On the dimension of higher secant varieties of Segre varieties Pn×⋯×Pn

In this paper we are concerned with the problem of the dimension of the higher secant variety of the Segre variety Pn×⋯×Pn (t-times, t≥3 and n>1). In order to determine the dimensions of these varieties we construct a specific subscheme W of Pnt which is a generic configuration of the union of t linear subspaces of dimension n−1 and s double points, and then we compute the value of Hilbert function of this scheme at degree t. We show that W has the expected Hilbert function at degree t, except possibly for certain n values of s.