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

${{𝒬}{𝓊}{ℴ}{𝓉}}_{{ℰ}/X/S}:(Sch/S{)}^{op}\to \text{Sets}$

sending

$T\to S$

to

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

where

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

and

${{ℰ}}_T=p{r}_X^{*}{ℰ}$

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

$\mathrm{\Phi }$

.

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 ${\mathrm{\Phi }}_{{ℱ}}:\mathbb{\mathbb{N}}\to \mathbb{\mathbb{N}}$ 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

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

where

${{𝒬}{𝓊}{ℴ}{𝓉}}_{{ℰ}/X/S}^{\mathrm{\Phi },{ℒ}}(T)=\{({ℱ},q)\in {{𝒬}{𝓊}{ℴ}{𝓉}}_{{ℰ}/X/S}(T):{\mathrm{\Phi }}_{{ℱ}}=\mathrm{\Phi }\}$

The Hilbert polynomial

${\mathrm{\Phi }}_{{ℱ}}$

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 ${{𝒬}{𝓊}{ℴ}{𝓉}}_{{ℰ}/X/S}^{\mathrm{\Phi },{ℒ}}$ are all representable by projective schemes ${\text{Quot}}_{{ℰ}/X/S}^{\mathrm{\Phi }}$ 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

$\mathrm{\Phi }(\lambda )=k$

. This shows

$G(n,k)$

represents the quot functor

${{𝒬}{𝓊}{ℴ}{𝓉}}_{{{𝒪}}_{G(n,k)}^{\oplus (n)}/\text{Spec}(\mathbb{\mathbb{Z}})/\text{Spec}(\mathbb{\mathbb{Z}})}^{k,{{𝒪}}_{G(n,k)}}$

Projective space

As a special case, we can construct the project space

$\mathbb{\mathbb{P}}({ℰ})$

as the quot scheme

${{𝒬}{𝓊}{ℴ}{𝓉}}_{{ℰ}/X/S}^{1,{{𝒪}}_X}$

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

${\mathrm{\Phi }}_Z$

, there is an isomorphism of schemes

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

Example of a parameterization

If

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

and

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

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

$s\in \mathrm{\Gamma }({𝒪}(d))$

has vanishing locus

$Z=Z(s)$

with Hilbert polynomial

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

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

$a\in k^{*}$

gives the same vanishing locus, the scheme

$Q=\mathbb{\mathbb{P}}(\mathrm{\Gamma }({𝒪}(d)))$

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

${{𝒬}{𝓊}{ℴ}{𝓉}}_{{𝒪}/{\mathbb{\mathbb{P}}}^n/\text{Spec}(k)}^{{\mathrm{\Phi }}_Z}$

Quadrics in the projective plane

If

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

and

$s\in \mathrm{\Gamma }({𝒪}(2))$

, the Hilbert polynomial is

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

and

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

The universal quotient over

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

is given by

${𝒪}\to {𝒰}$

where the fiber over a point

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

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

${{𝒬}{𝓊}{ℴ}{𝓉}}_{{{𝒪}}_C^{\oplus N}/{𝒞}/\mathbb{\mathbb{Z}}}$

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

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

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

${{𝒬}{𝓊}{ℴ}{𝓉}}_{{{𝒪}}_C^{\oplus N}/{𝒞}/\mathbb{\mathbb{Z}}}^{{\mathrm{\Phi }}_{{ℱ}},{ℒ}}$

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/