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
whose set of T-points
is the set of isomorphism classes of the quotients of
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
gives a Hilbert scheme.)
Definition
For a scheme of finite type
over a Noetherian base scheme
, and a coherent sheaf
, there is a functor[2][3]
![{\displaystyle {\mathcal {Quot}}_{{\mathcal {E}}/X/S}:(Sch/S)^{op}\to {\text{Sets}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
sending
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 }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
and
under the projection
. There is an equivalence relation given by
if there is an isomorphism
commuting with the two projections
; that is,
![{\displaystyle {\begin{matrix}{\mathcal {E}}_{T}&{\xrightarrow {q}}&{\mathcal {F}}\\\downarrow {}&&\downarrow \\{\mathcal {E}}_{T}&{\xrightarrow {q'}}&{\mathcal {F}}'\end{matrix}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is a commutative diagram for
. Alternatively, there is an equivalent condition of holding
. 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
-scheme called the quot scheme associated to a Hilbert polynomial
.
Hilbert polynomial
For a relatively very ample line bundle
[4] and any closed point
there is a function
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})}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
which is a polynomial for
. This is called the Hilbert polynomial which gives a natural stratification of the quot functor. Again, for
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}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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\}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The Hilbert polynomial
is the Hilbert polynomial of
for closed points
. Note the Hilbert polynomial is independent of the choice of very ample line bundle
.
Grothendieck's existence theorem
It is a theorem of Grothendieck's that the functors
are all representable by projective schemes
over
.
Examples
Grassmannian
The Grassmannian
of
-planes in an
-dimensional vector space has a universal quotient
![{\displaystyle {\mathcal {O}}_{G(n,k)}^{\oplus k}\to {\mathcal {U}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
is the
-plane represented by
. Since
is locally free and at every point it represents a
-plane, it has the constant Hilbert polynomial
. This shows
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)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Projective space
As a special case, we can construct the project space
as the quot scheme
![{\displaystyle {\mathcal {Quot}}_{{\mathcal {E}}/X/S}^{1,{\mathcal {O}}_{X}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
for a sheaf
on an
-scheme
.
Hilbert scheme
The Hilbert scheme is a special example of the quot scheme. Notice a subscheme
can be given as a projection
![{\displaystyle {\mathcal {O}}_{X}\to {\mathcal {O}}_{Z}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and a flat family of such projections parametrized by a scheme
can be given by
![{\displaystyle {\mathcal {O}}_{X_{T}}\to {\mathcal {F}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Since there is a hilbert polynomial associated to
, denoted
, there is an isomorphism of schemes
![{\displaystyle {\text{Quot}}_{{\mathcal {O}}_{X}/X/S}^{\Phi _{Z}}\cong {\text{Hilb}}_{X/S}^{\Phi _{Z}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Example of a parameterization
If
and
for an algebraically closed field, then a non-zero section
has vanishing locus
with Hilbert polynomial
![{\displaystyle \Phi _{Z}(\lambda )={\binom {n+\lambda }{n}}-{\binom {n-d+\lambda }{n}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Then, there is a surjection
![{\displaystyle {\mathcal {O}}\to {\mathcal {O}}_{Z}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with kernel
. Since
was an arbitrary non-zero section, and the vanishing locus of
for
gives the same vanishing locus, the scheme
gives a natural parameterization of all such sections. There is a sheaf
on
such that for any
, there is an associated subscheme
and surjection
. This construction represents the quot functor
![{\displaystyle {\mathcal {Quot}}_{{\mathcal {O}}/\mathbb {P} ^{n}/{\text{Spec}}(k)}^{\Phi _{Z}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Quadrics in the projective plane
If
and
, 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}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The universal quotient over
is given by
![{\displaystyle {\mathcal {O}}\to {\mathcal {U}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where the fiber over a point
gives the projective morphism
![{\displaystyle {\mathcal {O}}\to {\mathcal {O}}_{Z}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
For example, if
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}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
then the universal quotient over
gives the short exact sequence
![{\displaystyle 0\to {\mathcal {O}}(-2){\xrightarrow {f}}{\mathcal {O}}\to {\mathcal {O}}_{Z}\to 0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Semistable vector bundles on a curve
Semistable vector bundles on a curve
of genus
can equivalently be described as locally free sheaves of finite rank. Such locally free sheaves
of rank
and degree
have the properties[5]
![{\displaystyle H^{1}(C,{\mathcal {F}})=0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is generated by global sections
for
. 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}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Then, the quot scheme
parametrizes all such surjections. Using the Grothendieck–Riemann–Roch theorem the dimension
is equal to
![{\displaystyle \chi ({\mathcal {F}})=d+n(1-g)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
For a fixed line bundle
of degree
there is a twisting
, shifting the degree by
, so
[5]
giving the Hilbert polynomial
![{\displaystyle \Phi _{\mathcal {F}}(\lambda )=n\lambda +d+n(1-g)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
which can be used to construct the moduli space
of semistable vector bundles using a GIT quotient.[5]
See also
References
- ^ 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
- ^ 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.
- ^ 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.
- ^ Meaning a basis
for the global sections
defines an embedding
for ![{\displaystyle N={\text{dim}}(\Gamma (X,{\mathcal {L}}))}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- ^ a b c Hoskins, Victoria. "Moduli Problems and Geometric Invariant Theory" (PDF). pp. 68, 74–85. Archived (PDF) from the original on 1 March 2020.
Further reading
- Notes on stable maps and quantum cohomology
- https://amathew.wordpress.com/2012/06/02/the-stack-of-coherent-sheaves/