On generic identifiability of symmetric tensors of subgeneric rank

We prove that the general symmetric tensor in S^d C^(n+1) of rank r is identifiable, provided that r is smaller than the generic rank. That is, its Waring decomposition as a sum of r powers of linear forms is unique. Only three exceptional cases arise, two of which were known classically. The third exceptional case is given by cubic tensors in 6 variables of rank 9, whose proof of nonidentifiability we could not find in the literature. Our original contribution regards only the case of cubics (d = 3), while for d ≥ 4 we rely on known results on weak defectivity by Ballico, Ciliberto, Chiantini, and Mella.