Identifiability for the k-secant variety of the Segre-Veronese varieties

Identifiability holds for the k-secant variety σk(X) of an embedded variety X ⊂ Pr if a general q ∈ σk(X) is in the linear span of a unique subset of X with cardinality k. We consider here the case in which X is a Segre-Veronese embedding of a multiprojective space, i.e. q corresponds to a partially symmetric tensor and X-rank is the partially symmetric tensor rank. To improve by 1 the known results we exclude the case of codimension one contact loci and handle cases in which the irreducible components of the tangential k-contact locus are linear spaces.