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

Morality is its own Reward

Traditionally, Kantian ethics has been thought hostile to agents' well-being. Recent commentators have rightly called this view into question, but they do not push their challenge far enough. For they leave in place a fundamental assumption on which the traditional view rests, viz., that happiness is all there is to well-being. This assumption is important, since, combined with Kant’s rationalism about morality and empiricism about happiness, it implies that morality and well-being are at best extrinsically related. Since morality can only make our lives go well by making us happy, and since morality can only make us happy by influencing our sensibility, morality is not its own reward--not really. It is simply the condition for some separate benefit. Drawing on Kant’s underappreciated discussion of self-contentment, an intellectual analog of happiness, I reconstruct an alternative account of morality’s relation to well-being. Morality does make our lives go well--and so is its own reward--not because it makes us happy but because it makes us self-contented