Rational curves and rational singularities

Abstract

We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a ℂ*-action. For varieties with an isolated singularity, covered by a family of rational curves with a general member not passing through the singular point, we show that this singularity is rational. In particular, this provides an explanation of classical results due to H. A. Schwartz and G. H. Halphen on polynomial solutions of the generalized Fermat equation.