In alphabetical order you find all Titles and Abstracts of the talks of the conferences. First however we include: Open Problem List
For the survey talks, the title links to notes taken during the talks by Kevin Tucker or Mircea Mustata.
Survey Talks
We review the use of vector bundles and their asymptotic
behavior under Frobenius pull-backs to questions from Hilbert-Kunz theory
and tight closure. We will also discuss the asymptotic behavior of the
symmetric products of vector bundles as a possible characteristic-free
approach to these theories.
Mel Hochster -- Origins of the study of F-purity and Frobenius splitting
The talk will discuss some early applications that helped motivate the development
of these techniques, including the Cohen-Macaulay property for rings of invariants of
linearly reductive groups in characteristic 0, the existence of small Cohen-Macaulay
modules, Reisner's characterization of when face rings are Cohen-Macaulay, and
others. The talk will include a discussion of some interactions with tight closure theory,
and of several problems in this area that remain open.
This talk is a one hour introduction to Frobenius splitting of the
variety of complete flags for the general linear group keeping the Lie
theory to a bare minimum. Emphasis will be on residual normal
crossing divisors ending with the (open) problem of exhibiting maximal
diagonal splittings.
The theory of F-modules has important applications to local cohomology and D-modules in characteristic p>0. We will survey both basic results of this theory and some very interesting recent developments.
In this talk I will define both test ideals and multiplier ideals and describe the relation between the two objects.
In this talk, I will explain how to understand the singularities arising in the minimal model program in terms of the Frobenius morphism. There is a
correspondence between the singularities in characteristic zero and the singularities in positive characteristic, called F-singularities, defined using
the Frobenius morphism, some of which are obtained from the tight closure theory. I will also discuss the case of singularities associated to pairs (X,
\Delta) where \Delta is an effective \Q-divisor on a normal variety X.
Proving normality of Schubert varieties in arbitrary characteristic was one of the first big successes of Frobenius splitting. Historically
normality was important as it implies the Demazure character formula. By now there are several, more or less related, proofs of normality using
F-splitting techniques and in this talk I will present yet another one. I will also discuss how to obtain other geometric properties of Schubert
varieties such as e.g. rational singularities.
Bhargav Bhatt -- Derived direct summands
A variety satisfies the direct summand condition if its structure
sheaf is a summand of that of a finite cover; in characteristic 0, this
condition reduces to normality. We will discuss a derived category
enrichment of this condition. The two conditions diverge in characteristic 0
(the latter characterises rational singularities), but turn out to be very
closely related in positive and mixed characteristic. Time permitting, I
will explain how to use this relation to extend standard cohomological
vanishing results beyond the ample cone in positive characteristic algebraic
geometry.
Manuel Blickle -- Cartier Modules and applications to test ideals
In recent joint work with Gebhard Böckle we introduced a category of (quasi-)coherent sheaves M which are equipped with an additive map C: M ---> M which satisfies C(r^{p^e}m)=rC(m). Any splitting of the Frobenius is such a map. The standard example, however, is the Cartier morphism on the top differentials of a smooth variety. The resulting category is called Cartier Modules. An important structural result that we obtained is that, up to nilpotent actions, every object in the category of Cartier Modules has finite length. In my talk I will explain this result and, in the case of affine $k$-algebras, give a completely elementary proof of it. Furthermore I want to draw attention to some of the applications of the theory, in particular to test ideals.
Charles Hague -- On the B-canonical splittings of flag varieties
Let G be a semisimple algebraic group over an algebraically closed field of positive characteristic. In this talk, I will discuss the unique canonical Frobenius splitting of the full flag variety X of G. The main result is that an irreducible closed subvariety of X is compatibly split by the canonical splitting if and only if it is a Richardson variety, i.e. an intersection of a Schubert and an opposite Schubert variety.
Mitsuyasu Hashimoto -- Good filtrations and the strong $F$-regularity of the ring of $U$-invariants
Xuhua He -- Frobenius splitting on orbit closures of some spherical subgroups in flag varieties
Let $G$ be a connected reductive algebraic group and $H$ be a closed connected reductive subgroup that has only finitely many orbits on the flag variety of $G$. It is known that for the spherical pair $(G, H)=(K \times K, K_{diag})$, where $K$ is a connected reductive group, there exits a Frobenius splitting on the flag variety of $G$ that compatibly splits all the $H$-orbit closures. We will show that this is also true for some other spherical pairs. If time allows, we will discuss its applications to representation theory. This is a joint work with J. F. Thomsen.
