Classification of affine varieties being cones over nonsingular projective varieties: Hypersurface case

摘要

Let X be a nonsingular projective variety in CPn. Then the cone over X in Cn+1 is an affine variety V with an isolated singularity at the origin. It is a very natural question to ask when an affine variety with an isolated singularity at the origin is a cone over nonsingular projective variety.In this paper we shall treat the hypersurface case. Specifically, we will prove the Yau conjecture on 3-dimensional weighted homogeneous hypersurface singularities. In particular, we have obtained sharp upper estimate of geometric genus in terms of Milnor number and multiplicity of the singularity. An important corollary that we obtain is a numerical characterization when an affine hypersurface in C-4 with only isolated critical point at the origin is a cone over nonsingular hypersurface in CP3 after biholomorphic change of coordinates.