Elliptic fibrations on a generic Jacobian Kummer surface

TL;DR

This paper classifies all elliptic fibrations with sections on a generic Jacobian Kummer surface, providing explicit equations and analyzing their fiber structures, thus resolving a question from 2008.

Contribution

It completely describes all elliptic fibrations with sections on a generic Jacobian Kummer surface, including explicit equations and fiber analysis.

Findings

25 different elliptic fibrations identified

Explicit elliptic parameters and Weierstrass equations provided

Analysis of reducible fibers and Mordell-Weil lattices included

Abstract

We describe all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X and the symmetric group on the Weierstrass points of C. In particular, we compute elliptic parameters and Weierstrass equations for the 25 different fibrations and analyze the reducible fibers and Mordell-Weil lattices. This answers completely a question posed by Kuwata and Shioda in 2008.