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 $\mathrm{\Phi }$.

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

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

where

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

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 ${𝒬𝓊ℴ𝓉}_{ℰ/X/S}^{\mathrm{\Phi },ℒ}$ 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

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

Projective space

As a special case, we can construct the projective bundle $\mathbb{P}(ℰ)$ over $X$ 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 ${\mathrm{\Phi }}_Z$, 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 $X={\mathbb{P}}_k^n$ 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 $a\in k^{\ast }$ gives the same vanishing locus, the scheme $Q=\mathbb{P}(\mathrm{\Gamma }(𝒪(d)))$ 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

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

Quadrics in the projective plane

If $X={\mathbb{P}}^2$ and $s\in \mathrm{\Gamma }(𝒪(2))$, the Hilbert polynomial is

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

and

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

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

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

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

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

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/