Multiple points in $\mathbf{P}^2$ 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 $m_1,\dots,m_n$ at $n$ general points (Harbourne (1986), Hirschowitz (1985)). We propose a new method, based on the work of Ciliberto and Miranda (2000, 2003), by specializing the general points to an elliptic curve in $\mathbf{P}^2$.