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 $\operatorname{Quot}_F(X)$ 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.Grothendieck, Alexander. [http://www.numdam.org/item/?id=SB_1960-1961__6__249_0 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

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{{cite book

| last=Nitsure | first=Nitin

| date=2005

| chapter=Construction of Hilbert and Quot Schemes

| arxiv=math/0504590

| title=Fundamental algebraic geometry: Grothendieck’s FGA explained

| series=Mathematical Surveys and Monographs

| volume=123

| publisher=American Mathematical Society

| isbn=978-0-8218-4245-4

| pages=105–137}}{{cite journal |last1= Altman|first1=Allen B. |last2=Kleiman |first2=Steven L. |date=1980 |title=Compactifying the Picard scheme |journal=Advances in Mathematics |volume=35 |issue=1 |pages=50–112 |doi=10.1016/0001-8708(80)90043-2 |doi-access=free |issn=0001-8708}}

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

sending

T \to S

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

where

X_T = X\times_ST

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 $\mathcal{L} \in \text{Pic}(X)$ Meaning a basis $s_i$ for the global sections $\Gamma(X,\mathcal{L})$ defines an embedding $\mathbb{s}:X \to \mathbb{P}^N_S$ for $N = \text{dim}(\Gamma(X,\mathcal{L}))$ 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

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

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 $X = \mathbb{P}^n_{k}$ 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}

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

$\begin{align}
\Phi_Z(\lambda) &= \binom{2 + \lambda}{2} - \binom{2 - 2 + \lambda}{2} \\
&= \frac{(\lambda + 2)(\lambda + 1)}{2} - \frac{\lambda(\lambda - 1)}{2} \\
&= \frac{\lambda^2 + 3\lambda + 2}{2} - \frac{\lambda^2 - \lambda}{2} \\
&= \frac{2\lambda + 2}{2} \\
&= \lambda + 1
\end{align}$

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

$f = a_0x^2 + a_1xy + a_2xz + a_3y^2 + a_4yz + a_5z^2$

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{{Cite web|url=https://userpage.fu-berlin.de/hoskins/M15_Lecture_notes.pdf|title=Moduli Problems and Geometric Invariant Theory|last=Hoskins|first=Victoria|date=|website=|pages=68, 74–85|url-status=live|archive-url=https://web.archive.org/web/20200301001913/https://userpage.fu-berlin.de/hoskins/M15_Lecture_notes.pdf|archive-date=1 March 2020|access-date=}}

$H^1(C,\mathcal{F}) = 0$
$\mathcal{F}$ 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

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

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.

Quot scheme

Definition

= Hilbert polynomial =

= Grothendieck's existence theorem =

Examples

= Grassmannian =

== Projective space ==

= Hilbert scheme =

== Example of a parameterization ==

== Quadrics in the projective plane ==

= Semistable vector bundles on a curve =

See also

References

Further reading