Benutzer:Bocardodarapti/Talk in Essen 2008
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 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.
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.
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 .
, , 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
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 .
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.
(relatively over the affine line) variety and an effective divisor and a morphism
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.