The projectivization of the space of matrices of rank one coincides with the image of the Segre embedding of a product of two projective spaces. Its variety of secant (r−1)–planes is the space of matrices of rank at most r, whose equations are given by the (r+1)× (r + 1) minors of a generic matrix. A fundamental problem, with applications in complexity theory and algebraic statistics, is to understand rank varieties of higher order tensors. This is a very complicated problem in general, even for the relatively small case of σ4(P × P3 × P3), the variety of secant 3–planes to the Segre product of three projective 3–spaces (also known as the Salmon Problem). Inspired by experiments related to Bayesian networks, Garcia, Stillman and Sturmfels ([GSS05]) gave a conjectural description of the ideal of the variety of secant lines to a Segre product of projective spaces. The case of an n–factor Segre product has been obtained for n ≤ 5 in a series of papers ([LM04],[LW07],[AR08]). We have proved the general case of the conjecture in ([Rai10]).
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