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In this article we address the problem of computing the dimension of thespace 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-RochTheorem. Also, systems for which the dimension is larger than expected shouldhave a fixed part containing a multiple $(-1)$-curve. We reformulate thisconjecture 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 theconjecture holds for all $m \leq 12$.

Linear systems of plane curves with base points of equal multiplicity