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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8efa500bf756aa98765eb7dc961d4e198166145f)
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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f370617770e8586603906ca44a3402ec40c3a487)
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7517a83c8fcb61aac6fb52153bfead03e4d0c2)
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
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))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/f890abd582faea3b13b26c15096e25304eb3b203)
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\))](https://wikimedia.org/api/rest_v1/media/math/render/svg/df1cd8b1cb89bba5313e3d653bbaad0f10511a85)
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/381ba03ea086b31f99379c6336c805fba48f9de2)
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)))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f99988626dd372fe17d0457970354767416997c)
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9f0fad392688c9851fe46eee16fa8b7ab07359)
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))](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c7e0875569c0e059628dcf0087acf68e1ae72ff)
and a flat family of such projections parametrized by a scheme
can be given by
![{\displaystyle {\mathcal {O))_{X_{T))\to {\mathcal {F))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2267aa2efb0a77d84b4eb60ba6d99b7036dad1c)
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e142fdd0d390f43630ec21c818af64e06abf844d)
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1d0f40c89e3ea441319abe46d4e06e9f0ade20)
Then, there is a surjection
![{\displaystyle {\mathcal {O))\to {\mathcal {O))_{Z))](https://wikimedia.org/api/rest_v1/media/math/render/svg/73fca9a8c6dea75242c9dc1bd2b350fddc8285c3)
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6023e9316d742119a577bf852df08f481086e633)
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dee1f23e895dd61eff49c3baf50ed351ff486f5)
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))](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd03bd39599035cd0141292a41dec026027c28aa)
The universal quotient over
is given by
![{\displaystyle {\mathcal {O))\to {\mathcal {U))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f960b5afe6db4c56b2a1f42a8b8e7c7cbb80e2d1)
where the fiber over a point
gives the projective morphism
![{\displaystyle {\mathcal {O))\to {\mathcal {O))_{Z))](https://wikimedia.org/api/rest_v1/media/math/render/svg/73fca9a8c6dea75242c9dc1bd2b350fddc8285c3)
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))](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e7caf76d933cfda932c7c55f724ab3323295b2a)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fa2ac5bc70b1ab54fbc9eeb3f61172c5088941d)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb97f0167dfc9a79d9fe3069fe57104298c0472d)
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42d07e8122dfe9d6b996d93a2bc5e298ac9a283a)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a210457d2432fb828c981f69d0b59dd751f0046)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b223b68a01aafc17f60416bd05193fa0a6240a)
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))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/a552c371fb16d058bd0a152ff3cb709fb96dfec7)
which can be used to construct the moduli space
of semistable vector bundles using a GIT quotient.[5]