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 ${\displaystyle \operatorname {Quot} _{F}(X)}${\displaystyle \operatorname {Quot} _{F}(X)} whose set of T-points ${\displaystyle \operatorname {Quot} _{F}(X)(T)=\operatorname {Mor} _{S}(T,\operatorname {Quot} _{F}(X))}${\displaystyle \operatorname {Quot} _{F}(X)(T)=\operatorname {Mor} _{S}(T,\operatorname {Quot} _{F}(X))} is the set of isomorphism classes of the quotients of ${\displaystyle F\times _{S}T}${\displaystyle F\times _{S}T} that are flat over T. The notion was introduced by Alexander Grothendieck.^[1]

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 ${\displaystyle {\mathcal {O}}_{X}}${\displaystyle {\mathcal {O}}_{X}} gives a Hilbert scheme.)

For a scheme of finite type ${\displaystyle X\to S}${\displaystyle X\to S} over a Noetherian base scheme ${\displaystyle S}${\displaystyle S}, and a coherent sheaf ${\displaystyle {\mathcal {E}}\in {\text{Coh}}(X)}${\displaystyle {\mathcal {E}}\in {\text{Coh}}(X)}, there is a functor^{[2] [3]}

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

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

${\displaystyle {\mathcal {Quot}}_{{\mathcal {E}}/X/S}(T)=\left\{({\mathcal {F}},q):{\begin{matrix}{\mathcal {F}}\in {\text{QCoh}}(X_{T})\\{\mathcal {F}}\ {\text{finitely presented over}}\ X_{T}\\{\text{Supp}}({\mathcal {F}}){\text{ is proper over }}T\\{\mathcal {F}}{\text{ is flat over }}T\\q:{\mathcal {E}}_{T}\to {\mathcal {F}}{\text{ surjective}}\end{matrix}}\right\}/\sim }${\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 }

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

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

For a relatively very ample line bundle ${\displaystyle {\mathcal {L}}\in {\text{Pic}}(X)}${\displaystyle {\mathcal {L}}\in {\text{Pic}}(X)}^[4] and any closed point ${\displaystyle s\in S}${\displaystyle s\in S} there is a function ${\displaystyle \Phi _{\mathcal {F}}:\mathbb {N} \to \mathbb {N} }${\displaystyle \Phi _{\mathcal {F}}:\mathbb {N} \to \mathbb {N} } sending

${\displaystyle m\mapsto \chi ({\mathcal {F}}_{s}(m))=\sum _{i=0}^{n}(-1)^{i}{\text{dim}}_{\kappa (s)}H^{i}(X,{\mathcal {F}}_{s}\otimes {\mathcal {L}}_{s}^{\otimes m})}${\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 ${\displaystyle m>>0}${\displaystyle m>>0}. This is called the Hilbert polynomial which gives a natural stratification of the quot functor. Again, for ${\displaystyle {\mathcal {L}}}${\displaystyle {\mathcal {L}}} fixed there is a disjoint union of subfunctors

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

where

${\displaystyle {\mathcal {Quot}}_{{\mathcal {E}}/X/S}^{\Phi ,{\mathcal {L}}}(T)=\left\{({\mathcal {F}},q)\in {\mathcal {Quot}}_{{\mathcal {E}}/X/S}(T):\Phi _{\mathcal {F}}=\Phi \right\}}${\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 ${\displaystyle \Phi _{\mathcal {F}}}${\displaystyle \Phi _{\mathcal {F}}} is the Hilbert polynomial of ${\displaystyle {\mathcal {F}}_{t}}${\displaystyle {\mathcal {F}}_{t}} for closed points ${\displaystyle t\in T}${\displaystyle t\in T}. Note the Hilbert polynomial is independent of the choice of very ample line bundle ${\displaystyle {\mathcal {L}}}${\displaystyle {\mathcal {L}}}.

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

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

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

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

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

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

${\displaystyle {\mathcal {Quot}}_{{\mathcal {E}}/X/S}^{1,{\mathcal {O}}_{X}}}${\displaystyle {\mathcal {Quot}}_{{\mathcal {E}}/X/S}^{1,{\mathcal {O}}_{X}}}

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

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

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

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

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

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

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

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

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

Then, there is a surjection

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

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

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

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

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

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

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

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

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

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

${\displaystyle f=a_{0}x^{2}+a_{1}xy+a_{2}xz+a_{3}y^{2}+a_{4}yz+a_{5}z^{2}}${\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 ${\displaystyle [Z]}${\displaystyle [Z]} gives the short exact sequence

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

Semistable vector bundles on a curve ${\displaystyle C}${\displaystyle C} of genus ${\displaystyle g}${\displaystyle g} can equivalently be described as locally free sheaves of finite rank. Such locally free sheaves ${\displaystyle {\mathcal {F}}}${\displaystyle {\mathcal {F}}} of rank ${\displaystyle n}${\displaystyle n} and degree ${\displaystyle d}${\displaystyle d} have the properties^[5]

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

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

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

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

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

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

${\displaystyle \chi ({\mathcal {F}}(m))=mn+d+n(1-g)}${\displaystyle \chi ({\mathcal {F}}(m))=mn+d+n(1-g)}^[5]

giving the Hilbert polynomial

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

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

${\displaystyle {\mathcal {Quot}}_{{\mathcal {O}}_{C}^{\oplus N}/{\mathcal {C}}/\mathbb {Z} }^{\Phi _{\mathcal {F}},{\mathcal {L}}}}${\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 ${\displaystyle {\mathcal {M}}_{C}(n,d)}${\displaystyle {\mathcal {M}}_{C}(n,d)} of semistable vector bundles using a GIT quotient.^[5]

• 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/

• Algebraic geometry