The Euler Series of Restricted Chow Varieties

Let X be an algebraic projective variety in {\bf P}^n. Denote by {\cal C}_{\lambda} the space of all effective cycles on X whose homology class is \lambda \in H_{2p} (X,{\bf Z}). It is easy to show that {\cal C}_{\lambda} is an algebraic projective variety. Let \chi ({\cal C}_{\lambda} be its Euler characteristic. Define the Euler series of X by E_{p} = \sum_{\lambda\in{C}} \, \chi({\cal C}_{\lambda} \lambda \in \, {\bf Z}[[C]] where {\bf Z}[[C]] is the full algebra over {\bf Z} of the monoid C of all homology classes of effective p-cyles on X. This algebra is the ring of function (with respect the convolution product) over C. Denote by {\bf Z}[C] the ring of functions with finite support on C. We say that an element of {\bf Z}[[C]] is rational if it is the quotient of two elements in {\bf Z}[C]. If a basis for homology is fixed we can associated to any rationa element a rational function and therefore compute the Euler characteristic of {\cal C}_{\lambda}. We prove that E_p is rational for any projective variety endowed with an algebraic torus action in such a way that there are finitely many irreducible invariant subvarieties. If it is smooth we also define the equivariant Euler series and proved it is rational, we relate both series and compute some classical examples. The projective space {\bf P}^n, the blow up of {\bf P}^n at a point, Hirzebruch surfaces, the product of {\bf P}^n with {\bf P}^m.Comment: 15 pages, LaTe

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

In this article we prove, in a simple way, 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.Comment: To appear in Proceedings of the AMS, one figure, 5 pages, Author-supplied DVI file available at: http://calli.matem.unam.mx/investigadores/javier/investigacion.html LaTeX2e Sub-Class: 14C20 14M2

Rationality of Euler-Chow series and finite generation of Cox rings

In this paper we work with a series whose coefficients are the Euler characteristic of Chow varieties of a given projective variety. For varieties where the Cox ring is defined, it is easy to see that in this case the ring associated to the series is the Cox ring. If this ring is noetherian then the series is rational. It is an open question whether the converse holds. In this paper we give an example showing the converse fails. However we conjecture that it holds when the variety is rationally connected. As an evidence of this conjecture, It is proved that the series is not rational, and in a sense defined, not algebraic, in the case of the blowup of the projective plane at nine or more points in general position. Furthermore, we also treat some other examples of varieties with infinitely generated Cox ring, studied by Mukai and Hassett-Tschinkel. These are the first examples known where the series is not rational. We also compute the series for Del Pezzo surfaces.Comment: 26 pages. In this last version we correct many typos and add a cite of a work of Artebani and Laface in Theorem 1.6 which was brought to our attention. More typo correction

On the motive of certain subvarieties of fixed flags

We compute de Chow motive of certain subvarieties of the flags manifold and show that it is an Artin motive.Comment: 8 pages, 2 figure

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.Comment: This is the final version that will appear in the Journal of Algebra. 11 pages. LaTe

Spacetime foam as a quantum thermal bath

An effective model for the spacetime foam is constructed in terms of nonlocal interactions in a classical background. In the weak coupling approximation, the evolution of the low-energy density matrix is determined by a master equation that predicts loss of quantum coherence. Moreover, spacetime foam can be described by a quantum thermal field that, apart from inducing loss of coherence, gives rise to effects such as gravitational Lamb and Stark shifts as well as quantum damping in the evolution of the low-energy observables