Tight closure/Two-dimensional/Semistable vector bundle/Introduction/Section

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In the situation of a forcing algebra of homogeneous elements, this torsor can also be obtained as , where is the (not necessarily positively) graded forcing algebra. In particular, it follows that the containment is equivalent to the property that is not an affine variety. For this properties, positivity (ampleness) properties of the syzygy bundle are crucial. We need the concept of (Mumford-)- semistability.

Let be a vector bundle on a smooth projective curve . It is called semistable, if for all subbundles .

Suppose that the base field has positive characteristic . Then is called strongly semistable, if all (absolute)

Frobenius pull-backs are semistable.

An important property of a semistable bundle of negative degree is that it can not have any global section . Note that a semistable vector bundle need not be strongly semistable, the following is probably the simplest example.

Let be the smooth Fermat quartic given by , and consider on it the syzygy bundle (which is also the restricted cotangent bundle from the projective plane). This bundle is semistable. Suppose that the characteristic is . Then its Frobenius pull-back is . The curve equation gives a global non-trivial section of this bundle of total degree . But the degree of is negative, hence it can not be semistable anymore.

The following example is related to example.

Let , where is a field of positive characteristic , , and

The equation yields the short exact sequence

This shows that is strongly semistable.

For a strongly semistable vector bundle on and a cohomology class with corresponding torsor we obtain the following affineness criterion.

Let denote a smooth projective curve over an algebraically closed field and let be a strongly semistable vector bundle over together with a cohomology class . Then the torsor is an affine scheme if and only if and ( for all in positive characteristic[1]).

This result rests on the ampleness of occuring in the dual exact sequence given by (this rests on work of Gieseker and Hartshorne). It implies for a strongly semistable syzygy bundle the following degree formula for tight closure.

Suppose that is strongly semistable. Then

If we take on the right hand side , the Frobenius closure of the ideal, instead of , then this statement is true for all characteristics. As stated, it is true in a relative setting for large enough.

We indicate the proof of the inclusion result. The degree condition implies that is such that has non-negative degree. Then also all Frobenius pull-backs have non-negative degree. Let be a twist of the tautological line bundle on such that its degree is larger than the degree of , the dual of the canonical sheaf. Let be a non-zero element. Then , and by Serre duality we have

On the right hand side we have a semistable sheaf of negative degree, which can not have a non-trivial section. Hence

and therefore belongs to the tight closure.

  1. Here one has to check only finitely many s and there exist good estimates how far one has to go. Also, in a relative situation, this is only an extra condition for finitely many prime numbers.