(Mixed) Volumes of Lattice Polytopes and Divisors on Toric Varieties

In this talk, we describe a connection between full dimensional lattice polytopes and torus invariant Cartier Divisors on toric varieties. We will introduce a result connecting the volume of a polytope with the self-intersection number of its corresponding divisor. This result allows us to compute volumes of polytopes using intersection theory, prove properties of mixed volumes using multilinearality of intersection pairings, and even deduce the isoperimetric inequality from Hodge Index Theorem. Finally, we will introduce the Bernstein-Khovanskii- Kushnirenko theorem, which roughly says the solutions to a system of Laurent polynomial equations in a certain toric variety is generically counted by the mixed volume of polytopes associated with the equations. The talk will focus on illustrating the theorems with examples, rather than going through the proofs.

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Date published: Wednesday, April 10, 2024