Curves having one place at infinity and linear systems on rational surfaces

Denoting by image the linear system of plane curves of degree d passing through r+1 generic points p0,p1,…,pr of the projective plane with multiplicity mi (or larger) at each pi, we prove the Harbourne–Hirschowitz Conjecture for linear systems image determined by a wide family of systems of multiplicities image and arbitrary degree d. Moreover, we provide an algorithm for computing a bound for the regularity of an arbitrary system image, and we give its exact value when image is in the above family. To do that, we prove an H1-vanishing theorem for line bundles on surfaces associated with some pencils “at infinity”.