Non-defectivity of Segre-Veronese varieties (Q6652099)

Let \(X\subset \mathbb{P}^N\), \(N+1 = \prod_{i=1}^{k} \binom{n_i+d_i}{n_i}\), be the embedding of the multiprojective space \(\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_k}\), \(k\ge 2\), of multidegree \((d_1,\dots ,d_k)\). Knowing the dimensions of all secant varieties of \(X\) is equivalent to knowing the dimension of the set of all partially symmetric tensors with a prescribed format. There is an expected dimension (an upper bound for the true dimension) and conjecturally these numbers coincides except in a few known cases. The paper under review gives a YES solution to this conjecture if \(d_i\ge 3\) for all \(i\) and in a few other cases. The proofs use (with great skills) a 3-step Horace method used by the first two of the authors [Ann. Mat. Pura Appl. (4) 192, No. 1, 61--92 (2013; Zbl 1262.14065)]. They quote papers overlapping with their results and using the Differential Horace Method in a refined way [\textit{E. Ballico}, Math. Z. 308, No. 1, Paper No. 6, 15 p. (2024; Zbl 1548.14164)]. The case \(k=2\) (a key case, because all papers used induction on the integer \(k\)) was proved by \textit{F. Galuppi} and \textit{A. Oneto} [Adv. Math. 409, Part B, Article ID 108657, 58 p. (2022; Zbl 1503.14049)] using collision of fat points. The introduction of the paper and the last quoted one give an overview of the literature and the main reasons for such study.