2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65242A classification of normal affine surfaces admitting a $\bf C^*$-action was given in the work of Białynicki-Birula, Fieseler and L. Kaup, Orlik and Wagreich, Rynes and others. We provide a simple alternative description of such surfaces in terms of their graded rings as well as by defining equations. This is based on a generalization of the Dolgachev-Pinkham-Demazure construction in the case of a hyperbolic grading. As an apllication we determine the structure of singularities, of the orbits and the divisor class groups for such surfaces.Uses P.Taylor's diagrams.texAlgebraic GeometryCommutative Algebra13A02, 13F15, 14R05, 14L30text