On the ideals of secant varieties of Segre varieties

We establish basic techniques for determining the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman, and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.