## 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
Full Text: