Families of schemes

Introduction to Flatness and the Quot scheme

Sheffield, Reading seminar on DT theory, 308/04/2020

Anna Barbieri

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Today

0. intro1. flatness of sheaves/schemes2. the Quot scheme

Bibliography- Huybrechts, Lehn, The geometry of moduli spaces of sheaves, Ch. 2

- Eisenbud, Harris, The geometry of schemes

- Nitsure, Construction of Hilbert and Quot scheme, in “Fundamentalalgebraic geometry: Grothendieck’s FGA explained”

- Cristina!

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Introduction

X a scheme, X LOC=(

Spec R,OX

)(AffSch)op ' (Rings)

(Sch) (Schk ) (SchS)

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Introduction

X ∈ (SchS)f : T → S, morphism of schemes

Base-change

XT := X ×S Tfibres over T

XT πX//

πT��

X

��T // S

If S = Spec R, X = Spec R1, T = Spec R2, then XT = Spec(R1 ⊗R R2).

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1. Flatness

Flatness for modulesDef.: M ∈ R-mod flat iff

∀ M1 ↪→ M2 injectivethen M1 ⊗R M ↪→ M2 ⊗R M injective.

Examples- free modules are flat;- flat modules are torsion-free, silly ex:

0→ Z ·2−→Z→ Z/2Z→ 0 ⊗ZZ/2Z

Z/2Z ·2−→Z/2Z

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Flatness for coh sheaves

Def.: F ∈ Coh(X ) is flat if

it is a sheaf of OX -flat modules.

Equiv.:every stalk Fx is a flat OX ,x -module.

Example: Locally free sheaves are flat

OX (D) is a flat OX -sheaf

Non-example: torsion sheaves

D a divisor on X , OD is not flat over X .

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Flatness for schemes / sheaves on S-schemes

Def.: A map f : X → S is flat if

OX is flat as an OS-module.

Recall: a map of sch.s f : X → S induces a map of sheaves f # : OS → OX .

By the (non)example above, pt → CP1 is not flat.

Def.: A sheaf F on X ∈ (SchS) is flat over S if

Fx is flat as an Of (x),S-module.

Remark: Given f : X → S, a sheaf F ∈ Coh(X ) which is flatover S is also called a

flat family of coherent sheaves on the fibres of f .

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Some general facts

• Open embeddings are flat (maps)

• X a Noetherian scheme, then

F flat over X ⇐⇒ locally free.

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Some general factsf : X → S, F ∈ Coh(X ) S-flat

• Base change

• transitivity: g : S → S′ flat⇒ F is S′-flat too• behaviour in short exact sequences

0→ A→ B → FF → 0

A is S-flat⇔ B is S-flat.

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Flat families of schemes

X (flat) S-scheme, akaπ : X → S (flat) family of schemes

Flat families provide a notion of “continuously varying” family.

This will be better understood if we look at one-parameter families⇒ geometric characterization of flatness.

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Geometric characterization of flatness for maps ofschemes, 1

Dimension 1

Propositionf : X → S a morphism of schemes,where S = Spec R, smooth, 1-dimensional affine scheme.

map f flat ⇐⇒ no irreducible nor embedded componentsare contracted.

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Proof

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Proof

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Flatness is a local property!

Dimension 1

PropositionS = Spec R 1-dimensional affine smooth scheme,X o ⊂ An

B a flat family of closed schemes over So = S \ {b∗}.

There is a unique way of completing X o to a flat family over B.

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Flatness is a local property!

Obtained by taking the closure X o inside AnB.

Added fibre: limit of X o at b∗.

Depends very much on the embedding.

Does not admit generalization to any dimensional bases.

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Geometric characterization of flatness for maps ofschemes, 2

Any dimension

PropositionS be a reduced scheme

f : X → S is flat ⇐⇒ ∀C ↪→ S a curve,fC is flat

XC //

fC��

X

f��

C // S

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Geometric meaning of flatness

Consequence: all fibres must have the same dimension. So

1. SpecC[x , y ]/(xy , x2)→ SpecC[x , y ]/(x) is not flat

2. BlpP2 → P2 is not flat

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RemarkWe are not able to speak about limit fibre of a punctured sheafis S is not of dimension 1.

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Open condition

PropositionX ,S Noetherian schemes.

Flatness of f : X → S (resp. of a sheaf F ∈ Coh(X )) is an opencondition.

I.e.

∀V ⊂ S, there is a maximal open U ⊂ V ⊂ Xwhere f (resp. F ) is flat.

As a corollary, being locally free is an open condition as well.

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Hilbert polynomial of flat families

f : X → S a projective family

Def.: L = O(1) is an f -ample line bundle on X if L|Xs is amplefor any s ∈ S.

Hilbert polynomial of a projective family f : X → S{φL|Xs (Xs) : s ∈ S

}

• it is locally constant as a function of s ∈ S;• it is constant⇔ if f is flat and B is reduced;• finitely many possible Hilbert polynomials of the fibres;

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Hilbert polynomial of flat families

Hilbert polynomial of a projective family f : X → S{φL|Xs (Xs) : s ∈ S

}

• induces a stratification of S by

φs = φL|Xs (Xs).

This decomposition is called a flattening stratification of S forf and it is unique.

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2. The Quot functor

Recall the Grassmannian of a vector space. . .

V v.s. of dimension n > 0, /k ,0 ≤ r ≤ n

Gr (r ,n) = {v.s. W ⊆ V , dim W = r} ' Gr (n − r ,n)

via V = W ⊕W⊥ = V ⊕ V/W .

