## Advanced Studies in Pure Mathematics

### On classes in the classification of curves on rational surfaces with respect to logarithmic plurigenera

Hirotaka Ishida

#### 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 $\Sigma_d$.

#### Article information

Dates
Revised: 28 September 2012
First available in Project Euclid: 19 October 2018

https://projecteuclid.org/ euclid.aspm/1539916282

Digital Object Identifier
doi:10.2969/aspm/06610093

Mathematical Reviews number (MathSciNet)
MR3382045

Zentralblatt MATH identifier
1360.14039

#### Citation

Ishida, Hirotaka. On classes in the classification of curves on rational surfaces with respect to logarithmic plurigenera. Singularities in Geometry and Topology 2011, 93--110, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06610093. https://projecteuclid.org/euclid.aspm/1539916282