Big rational surfaces

TL;DR

This paper proves the finite generation of the Cox ring for certain smooth rational surfaces with big anticanonical class and classifies those obtained by blowing up the plane at points on a cubic.

Contribution

It establishes finite generation of Cox rings for a class of rational surfaces and classifies specific blow-up configurations.

Findings

Cox ring of these surfaces is finitely generated.

Classification of surfaces obtained by blowing up points on a cubic.

Identification of conditions for big anticanonical class.

Abstract

We prove that the Cox ring of a smooth rational surface with big anticanonical class is finitely generated. We classify surfaces of this type that are blow-ups of the plane at distinct points lying on a (possibly reducible) cubic.