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Concept
Quot scheme
Summary
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.
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.)
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 .
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 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 .
It is a theorem of Grothendieck's that the functors are all representable by projective schemes over .
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
As a special case, we can construct the project space as the quot schemefor a sheaf on an -scheme .
Official source
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