A Geometric Criterion for the Finite Generation of the Cox Ring of Projective Surfaces

TL;DR

This paper provides a geometric criterion for when the Cox ring of certain rational surfaces is finitely generated, linking it to the structure of effective divisors and specific curve configurations.

Contribution

It introduces a geometric characterization of Cox ring finite generation for anticanonical rational surfaces, relating it to the effective monoid and curves of self-intersection -1.

Findings

Finite generation linked to the effective monoid.

Finite generation equivalent to finitely many -1 curves in specific surfaces.

Applicable over algebraically closed fields of any characteristic.

Abstract

The aim is to give a geometric characterization of the finite generation of the Cox ring of anticanonical rational surfaces. This characterization is encoded in the finite generation of the effective monoid. Furthermore, we prove that in the case of a smooth projective rational surface having a negative multiple of its canonical divisor with only two linearly independent global sections (e.g., an elliptic rational surface), the finite generation is equivalent to the fact that there are only a finite number of smooth projective rational curves of self-intersection -1. The ground field is assumed to be algebraically closed of arbitrary characteristic.