Rational curves and rational singularities

We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a ${\text{C}}^{\ast }$-action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the singular point strongly affects the character of the singularity. This provides an explanation of classical results due to H.A. Schwartz and G.H. Halphen on polynomial solutions of the generalized Fermat equation.