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

Denoting by Ld(m0,m1,…,mr)Ld(m0,m1,…,mr) the linear system of plane curves of degree dd passing through r+1r+1 generic points p0,p1,…,prp0,p1,…,pr of the projective plane with multiplicity mimi (or larger) at each pipi, we prove the Harbourne–Hirschowitz Conjecture for linear systems Ld(m0,m1,…,mr)Ld(m0,m1,…,mr) determined by a wide family of systems of multiplicities m=(mi)i=0r and arbitrary degree dd. Moreover, we provide an algorithm for computing a bound for the regularity of an arbitrary system m, and we give its exact value when m is in the above family. To do that, we prove an H1H1-vanishing theorem for line bundles on surfaces associated with some pencils “at infinity”.