Ranks of tensors, secant varieties of Segre varieties and fat points

A classical unsolved problem of projective geometry is that of finding the dimensions of all the (higher) secant varieties of the Segre embeddings of an arbitrary product of projective spaces. An important subsidiary problem is that of finding the smallest integer t for which the secant variety of projective t-spaces fills the ambient projective space. In this paper we give a new approach to these problems. The crux of our method is the translation of a well-known lemma of Terracini into a question concerning the Hilbert function of "fat points" in a multiprojective space. Our approach gives much new information on the classical problem even in the case of three factors (a case also studied in the area of Algebraic Complexity Theory).