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Locally nilpotent derivations on affine surfaces with a \(\mathbb C^*\)-action. (English) Zbl 1140.14323

Summary: We give a classification of normal affine surfaces admitting an algebraic group action with an open orbit. In particular an explicit algebraic description of the affine coordinate rings and the defining equations of such varieties is given. By our methods we recover many known results, e.g. the classification of normal affine surfaces with a ‘big’ open orbit of M. H. Gizatullin [Math. USSR, Izv. 5(1971), 1057–1081 (1972; Zbl 0249.14010)] and V. L. Popov [Izv. Akad. Nauk SSSR, Ser. Mat. 37, 1038–1055 (1973; Zbl 0251.14018)] or some of the classification results of V. I. Danilov-M.H. Gizatullin [Math. USSR, Izv. 9(1975), 493–534 (1976; Zbl 0331.14007)], J. Bertin [J. Reine Angew. Math. 341, 32–53 (1983; Zbl 0501.14028)] and others.

MSC:

14R20 Group actions on affine varieties
14R05 Classification of affine varieties
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