Cox rings of cubic surfaces and Fano threefolds

TL;DR

This paper computes Cox rings for specific algebraic varieties, including certain cubic surfaces, their resolutions, and Fano threefolds, advancing understanding of their algebraic structure.

Contribution

It provides explicit descriptions of Cox rings for minimal resolutions of cubic surfaces with rational double points and for certain Fano threefolds, which was previously unknown.

Findings

Cox rings of minimal resolutions of cubic surfaces with rational double points are determined.

Cox rings of blow-ups of the projective plane at specific point configurations are computed.

Cox rings of certain smooth Fano threefolds with low Picard number are explicitly described.

Abstract

We determine the Cox rings of the minimal resolutions of cubic surfaces with at most rational double points, of blow ups of the projective plane at non-general configurations of six points and of three dimensional smooth Fano varieties of Picard numbers one and two.