57 research outputs found
Vanishing theorems for linearly obstructed divisors
We study divisors in the blow-up of 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
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 -th powers of forms of degree . 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 -th powers of
degree 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
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