Linear systems of plane curves with imposed multiple points

Abstract

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