The degree of a projective variety of dimension r (as in B´ezout’stheorem) is r! times the leading coefficient of pX (m).

Rosa Schwarz The Hilbert scheme Hilbert polynomial

Remarks: The degree of a projective variety of dimension r (as in B´ezout’stheorem) is r! times the leading coefficient of pX (m). Other definitions: n Let X ⊂ Pk be a projective scheme. The Hilbert polynomial is 0 the unique polynomial such that p(m) = dimk H (X , OX (m)) for sufficiently large m. (Koll´ar) Or for F a coherent sheaf on X as the Euler characteristic ∞ X i i χ(X , F (m)) = (−1) dimk H (X , F (m)) i=0 (Fantechi e.a.)

Rosa Schwarz The Hilbert scheme Answer (see for example Emily Clader’s notes), pX (m) = nm + 1.

Hilbert polynomial

Example: 1 n Let νn : Pk → Pk be the n-th Veronese embedding: (x : y) 7→ (xn : xn−1y : ... : xy n−1 : y n)

to all monomials of total degree n in variables x and y. Let 1 X = νn(P ), what is pX (m)?

Rosa Schwarz The Hilbert scheme Hilbert polynomial

Example: 1 n Let νn : Pk → Pk be the n-th Veronese embedding: (x : y) 7→ (xn : xn−1y : ... : xy n−1 : y n)

to all monomials of total degree n in variables x and y. Let 1 X = νn(P ), what is pX (m)? Answer (see for example Emily Clader’s notes), pX (m) = nm + 1.

Rosa Schwarz The Hilbert scheme m+n m+n−d Answer: pX (m) = n − n .

Hilbert polynomial

Example: Let A = k[x0, ..., xn]d and let f ∈ A be a homogeneous polynomial n of degree d. Then X = V (f ) ⊂ Pk is a degree-d hypersurface; what is pX (m)?

Rosa Schwarz The Hilbert scheme Hilbert polynomial

Example: Let A = k[x0, ..., xn]d and let f ∈ A be a homogeneous polynomial n of degree d. Then X = V (f ) ⊂ Pk is a degree-d hypersurface; what is pX (m)? m+n m+n−d Answer: pX (m) = n − n .

Rosa Schwarz The Hilbert scheme Hilbert functor

Hartshorne works over S = Spec(k). Definition n Let Y ⊂ PS be a closed subscheme with Hilbert polynomial P. Define the Hilbert functor as the functor

n op HilbP (PS /S): SchS → Set  subsch Z ⊂ n × T flat over T  T 7→ PS S whose fibers have Hilbert poly P

Rosa Schwarz The Hilbert scheme Hilbert functor

Without specifying a Hilbert polynomial we have

n op Hilb(PS /S): SchS → Set n T 7→ {subschemes Z ⊂ PS ×S T flat over T }

and if T is connected then

n G n Hilb(PS /S)(T ) = HilbP (PS /S)(T ). P

Rosa Schwarz The Hilbert scheme Hilbert Scheme

Theorem n The functor HilbP (PS /S) is representable by a scheme n HilbP (PS /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. The functor is

op Grass(r, E): SchS → Set T 7→ {Subvector bundles of rank r of E ×S T }

Rosa Schwarz The Hilbert scheme Properties

n Let X ⊂ PS be a closed subscheme over S. The theorem implies the existence of a scheme HilbP (X /S). There is a natural injection

n Hilb(X /S) → Hilb(PS /S).

and (as in 1.8 step 4 Koll´ar)we can then represent Hilb(X /S) by n a subscheme of Hilb(PS /S).

Rosa Schwarz The Hilbert scheme ∼ ∼ 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 Examples of Hilbert Schemes.

Hilbert scheme - example

Consider the constant polynomial 1. Then what is Hilb1(X /S)? (And what is Hilb0(X /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 Examples of Hilbert Schemes.

Rosa Schwarz The Hilbert scheme n Hartshorne: if S is connected, then HilbP (PS /S) is connected. (Reference: , Connectedness of the Hilbert Scheme)

Properties

n Let X ⊂ PS be a closed subscheme over S. The scheme HilbP (X /S) is projective over S and Hilb(X /S) is a countable disjoint union of the projective schemes HilbP (X /S).

Rosa Schwarz The Hilbert scheme Properties

n Let X ⊂ PS be a closed subscheme over S. The scheme HilbP (X /S) is projective over S and Hilb(X /S) is a countable disjoint union of the projective schemes HilbP (X /S). n Hartshorne: if S is connected, then HilbP (PS /S) is connected. (Reference: Robin Hartshorne, Connectedness of the Hilbert Scheme)

Rosa Schwarz The Hilbert scheme (Fantechi e.a., number (4) in section 5.1.5) n+t n−d+t For p(t) = n − n have

n ∼ (n+d)−1 Hilbp(t)(P ) = P d

Hilbert scheme - example

Example of a nice Hilbert scheme (see Hartshorne exercise 1): Curves in 2 of degree d are parametrized by a Hilbert scheme Pk¯ d+2 that is a 2 − 1-dimensional projective space.

