  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.
