Rational curves and rational singularities. (English) Zbl 1043.14008

Flenner, Hubert; Zaidenberg, Mikhail

Math. Z. 244, No. 3, 549-575 (2003).

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.

Reviewer: Eugenii I. Shustin (Tel Aviv)

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

Keywords:

affine algebraic surfaces; quasihomogeneous varieties; rational singularities; quotient singularities; rational curves