Classification of planar rational cuspidal curves. I. C**-fibrations

TL;DR

This paper classifies certain rational cuspidal curves in the projective plane whose complements are fibered by the complex punctured line, revealing that most known examples fit this classification and introducing a new family of bicuspidal curves.

Contribution

It provides a classification of rational cuspidal curves with $ ext{C}^{**}$-fibered complements and introduces a new infinite family of bicuspidal curves with unique properties.

Findings

Most known rational cuspidal curves with log general type complements are $ ext{C}^{**}$-fibered.

A new infinite family of bicuspidal curves with unusual properties is discovered.

The classification is up to projective equivalence.

Abstract

To classify complex rational cuspidal curves $E\subseteq \mathbb{P}^2$ it remains to classify the ones with complement of log general type, i.e. the ones for which $\kappa(K_X+D)=2$, where $(X,D)$ is a log resolution of $(\mathbb{P}^2,E)$. It is conjectured that $\kappa(K_X+\frac{1}{2}D)=-\infty$ and hence $\mathbb{P}^2\setminus E$ is $\mathbb{C}^{**}$-fibered, where $\mathbb{C}^{**}=\mathbb{C}^1\setminus\{0,1\}$, or $-(K_X+\frac{1}{2}D)$ is ample on some minimal model of $(X,\frac{1}{2}D)$. Here we classify, up to a projective equivalence, those rational cuspidal curves for which the complement is $\mathbb{C}^{**}$-fibered. From the rich list of known examples only very few are not of this type. We also discover a new infinite family of bicuspidal curves with unusual properties.