Quot scheme

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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 $\operatorname {Quot} _{F}(X)$ whose set of T-points $\operatorname {Quot} _{F}(X)(T)=\operatorname {Mor} _{S}(T,\operatorname {Quot} _{F}(X))$ 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.^[1]

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 ${\mathcal {O}}_{X}$ gives a Hilbert scheme.)

Definition

For a scheme of finite type $X\to S$ over a Noetherian base scheme $S$ , and a coherent sheaf ${\mathcal {E}}\in {\text{Coh}}(X)$ , there is a functor^[2]^[3]

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

sending $T\to S$ to

${\mathcal {Quot}}_{{\mathcal {E}}/X/S}(T)=\left\{({\mathcal {F}},q):{\begin{matrix}{\mathcal {F}}\in {\text{QCoh}}(X_{T})\\{\mathcal {F}}\ {\text{finitely presented over}}\ X_{T}\\{\text{Supp}}({\mathcal {F}}){\text{ is proper over }}T\\{\mathcal {F}}{\text{ is flat over }}T\\q:{\mathcal {E}}_{T}\to {\mathcal {F}}{\text{ surjective}}\end{matrix}}\right\}/\sim$

where $X_{T}=X\times _{S}T$ and ${\mathcal {E}}_{T}=pr_{X}^{*}{\mathcal {E}}$ under the projection $pr_{X}:X_{T}\to X$ . There is an equivalence relation given by $({\mathcal {F}},q)\sim ({\mathcal {F}}',q')$ if there is an isomorphism ${\mathcal {F}}\to {\mathcal {F}}'$ commuting with the two projections $q,q'$ ; that is,

${\begin{matrix}{\mathcal {E}}_{T}&{\xrightarrow {q}}&{\mathcal {F}}\\\downarrow {}&&\downarrow \\{\mathcal {E}}_{T}&{\xrightarrow {q'}}&{\mathcal {F}}'\end{matrix}}$

is a commutative diagram for ${\mathcal {E}}_{T}{\xrightarrow {id}}{\mathcal {E}}_{T}$ . Alternatively, there is an equivalent condition of holding ${\text{ker}}(q)={\text{ker}}(q')$ . 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 ${\mathcal {L}}\in {\text{Pic}}(X)$ ^[4] and any closed point $s\in S$ there is a function $\Phi _{\mathcal {F}}:\mathbb {N} \to \mathbb {N}$ sending

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

which is a polynomial for $m>>0$ . This is called the Hilbert polynomial which gives a natural stratification of the quot functor. Again, for ${\mathcal {L}}$ fixed there is a disjoint union of subfunctors

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

where

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

The Hilbert polynomial $\Phi _{\mathcal {F}}$ is the Hilbert polynomial of ${\mathcal {F}}_{t}$ for closed points $t\in T$ . Note the Hilbert polynomial is independent of the choice of very ample line bundle ${\mathcal {L}}$ .

Grothendieck's existence theorem

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

Examples

Grassmannian

The Grassmannian $G(n,k)$ of $k$ -planes in an $n$ -dimensional vector space has a universal quotient

${\mathcal {O}}_{G(n,k)}^{\oplus k}\to {\mathcal {U}}$

where ${\mathcal {U}}_{x}$ is the $k$ -plane represented by $x\in G(n,k)$ . Since ${\mathcal {U}}$ 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

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

Projective space

As a special case, we can construct the project space $\mathbb {P} ({\mathcal {E}})$ as the quot scheme

${\mathcal {Quot}}_{{\mathcal {E}}/X/S}^{1,{\mathcal {O}}_{X}}$

for a sheaf ${\mathcal {E}}$ 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

${\mathcal {O}}_{X}\to {\mathcal {O}}_{Z}$

and a flat family of such projections parametrized by a scheme $T\in Sch/S$ can be given by

${\mathcal {O}}_{X_{T}}\to {\mathcal {F}}$

Since there is a hilbert polynomial associated to $Z$ , denoted $\Phi _{Z}$ , there is an isomorphism of schemes

${\text{Quot}}_{{\mathcal {O}}_{X}/X/S}^{\Phi _{Z}}\cong {\text{Hilb}}_{X/S}^{\Phi _{Z}}$

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 $s\in \Gamma ({\mathcal {O}}(d))$ has vanishing locus $Z=Z(s)$ with Hilbert polynomial

