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.[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]

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 [3] 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

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[4]

  1. 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

[4]

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.[4]

See also

References

  1. 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
  2. Nitsure, Nitin (2005-04-29). "Construction of Hilbert and Quot Schemes". arXiv:math/0504590.
  3. Meaning a basis for the global sections defines an embedding for
  4. Hoskins, Victoria. "Moduli Problems and Geometric Invariant Theory" (PDF). pp. 68, 74–85. Archived (PDF) from the original on 1 March 2020.
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