Vanishing theorems for linearly obstructed divisors

We study divisors in the blow-up of $\mathbb{P}^n$ at points in general position that are non-special with respect to the notion of linear speciality introduced in [5]. We describe the cohomology groups of their strict transforms via the blow-up of the space along their linear base locus. We extend the result to non-effective divisors that sit in a small region outside the effective cone. As an application, we describe linear systems of divisors in $\mathbb{P}^n$ blown-up at points in star configuration and their strict transforms via the blow-up of the linear base locus

Secant degree of toric surfaces and delightful planar toric degenerations

The k-secant degree is studied with a combinatorial approach. A planar toric degeneration of any projective toric surface X corresponds to a regular unimodular triangulation D of the polytope defining X. If the secant ideal of the initial ideal of X with respect to D coincides with the initial ideal of the secant ideal of X, then D is said to be delightful and the k-secant degree of X is easily computed. We establish a lower bound for the 2- and 3-secant degree, by means of the combinatorial geometry of non-delightful triangulations. © de Gruyter 2013

A new proof of the Alexander-Hirschowitz interpolation theorem

The classical polynomial interpolation problem in several variables can be generalized to the case of points with greater multiplicities. What is known so far is essentially concentrated in the Alexander-Hirschowitz Theorem which says that a general collection of double points in P r gives independent conditions on the linear system L of the hypersurfaces of degree d, with a well known list of exceptions. We present a new proof of this theorem which consists in performing degenerations of P r and analyzing how L degenerates. © 2010 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag

On a notion of speciality of linear systems in P^n

Given a linear system in P^n with assigned multiple general points we compute the cohomology groups of its strict transforms via the blow-up of its linear base locus. This leads us to give a new definition of expected dimension of a linear system, which takes into account the contribution of the linear base locus, and thus to introduce the notion of linear speciality. We investigate such a notion giving sufficient conditions for a linear system to be linearly non-special for arbitrary number of points, and necessary conditions for small numbers of points.Comment: 26 pages. Minor changes, Definition 3.2 slightly extended. Accepted for publication in Transactions of AM