## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Galois–Teichmüller Theory and Arithmetic Geometry, H. Nakamura, F. Pop, L. Schneps and A. Tamagawa, eds. (Tokyo: Mathematical Society of Japan, 2012), 565 - 577

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

#### 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**

Received: 26 April 2011

Revised: 10 October 2011

First available in Project Euclid:
24 October 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/06310565

**Mathematical Reviews number (MathSciNet)**

MR3051255

**Zentralblatt MATH identifier**

1325.14028

**Subjects**

Primary: 14E20: Coverings [See also 14H30] 14G99: None of the above, but in this section

**Keywords**

Quadratatic residue curve Mordell–Weil group

#### 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