Groebner bases, monomial group actions, and the Cox rings of Del Pezzo surfaces

TL;DR

This paper introduces monomial group actions and applies them to prove a conjecture about the structure of Cox rings of Del Pezzo surfaces, showing they are generated by quadrics.

Contribution

It defines monomial group actions and uses this concept to prove a conjecture on Cox rings of Del Pezzo surfaces, linking group actions to algebraic geometry.

Findings

Cox rings of Del Pezzo surfaces are quotients of polynomial rings by quadratic ideals.

Monomial group actions influence Groebner basis theory.

The conjecture of Batyrev and Popov is confirmed for degrees at least 3.

Abstract

We introduce the notion of monomial group action and study some of its consequences for Groebner basis theory. As an application we prove a conjecture of V. Batyrev and O. Popov describing the Cox rings of Del Pezzo surfaces (of degree at least 3) as quotients of a polynomial ring by an ideal generated by quadrics.