Secant cumulants and toric geometry

TL;DR

This paper investigates the secant line variety of Segre products using cumulant coordinates, revealing its toric structure, singularities, and Gorenstein cases, with implications for algebraic geometry.

Contribution

It introduces a novel approach using cumulant coordinates to analyze secant varieties, showing they are covered by toric varieties and characterizing their singularities and Gorenstein properties.

Findings

Secant variety is covered by open normal toric varieties.

Ideal generated by binomial quadrics in cumulant coordinates.

Secant variety has rational singularities and a classified Gorenstein case.

Abstract

We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant coordinates its ideal is generated by binomial quadrics. We present new results on the local structure of the secant variety. In particular, we show that it has rational singularities and we give a description of the singular locus. We also classify all secant varieties that are Gorenstein. Moreover, generalizing (Sturmfels and Zwiernik 2012), we obtain analogous results for the tangential variety.