Secant Varieties of Segre--Veronese Varieties

Understand

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.