On the secant varieties of tangential varieties

Abstract

Let X ⊂ P r be an integral and non-degenerate variety. Let σ a , b ( X ) ⊆ P r , ( a , b ) ∈ N 2 , 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 ⁡ σ a , 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 ⁡ σ 0 , 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 d i ≥ 3 , 1 ≤ i ≤ n , and ( d 1 + 1 ) ( d 2 + 1 ) ≥ 38 the tangential variety of ( P 1 ) n embedded by | O ( P 1 ) n ( d 1 , … , d n ) | 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 k 0 such that for all t ≥ k 0 the tangential variety of X with respect to | L ⊗ t | is not defective.