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Linear Systems of Plane Curves with Base Points of Equal Multiplicity

Ciro Ciliberto and Rick Miranda
Transactions of the American Mathematical Society
Vol. 352, No. 9 (Sep., 2000), pp. 4037-4050
Stable URL: http://www.jstor.org/stable/118173
Page Count: 14
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Linear Systems of Plane Curves with Base Points of Equal Multiplicity
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Abstract

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≥ 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 to show that the conjecture holds for all m≤ 12.

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