Quot scheme

In algebraic geometry, the Quot scheme is a scheme parametrizing 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.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

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 functorsending towhere 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 Meaning a basis for the global sections defines an embedding for 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 subfunctorswhereThe 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 quotientwhere 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

Projective space As a special case, we can construct the projective bundle over as the quot schemefor 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 projectionand a flat family of such projections parametrized by a scheme can be given bySince 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 polynomialThen, there is a surjectionwith 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 isandThe universal quotient over is given bywhere the fiber over a point gives the projective morphismFor 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

is generated by global sections

for . This implies there is a surjectionThen, the quot scheme parametrizes all such surjections. Using the Grothendieck–Riemann–Roch theorem the dimension is equal toFor a fixed line bundle of degree there is a twisting , shifting the degree by , sogiving the Hilbert polynomialThen, the locus of semi-stable vector bundles is contained inwhich can be used to construct the moduli space of semistable vector bundles using a GIT quotient.

Quot scheme

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