A primal Barvinok algorithm based on irrational decompositions

We introduce variants of Barvinok's algorithm for counting lattice points in polyhedra. The new algorithms are based on irrational signed decomposition in the primal space and the construction of rational generating functions for cones with low index. We give computational results that show that the new algorithms are faster than the existing algorithms by a large factor.Comment: v3: New all-primal algorithm. v4: Extended introduction, updated computational results. To appear in SIAM Journal on Discrete Mathematic

Spherical orbit closures in simple projective spaces and their normalizations

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module of finite dimension. If G/H \subset P(V) is a spherical orbit and if X is its closure, then we describe the orbits of X and those of its normalization. If moreover the wonderful completion of G/H is strict, then we give necessary and sufficient combinatorial conditions so that the normalization morphism is a homeomorphism. Such conditions are trivially fulfilled if G is simply laced or if H is a symmetric subgroup.Comment: 24 pages, LaTeX. v4: Final version, to appear in Transformation Groups. Simplified some proofs and corrected minor mistakes, added references. v3: major changes due to a mistake in previous version

A combinatorial smoothness criterion for spherical varieties

We suggest a combinatorial criterion for the smoothness of an arbitrary spherical variety using the classification of multiplicity-free spaces, generalizing an earlier result of Camus for spherical varieties of type $A$.Comment: 14 pages, 2 table

Error correction in ensemble registers for quantum repeaters and quantum computers

We propose to use a collective excitation blockade mechanism to identify errors that occur due to disturbances of single atoms in ensemble quantum registers where qubits are stored in the collective population of different internal atomic states. A simple error correction procedure and a simple decoherence-free encoding of ensemble qubits in the hyperfine states of alkali atoms are presented.Comment: 4 pages, 2 figure

A Frobenius variant of Seshadri constants

We define and study a version of Seshadri constant for ample line bundles in positive characteristic. We prove that lower bounds for this constant imply the global generation or very ampleness of the corresponding adjoint line bundle. As a consequence, we deduce that the criterion for global generation and very ampleness of adjoint line bundles in terms of usual Seshadri constants holds also in positive characteristic.Comment: 16 page

K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$, where G = GL(n,\C), and where $K$ is one of the symmetric subgroups O(n,\C) or Sp(n,\C). We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formulas for the classes of such loci in terms of the Chern classes of the various bundles.Comment: Minor revisions and corrections suggested by referees. Final version, to appear in Transformation Group