Secant Varieties of Segre--Veronese Varieties

TL;DR

This paper proves that the ideal of secant lines to Segre--Veronese varieties is generated in degree three by minors of flattenings, and describes the decomposition of the coordinate ring into irreducible representations.

Contribution

It establishes the degree and generators of the ideal of secant lines to Segre--Veronese varieties and details the representation-theoretic decomposition of its coordinate ring.

Findings

Ideal of secant lines generated in degree three by minors of flattenings

Confirmed conjecture for Segre varieties related to algebraic statistics

Decomposition of coordinate ring into irreducible representations

Abstract

We prove that the ideal of the variety of secant lines to a Segre--Veronese variety is generated in degree three by minors of flattenings. In the special case of a Segre variety this was conjectured by Garcia, Stillman and Sturmfels, inspired by work on algebraic statistics, as well as by Pachter and Sturmfels, inspired by work on phylogenetic inference. In addition, we describe the decomposition of the coordinate ring of the secant line variety of a Segre--Veronese variety into a sum of irreducible representations under the natural action of a product of general linear groups.