On the secant varieties of tangential varieties (Q2154264)

Let \(X \subset \mathbb P^r_k\) be an integral non-degenerate variety of dimension \(n\) over an algebraically closed field \(k\) of characteristic zero. The \(a\)th secant variety \(\sigma_a (X)\) is the closure of the union of all linear subspaces of \(\mathbb P^r\) spanned by \(a\) points of \(X\). The variety \(X\) is \textit{not defective} if \(\sigma_a (X)\) has the expected dimension \(\min\{r,a(n+1)-1\}\) for all \(a > 0\). The \textit{tangential variety} \(\tau (X) \subset X\) is the closure of the union of all Zariski tangent spaces \(T_q X \subset \mathbb P^r\) taken over \(q \in X_{\text{reg}}\) and is a variety of dimension at most \(2n\). \textit{Abo and Vannieuvenhoven} described all integers \(\dim \sigma_a (\tau (X))\) for the image \(X \subset \mathbb P^r\) of the \(d\)-Veronese embedding of \(\mathbb P^n\) [\textit{H. Abo} and \textit{N. Vannieuwenhoven}, Trans. Am. Math. Soc. 370, No. 1, 393--420 (2018; Zbl 1387.14136)], proving a conjecture of \textit{A. Bernardi} et al. [J. Algebra 321, No. 3, 982--1004 (2009; Zbl 1226.14065)]. In the paper under review, the author uses results of \textit{J. Alexander} and \textit{A. Hirschowitz} [J. Algebr. Geom. 4, No. 2, 201--222 (1995; Zbl 0829.14002); Invent. Math. 140, 303--325 (2000; Zbl 0973.14026)] and methods from Abo and Vannieuvenhoven [loc. cit.] to compute the dimension of the join \(\sigma_{a,b} (X)\) of \(a\) copies of \(X\) and \(b\) copies of \(\tau (X)\) when \(X \subset \mathbb P^r\) is the degree \(d\) Veronese embedding of \(\mathbb P^n\). After proving some conditions under which a product of a variety and a curve under a Segre embedding are not defective, he proves that the tangential variety of \((\mathbb P^1)^n\) embedding by the linear system \(|{\mathcal O}_{(\mathbb P^1)^n} (d_1, \dots, d_n)|\) is not defective if \(d_i \geq 3\) and \((d_1+1)/(d_2+1) \geq 38\) and a similar statement for \(\mathbb P^n \times \mathbb P^1\). He also proves some asymptotic results giving generic non-defective behavior.