Elliptic fibrations on a generic Jacobian Kummer surface. (English) Zbl 1304.14045

Kumar, Abhinav

J. Algebr. Geom. 23, No. 4, 599-667 (2014).

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 :
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.
The motivation for this paper is due to Problem 5 of [M. Kuwata and 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.

Reviewer: Paola Comparin (Consepción)

MSC:

14J27	Elliptic surfaces, elliptic or Calabi-Yau fibrations
14J28	\(K3\) surfaces and Enriques surfaces

Keywords:

elliptic fibrations; Kummer surfaces; Weierstrass equation; , Mordell-Weil lattice; Jacobian of a curve

Citations:

Zbl 1139.14032

Software:

SymPol; PermLib

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