Allen Knutson -- Point-counting and degeneration of Frobenius splittings
Let f be a polynomial in F_p[x_1..x_n] of degree n > 0 (yes, same n).
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If the number of points of the hypersurface {f=0} is not a multiple of p, or
- If the initial term of f (w.r.t. some term order) is the product of the variables,
then there is a Frobenius splitting of affine space that compatibly splits the hypersurface {f=0}.
In fact (2) implies (1), and (2) also implies that init Y is reduced for any other compatibly split subscheme. I'll explain the combinatorial implications of this for affine patches on Schubert varieties.
V. Lakshmibai -- Some geometric applications of Frobenius splitting
Adrian Langer -- Fundamental groups of algebraic varieties in positive characteristic
I would like to give survey some of recent results concerning fundamental group schemes associated to algebraic varieties in positive characteristic. There are essentially two approaches: one via flat vector bundles (D-modules or stratified sheaves) and another via numerically flat vector bundles (strongly semistable sheaves with vanishing Chern classes). They generalize Grothendieck's etale fundamental group and Nori's fundamental group scheme respectively. When describing properties I will focus mostly on the approach via numerically flat bundles and show some proofs (using restriction theorems, some vanishing theorems Frobenius morphism and its action on semistable bundles).
Sam Payne -- Frobenius splittings of toric varieties
I will discuss the space of Frobenius splittings of a toric variety X and present a polyhedral criterion for the diagonal to be compatibly split in X x X. This criterion can be applied to show that section rings of nef line bundles on diagonally split toric varieties are normal and Koszul, and that Schubert varieties are not diagonally split in general.
Jenna Rajchgot -- Looking for compatibly split subschemes of the Hilbert scheme of points in the plane
The Hilbert scheme of n points in the affine plane is known to be Frobenius split, compatibly with the divisor ``at least one point is on an axis''. This fact may prompt one to ask, ``What are all of the compatibly split subvarieties?''
I'll discuss an approach for studying this question and present the answer for some small values of n.
Alexander Samokhin -- Frobenius morphism on homogeneous spaces
Let G be a semisimple algebraic group over a field of positive characteristic, P a parabolic subgroup of G, and G/P the grassmannian.
Bezrukvanikov et al. construct a non-standard t-structure on the derived category of coherent sheaves on T*(G/P), the cotangent
bundle of G/P. This t-structure comes
from an equivalence between the above category and the derived category of modules over some non-commutative algebra (tilting
equivalence).
Tilting equivalences (maybe in a slightly weaker form) shoud exist for homogeneous spaces G/P themselves. One can restrict the
tilting
generator on T*(G/P) - in fact, a vector bundle - to G/P, embedded into T*(G/P) as the zero section. A priori, there is no reason
why the restricted bundle
will remain tilting on G/P.
For some specific G and P one can prove nevertheless that this restriction gives a tilting bundle on G/P too. Essentially, what one
needs to prove is some
vanishing statement. Hypothetically, it should hold for any G/P.
In my talk I will discuss this vanishing and show how it is related to abelian $D$-affinity of G/P with respect to big differential
operators and, therefore,
to abelian localization of modules for semisimple Lie algebras in positive characteristic. This contrasts with the derived
localization theorem for crystalline
differential operators proven by Bezrukavnikov, Mirkovic and Rumynin.
Time permitting, I will mention further relations to Bezrukavnikov & Kaledin's work on t-structures on derived categories of
coherent sheaves that arise from representation theory.
Tadakazu Sawada -- Splitting of Frobenius sandwiches
In this talk, we present a classification of globally
F-regular Frobenius sandwiches of the projective plane and Hirzebruch surfaces. Let X be a smooth variety over an
algebraically closed field of positive characteristic. A Frobenius
sandwich of X is a normal variety Y through which the
Frobenius morphism of X factors. Bloch showed that a Frobenius
sandwich of the projective plane is singular, and Ganong
and Russell showed that for each Hirzebruch surface there
are at most two smooth Frobenius sandwiches. We study
Frobenius sandwiches of the projective plane and Hirzebruch
surfaces from the different viewpoint of Frobenius splitting.
