Linear systems associated to unicuspidal rational plane curves

Abstract

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 $\tilde{\nu}(C) \ge 0$.