On defectivity of families of full-dimensional point configurations

Abstract

The mixed discriminant of a family of point configurations can be considered as a generalization of the A-discriminant of one Laurent polynomial to a family of Laurent polynomials. Generalizing the concept of defectivity, a family of point configurations is called defective if the mixed discriminant is trivial. Using a recent criterion by Furukawa and Ito we give a necessary condition for defectivity of a family in the case that all point configurations are full-dimensional. This implies the conjecture by Cattani, Cueto, Dickenstein, Di Rocco, and Sturmfels that a family of n full-dimensional configurations in Zn is defective if and only if the mixed volume of the convex hulls of its elements is 1.