  We construct log resolutions of pairs on the blow-up of the projective space
in an arbitrary number of general points and we discuss the semi-ampleness of
the strict transforms. As an application we prove that the abundance conjecture
holds for an infinite family of such pairs. For $n+2$ points, these strict
transforms are F-nef divisors on the moduli space
$\overline{\mathcal{M}}_{0,n+3}$ in a Kapranov's model: we show that all of
them are nef.
