On the secant varieties of tangential varieties

2021/06/01 by Ballico, Edoardo
#14N05 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.00450

Abstract

Let X⊂ ℙ^r be an integral and non-degenerate variety. Let σ_a,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 σ_a,b(X) in many cases when X is the d-Veronese embedding of ℙⁿ. 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× ℙ¹ has a suitable Segre-Veronese style embedding in ℙ^r. As a corollary we prove that if d_i≥ 3, 1≤ i ≤ n, and (d₁+1)(d₂+1)≥ 38 the tangential variety of (ℙ¹)ⁿ embedded by |O _{(ℙ ¹)ⁿ}(d₁,… ,d_n)| is not defective and a similar statement for ℙⁿ× ℙ¹. 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.

Related

• Secant varieties of Segre-Veronese varieties ℙ^m×ℙⁿ embedded by O(1,2) are non-defective for n≫ m³, m≥3
• Secant varieties of Segre-Veronese embeddings of (P¹)^r
• Tangential varieties of Segre-Veronese varieties
• On Degenerate Secant and Tangential Varieties and Local Differential Geometry
• On non-secant defectivity of Segre-Veronese varieties