On the ideals and singularities of secant varieties of Segre varieties

We find minimal generators for the ideals of secant varieties of Segre varieties in the cases Of sigma(k)(P-1 x P-n x pm) for all k, n, m, sigma(2) (P-n x P-m x P-p x P-r) for all n, m, p, r (GSS conjecture for four factors), and sigma(3) (P-n x P-rn x P-p) for all n, m, p and prove they are normal with rational singularities in the first case and arithmetically CohenMacaulay in the second two cases.