Quot scheme


In algebraic geometry, the Quot scheme is a scheme parametrizing locally free 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.
It is typically used to construct another scheme parametrizing geometric objects that are of interest such as a Hilbert scheme.

Definition

For a scheme of finite type over a Noetherian base scheme, and a coherent sheaf, there is a functor
sending to
where and under the projection. There is an equivalence relation given by if there is an isomorphism commuting with the two projections ; that is,
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 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
where
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
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

Hilbert scheme

The Hilbert scheme is a special example of the quot scheme. Notice a subscheme can be given as a projection
and a flat family of such projections parametrized by a scheme can be given by
Since there is a hilbert polynomial associated to, denoted, there is an isomorphism of schemes

Example of a parameterization

If and for an algebraically closed field, then a non-zero section has vanishing locus with Hilbert polynomial
Then, there is a surjection
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

Quadrics in the projective plane

If and, the Hilbert polynomial is
and
The universal quotient over is given by
where the fiber over a point gives the projective morphism
For example, if represents the coefficients of
then the universal quotient over gives the short exact sequence

Semistable vector bundles on a curve

s 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
  2. is generated by global sections
for. This implies there is a surjection
Then, the quot scheme parametrizes all such surjections. Using the Grothendieck–Riemann–Roch theorem the dimension is equal to
For a fixed line bundle of degree there is a twisting, shifting the degree by, so
giving the Hilbert polynomial
Then, the locus of semi-stable vector bundles is contained in
which can be used to construct the moduli space of semistable vector bundles using a GIT quotient.