# Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

## TL;DR

This paper explores the relationship between rational elliptic surfaces and K3 surfaces, focusing on how linear systems on the former induce various elliptic fibrations on the latter, including classification of singular fibers.

## Contribution

It characterizes all elliptic fibrations on K3 surfaces arising from rational elliptic surfaces and classifies their singular fibers based on linear systems on the rational surfaces.

## Key findings

- Every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface.
- Classification of singular fibers of the genus 1 fibration in terms of singular fibers on the rational surface.
- Description of special linear systems inducing elliptic fibrations on K3 surfaces.

## Abstract

We consider K3 surfaces which are double cover of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.