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Linear systems of rational curves on rational surfaces

Daniel Daigle^a, Alejandro Melle-Hernández<sup>b

a</sup> Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada K1N 6N5
^b ICMAT (CSIC-UAM-UC3M-UCM) Dept. of Algebra, Facultad de Matemáticas, Universidad Complutense, 28040, Madrid, Spain

Abstract: Given a curve $C$ on a projective nonsingular rational surface $S$, over an algebraically closed field of characteristic zero, we are interested in the set $\Omega_{C}$ of linear systems $\mathbb{L}$ on $S$ satisfying $C \in \mathbb{L}$, $\dim \mathbb{L} \ge1$, and the general member of $\mathbb{L}$ is a rational curve. The main result of the paper gives a complete description of $\Omega_{C}$ and, in particular, characterizes the curves $C$ for which $\Omega_{C}$ is non empty.

Key words and phrases: rational curves, rational surfaces, linear systems, weighted cluster of singular points.

MSC: 14C20, 14J26

Received: July 19, 2011; in revised form December 29, 2011

Language: English

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