Convex bodies associated to linear series -- Rob Lazarsfeld,
December 5, 2008

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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.