Convex Bodies Associated to Linear Series

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.

51 pages, 4 figures

References in corpus (5)

• The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension
• Convex bodies and algebraic equations on affine varieties
• Continuity of volumes on arithmetic varieties
• Cones and Asymptotic Invariants of Multigraded Systems of Ideals
• Units of $p$-power order in principal $p$-blocks of $p$-constrained groups