Abstract:

Given a general configuration of integer points one can associate to it (via toric geometry) an irreducible homogeneous polynomial, called the discriminant. In the exceptional cases the discriminant is set to be 1. It is a classical problem in combinatorics and algebraic geometry to give a classification or a characterization of the exceptional cases. I will illustrate how the interaction between combinatorics and geometry has been successful in studying such problems. In particular, when the convex hull of the integer points includes only the given points and is a simple polytope, the discriminant is 1 if and only if the volume of the faces satisfy a "Pick's type" formula.