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.

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 "$\mathcal{Quot}_{\mathcal{E}/X/S}: (Sch/S)^{op} \to \text{Sets}$"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 $$X_T = X\times_ST$$ 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 $$\mathcal{L} \in \text{Pic}(X)$$ 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"$\mathcal{Quot}_{\mathcal{E}/X/S} = \coprod_{\Phi \in \mathbb{Q}[t]} \mathcal{Quot}_{\mathcal{E}/X/S}^{\Phi,\mathcal{L}}$|undefined"where"$\mathcal{Quot}_{\mathcal{E}/X/S}^{\Phi,\mathcal{L}}(T) = \left\{ (\mathcal{F},q) \in \mathcal{Quot}_{\mathcal{E}/X/S}(T) : \Phi_\mathcal{F} = \Phi \right\}$|undefined"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 $$S$$.

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

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


 * 1) $$H^1(C,\mathcal{F}) = 0$$
 * 2) $$\mathcal{F}$$ 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"$\mathcal{Quot}_{\mathcal{O}_C^{\oplus N}/\mathcal{C}/\mathbb{Z}}^{\Phi_\mathcal{F}, \mathcal{L}}$|undefined"which can be used to construct the moduli space $$\mathcal{M}_C(n,d)$$ of semistable vector bundles using a GIT quotient.