  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.
