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VOL. 66 | 2015 On classes in the classification of curves on rational surfaces with respect to logarithmic plurigenera

## Abstract

Let $C$ be a nonsingular curve on a rational surface $S$. In the case when the logarithmic 2 genus of $C$ is equal to two, Iitaka proved that the geometric genus of $C$ is either zero or one and classified such pairs $(S, C)$. In this article, we prove the existence of these classes with geometric genus one in Iitaka's classification. The curve in the class is a singular curve on $\mathbb{P}^2$ or the Hirzebruch surface $\Sigma_d$ and its singularities are not in general position. For this purpose, we provide the arrangement of singular points by considering invariant curves under a certain automorphism of $\Sigma_d$.

## Information

Published: 1 January 2015
First available in Project Euclid: 19 October 2018

zbMATH: 1360.14039
MathSciNet: MR3382045

Digital Object Identifier: 10.2969/aspm/06610093

Subjects:
Primary: 14E20, 14H45, 14J26