Quot scheme

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.