An asymptotic existence theorem for plane curves with prescribed singularities

Let $d,m_1,...,m_r$ be ($r+1$) positive integers, and $P_1,...,P_r$ be $r$ general points in the projective plane ; let $m$ be a positive integer. We prove that there exists a bound $d_0(m)$ such that : If $m_i < m$ ($0<i<r+1$), and $d > d_0(m)$ then the linear system $L$ of plane curves of degree $d$ having a multiplicity at least $m_i$ at each point $P_i$ has the expected dimension ; moreover, if $L$ is not empty, there exists an irreducible plane curve of degree $d$, smooth away from the $r$ points $P_i$, and having an ordinary singularity of the prescribed multiplicity $m_i$ at each point $P_i$. This curve may be isolated in $L$.