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.

Abstract. We compute the facets of the effective and movable cones of divisors on the blow-up of P n at n + 3 points in general position. Given any linear system of hypersurfaces of P n based at n + 3 multiple points in general position, we prove that the secant varieties to the rational normal curve of degree n passing through the points, as well as their joins with linear subspaces spanned by some of the points, are cycles of the base locus and we compute their multiplicity. We conjecture that a linear system with n + 3 points is linearly special only if it contains such subvarieties in the base locus and we give a new formula for the expected dimension.

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

Abstract. We use a double degeneration technique to calculate the dimension of the secant variety of any Segre-Veronese embedding of (P 1 ) r .