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arXiv · 1801.07467

On defectivity of families of full-dimensional point configurations

Christopher Borger·Benjamin Nill
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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 $1$.

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Christopher Borger, Benjamin Nill. 2018-01-23. On defectivity of families of full-dimensional point configurations. https://arxiv.org/abs/1801.07467

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