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 [math]\displaystyle{ \operatorname{Quot}_F(X) }[/math] whose set of T-points [math]\displaystyle{ \operatorname{Quot}_F(X)(T) = \operatorname{Mor}_S(T, \operatorname{Quot}_F(X)) }[/math] is the set of isomorphism classes of the quotients of [math]\displaystyle{ F \times_S T }[/math] 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 [math]\displaystyle{ \mathcal{O}_X }[/math] gives a Hilbert scheme.)

Definition

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

[math]\displaystyle{ \mathcal{Quot}_{\mathcal{E}/X/S}: (Sch/S)^{op} \to \text{Sets} }[/math]

sending [math]\displaystyle{ T \to S }[/math] to

[math]\displaystyle{ \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 }[/math]

where [math]\displaystyle{ X_T = X\times_ST }[/math] and [math]\displaystyle{ \mathcal{E}_T = pr_X^*\mathcal{E} }[/math] under the projection [math]\displaystyle{ pr_X: X_T \to X }[/math]. There is an equivalence relation given by [math]\displaystyle{ (\mathcal{F},q) \sim (\mathcal{F}',q') }[/math] if there is an isomorphism [math]\displaystyle{ \mathcal{F} \to \mathcal{F}'' }[/math] commuting with the two projections [math]\displaystyle{ q, q' }[/math]; that is,

[math]\displaystyle{ \begin{matrix} \mathcal{E}_T & \xrightarrow{q} & \mathcal{F} \\ \downarrow{} & & \downarrow \\ \mathcal{E}_T & \xrightarrow{q'} & \mathcal{F}' \end{matrix} }[/math]

is a commutative diagram for [math]\displaystyle{ \mathcal{E}_T \xrightarrow{id} \mathcal{E}_T }[/math] . Alternatively, there is an equivalent condition of holding [math]\displaystyle{ \text{ker}(q) = \text{ker}(q') }[/math]. 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 [math]\displaystyle{ S }[/math]-scheme called the quot scheme associated to a Hilbert polynomial [math]\displaystyle{ \Phi }[/math].

Hilbert polynomial

For a relatively very ample line bundle [math]\displaystyle{ \mathcal{L} \in \text{Pic}(X) }[/math] and any closed point [math]\displaystyle{ s \in S }[/math] there is a function [math]\displaystyle{ \Phi_\mathcal{F}: \mathbb{N} \to \mathbb{N} }[/math] sending

[math]\displaystyle{ 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}) }[/math]

which is a polynomial for [math]\displaystyle{ m \gt \gt 0 }[/math]. This is called the Hilbert polynomial which gives a natural stratification of the quot functor. Again, for [math]\displaystyle{ \mathcal{L} }[/math] fixed there is a disjoint union of subfunctors

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

where

[math]\displaystyle{ \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\} }[/math]

The Hilbert polynomial [math]\displaystyle{ \Phi_\mathcal{F} }[/math] is the Hilbert polynomial of [math]\displaystyle{ \mathcal{F}_t }[/math] for closed points [math]\displaystyle{ t \in T }[/math]. Note the Hilbert polynomial is independent of the choice of very ample line bundle [math]\displaystyle{ \mathcal{L} }[/math].

Grothendieck's existence theorem

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

Examples

Grassmannian

The Grassmannian [math]\displaystyle{ G(n,k) }[/math] of [math]\displaystyle{ k }[/math]-planes in an [math]\displaystyle{ n }[/math]-dimensional vector space has a universal quotient

[math]\displaystyle{ \mathcal{O}_{G(n,k)}^{\oplus k} \to \mathcal{U} }[/math]

where [math]\displaystyle{ \mathcal{U}_x }[/math] is the [math]\displaystyle{ k }[/math]-plane represented by [math]\displaystyle{ x \in G(n,k) }[/math]. Since [math]\displaystyle{ \mathcal{U} }[/math] is locally free and at every point it represents a [math]\displaystyle{ k }[/math]-plane, it has the constant Hilbert polynomial [math]\displaystyle{ \Phi(\lambda) = k }[/math]. This shows [math]\displaystyle{ G(n,k) }[/math] represents the quot functor

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

Projective space

As a special case, we can construct the project space [math]\displaystyle{ \mathbb{P}(\mathcal{E}) }[/math] as the quot scheme

[math]\displaystyle{ \mathcal{Quot}^{1,\mathcal{O}_X}_{\mathcal{E}/X/S} }[/math]

for a sheaf [math]\displaystyle{ \mathcal{E} }[/math] on an [math]\displaystyle{ S }[/math]-scheme [math]\displaystyle{ X }[/math].

