Cox rings of rational surfaces and flag varieties of ADE-types

TL;DR

This paper generalizes Cox rings to G-surfaces associated with Lie groups of types A, D, and E, showing their structure relates to irreducible representations and quadratic equations in projective space.

Contribution

It introduces a new class of G-surfaces and describes their Cox rings in terms of irreducible representations and geometric properties.

Findings

Cox rings of G-surfaces are generated by degree one elements.

The Proj of the Cox ring is a sub-variety of the orbit of a highest weight vector.

These sub-varieties are defined by quadratic equations in projective space.

Abstract

The Cox rings of del Pezzo surfaces are closely related to the Lie groups E_n. In this paper, we generalize the definition of Cox rings to G- surfaces defined by us earlier, where the Lie groups G=A_n, D_n or E_n. We show that the Cox ring of a G-surface S is almost determined by an irreducible representation V of G, and is generated by degree one elements. The Proj of this ring is a sub-variety of the orbit of the highest weight vector in V, and both are closed sub-varieties of the projective space P(V) defined by quadratic equations.