# Kurs:Vector bundles, forcing algebras and local cohomology (Medellin 2012)/Lecture 8

In the remaining lectures we will continue with the question when are the torsors given by a forcing algebras over a two-dimensional ring affine? We will look at the graded situation to be able to work on the corresponding projective curve.

In particular we want to address the following questions

- Is there a procedure to decide whether the torsor is affine?
- Is it non-affine if and only if there exists a geometric reason for it not to be affine (because the superheight is too large)?
- How does the affineness vary in an arithmetic family, when we vary the prime characteristic?
- How does the affineness vary in a geometric family, when we vary the base ring?

In terms of tight closure, these questions are directly related to the tantalizing question of tight closure (is it the same as plus closure), the dependence of tight closure on the characteristic and the localization problem of tight closure.

*Geometric interpretation in dimension two*

We will restrict now to the two-dimensional homogeneous case in order to work on the corresponding projective curve. We want to find an object over the curve which corresponds to the forcing algebra or its induced torsor.

Let be a two-dimensional standard-graded normal domain over an algebraically closed field . Let be the corresponding smooth projective curve and let

be an -primary homogeneous ideal with generators of degrees . Then we get on the short exact sequence

Here is a vector bundle, called the *syzygy bundle*, of rank and of degree

Recall that the degree of a vector bundle on a projective curve is defined as the degree of the invertible sheaf , where is the rank of . The degree is additive on short exact sequences.

A homogeneous element of degree defines an element in and thus a cohomology class , so this defines a torsor over the projective curve. We mention an alternative description of the torsor corresponding to a first cohomology class in a locally free sheaf which is better suited for the projective situation.

Let denote a locally free sheaf on a scheme . For a cohomology class one can construct a geometric object: Because of , the class defines an extension

This extension is such that under the connecting homomorphism of cohomology, is sent to . The extension yields a projective subbundle

If is the corresponding geometric vector bundle of , one may think of as which consists for every base point of all the lines in the fiber passing through the origin. The projective subbundle has codimension one inside , for every point it is a projective space lying (linearly) inside a projective space of one dimension higher. The complement is then over every point an affine space. One can show that the global complement

is another model for the torsor given by the cohomology class. The advantage of this viewpoint is that we may work, in particular when is projective, in an entirely projective setting.

*Semistability of vector bundles*

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 Beispiel *****.

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). **

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.

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.

*Harder-Narasimhan filtration*

In general, there exists an exact criterion for the affineness of the torsor depending on and the *strong Harder-Narasimhan filtration* of . For this we give the definition of the Harder-Narasimhan filtration.

Let be a vector bundle on a smooth projective curve over an algebraically closed field . Then the (uniquely determined) filtration

of subbundles such that all quotient bundles are semistable with decreasing slopes
,
is called the *Harder-Narasimhan filtration* of .

The Harder-Narasimhan filtration exists uniquely
(by a Theorem of Harder and Narasimhan).
A Harder-Narasimhan filtration is called *strong* if all the quotients are strongly semistable. A Harder-Narasimhan filtration is not strong in general, however, by a Theorem of A. Langer, there exists some Frobenius pull-back such that its Harder-Narasimhan filtration is strong.

Let denote a smooth projective curve over an algebraically closed field and let be a vector bundle over together with a cohomology class . Let

be a strong Harder-Narasimhan filtration. We choose such that has degree and that has degree . We set .

** Then the following are equivalent.
**

- The torsor is not an affine scheme.
- Some Frobenius power of the image of inside is .