Linear systems of plane curves with base points of equal multiplicity

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 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...

Stanford Algebraic Geometry — Seminar — LINEAR SYSTEMS OF PLANE CURVES WITH BASE POINTS OF BOUNDED MULTIPLICITY

We address the problem of computing the dimension of the space of plane curves of fixed degree and general multiple base points. A conjecture of Harbourne and Hirschowitz gives geometric meaning to when this dimension is larger than the expected dimension obtained from Riemann-Roch; specifically, the dimension is larger than expected if and only if the system has a multiple (−1)-curve in its ba...

Linear Systems in P2 with Base Points of Bounded Multiplicity

We present a proof of the Harbourne-Hirschowitz conjecture for linear systems with multiple points of order 7 or less. This uses a well-known degeneration of the plane developed by Ciliberto and Miranda as well as a combinatorial game that arises from specializing points onto lines.

On the Dimension of Linear Systems of Plane Curves with General Multiple Base Points

Fixing n general points pi in the plane, what is the dimension of the space of plane curves of degree d having multiplicity mi at pi for each i? In this article we propose an approach to attack this problem, and demonstrate it by successfully computing this dimension for all n and for mi constant, at most 3. Our approach is based on an analysis of the corresponding linear system on a degenerati...

Linear Systems of Plane Curves