Cox rings of extremal rational elliptic surfaces

TL;DR

This paper determines minimal generators for the Cox rings of extremal rational elliptic surfaces and develops a technique to compute relations, enabling presentations of these rings in most cases.

Contribution

It introduces a method for computing Cox ring relations and provides explicit presentations for extremal rational elliptic surfaces.

Findings

Minimal generating sets for Cox rings identified

A new technique for computing relations developed

Explicit Cox ring presentations obtained in most cases

Abstract

In this paper we determine a minimal set of generators for the Cox rings of extremal rational elliptic surfaces. Moreover, we develop a technique for computing the ideal of relations between them which allows, in all but three cases, to provide a presentation of the Cox ring.