Ryo Kawaguchi (Nara Medical University)

The volume of polytopes associated to Castelnuovo varieties

Abstract

A pair of a projective variety and an ample line bundle is called a polarized variety. For polarized varieties, it is known that the sectional genus has a upper bound, which is a higher dimensional extension of the Castelnuovo bound for the genus of a projective curve. In the case where the ambient space is a toric variety, there is a polytope associated to the line bundle. In the theory of polytopes, it is known that the volume of a convex polytope has a lower bound. In this talk, we will see the equivalency of these two bounds. Namely, we prove that the sectional genus of a polarized toric variety achieves the upper bound if and only if the volume of its associated polytope achieves the lower bound.