Hilbert scheme

The Hilbert scheme is a special example of the quot scheme. Notice a subscheme [math]\displaystyle{ Z \subset X }[/math] can be given as a projection

[math]\displaystyle{ \mathcal{O}_X \to \mathcal{O}_Z }[/math]

and a flat family of such projections parametrized by a scheme [math]\displaystyle{ T \in Sch/S }[/math] can be given by

[math]\displaystyle{ \mathcal{O}_{X_T} \to \mathcal{F} }[/math]

Since there is a hilbert polynomial associated to [math]\displaystyle{ Z }[/math], denoted [math]\displaystyle{ \Phi_Z }[/math], there is an isomorphism of schemes

[math]\displaystyle{ \text{Quot}_{\mathcal{O}_X/X/S}^{\Phi_Z} \cong \text{Hilb}_{X/S}^{\Phi_Z} }[/math]

Example of a parameterization

If [math]\displaystyle{ X = \mathbb{P}^n_{k} }[/math] and [math]\displaystyle{ S = \text{Spec}(k) }[/math] for an algebraically closed field, then a non-zero section [math]\displaystyle{ s \in \Gamma(\mathcal{O}(d)) }[/math] has vanishing locus [math]\displaystyle{ Z = Z(s) }[/math] with Hilbert polynomial

[math]\displaystyle{ \Phi_Z(\lambda) = \binom{n+\lambda}{n} - \binom{n-d+\lambda}{n} }[/math]

Then, there is a surjection

[math]\displaystyle{ \mathcal{O} \to \mathcal{O}_Z }[/math]

with kernel [math]\displaystyle{ \mathcal{O}(-d) }[/math]. Since [math]\displaystyle{ s }[/math] was an arbitrary non-zero section, and the vanishing locus of [math]\displaystyle{ a\cdot s }[/math] for [math]\displaystyle{ a \in k^* }[/math] gives the same vanishing locus, the scheme [math]\displaystyle{ Q=\mathbb{P}(\Gamma(\mathcal{O}(d))) }[/math] gives a natural parameterization of all such sections. There is a sheaf [math]\displaystyle{ \mathcal{E} }[/math] on [math]\displaystyle{ X\times Q }[/math] such that for any [math]\displaystyle{ [s] \in Q }[/math], there is an associated subscheme [math]\displaystyle{ Z \subset X }[/math] and surjection [math]\displaystyle{ \mathcal{O} \to \mathcal{O}_Z }[/math]. This construction represents the quot functor

[math]\displaystyle{ \mathcal{Quot}_{\mathcal{O}/\mathbb{P}^n/\text{Spec}(k)}^{\Phi_Z} }[/math]

Quadrics in the projective plane

If [math]\displaystyle{ X = \mathbb{P}^2 }[/math] and [math]\displaystyle{ s \in \Gamma(\mathcal{O}(2)) }[/math], the Hilbert polynomial is

[math]\displaystyle{ \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} }[/math]

and

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

The universal quotient over [math]\displaystyle{ \mathbb{P}^5\times\mathbb{P}^2 }[/math] is given by

[math]\displaystyle{ \mathcal{O} \to \mathcal{U} }[/math]

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

[math]\displaystyle{ \mathcal{O} \to \mathcal{O}_Z }[/math]

For example, if [math]\displaystyle{ [Z] = [a_{0}:a_{1}:a_{2}:a_{3}:a_{4}:a_{5}] }[/math] represents the coefficients of

[math]\displaystyle{ f = a_0x^2 + a_1xy + a_2xz + a_3y^2 + a_4yz + a_5z^2 }[/math]

then the universal quotient over [math]\displaystyle{ [Z] }[/math] gives the short exact sequence

[math]\displaystyle{ 0 \to \mathcal{O}(-2)\xrightarrow{f}\mathcal{O} \to \mathcal{O}_Z \to 0 }[/math]

Semistable vector bundles on a curve

Semistable vector bundles on a curve [math]\displaystyle{ C }[/math] of genus [math]\displaystyle{ g }[/math] can equivalently be described as locally free sheaves of finite rank. Such locally free sheaves [math]\displaystyle{ \mathcal{F} }[/math] of rank [math]\displaystyle{ n }[/math] and degree [math]\displaystyle{ d }[/math] have the properties

1. [math]\displaystyle{ H^1(C,\mathcal{F}) = 0 }[/math]
2. [math]\displaystyle{ \mathcal{F} }[/math] is generated by global sections

for [math]\displaystyle{ d \gt n(2g-1) }[/math]. This implies there is a surjection

[math]\displaystyle{ H^0(C,\mathcal{F})\otimes \mathcal{O}_C \cong \mathcal{O}_C^{\oplus N} \to \mathcal{F} }[/math]

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

[math]\displaystyle{ \chi(\mathcal{F}) = d + n(1-g) }[/math]

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

[math]\displaystyle{ \chi(\mathcal{F}(m)) = mn + d + n(1-g) }[/math]

giving the Hilbert polynomial

[math]\displaystyle{ \Phi_\mathcal{F}(\lambda) = n\lambda + d + n(1-g) }[/math]

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

[math]\displaystyle{ \mathcal{Quot}_{\mathcal{O}_C^{\oplus N}/\mathcal{C}/\mathbb{Z}}^{\Phi_\mathcal{F}, \mathcal{L}} }[/math]

which can be used to construct the moduli space [math]\displaystyle{ \mathcal{M}_C(n,d) }[/math] 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/