On defectivity of families of full-dimensional point configurations

TL;DR

This paper investigates the defectivity of families of full-dimensional point configurations by establishing a necessary condition for defectivity, confirming a conjecture relating defectivity to mixed volume in these configurations.

Contribution

It provides a necessary condition for defectivity of full-dimensional point configuration families, confirming a conjecture linking defectivity to mixed volume.

Findings

A necessary condition for defectivity in full-dimensional configurations.

Confirmation of the conjecture that defectivity occurs iff mixed volume is 1.

Application of Furukawa and Ito's criterion to this problem.

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 $\mathbb{Z}^n$ is defective if and only if the mixed volume of the convex hulls of its elements is $1$.