Communications in Analysis and Geometry

Volume 23 (2015)

Number 2

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

Pages: 293 – 317



Naichung Conan Leung (Institute of Mathematical Sciences and Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong)

Jiajin Zhang (Department of Mathematics, Sichuan University, Chengdu, China)


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 closely related to an irreducible representation $V$ of $G$, and is generated by degree one elements. The Proj of the Cox ring of $S$ is a sub-variety of the orbit of the highest weight vector in $V$, and both are closed sub-varieties of $\mathbb{P}(V)$ defined by quadratic equations. The GIT quotient of the Spec of such a Cox ring by a natural torus action is considered.

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