Multigraded Regularity and Asymptotic Invariants.

Abstract

This thesis consists of two parts which are mutually independent from each other. In the first part, our main concern is to study multigraded regularity on multiprojective spaces and we describe in passing all those coherent sheaves whose regularity set is finitely generated. The principal result here is a bound on regularity for smooth irreducible curves of biprojective spaces in terms of their bidegree. In the second part of this thesis we study asymptotic invariants associated to linear series: this is joint work with A. K" uronya and C. Maclean. First we consider convex bodies, called Okounkov bodies, associated to any line bundle on a projective variety, which were constructed recently by Lazarsfeld and Mustac tu a. We show that in general there are only countably many of these convex bodies and then give a complete characterization of those appearing in the two dimensional case. Secondly, we deal with the volume function associated to a multigraded linear series. We show that in the global case, of complete linear series, only countably many functions appear. By contrast, we prove that any continuous, homogeneous and log-concave function can occur (up to scaling) as the volume function of a multigraded linear series.