A generic uniqueness result for an interpolation problem for the join of a tangential variety gamma(x) and several copies of the variety X in P^2

Abstract

Let X ⊂ ℙ^r be an integral and non-degenerate variety non-singular in codimension 1. Let τ(X,2) =τ(X) ⊂ ℙ^r be the tangential variety of X. Set n := dim(X). For any integer b>2 let τ(X,b)$ be the join of τ(X) and b-2 copies of X. Mimicking the notion of weakly k-degenerate varieties introduced by Chiantini and Ciliberto we give a conditions on X assuring that a general q ∈ τ(X,b) is in the linear span of a unique scheme Z ⊂ X with Z = v ∪ {p₁, ... , p_b-2} with v connected of degree 2 and v ⊂ X_reg.