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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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


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