Quot scheme

For a relatively very ample line bundle ${\displaystyle {\mathcal {L}}\in {\text{Pic}}(X)}$[3] and any closed point ${\displaystyle s\in S}$ there is a function ${\displaystyle \phi$ :\mathbb {N} \to \mathbb {N} } sending

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

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

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

where

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Then, there is a surjection

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

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

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

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

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

and

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

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

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

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

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

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

${\displaystyle f=a_{0}x^{2}+a_{1}xy+a_{2}xz+a_{3}y^{2}+a_{4}yz+a_{5}z^{2}}$

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

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

Semistable vector bundles on a curve ${\displaystyle C}$ of genus ${\displaystyle g}$ can equivalently be described as locally free sheaves of finite rank. Such locally free sheaves ${\displaystyle {\mathcal {F}}}$ of rank ${\displaystyle n}$ and degree ${\displaystyle d}$ have the properties[4]

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

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

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

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

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

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

${\displaystyle \chi ({\mathcal {F}}(m))=mn+d+n(1-g)}$[4]

giving the Hilbert polynomial

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

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

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

which can be used to construct the moduli space ${\displaystyle {\mathcal {M}}_{C}(n,d)}$ of semistable vector bundles using a GIT quotient.[4]