On the secant varieties of tangential varieties

Let X ⊂ P r be an integral and non-degenerate variety. Let σa,b(X) ⊆ P r , (a, b) ∈ N2, be the join of a copies of X and b copies of the tangential variety of X. Using the classical Alexander-Hirschowitz theorem (case b = 0) and a recent paper by H. Abo and N. Vannieuwenhoven (case a = 0) we compute dim σa,b(X) in many cases when X is the d-Veronese embedding of P n. This is related to certain additive decompositions of homogeneous polynomials. We give a general theorem proving that dim σ0,b(X) is the expected one when X = Y ×P 1 has a suitable Segre-Veronese style embedding in P r . As a corollary we prove that if di ≥ 3, 1 ≤ i ≤ n, and (d1 + 1)(d2 + 1) ≥ 38 the tangential variety of (P 1)n embedded by |O(P 1 )n (d1, . . . , dn)| is not defective and a similar statement for P n × P 1. For an arbitrary X and an ample line bundle L on X we prove the existence of an integer k0 such that for all t ≥ k0 the tangential variety of X with respect to |L⊗t| is not defective. © 2022 Elsevier B.V. All rights reserved.