Rodney Sharp -- Graded annihilators and tight closure
This lecture is concerned with big tight closure test elements for a commutative Noetherian ring R of prime characteristic. A big test element for R is an element c of R which lies outside every minimal prime ideal of R and which can be used in every tight closure membership test for every R-module. Some results will be presented about interrelationships between big test elements and the R-annihilators of certain left modules over the Frobenius skew polynomial ring R[x; f] associated to R and the Frobenius ring homomorphism f : R -> R. One consequence is a necessary and sufficient condition for R to possess a big test element; another is the result that, if R is local and E denotes the injective envelope of the simple R-module, and if it is possible to endow E with a structure as left R[x; f]-module that extends its R-module structure in a sufficiently non-trivial way so that the R-annihilator of its x-torsion submodule has positive height, then R has a big test element. This leads on to the results that, (i) if R is reduced, local and excellent, then it has a big test element, and (ii) if R is excellent and F-pure (but not necessarily local), then it has a big test element.
The ideas underlying this lecture also lead to the results that, whenever c is a big test element for R, then, for each prime ideal P of R, the image c/1 of c in the localization R_P of R at P is automatically a big test element for R_P , and, if R is excellent, c/1 is also a big test element for the completion of R_P .
Kei-ichi Watanabe -- Some topics on F-thresholds
Yongwei Yao -- The lower semicontinuity of the Frobenius splitting numbers
Let $R$ be an F-finite reduced Noetherian ring of prime characteristic $p$ and let $q = p^e$. Then, for every $P \in \operatorname{Spec}(R)$, the $e$-th \emph{Frobenius splitting number} of $R$ at $P$, denoted
$a_e(P)$ or $a_e(R_P)$ is the maximal rank of free $R_P$-modules that are direct summands of $(R_P)^{1/q}$. In this case, it is not hard to see that $P \mapsto a_e(P)$ is a lower semicontinuous function from $\operatorname{Spec}(R)$ to $\mathbb Z$ if $R$ is locally equidimensional.
However, given a general Noetherian ring $R$ of characteristic $p$, the above definition of $a_e(P)$ does not apply. Nevertheless, by considering the bimodule structure induced by the Frobenius map, one can define the $e$-th \emph{normalized Frobenius splitting number} of $R$ at a prime $P$, which we denote by $s_e(P)$. We show that, under mild conditions, the $e$-th normalized Frobenius splitting numbers (as a function $P \mapsto s_e(P)$) are
lower semicontinuous.
This is joint work with Florian Enescu.
We are going to discuss some very recent results on local
cohomology modules of a polynomial ring in a finite number of variables over
a field of positive characteristic. It is very interesting that these
results hold (we believe) in characteristic zero as well, but we are unaware
of any technique that would allow us to prove this.
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Ċ ď Karl Schwede, Feb 17, 2012 8:18 PM Ċ ď Manuel Blickle, Apr 4, 2010 8:49 AM Ċ ď Karl Schwede, Jun 12, 2010 3:03 AM Ċ ď Karl Schwede, Jun 12, 2010 3:04 AM Ċ ď Manuel Blickle, May 27, 2010 12:27 AM Ċ ď Manuel Blickle, May 18, 2010 1:51 PM Ċ ď Manuel Blickle, May 18, 2010 1:51 PM Ċ ď Manuel Blickle, May 18, 2010 1:51 PM Ċ ď Manuel Blickle, May 18, 2010 5:33 AM Ċ ď Manuel Blickle, May 18, 2010 1:52 PM Ċ ď Manuel Blickle, May 18, 2010 1:52 PM Ċ ď Manuel Blickle, May 18, 2010 1:52 PM Ċ ď Manuel Blickle, May 27, 2010 12:27 AM Ċ ď Manuel Blickle, May 19, 2010 8:05 PM Ċ ď Manuel Blickle, May 27, 2010 12:27 AM Ċ ď Manuel Blickle, May 27, 2010 12:27 AM Ċ ď Manuel Blickle, May 27, 2010 12:27 AM Ċ ď Karl Schwede, Apr 29, 2010 8:58 AM Ċ ď Karl Schwede, Apr 29, 2010 5:24 AM
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