140 research outputs found
Embeddings of homogeneous spaces in prime characteristics
Let be a reductive linear algebraic group. The simplest example of a
projective homogeneous -variety in characteristic , not isomorphic to a
flag variety, is the divisor in , which is modulo a non-reduced stabilizer containing the upper
triangular matrices. In this paper embeddings of projective homogeneous spaces
viewed as , where is any subgroup scheme containing a Borel subgroup,
are studied. We prove that can be identified with the orbit of the
highest weight line in the projective space over the simple -representation
of a certain highest weight . This leads to some strange
embeddings especially in characteristic , where we give an example in the
-case lying on the boundary of Hartshorne's conjecture on complete
intersections. Finally we prove that ample line bundles on are very
ample. This gives a counterexample to Kodaira type vanishing with a very ample
line bundle, answering an old question of Raynaud.Comment: 10 pages, AMS-LaTe
Local cohomology and D-affinity in positive characteristic
Comparing vanishing of local cohomology in zero and positive characteristic,
we give an example of a D-module on a Grassmann variety in positive
characteristic with non-vanishing first cohomology group. This is a
counterexample to D-affinity and the Beilinson-Bernstein equivalence for flag
manifolds in positive characteristic
Maximal compatible splitting and diagonals of Kempf varieties
Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal
multiplicity vanishing in Frobenius splitting. In this paper we define the
algebraic analogue of this concept and construct a Frobenius splitting
vanishing with maximal multiplicity on the diagonal of the full flag variety.
Our splitting induces a diagonal Frobenius splitting of maximal multiplicity
for a special class of smooth Schubert varieties first considered by Kempf.
Consequences are Frobenius splitting of tangent bundles, of blow-ups along the
diagonal in flag varieties along with the LMP and Wahl conjectures in positive
characteristic for the special linear group.Comment: Revised according to referee suggestions. To appear in Annales de
l'Institut Fourie
Frobenius splitting of cotangent bundles of flag varieties and geometry of nilpotent cones
We use the G-invariant non-degenerate form on the Steinberg module to
Frobenius split the cotangent bundle of a flag variety in good prime
characteristics. This was previously only known for the general linear group.
Applications are a vanishing theorem for pull back of line bundles to the
cotangent bundle (proved for the classical groups and G_2 by Andersen and
Jantzen and in characteristic zero by B. Broer (for all groups)), normality and
rational singularities for the subregular nilpotent variety and good
filtrations of the global sections of pull backs of line bundles to the
cotangent bundle, which in turn implies good filtrations of cohomology of
induced representations.Comment: LaTeX (amsart, amsmath, xypic), 14 page
Exact Algorithms for Solving Stochastic Games
Shapley's discounted stochastic games, Everett's recursive games and
Gillette's undiscounted stochastic games are classical models of game theory
describing two-player zero-sum games of potentially infinite duration. We
describe algorithms for exactly solving these games
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