Linear families of polytopes and a notion of anti-canonical class

A linear family of polytopes is a linear map from a (real) vector space V to the space of polytope in R^n (with Minkowski addition). There are important classes of projective algebraic varieties whose cone of (ample) line bundles gives such linear families of polytopes. Examples include toric varieites and the complete flag variety.

In the first hour I will cover some background material about toric varieties, flag varieties and their Gelfand-Zetlin polytopes. This should be accessible to most graduate students. In the next hour I will talk about a notion of anti-canonical class for linear families of polytopes. This approach generalizes the formula for the anti-canonical class of a projective toric variety and allows to compute anti-canonical class of some other classes of varieties (e.g. flag varieties) using only convex geometry. This is joint work with Elise Villella.