Quasihomogeneous singularities
If one has an ample line bundle over a projective variety \(X\) with a group action, one may contract the zero section and then divide the cone by the group-action. These varieties are called cone quotients of \(X\). \textit{Pinkham} showed that analytically every quasihomogeneous normal surface singularity is a cone quotient of a curve and that the resolution graph is star-shaped. This holds also algebraically for any characteristic and the group may be chosen abelian. One gets a bijection between quasihomogeneous normal surface singularities and the data of the resolution in the algebraic category.
quasihomogeneous normal surface singularity
scientific article; zbMATH DE number 4014889