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 ${\displaystyle \operatorname {Quot} _{F}(X)}$whose set of T-points ${\displaystyle \operatorname {Quot} _{F}(X)(T)=\operatorname {Mor} _{S}(T,\operatorname {Quot} _{F}(X))}$is the set of isomorphism classes of the quotients of ${\displaystyle F\times _{S}T}$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 ${\displaystyle {\mathcal {O}}_{X}}$gives a Hilbert scheme.)

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

sending ${\displaystyle T\to S}$to

where ${\displaystyle X_{T}=X\times _{S}T}$and ${\displaystyle {\mathcal {E}}_{T}=pr_{X}^{*}{\mathcal {E}}}$under the projection ${\displaystyle pr_{X}:X_{T}\to X}$. There is an equivalence relation given by ${\displaystyle ({\mathcal {F}},q)\sim ({\mathcal {F}}',q')}$if there is an isomorphism ${\displaystyle {\mathcal {F}}\to {\mathcal {F}}'}$commuting with the two projections ${\displaystyle q,q'}$; that is,

is a commutative diagram for ${\displaystyle {\mathcal {E}}_{T}{\xrightarrow {id}}{\mathcal {E}}_{T}}$. Alternatively, there is an equivalent condition of holding ${\displaystyle {\text{ker}}(q)={\text{ker}}(q')}$. 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 ${\displaystyle S}$-scheme called the quot scheme associated to a Hilbert polynomial ${\displaystyle \Phi }$.

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

where

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

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

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

for a sheaf ${\displaystyle {\mathcal {E}}}$on an ${\displaystyle S}$-scheme ${\displaystyle X}$.

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

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

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

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

Then, there is a surjection

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

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

and

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

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

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

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

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

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

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

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

giving the Hilbert polynomial

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

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

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

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