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Benutzer:Bocardodarapti/Talk in Essen 2008

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ARITHMETIC AND GEOMETRIC DEFORMATIONS OF STRONG SEMISTABILITY AND OF TIGHT CLOSURE



DEFORMATIONS OF STRONG 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.


Let be a smooth projective relative curve, where is a domain. A vector bundle over induces on every fiber a vector bundle. If is semistable in the generic fiber, then it is also semistable for the fibers over a non-empty open set of . What can we say about strong semistability?

We have to distinguish two forms of deformations:


  • Arithmetic deformations: contains . Here the generic fiber is in characteristic zero, and suppose the bundle is semistable on the generic fiber. What can we say about strong semistability on the special fibers?

A problem of Miyaoka (85) asks in this setting (for of dimension one): does there exist infinitely many closed points in such that is strongly semistable on ?

This is still open, but the stronger question whether for almost all closed points the restricition is strongly semistable (asked by Shepherd-Barron) has a negative answer.


  • Geometric deformations: contains . Here suppose that the bundle on the generic fiber is strongly semistable. Then makes sense everywhere and it is semistable on the fibers over an open subset , however this subset depends on and the intersection might become smaller and smaller.


In retrospective, Paul Monsky provided for both questions counter-examples. His results were formulated in terms of Hilbert-Kunz multiplicity, but this can be translated to strong semistability by work of Brenner and Trivedi.



Before we discuss such examples, I want to relate strong semistability to the notion of tight closure from commutative algebra (Hochster/Huneke).


TIGHT CLOSURE

Let be a projective variety over a field of positive characteristic , a coherent sheaf on and consider

.

Do there exist any interesting -subspaces of ?

  • Classes which are annihilated by some Frobenius power.
  • Classes which are annihilated under some finite extension .
  • Classes with the property: there exists a , , such that
for all .
  • For and a vector bundles one can also look for properties of the geometric torsor defined by a class . The class defines an extension

and hence projective bundles

and the corresponding torsor

.

Are there subspaces related to properties of ?


Let be a noetherian domain of positive characteristic, let

be the Frobenius homomorphism and

(mit ) its th iteration. Let be an ideal and set

Then define the tight closure of to be the ideal



LOCALIZATION PROBLEM OF TIGHT CLOSURE

Let be a multiplicative system and an ideal in . Then the localization problem of tight closure is the question whether the identity

holds.

Here the inclusion is always true and is the problem. The problem means explicitly:

if , can we find an such that holds in ?


Why does localization often holds?

Because an element belongs to the tight closure usually because of a certain reason, and this reason often localizes/globalizes.

A typical reason for belonging to the tight closure is through plus closure.


For an ideal in a domain define its plus closure by

Equivalent: Let be the absolute integral closure of . This is the integral closure of in an algebraic closure of the quotient field (first considered by Artin). Then

The plus closure commutes with localization.

We also have the inclusion . Here the question arises:

Question: Is ?

This question is known as the tantalizing question in tight closure theory.


One should think of the left hand side as a geometric condition and of the right hand side as a cohomological conditon (later). The two main results on this question are the following:



Let be local and excellent. If is a parameter ideal (generated by a (sub-)system of parameters), then
and the tight closure of commutes with localization.



Let be a standard-graded, two-dimensional normal domain over (the algebraic closure of) a finite field. Let be an -primary graded ideal.

Then

In dimension two tight closure always localizes. However, a positive answer to the tantalizing question in dimenson two in general would imply the localization property in dimension three.

The result just mentioned rests on a detailed study of tight closure in dimension two with the help of vector bundles on the corresponding curve.



GEOMETRIC INTERPRETATION IN DIMENSION TWO

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

An element

defines a cohomology class

With this notation we have

(homogeneous of some degree ) such that for all . This cohomology class lives in

For the plus closure we have a similar correspondence:

if and only if there exists a curve such that the pull-back of the cohomology class vanishes.



Suppose that is strongly semistable. Then


In general, there exists an exact criterion depending on and the strong Harder-Narasimhan filtration of .

