Multiple points in P^2 and degenerations to elliptic curves

We consider the problem of bounding the dimension of the linear system of curves in ${\mathbf{P}}^2$ of degree $d$ with prescribed multiplicities $m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_n$ at $n$ general points (\cite{Hir1},\cite{Hir2}). We propose a new method, based on the work of Ciliberto and Miranda (\cite {CM1}, \cite {CM2}), by specializing the general points to an elliptic curve in ${\mathbf{P}}^2$.