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Definition

For a scheme of finite type X → S {\displaystyle X\to S} over a Noetherian base scheme S {\displaystyle S} , and a coherent sheaf E ∈ Coh ( X ) {\displaystyle {\mathcal {E}}\in {\text{Coh}}(X)} , there is a functor²³

Q u o t E / X / S : ( S c h / S ) o p → Sets {\displaystyle {\mathcal {Quot}}_{{\mathcal {E}}/X/S}:(Sch/S)^{op}\to {\text{Sets}}}

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

Q u o t E / X / S ( T ) = { ( F , q ) : F ∈ QCoh ( X T ) F   finitely presented over   X T Supp ( F )  is proper over  T F  is flat over  T q : E T → F  surjective } / ∼ {\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 X T = X × S T {\displaystyle X_{T}=X\times _{S}T} and E T = p r X ∗ E {\displaystyle {\mathcal {E}}_{T}=pr_{X}^{*}{\mathcal {E}}} under the projection p r X : X T → X {\displaystyle pr_{X}:X_{T}\to X} . There is an equivalence relation given by ( F , q ) ∼ ( F ′ , q ′ ) {\displaystyle ({\mathcal {F}},q)\sim ({\mathcal {F}}',q')} if there is an isomorphism F → F ′ {\displaystyle {\mathcal {F}}\to {\mathcal {F}}'} commuting with the two projections q , q ′ {\displaystyle q,q'} ; that is,

E T → <605 width="+0.611em" citerefrathads2007="0.278em" ref_2=".15em"> q F ↓ ↓ E T → <605 width="+0.611em" citerefrathads2007="0.278em" ref_2=".15em"> q ′ F ′ {\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 E T → <605 width="+0.611em" citerefrathads2007="0.278em" ref_2=".15em"> i d E T {\displaystyle {\mathcal {E}}_{T}{\xrightarrow {id}}{\mathcal {E}}_{T}} . Alternatively, there is an equivalent condition of holding ker ( q ) = 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 S {\displaystyle S} -scheme called the quot scheme associated to a Hilbert polynomial Φ {\displaystyle \Phi } .

Hilbert polynomial

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

m ↦ χ ( F s ( m ) ) = ∑ i = 0 n ( − 1 ) i dim κ ( s ) H i ( X , F s ⊗ L s ⊗ 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 m >> 0 {\displaystyle m>>0} . This is called the Hilbert polynomial which gives a natural stratification of the quot functor. Again, for L {\displaystyle {\mathcal {L}}} fixed there is a disjoint union of subfunctors

Q u o t E / X / S = ∐ Φ ∈ Q [ t ] Q u o t E / X / S Φ , L {\displaystyle {\mathcal {Quot}}_{{\mathcal {E}}/X/S}=\coprod _{\Phi \in \mathbb {Q} [t]}{\mathcal {Quot}}_{{\mathcal {E}}/X/S}^{\Phi ,{\mathcal {L}}}}

where

Q u o t E / X / S Φ , L ( T ) = { ( F , q ) ∈ Q u o t E / X / S ( T ) : Φ F = Φ } {\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 Φ F {\displaystyle \Phi _{\mathcal {F}}} is the Hilbert polynomial of F t {\displaystyle {\mathcal {F}}_{t}} for closed points t ∈ T {\displaystyle t\in T} . Note the Hilbert polynomial is independent of the choice of very ample line bundle L {\displaystyle {\mathcal {L}}} .

Grothendieck's existence theorem

It is a theorem of Grothendieck's that the functors Q u o t E / X / S Φ , L {\displaystyle {\mathcal {Quot}}_{{\mathcal {E}}/X/S}^{\Phi ,{\mathcal {L}}}} are all representable by projective schemes Quot E / X / S Φ {\displaystyle {\text{Quot}}_{{\mathcal {E}}/X/S}^{\Phi }} over S {\displaystyle S} .