If is finite, then the same criterion holds for plus closure. This gives the theorem mentioned above.


We can also give a geometric interpretation of what it means for to belong to the tight closure of the ideal in terms of the torsor defined by :

if and only if is not an affine variety.

Moreover, if and only if the torsor contains projective curves (geometric and cohomological properties).



DEFORMATIONS OF 2-DIMENSIONAL TIGHT CLOSURE PROBLEMS

Let flat, and in . For every , a field, we can consider and in . In particular for , .

How does the property

vary with ?

There are two cases:

    • contains . This is a geometric or equicharacteristic deformation. Example .
    • contains . This is an arithmetic or mixed characteristic deformation. Example .


    We deal first with the arithmetic situation.



    Consider and take the ideal and the element . Consider reductions . Then

    and

    In particular, the bundle is semistable in the generic fiber, but not strongly semistable for any reduction . The corresponding torsor is an affine scheme for infinitely many prime reductions and not an affine scheme for infinitely many prime reductions.


    We now show that the geometric deformations are a special case of the localization problem.



    Let

    be a one-dimensional domain and of finite type, and an ideal in . Suppose that localization holds and that

    ( is the multiplicative system). Then holds in for almost all in Spec .

    By localization, there exists

    , , such that .

    By persistence of tight closure (under a ring homomorphism) we get

    The element does not belong to for almost all , so is a unit in and hence

    for almost all .


    We will look for the easiest possibilty of such a deformation:

    where has degree and have degree one and is homogeneous. Then (for )

    is a two-dimensional standard-graded ring over . For residue class fields of points of we have basically two possibilities.

    • , the function field. This is the generic or transcendental case.
    • , a finite field. This is the special or algebraic case.

    Note that in the second case tight closure is plus closure. To analyze the behavior of tight closure in such a family we can use what we know in the two-dimensional standard-graded situation.



    A COUNTEREXAMPLE TO THE LOCALIZATION PROPERTY

    In order to establish an example where tight closure does not behave uniformly under a geometric deformation we first need a situation where strong semistability does not behave uniformly.



    Let

    Consider

    Then Monsky proved the following results on the Hilbert-Kunz multiplicity of the maximal ideal in , a field:


    By the geometric interpretation of Hilbert-Kunz theory (Brenner, Trivedi) this means that the restricted cotangent bundle

    {{ Math/display|term= Syz(x,y,z) = (\Omega_{{\mathbb P}^2})|_C |SZ= }}

    is strongly semistable in the transcendental case, but not strongly semistable in the algebraic case. In fact, for , , the -th Frobenius pull-back destabilizes.


    The maximal ideal can not be used directly. However, we look at the second Frobenius pull-back which is (characteristic two) just

    The bundle has (up to counting) the same stability properties as .

    By the degree formula we have to look for an element of degree . Let's take

    .

    This is our example! First, by strong semistability in the transcendental case we have

    in

    by the degree formula. If localization would hold, then would also belong to the tight closure of for almost all algebraic instances , . Contrary to that we will show that for all algebraic instances the element does not belong to the tight closure of .



    Let

    , , . Set . Then

    This is an elementary but tedious computation.




    Tight closure does not commute with localization.
    One knows in our situation that is a so-called test element. Hence the previous Lemma shows that .




    Tight closure is not plus closure in graded dimension two for fields with transcendental elements.
    Consider

    In this ring ,

    but it can not belong to the plus closure. Else there would be a curve morphism which annihilates the cohomology class and this would extend to a morphism of relative curves almost everywhere.




    There is an example of a smooth projective

    (relatively over the affine line) variety and an effective divisor and a morphism

    such that is not an affine variety over the generic point , but for every algebraic point the fiber is an affine variety.
    Take

    to be the Monsky quartic and consider the syzygy bundle

    together with the cohomology class determined by . This class defines an extension

    and hence .

    Then is an example with the stated properties by the previous results.


    A similar interpretation holds in the arithmetic situation.