Quot Scheme

In algebraic geometry, the Quot scheme is a scheme parametrizing locally free 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 $\operatorname_F(X)$ whose set of ''T''-points $\operatorname_F(X)(T) = \operatorname_S(T, \operatorname_F(X))$ 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 $\mathcal_X$ gives a Hilbert scheme.)

Definition

For a scheme of finite type $X \to S$ over a Noetherian base scheme $S$, and a coherent sheaf $\mathcal \in \text(X)$, there is a functor

$\mathcal_: (Sch/S)^ \to \text$

sending $T \to S$ to

$\mathcal_(T) = \left\/ \sim$

where $X_T = X\times_ST$ and $\mathcal_T = pr_X^*\mathcal$ under the projection $pr_X: X_T \to X$. There is an equivalence relation given by $(\mathcal,q) \sim (\mathcal',q')$ if there is an isomorphism $\mathcal \to \mathcal''$ commuting with the two projections $q, q'$; that is,

$\begin \mathcal_T & \xrightarrow & \mathcal \\ \downarrow & & \downarrow \\ \mathcal_T & \xrightarrow & \mathcal' \end$

is a commutative diagram for $\mathcal_T \xrightarrow \mathcal_T$ . Alternatively, there is an equivalent condition of holding $\text(q) = \text(q')$. 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 $\mathcal \in \text(X)$Meaning a basis $s_i$ for the global sections $\Gamma(X,\mathcal)$ defines an embedding $\mathbb:X \to \mathbb^N_S$ for $N = \text(\Gamma(X,\mathcal))$ and any closed point $s \in S$ there is a function $\phi: \mathbb \to \mathbb$ sending

$m \mapsto \chi(\mathcal_s(m)) = \sum_^n (-1)^i\text_H^i(X,\mathcal_s\otimes \mathcal_s^)$

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

$\mathcal_ = \coprod_ \mathcal_^$

where

$\mathcal_^(T) = \left\$

The Hilbert polynomial $\Phi_\mathcal$ is the Hilbert polynomial of $\mathcal_t$ for closed points $t \in T$. Note the Hilbert polynomial is independent of the choice of very ample line bundle $\mathcal$.

Grothendieck's existence theorem

It is a theorem of Grothendieck's that the functors $\mathcal_^$ are all representable by projective schemes $\text_^$ over $S$.

Examples

Grassmannian

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

$\mathcal_^ \to \mathcal$

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

$\mathcal_^$

Projective space

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

$\mathcal^_$

for a sheaf $\mathcal$ 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

$\mathcal_X \to \mathcal_Z$

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

$\mathcal_ \to \mathcal$

Since there is a hilbert polynomial associated to $Z$, denoted $\Phi_Z$, there is an isomorphism of schemes

$\text_^ \cong \text_^$

Example of a parameterization

If $X = \mathbb^n_$ and $S = \text(k)$ for an algebraically closed field, then a non-zero section $s \in \Gamma(\mathcal(d))$ has vanishing locus $Z = Z(s)$ with Hilbert polynomial

$\Phi_Z(\lambda) = \binom - \binom$

Then, there is a surjection

$\mathcal \to \mathcal_Z$

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

$\mathcal_^$

Quadrics in the projective plane

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

$\begin \Phi_Z(\lambda) &= \binom - \binom \\ &= \frac - \frac \\ &= \frac - \frac \\ &= \frac \\ &= \lambda + 1 \end$

and

$\text_^ \cong \mathbb(\Gamma(\mathcal(2))) \cong \mathbb^$

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

$\mathcal \to \mathcal$

where the fiber over a point \in \text_^ gives the projective morphism

$\mathcal \to \mathcal_Z$

For example, if = _:a_:a_:a_:a_:a_/math> 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 /math> gives the short exact sequence

$0 \to \mathcal(-2)\xrightarrow\mathcal \to \mathcal_Z \to 0$

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 $\mathcal$ of rank $n$ and degree $d$ have the properties

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

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

$H^0(C,\mathcal)\otimes \mathcal_C \cong \mathcal_C^ \to \mathcal$

Then, the quot scheme $\mathcal_$ parametrizes all such surjections. Using the Grothendieck–Riemann–Roch theorem the dimension $N$ is equal to

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

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

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

giving the Hilbert polynomial

$\Phi_\mathcal(\lambda) = n\lambda + d + n(1-g)$

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

$\mathcal_^$

which can be used to construct the moduli space $\mathcal_C(n,d)$ of semistable vector bundles using a GIT quotient.

See also

* Hilbert polynomial
* Flat morphism
* Hilbert scheme
* Moduli space
* GIT quotient

References

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Further reading

* Notes on stable maps and quantum cohomology
*https://amathew.wordpress.com/2012/06/02/the-stack-of-coherent-sheaves/

Algebraic geometry