The Hilbert Scheme - Universiteit...
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The Hilbert SchemeTopics in Algebraic Geometry
Rosa Schwarz
Universiteit Leiden
20 februari 2019
Rosa Schwarz The Hilbert scheme
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Overview
Hilbert polynomial (and examples)
Hilbert functor
Hilbert scheme (and examples)
Properties
Applications: the existence of a Hom scheme
Rosa Schwarz The Hilbert scheme
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Hilbert Scheme
Theorem
The functor HilbP(PnS/S) is representable by a scheme
HilbP(PnS/S).
Rosa Schwarz The Hilbert scheme
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Hilbert Scheme
We may relate this statement to Theorem 1.1(a) in Hartshorne
Rosa Schwarz The Hilbert scheme
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“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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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