Quot scheme

In algebraic geometry, the Quot scheme is a scheme parametrizing sheaves on a projective scheme. More specifically, if X is a projective scheme over a Noetherian scheme S and if F is a coherent sheaf on X, then there is a scheme ${\mathrm{Quot}}_F(X)$ whose set of T-points ${\mathrm{Quot}}_F(X)(T)={\mathrm{Mor}}_S(T,{\mathrm{Quot}}_F(X))$ is the set of isomorphism classes of the quotients of $F{\times }_{S}T$ that are flat over T. The notion was introduced by Alexander Grothendieck.

It is typically used to construct another scheme parametrizing geometric objects that are of interest such as a Hilbert scheme. (In fact, taking F to be the structure sheaf ${𝒪}_X$ gives a Hilbert scheme.)

Definition

For a scheme of finite type

$X\to S$

over a Noetherian base scheme

$S$

, and a coherent sheaf

$ℰ\in \text{Coh}(X)$

, there is a functor

${𝒬𝓊ℴ𝓉}_{ℰ/X/S}:(Sch/S{)}^{op}\to \text{Sets}$

sending

$T\to S$

to

${𝒬𝓊ℴ𝓉}_{ℰ/X/S}(T)=\{(ℱ,q):\begin{array}{c}ℱ\in \text{QCoh}(X_T)\\ ℱ\text{finitely presented over}{X}_T\\ \text{Supp}(ℱ)\text{ is proper over }T\\ ℱ\text{ is flat over }T\\ q:{ℰ}_T\to ℱ\text{ surjective}\end{array}\}/\sim$

where

$X_T=X{\times }_{S}T$

and

${ℰ}_T=p{r}_X^{\ast }ℰ$

under the projection

$p{r}_X:X_T\to X$

. There is an equivalence relation given by

$(ℱ,q)\sim ({ℱ}^{\prime },q^{\prime })$

if there is an isomorphism

$ℱ\to {ℱ}^{\prime }$

commuting with the two projections

$q,q^{\prime }$

; that is,

$\begin{array}{ccc}{ℰ}_T& \stackrel{q}{\to }& ℱ\\ \downarrow & & \downarrow \\ {ℰ}_T& \stackrel{q^{\prime }}{\to }& {ℱ}^{\prime }\end{array}$

is a commutative diagram for

${ℰ}_T\stackrel{id}{\to }{ℰ}_T$

. Alternatively, there is an equivalent condition of holding

$\text{ker}(q)=\text{ker}(q^{\prime })$

. This is called the quot functor which has a natural stratification into a disjoint union of subfunctors, each of which is represented by a projective

$S$

-scheme called the quot scheme associated to a Hilbert polynomial

$\Phi$

.

Hilbert polynomial

For a relatively very ample line bundle $ℒ\in \text{Pic}(X)$ and any closed point $s\in S$ there is a function ${\Phi }_{ℱ}:\mathbb{N}\to \mathbb{N}$ sending

$m\mapsto \chi ({ℱ}_s(m))=\sum _{i=0}^n(-1{)}^i{\text{dim}}_{\kappa (s)}{H}^i(X,{ℱ}_s\otimes {ℒ}_s^{\otimes m})$

which is a polynomial for

$m>>0$

. This is called the Hilbert polynomial which gives a natural stratification of the quot functor. Again, for

$ℒ$

fixed there is a disjoint union of subfunctors

${𝒬𝓊ℴ𝓉}_{ℰ/X/S}=\coprod _{\Phi \in \mathbb{Q}[t]}{𝒬𝓊ℴ𝓉}_{ℰ/X/S}^{\Phi ,ℒ}$

where

${𝒬𝓊ℴ𝓉}_{ℰ/X/S}^{\Phi ,ℒ}(T)=\{(ℱ,q)\in {𝒬𝓊ℴ𝓉}_{ℰ/X/S}(T):{\Phi }_{ℱ}=\Phi \}$

The Hilbert polynomial

${\Phi }_{ℱ}$

is the Hilbert polynomial of

${ℱ}_t$

for closed points

$t\in T$

. Note the Hilbert polynomial is independent of the choice of very ample line bundle

$ℒ$

.

Grothendieck's existence theorem

It is a theorem of Grothendieck's that the functors ${𝒬𝓊ℴ𝓉}_{ℰ/X/S}^{\Phi ,ℒ}$ are all representable by projective schemes ${\text{Quot}}_{ℰ/X/S}^{\Phi }$ over $S$.

Examples

Grassmannian

The Grassmannian

$G(n,k)$

of

$k$

-planes in an

$n$

-dimensional vector space has a universal quotient

${𝒪}_{G(n,k)}^{\oplus k}\to 𝒰$

where

${𝒰}_x$

is the

$k$

-plane represented by

$x\in G(n,k)$

. Since

$𝒰$

is locally free and at every point it represents a

$k$

-plane, it has the constant Hilbert polynomial

$\Phi (\lambda )=k$

. This shows

$G(n,k)$

represents the quot functor

${𝒬𝓊ℴ𝓉}_{{𝒪}_{G(n,k)}^{\oplus (n)}/\text{Spec}(\mathbb{Z})/\text{Spec}(\mathbb{Z})}^{k,{𝒪}_{G(n,k)}}$

Projective space

As a special case, we can construct the project space

$\mathbb{P}(ℰ)$

as the quot scheme

${𝒬𝓊ℴ𝓉}_{ℰ/X/S}^{1,{𝒪}_X}$

for a sheaf

$ℰ$

on an

$S$

-scheme

$X$

.

