Sets computing the symmetric tensor rank

Abstract

Let n_d denote the degree d Veronese embedding of a projective space P^r. For any symmetric tensor P, the 'symmetric tensor rank' sr(P) is the minimal cardinality of a subset A of P^r, such that n_d(A) spans P. Let S(P) be the space of all subsets A of P^r, such that n_d(A) computes sr(P). Here we classify all P in P^n such that sr(P) < 3d/2 and sr(P) is computed by at least two subsets. For such tensors P, we prove that S(P) has no isolated points.