# Secant varieties of Segre-Veronese varieties

# Secant varieties of Segre-Veronese varieties

Secant varieties of Segre and Veronese varieties are classical objects that go back to the Italian school in the nineteen century. Surprisingly, very little is known about their equations. Inspired by experiments related to algebraic statistics, Garcia, Stillman and Sturmfels gave a conjectural description of the generators of the ideal of the secant line variety $Sec(X)$ of a Segre variety $X$. This generalizes the familiar result which states that matrices of rank two are defined by the vanishing of their $3\times 3$ minors. For a Veronese variety $X$, it was known by work of Kanev that the ideal of $Sec(X)$ is generated in degree three by minors of catalecticant matrices. I will introduce a new technique for studying the equations of the secant varieties of Segre-Veronese varieties, based on the usual representation theoretic approach to this problem. I will explain how this technique applies to show that for $X$ a Segre-Veronese variety, the ideal of $Sec(X)$ is generated in degree three by minors of matrices of flattenings, and to give a description of the decomposition into irreducible representations of the homogeneous coordinate ring of $Sec(X)$. This will recover as special cases the conjecture of Garcia, Stillman and Sturmfels, and the result of Kanev.