$\Phi _{Z}(\lambda )={\binom {n+\lambda }{n}}-{\binom {n-d+\lambda }{n}}$

Then, there is a surjection

${\mathcal {O}}\to {\mathcal {O}}_{Z}$

with kernel ${\mathcal {O}}(-d)$ . 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 $Q=\mathbb {P} (\Gamma ({\mathcal {O}}(d)))$ gives a natural parameterization of all such sections. There is a sheaf ${\mathcal {E}}$ on $X\times Q$ such that for any $[s]\in Q$ , there is an associated subscheme $Z\subset X$ and surjection ${\mathcal {O}}\to {\mathcal {O}}_{Z}$ . This construction represents the quot functor

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

Quadrics in the projective plane

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

${\begin{aligned}\Phi _{Z}(\lambda )&={\binom {2+\lambda }{2}}-{\binom {2-2+\lambda }{2}}\\&={\frac {(\lambda +2)(\lambda +1)}{2}}-{\frac {\lambda (\lambda -1)}{2}}\\&={\frac {\lambda ^{2}+3\lambda +2}{2}}-{\frac {\lambda ^{2}-\lambda }{2}}\\&={\frac {2\lambda +2}{2}}\\&=\lambda +1\end{aligned}}$

and

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

The universal quotient over $\mathbb {P} ^{5}\times \mathbb {P} ^{2}$ is given by

${\mathcal {O}}\to {\mathcal {U}}$

where the fiber over a point $[Z]\in {\text{Quot}}_{{\mathcal {O}}/\mathbb {P} ^{2}/{\text{Spec}}(k)}^{\lambda +1}$ gives the projective morphism

${\mathcal {O}}\to {\mathcal {O}}_{Z}$

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

$0\to {\mathcal {O}}(-2){\xrightarrow {f}}{\mathcal {O}}\to {\mathcal {O}}_{Z}\to 0$

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 ${\mathcal {F}}$ of rank $n$ and degree $d$ have the properties^[5]

$H^{1}(C,{\mathcal {F}})=0$
${\mathcal {F}}$ is generated by global sections

for $d>n(2g-1)$ . This implies there is a surjection

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

Then, the quot scheme ${\mathcal {Quot}}_{{\mathcal {O}}_{C}^{\oplus N}/{\mathcal {C}}/\mathbb {Z} }$ parametrizes all such surjections. Using the Grothendieck–Riemann–Roch theorem the dimension $N$ is equal to

$\chi ({\mathcal {F}})=d+n(1-g)$

For a fixed line bundle ${\mathcal {L}}$ of degree $1$ there is a twisting ${\mathcal {F}}(m)={\mathcal {F}}\otimes {\mathcal {L}}^{\otimes m}$ , shifting the degree by $nm$ , so

$\chi ({\mathcal {F}}(m))=mn+d+n(1-g)$ ^[5]

giving the Hilbert polynomial

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

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

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

which can be used to construct the moduli space ${\mathcal {M}}_{C}(n,d)$ of semistable vector bundles using a GIT quotient.^[5]

References

^ Grothendieck, Alexander. Techniques de construction et théorèmes d'existence en géométrie algébrique IV : les schémas de Hilbert. Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Talk no. 221, p. 249-276
^ Nitsure, Nitin (2005). "Construction of Hilbert and Quot Schemes". Fundamental algebraic geometry: Grothendieck’s FGA explained. Mathematical Surveys and Monographs. Vol. 123. American Mathematical Society. pp. 105–137. arXiv:math/0504590. ISBN 978-0-8218-4245-4.
^ Altman, Allen B.; Kleiman, Steven L. (1980). "Compactifying the Picard scheme". Advances in Mathematics. 35 (1): 50–112. doi:10.1016/0001-8708(80)90043-2. ISSN 0001-8708.
^ Meaning a basis $s_{i}$ for the global sections $\Gamma (X,{\mathcal {L}})$ defines an embedding $\mathbb {s} :X\to \mathbb {P} _{S}^{N}$ for $N={\text{dim}}(\Gamma (X,{\mathcal {L}}))$
^ ^a ^b ^c Hoskins, Victoria. "Moduli Problems and Geometric Invariant Theory" (PDF). pp. 68, 74–85. Archived (PDF) from the original on 1 March 2020.