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

We introduce the notion of monomial group action and study some of its consequences for Gröbner 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 3) as quotients of a polynomial ring by an ideal generated by quadrics.