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Rational curves and rational singularities. (English) Zbl 1043.14008

The paper gives a rationality criterion for isolated singular points of complex normal affine algebraic surfaces admitting a \(\mathbb C^*\)-action, via the existence of rational curves which do not pass through the singular point. The authors provide a number of refinements and generalizations of the criterion, as well as a nice explanation of the classical Schwartz-Halphen theorem of polynomial solutions of the generalized Fermat equation.

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
14L30 Group actions on varieties or schemes (quotients)
14R20 Group actions on affine varieties
14M20 Rational and unirational varieties
11D41 Higher degree equations; Fermat’s equation
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