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.)
{\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 }
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
For a relatively very ample line bundle
This is called the Hilbert polynomial which gives a natural stratification of the quot functor.
fixed there is a disjoint union of subfunctors
Note the Hilbert polynomial is independent of the choice of very ample line bundle
-dimensional vector space has a universal quotient
is locally free and at every point it represents a
-plane, it has the constant Hilbert polynomial
As a special case, we can construct the project space
and a flat family of such projections parametrized by a scheme
for an algebraically closed field, then a non-zero section
was an arbitrary non-zero section, and the vanishing locus of
gives the same vanishing locus, the scheme
gives a natural parameterization of all such sections.
This construction represents the quot functor
gives the short exact sequence
Semistable vector bundles on a curve
can equivalently be described as locally free sheaves of finite rank.
For a fixed line bundle
[5]giving the Hilbert polynomial
Then, the locus of semi-stable vector bundles is contained in
of semistable vector bundles using a GIT quotient.