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

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

Waring identifiability for powers of forms via degenerations

We discuss an approach to the secant non-defectivity of the varieties parametrizing $k$-th powers of forms of degree $d$. It employs a Terracini type argument along with certain degeneration arguments, some of which are based on toric geometry. This implies a result on the identifiability of the Waring decompositions of general forms of degree kd as a sum of $k$-th powers of degree $d$ forms, for which an upper bound on the Waring rank was proposed by Fr\"oberg, Ottaviani and Shapiro.Comment: 26 pages, 2 figures. Fixed a typo in the statement of Theorem 1.2 and Corollary 5.

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