Set-theoretic defining equations of the tangential variety of the Segre variety

TL;DR

This paper proves a set-theoretic version of a conjecture on the defining equations of the tangential variety of Segre varieties, introducing the concept of exclusive rank and linking it to principal minors of symmetric matrices.

Contribution

It establishes the set-theoretic defining equations for the tangential variety of Segre varieties and introduces the novel concept of exclusive rank.

Findings

Proves the Landsberg--Weyman Conjecture set-theoretically.

Connects the tangential variety to principal minors of symmetric matrices.

Introduces the concept of exclusive rank.

Abstract

We prove a set-theoretic version of the Landsberg--Weyman Conjecture on the defining equations of the tangential variety of a Segre product of projective spaces. We introduce and study the concept of exclusive rank. For the proof of this conjecture we use a connection to the author's previous work \cite{oeding_pm_paper, oeding_thesis} and re-express the tangential variety as the variety of principal minors of symmetric matrices that have exclusive rank no more than one.