Elliptic fibrations on K3 surfaces with a non-symplectic involution fixing rational curves and a curve of positive genus

TL;DR

This paper completes the classification of elliptic fibrations on K3 surfaces with a specific non-symplectic involution, providing geometric constructions and explicit Weierstrass equations, especially when the involution fixes a genus 1 curve.

Contribution

It offers a complete classification and geometric construction of elliptic fibrations on K3 surfaces with a non-symplectic involution acting trivially on the Néron–Severi group, including explicit equations.

Findings

Classification of elliptic fibrations with the involution

Explicit Weierstrass equations for these fibrations

Relation to fibrations on rational elliptic surfaces

Abstract

In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the N\'eron--Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibrations on the K3 surfaces explicitly and to study their specializations.