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

In this note we consider multiples aD, where D is a divisor of the blow-up of 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.Comment: 10 pages, LaTe

Cox ring of the generic fiber

Given a surjective morphism $\pi\colon X\to Y$ of normal varieties satisfying some regularity hypotheses we prove how to recover a Cox ring of the generic fiber of $\pi$ from the Cox ring of $X$. As a corollary we show that in some cases it is also possible to recover the Cox ring of a very general fiber, and finally we give an application in the case of the blowing-up of a toric fiber space

A counterexample to a conjecture on linear systems on P^3

In this note we give a counterexample to a conjecture proposed by Ciliberto about special linear systems of P^n through multiple base points.Comment: 6 pages, LaTe