Computing Cox rings

TL;DR

This paper develops algorithms to analyze how modifications like blow ups affect Cox rings of Mori dream spaces, with applications to Gorenstein log del Pezzo surfaces, rational surfaces, and blown-up projective spaces.

Contribution

It introduces computational methods for Cox rings under modifications and explicitly computes Cox rings for various classes of algebraic surfaces.

Findings

Cox rings of all Gorenstein log del Pezzo surfaces of Picard number one are computed.

All smooth rational surfaces with Picard number ≤6 are Mori dream surfaces.

Explicit Cox ring presentations are provided for certain blown-up projective spaces.

Abstract

We consider modifications, for example blow ups, of Mori dream spaces and provide algorithms for investigating the effect on the Cox ring, e.g. testing finite generation or computing an explicit presentation in terms of generators and relations. As a first application, we compute the Cox rings of all Gorenstein log del Pezzo surfaces of Picard number one. Moreover, we show computationally that all smooth rational surfaces of Picard number at most six are Mori dream surfaces and we provide explicit presentations of the Cox ring for those not admitting a torus action. Finally, we provide the Cox rings of projective spaces blown up at a certain special point configurations.