Linear systems of plane curves with base points of equal multiplicity

In this article we address the problem of computing the dimension of the space of plane curves of degree with general points of multiplicity . A conjecture of Harbourne and Hirschowitz implies that when , 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 -curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all .