Rosa Schwarz The Hilbert scheme Hilbert scheme - example

Example of a nice Hilbert scheme (see Hartshorne exercise 1): Curves in 2 of degree d are parametrized by a Hilbert scheme Pk¯ d+2 that is a 2 − 1-dimensional projective space. (Fantechi e.a., number (4) in section 5.1.5) n+t n−d+t For p(t) = n − n have

n ∼ (n+d)−1 Hilbp(t)(P ) = P d

Rosa Schwarz The Hilbert scheme Properties

Hilberts schemes can be nice sometimes, but generally horrible: Murphy’s law (Vakil, Mumford) Arbitrarily bad singularities occur in Hilbert schemes. Reference: Vakil, Murphy’s law in algebraic geometry, badly behaved deformation spaces.

Rosa Schwarz The Hilbert scheme Properties

n Let Z → S be a morphism and X ⊂ PS closed subscheme, then we have ∼ Hilb(X ×S Z/Z) = Hilb(X /S) ×S Z.

Rosa Schwarz The Hilbert scheme Then the Hilbert scheme Hilbm(C) is the collection of degree m subschemes 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, see Fantechi, ea ..., Fundamental Algebraic Geometry, Grothendieck’s FGA explained, chapter 7.3 Examples of Hilbert Schemes.

Hilbert Scheme - example

Let C be a smooth curve over a field k¯. Consider the Hilbert scheme Hilbm(C) for m ∈ Z>0.

Rosa Schwarz The Hilbert scheme Hilbert Scheme - example

Let C be a smooth curve over a field k¯. Consider the Hilbert scheme Hilbm(C) for m ∈ Z>0. Then the Hilbert scheme Hilbm(C) is the collection of degree m subschemes 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, see Fantechi, ea ..., Fundamental Algebraic Geometry, Grothendieck’s FGA 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 Hilbert scheme. For example: the Hom scheme. Definition Let X /S and Y /S be schemes. Define the functor op HomS (X , Y ): SchS → 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 X is flat over S. Then HomS (X , Y ) is represented by an open subscheme HomS (X , Y ) ⊂ Hilb(X ×S Y /S).

Rosa Schwarz The Hilbert scheme Then

Graph map is a closed immersion so Γf ⊂ X ×S Y ×S T is a closed subscheme.

X is flat over S and so X ×S T is flat over T, and so ∼ Γf = X ×S T is flat over T .

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 ) in X ×S Y ×S T :

(id,f ) ∼ X ×S T → X ×S T ×T Y ×S T → X ×S Y ×S 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 ) in X ×S Y ×S T :

(id,f ) ∼ X ×S T → 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 a closed subscheme.

X is flat over S and so X ×S T is flat over T, and so ∼ Γf = 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 an isomorphism. Therefore we can consider HomS (X , Y ) as subfunctor of Hilb(X ×s Y /S).

Rosa Schwarz The Hilbert scheme 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 an isomorphism).

Application - Hom scheme

Now we want to show that HomS (X , Y ) is an open subfunctor of Hilb(X ×s Y /S). That means, for all S-schemes T and maps T → Hilb(X ×S Y /S) the fiber product

T ×Hilb Hom T

γ HomS (X , Y ) Hilb(X ×S Y /S)

is represented by an open subscheme of T .

Rosa Schwarz The Hilbert scheme Application - Hom scheme

Now we want to show that HomS (X , Y ) is an open subfunctor of Hilb(X ×s Y /S). That means, for all S-schemes T and maps T → Hilb(X ×S Y /S) the fiber product

T ×Hilb Hom T

γ HomS (X , Y ) Hilb(X ×S Y /S)

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 an isomorphism).

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 ), 0 then the fiber product T ×Hilb Hom is given at T → T by pairs

0 0 0 ∗  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 a graph. 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 base scheme 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 an isomorphism is an open subset U of S, and f is an isomorphism on the preimage of U.

Lemma Let 0 ∈ T be the spectrum of a local ring. Let U/T be flat and proper and V /T arbitrary. Let p : U → V be a morphism over T . If p0 : U0 → V0 is a closed immersion (resp. an isomorphism), then p 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 S the functor

op IsomS (X , Y ): SchS → 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 the Aut scheme.

Rosa Schwarz The Hilbert scheme Application - Cartier Divisors

Let f : X → S be flat, then D ⊂ X is an effective Cartier divisor if for every x ∈ X there is an fx ∈ OX ,x which is not a zero divisor such that D = Spec(OX ,x /(fx )) in a neighborhood of x. Let X /S be flat. Consider the functor

op CDiv(X /S): SchS → Set

CDiv(X /S)(T ) = {relative effective Cartier divisors V ⊂ X ×S T } .

Theorem (Theorem 1.13.1 Koll´ar)Let X be a scheme, flat and projective over S. Then CDiv(X /S) is representable by an open subscheme CDiv(X /S) ⊂ Hilb(X /S).

Rosa Schwarz The Hilbert scheme References

Robin Hartshorne, Deformation Theory, Springer, 2010, chapters 1 and 24. J´anosKoll´ar,Rational Curves on Algebraic Varieties, Springer (corrected second printing 1999), chapter I.1. Emily Clader, Hilbert polynomials and the degree of a projective variety, notes available on http://www-personal. umich.edu/~eclader/HilbertPolynomials.pdf. Fantechi, ea ..., Fundamental Algebraic Geometry, Grothendieck’s FGA explained, AMS, 2005, chapter 7.3 Examples of Hilbert Schemes. Brian Osserman, A Pithy Look at the Quot, Hilbert and Hom Schemes.

Rosa Schwarz The Hilbert scheme