Hilbert scheme

The Hilbert scheme is a special example of the quot scheme. Notice a subscheme

$Z\subset X$

can be given as a projection

${𝒪}_X\to {𝒪}_Z$

and a flat family of such projections parametrized by a scheme

$T\in Sch/S$

can be given by

${𝒪}_{X_T}\to ℱ$

Since there is a hilbert polynomial associated to

$Z$

, denoted

${\Phi }_Z$

, there is an isomorphism of schemes

${\text{Quot}}_{{𝒪}_X/X/S}^{{\Phi }_Z}\cong {\text{Hilb}}_{X/S}^{{\Phi }_Z}$

Example of a parameterization

If

$X={\mathbb{P}}_k^n$

and

$S=\text{Spec}(k)$

for an algebraically closed field, then a non-zero section

$s\in \Gamma (𝒪(d))$

has vanishing locus

$Z=Z(s)$

with Hilbert polynomial

${\Phi }_Z(\lambda )=(\genfrac{}{}{0ex}{}{n+\lambda }{n})-(\genfrac{}{}{0ex}{}{n-d+\lambda }{n})$

Then, there is a surjection

$𝒪\to {𝒪}_Z$

with kernel

$𝒪(-d)$

. Since

$s$

was an arbitrary non-zero section, and the vanishing locus of

$a\cdot s$

for

$a\in k^{\ast }$

gives the same vanishing locus, the scheme

$Q=\mathbb{P}(\Gamma (𝒪(d)))$

gives a natural parameterization of all such sections. There is a sheaf

$ℰ$

on

$X\times Q$

such that for any

$[s]\in Q$

, there is an associated subscheme

$Z\subset X$

and surjection

$𝒪\to {𝒪}_Z$

. This construction represents the quot functor

${𝒬𝓊ℴ𝓉}_{𝒪/{\mathbb{P}}^n/\text{Spec}(k)}^{{\Phi }_Z}$

Quadrics in the projective plane

If

$X={\mathbb{P}}^2$

and

$s\in \Gamma (𝒪(2))$

, the Hilbert polynomial is

$\begin{array}{cc}{\Phi }_Z(\lambda )& =(\genfrac{}{}{0ex}{}{2+\lambda }{2})-(\genfrac{}{}{0ex}{}{2-2+\lambda }{2})\\ & =\frac{(\lambda +2)(\lambda +1)}{2}-\frac{\lambda (\lambda -1)}{2}\\ & =\frac{{\lambda }^2+3\lambda +2}{2}-\frac{{\lambda }^2-\lambda }{2}\\ & =\frac{2\lambda +2}{2}\\ & =\lambda +1\end{array}$

and

${\text{Quot}}_{𝒪/{\mathbb{P}}^2/\text{Spec}(k)}^{\lambda +1}\cong \mathbb{P}(\Gamma (𝒪(2)))\cong {\mathbb{P}}^5$

The universal quotient over

${\mathbb{P}}^5\times {\mathbb{P}}^2$

is given by

$𝒪\to 𝒰$

where the fiber over a point

$[Z]\in {\text{Quot}}_{𝒪/{\mathbb{P}}^2/\text{Spec}(k)}^{\lambda +1}$

gives the projective morphism

$𝒪\to {𝒪}_Z$

For example, if

$[Z]=[a_0:a_1:a_2:a_3:a_4:a_5]$

represents the coefficients of

$f=a_0{x}^2+a_{1}xy+a_{2}xz+a_3{y}^2+a_{4}yz+a_5{z}^2$

then the universal quotient over

$[Z]$

gives the short exact sequence

$0\to 𝒪(-2)\stackrel{f}{\to }𝒪\to {𝒪}_Z\to 0$

Semistable vector bundles on a curve

Semistable vector bundles on a curve $C$ of genus $g$ can equivalently be described as locally free sheaves of finite rank. Such locally free sheaves $ℱ$ of rank $n$ and degree $d$ have the properties

1. $H^1(C,ℱ)=0$
2. $ℱ$ is generated by global sections

for

$d>n(2g-1)$

. This implies there is a surjection

$H^0(C,ℱ)\otimes {𝒪}_C\cong {𝒪}_C^{\oplus N}\to ℱ$

Then, the quot scheme

${𝒬𝓊ℴ𝓉}_{{𝒪}_C^{\oplus N}/𝒞/\mathbb{Z}}$

parametrizes all such surjections. Using the Grothendieck–Riemann–Roch theorem the dimension

$N$

is equal to

$\chi (ℱ)=d+n(1-g)$

For a fixed line bundle

$ℒ$

of degree

$1$

there is a twisting

$ℱ(m)=ℱ\otimes {ℒ}^{\otimes m}$

, shifting the degree by

$nm$

, so

$\chi (ℱ(m))=mn+d+n(1-g)$

giving the Hilbert polynomial

${\Phi }_{ℱ}(\lambda )=n\lambda +d+n(1-g)$

Then, the locus of semi-stable vector bundles is contained in

${𝒬𝓊ℴ𝓉}_{{𝒪}_C^{\oplus N}/𝒞/\mathbb{Z}}^{{\Phi }_{ℱ},ℒ}$

which can be used to construct the moduli space

${ℳ}_C(n,d)$

of semistable vector bundles using a GIT quotient.

See also

• Hilbert polynomial
• Flat morphism
• Hilbert scheme
• Moduli space
• GIT quotient

Further reading

• Notes on stable maps and quantum cohomology
• https://amathew.wordpress.com/2012/06/02/the-stack-of-coherent-sheaves/