Linear systems of plane curves with imposed multiple points

A conjecture of Harbourne and Hirschowitz implies that r greater than or equal to 9 general points of multiplicity m impose independent conditions to the linear system of curves of degree d when d(d + 3) greater than or equal to rm(m + 1) - 2. In this paper we prove that the conditions are independent provided d + 2 greater than or equal to (m + 1) (rootr + 1.9 + pi/8).