General Topology-I

Le but de cet article est de definir et d'etudier systematiquement quelques invariants asymptotiques associes aux lieux de base des fibres en droites sur les varietes projectives lisses. Le comportement fonctionnel de ces invariants est lie au comportement ensembliste des lieux de base.

/pdf/asymptotic-invariants-of-base-loci-1y8ykweeer.pdf

Asymptotic invariants of base loci

The purpose of this paper is to define and study systematically some asymptotic invariants associated to base loci of line bundles on smooth projective varieties. We distinguish an open dense subset of the real big cone, called the stable locus, consisting of the set of classes on which the asymptotic base locus is locally constant. The asymptotic invariants define continuous functions on the big cone, whose vanishing characterizes, roughly speaking, the unstable locus. We show that for toric varieties at least, there exists a polyhedral decomposition of the big cone on which these functions are polynomial.

The total coordinate ring of a wonderful variety

This is the first chapter of an introductory text under construction; further chapters are available via the authors' web pages Our aim is to provide an elementary access to Cox rings and their applications in algebraic and arithmetic geometry Any comments and suggestions on this draft will be highly appreciated

Cox Rings

The total coordinate ring of a normal projective variety

The total coordinate ring TC(X) of a normal variety is a generalization of the ring introduced and studied by Cox in connection with a toric variety. Consider a normal projective variety X with divisor class group Cl(X), and let us assume that it is a finitely generated free abelian group. We define the total coordinate ring of X to be TC(X) = oplus_{D} H^0 (X, O_X (D)), where the sum as above is taken over all Weil divisors of X contained in a fixed complete system of representatives of Cl(X). We prove that for any normal projective variety X, TC(X) is a UFD, this is a corollary of a more general theorem that is proved in the paper. (Berchtold and Haussen proved the unique factorization for a smooth variety independently.) We also prove that for X, the blow up of P^2 along a finite number of collinear points, TC(X) is Noetherian. We also give an example that TC(X) is not Noetherian but oplus_n H^0 (X, O(nD)) is Noetherian for any Weil divisor D.

We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k-forms of projective spaces when K/k is cyclic, and we also study k-forms of surfaces.

/pdf/arithmetic-toric-varieties-79z28w0oju.pdf

Arithmetic Toric Varieties

Arithmetic toric varieties

In this article we prove that for any complete toric variety, and for any Cartier divisor, the ring of global sections of multiples of the line bundle associated to the divisor is finitely generated.

/pdf/the-ring-of-global-sections-of-multiples-of-a-line-bundle-on-514jdryflz.pdf

E. Javier Elizondo

Papers

The total coordinate ring of a normal projective variety

The total coordinate ring of a normal projective variety

Arithmetic Toric Varieties

Arithmetic toric varieties

The ring of global sections of multiples of a line bundle on a toric variety