Convex Bodies Associated to Linear Series

TL;DR

This paper develops a systematic theory connecting convex bodies to linear series on projective varieties, extending Okounkov's construction beyond ample line bundles to big divisors, and explores applications of this geometric perspective.

Contribution

It generalizes the association of convex bodies to linear series for arbitrary big divisors, providing new insights and tools for studying asymptotic invariants.

Findings

Convex bodies can be associated with linear series of big divisors.

The convex geometric approach simplifies understanding of asymptotic invariants.

Applications include new results and examples in algebraic geometry.

Abstract

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens the door to a number of extensions. The purpose of this paper is to initiate a systematic development of the theory, and to give a number of applications and examples.