Open Access
April 2014 Linear systems associated to unicuspidal rational plane curves
Daniel Daigle, Alejandro Melle-Hernández
Osaka J. Math. 51(2): 481-513 (April 2014).


Given a unicuspidal rational curve $C \subset \mathbb{P}^{2}$ with singular point $P$, we study the unique pencil $\Lambda_{C}$ on $\mathbb{P}^{2}$ satisfying $C \in \Lambda_{C}$ and $\mathrm{Bs}(\Lambda_{C})=\{P\}$. We show that the general member of $\Lambda_{C}$ is a rational curve if and only if $\tilde{\nu}(C) \ge 0$, where $\tilde{\nu}(C)$ denotes the self-intersection number of $C$ after the minimal resolution of singularities. We also show that if $\tilde{\nu}(C) \ge0$, then $\Lambda_{C}$ has a dicritical of degree $1$. Note that all currently known unicuspidal rational curves $C \subset \mathbb{P}^{2}$ satisfy $\tilde{\nu}(C) \ge 0$.


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Daniel Daigle. Alejandro Melle-Hernández. "Linear systems associated to unicuspidal rational plane curves." Osaka J. Math. 51 (2) 481 - 513, April 2014.


Published: April 2014
First available in Project Euclid: 8 April 2014

zbMATH: 1296.14004
MathSciNet: MR3192552

Primary: 14C20
Secondary: 14J26

Rights: Copyright © 2014 Osaka University and Osaka City University, Departments of Mathematics

Vol.51 • No. 2 • April 2014
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