  In this article we address the problem of computing the dimension of the
space of plane curves of degree $d$ with $n$ general points of multiplicity
$m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$,
the dimension is equal to the expected dimension given by the Riemann-Roch
Theorem. Also, systems for which the dimension is larger than expected should
have a fixed part containing a multiple $(-1)$-curve. We reformulate this
conjecture by explicitly listing those systems which have unexpected dimension.
Then we use a degeneration technique developed in a previous article
("Degenerations of Planar Linear Systems", alg-geom/9702015) to show that the
conjecture holds for all $m \leq 12$.
