Advanced Studies in Pure Mathematics

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

Hirotaka Ishida

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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

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

Received: 23 May 2012
Revised: 28 September 2012
First available in Project Euclid: 19 October 2018

Permanent link to this document euclid.aspm/1539916282

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Plane curve rational surface double cover


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.

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