Partially symmetric tensors and the non-defectivity of secant varieties of products with a projective line as a factor

We prove (with a mild restriction on the multidegrees) that all secant varieties of Segre– Veronese varieties with k > 2 factors, k − 2 of them being P1, have the expected dimension. This is equivalent to compute the dimension of the set of all partially symmetric tensors with a fixed rank and the same format. The proof uses the case k = 2 proved by Galuppi and Oneto. Our theorem is an easy consequence of a theorem proved here for arbitrary projective varieties with a projective line as a factor and with respect to complete linear systems.