Examples

Grassmannian

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

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

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

Q u o t O G ( n , k ) ⊕ ( n ) / Spec ( Z ) / Spec ( Z ) k , 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)}}}

Projective space

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

Q u o t E / X / S 1 , O X {\displaystyle {\mathcal {Quot}}_{{\mathcal {E}}/X/S}^{1,{\mathcal {O}}_{X}}}

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

Hilbert scheme

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

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

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

O X T → F {\displaystyle {\mathcal {O}}_{X_{T}}\to {\mathcal {F}}}

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

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

Example of a parameterization

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

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

Then, there is a surjection

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

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

Q u o t O / P n / Spec ( k ) Φ Z {\displaystyle {\mathcal {Quot}}_{{\mathcal {O}}/\mathbb {P} ^{n}/{\text{Spec}}(k)}^{\Phi _{Z}}}

Quadrics in the projective plane

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

Φ Z ( λ ) = ( 2 + λ 2 ) − ( 2 − 2 + λ 2 ) = ( λ + 2 ) ( λ + 1 ) 2 − λ ( λ − 1 ) 2 = λ 2 + 3 λ + 2 2 − λ 2 − λ 2 = 2 λ + 2 2 = λ + 1 {\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

Quot O / P 2 / Spec ( k ) λ + 1 ≅ P ( Γ ( O ( 2 ) ) ) ≅ 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 P 5 × P 2 {\displaystyle \mathbb {P} ^{5}\times \mathbb {P} ^{2}} is given by

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

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

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

For example, if [ 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

f = a 0 x 2 + a 1 x y + a 2 x z + a 3 y 2 + a 4 y z + 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 [ Z ] {\displaystyle [Z]} gives the short exact sequence

0 → O ( − 2 ) → <605 width="+0.611em" citerefrathads2007="0.278em" ref_2=".15em"> f O → O Z → 0 {\displaystyle 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 {\displaystyle C} of genus g {\displaystyle g} can equivalently be described as locally free sheaves of finite rank. Such locally free sheaves F {\displaystyle {\mathcal {F}}} of rank n {\displaystyle n} and degree d {\displaystyle d} have the properties⁵

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

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

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

Then, the quot scheme Q u o t O C ⊕ N / C / 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 N {\displaystyle N} is equal to

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

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

χ ( F ( m ) ) = m n + d + n ( 1 − g ) {\displaystyle \chi ({\mathcal {F}}(m))=mn+d+n(1-g)} ⁶

giving the Hilbert polynomial

Φ F ( λ ) = n λ + 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

Q u o t O C ⊕ N / C / Z Φ F , 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 M C ( n , d ) {\displaystyle {\mathcal {M}}_{C}(n,d)} of semistable vector bundles using a GIT quotient.⁷

See also

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

Further reading

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

References

1. Grothendieck, Alexander. Techniques de construction et théorèmes d'existence en géométrie algébrique IV : les schémas de Hilbert. Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Talk no. 221, p. 249-276 http://www.numdam.org/item/?id=SB_1960-1961__6__249_0 ↩

2. Nitsure, Nitin (2005). "Construction of Hilbert and Quot Schemes". Fundamental algebraic geometry: Grothendieck’s FGA explained. Mathematical Surveys and Monographs. Vol. 123. American Mathematical Society. pp. 105–137. arXiv:math/0504590. ISBN 978-0-8218-4245-4. 978-0-8218-4245-4 ↩

3. Altman, Allen B.; Kleiman, Steven L. (1980). "Compactifying the Picard scheme". Advances in Mathematics. 35 (1): 50–112. doi:10.1016/0001-8708(80)90043-2. ISSN 0001-8708. https://doi.org/10.1016%2F0001-8708%2880%2990043-2 ↩

4. Meaning a basis s i {\displaystyle s_{i}} for the global sections Γ ( X , L ) {\displaystyle \Gamma (X,{\mathcal {L}})} defines an embedding s : X → P S N {\displaystyle \mathbb {s} :X\to \mathbb {P} _{S}^{N}} for N = dim ( Γ ( X , L ) ) {\displaystyle N={\text{dim}}(\Gamma (X,{\mathcal {L}}))} ↩

5. Hoskins, Victoria. "Moduli Problems and Geometric Invariant Theory" (PDF). pp. 68, 74–85. Archived (PDF) from the original on 1 March 2020. https://userpage.fu-berlin.de/hoskins/M15_Lecture_notes.pdf ↩

6. Hoskins, Victoria. "Moduli Problems and Geometric Invariant Theory" (PDF). pp. 68, 74–85. Archived (PDF) from the original on 1 March 2020. https://userpage.fu-berlin.de/hoskins/M15_Lecture_notes.pdf ↩

7. Hoskins, Victoria. "Moduli Problems and Geometric Invariant Theory" (PDF). pp. 68, 74–85. Archived (PDF) from the original on 1 March 2020. https://userpage.fu-berlin.de/hoskins/M15_Lecture_notes.pdf ↩