# Convex bodies associated to linear series

# Convex bodies associated to linear series

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, as well as opening the door to a number of extensions. I will explain the construction, and give a number of examples, applications and open questions. If time permits, I will also mention briefly how Yuan has carried over the construction to the arithmetic setting.Location: Columbia University, Mathematics Hall 312