Generalize to trivial v.bundle over a base B, take sub-vectorbundles of a given rank r .

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OB ⊗k V , loc. free, B ∈ (Schk )

defineGrB(V , r )

as the set of all sub-sheaves K ⊂ OB ⊗k V with locally freequotient

F =(OB ⊗k V

)/K

of constant rank r ,i.e.

q : OB ⊗k V → F → 0.

Why quotient sheaves?- in general sequences do not split,- quotient sheaves behaves better than sub-sheaves.

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Sketchy Quot

Vaguely speaking, forf : X → S projective morphismH ∈ Coh(X ),P ∈ Q[z]

we want to parametrise flat coherent quotient sheaves

q : H → F → 0

with Hilbert polynomial P.

Work “relative“, and take Quot as the scheme that represents afunctor Quot.

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The functor of points

X a scheme, the functor of points of the scheme X is

hX : (Sch)op → (Sets)T 7→ Mor(T ,X )(

f : Y → Z)7→(hX (f ) : g 7→ g ◦ f

)g ∈ Mor(Z , X ), g ◦ f ∈ Mor(Y , X )

Def.: A covariant functor

H : (Sch)op → (Sets)

is representable if

∃X scheme, s.t. H = hX .

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The Quot functor, Pn

S a (locally) Noetherian scheme, PnS = Pn ×Z S, r ∈ N>0.

A family of quotients of O⊕rPn parametrised by S is⟨

F ,q⟩

(i) F ∈ Coh(PnS) flat over S,and

(ii) q : O⊕rPn

S→ F surjective and OPn

S-linear homomorphism,

up to⟨F ,q

⟩=⟨F ′,q′

⟩if ker(q) = ker(q′).

The equivalence relation is also given byF'→ F ′ fitting the commutative diagram:

F

'��

⊕rOPnS

q 44

q′

))F ′

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The Quot functor, Pn

Controvariant functor

QuotO⊕rPn

: (Sch)→ (Sets)

S 7→{〈F ,q〉 parametrised by S

}

〈F , q〉 family of quotients parametrised by S:F ∈ Coh(Pn

S) flat over S,q : O⊕r

PnS→ F surjective.

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If we wanted

OPnS

q // F

xx

OPn

xxS � // Pn

S//

��

Pn

��S // Z

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The functor QuotE /X/S

S a Noetherian schemeX → S a finite type schemeE ∈ Coh(X )

E

wwX

��S

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The functor QuotE /X/S

S a Noetherian schemeX → S a finite type schemeE ∈ Coh(X )

∀ T → S an S-scheme

ET := π∗X E

ET

E

xxXT

πT��

πX// X

��T // S

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The functor QuotE /X/S

S a Noetherian schemeX → S a finite type schemeE ∈ Coh(X )

∀ T → S an S-schemeET := π∗X E

ET

q // F

ww

E

wwXT πX//

πT��

X

��T // S

choose

〈F ,q〉F T -flat sheaf, q : ET → F surjective

up to equivalence relation given by ker(q) = ker(q′).

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The functor QuotE /X/S

S a Noetherian schemeX → S a finite type schemeE ∈ Coh(X )

∀ T → S an S-schemeET := π∗X E

ET

q // F

ww

E

wwXT πX//

πT��

X

��T // S

choose

〈F ,q〉shf T -flat sheaf, q : ET → F surjective

up to equivalence relation given by ker(q) = ker(q′).

QuotE /X/S : (SchS)→ (Sets)

T 7→{〈F ,q〉 parametrised by T

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The Hilbert functorA special Quot

HilbX/S := QuotOX/X/S

HilbX/S : (SchS)→ (Sets)

T 7→{closed Y ⊂ XT flat over T

}In particular

HilbPn := HilbPnZ/SpecZ

associates to S families of subschemes of Pn parametrised byS.

(closed subschemes Y ⊂ PnS flat /S same as OPn → OY on Pn

S flat /S)

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Stratification by Hilbert polynomial

f : X → SL ample line bundle on X relative SRecall the flattening stratification of S for f

Decomposition

QuotE/X/S =⊔φ,L

Quotφ,LE/X/S,

where, for any φ ∈ Q[λ],

Quotφ,LE/X/S : (T → S) 7→{〈F ,q〉 | ∀ t ∈ T φL∗T (Ft ) = φ

}.

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The Grassmannian as a Quot scheme

V v.s. dim V = n, 0 ≤ d ≤ n.The functor T 7→ GrT (V ,d) is

Grass(n,d) = Quotd ,OZO⊕nZ /Spec(Z)/Spec(Z)

,

abbreviated Quotd ,OZO⊕nZ /Z/Z. It maps

T 7→⟨F ∈ Coh(T ) l.f. rk d , q : O⊕n

T → F⟩

When d = 1, we know this is PnZ = ProjZ[x0, . . . , xn]. In

particular it is representable.

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The Grassmannian as a Quot scheme

E locally free OS-module of rank r

Grass(E ,d) = Quotd ,OSE/S/S.

is also representable. Grass(E ,d) parametrises sub-vectorbundles of E .

When d = 1,

Grass(E ,d) = P(E) = Proj SymOS(E).

The construction of the scheme Quot make sense even ifE ∈ Coh(S) is not locally free.

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Representability of Quot

The functor Quot is proven to be representable.

π : X → S, Noetherian.In the special case when

X = P(V ),E = π∗(W )

V ,W vector bundles /S

the representability is obtained by

Quotφ,LE /X/S → Grass(W ⊗OS Symr V , φ(r )

)for some r ∈ N>0

The general case then follows from base change and usingsome properties of flat families.

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Thanks

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