Elliptic fibrations on a generic Jacobian Kummer surface (Q2922500)

The paper is devoted to the classification of elliptic fibrations with section on the so-called ``generic Jacobian Kummer surface'': this is the name for \(\mathrm{Km}(\mathrm{Jac}(C))\), where \(C\) is a genus 2 curve whose Jacobian \(\mathrm{Jac}(C)\) has no extra endomorphisms. For this class of surfaces, the author states the following :NEWLINENEWLINE Theorem. There are exactly 25 different elliptic fibrations with section on a generic Jacobian Kummer surface \(\mathrm{Km}(\mathrm{Jac}(C))\) over an algebraically closed field of characteristic zero, modulo the action of the automorphism group of the surface and permutations of the Weierstrass points of \(C\). Then the author analyzes in details all the 25 elliptic fibrations in order to show explicitly, for each fibration, the elliptic parameter and the reducible fibers of the fibrations, the Weiestrass equation, torsion and non-torsion sections and a basis of the Mordell-Weil lattice.NEWLINENEWLINEThe motivation for this paper is due to Problem 5 of [\textit{M. Kuwata} and \textit{T. Shioda}, Adv. Stud. Pure Math. 50, 177--215 (2008; Zbl 1139.14032)], where the authors suggest the interest to study elliptic fibrations on a Kummer surface \(\mathrm{Km}(\mathrm{Jac}(C))\), where \(C\) is a genus 2 curve.