On a notion of speciality of linear systems in pn

Given a linear system in Pn 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 an arbitrary number of points and necessary conditions for a small number of points.