## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Singularities in Geometry and Topology 2011, V. Blanlœil and O. Saeki, eds. (Tokyo: Mathematical Society of Japan, 2015), 93 - 110

### 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 $\Sigma_d$.

#### Article information

**Source***Singularities in Geometry and Topology 2011*, V. Blanlœil and O. Saeki, eds. (Tokyo: Mathematical Society of Japan, 2015), 93-110

**Dates**

Received: 23 May 2012

Revised: 28 September 2012

First available in Project Euclid:
19 October 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/06610093

**Mathematical Reviews number (MathSciNet)**

MR3382045

**Zentralblatt MATH identifier**

1360.14039

**Subjects**

Primary: 14H45: Special curves and curves of low genus 14J26: Rational and ruled surfaces 14E20: Coverings [See also 14H30]

**Keywords**

Plane curve rational surface double cover

#### 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