On multiples of divisors associated to Veronese embeddings with defective secant variety

Abstract

In this note we consider multiples $aD$, where $D$ is a divisor of the blow-up of $\mathbb{P}^n$ along points in general position which appears in the Alexander and Hirschowitz list of Veronese embeddings having defective secant varieties. In particular we show that there is such a $D$ with $h^1(X,D) > 0$ and $h^1(X,2D) = 0$.