## Advanced Studies in Pure Mathematics

### A note on quadratic residue curves on rational ruled surfaces

Hiro-o Tokunaga

#### Abstract

Let $\Sigma$ be a smooth projective surface, let $f':S'\to\Sigma$ be a double cover of $\Sigma$ and let $\mu :S\to S'$ be the canonical resolution of $S'$. Put $f = f'\circ\mu$. An irreducible curve $D$ on $\Sigma$ is said to be a splitting curve with respect to $f$ if $f^*D$ is of the form $D^+ + D^- + E$, where $D^+ \neq D^-$, $D^- = \sigma_f^* D^+$, $\sigma_f$ being the covering transformation of $f$ and all irreducible components of $E$ are contained in the exceptional set of $\mu$. In this article, we consider "reciprocity" concerning splitting curves when $\Sigma$ is a rational ruled surface.

#### Article information

Dates
Revised: 10 October 2011
First available in Project Euclid: 24 October 2018

https://projecteuclid.org/ euclid.aspm/1540417830

Digital Object Identifier
doi:10.2969/aspm/06310565

Mathematical Reviews number (MathSciNet)
MR3051255

Zentralblatt MATH identifier
1325.14028

#### Citation

Tokunaga, Hiro-o. A note on quadratic residue curves on rational ruled surfaces. Galois–Teichmüller Theory and Arithmetic Geometry, 565--577, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06310565. https://projecteuclid.org/euclid.aspm/1540417830