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Cox rings of rational surfaces and flag varieties of ADE-types. (English) Zbl 1376.14011

Summary: 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.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14E30 Minimal model program (Mori theory, extremal rays)
14J26 Rational and ruled surfaces
14M15 Grassmannians, Schubert varieties, flag manifolds

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