Tropical duals of toric varieties

A, DICKENSTEIN

Warwick

Congreso; Tropical day; 2012

University of Warwick

Let X be a projective toric variety rationally parametrized by monomials with exponents in  a lattice configuration A of cardinality n. For any given positive integer k, the k-th dual variety X^(k) is defined as the closure in the dual projective space of all hyperplanes that intersect X at a smooth point and contain the k-th osculating space at this point, generalizing the classical definition of the projective dual for k=1. In this talk, I will present the following results.  In joint work with L. Tabera, we define tropical Euler derivatives to characterize those weights p in R^n such that the tropical polynomial f with support A and coefficients p, defines a singular tropical hypersurface. With this approach, we recover the description obtained in collaboration with E.M. Feichtner and B. Sturmfels of the tropicalization of the A-discriminant variety, and we locate the singular points.  With similar methods, in joint work with S. di Rocco and R. Piene we describe those weights p for which f lies in the tropicalization of X^(k) for any k.