Srinivas' Problem for Rational Double Points

For the completion B of a local geometric normal domain, V. Srinivas asked which subgroups of Cl B arise as the image of the map from Cl A to Cl B on class groups as A varies among normal geometric domains with B isomorphic to the completion of A. For two dimensional rational double point singularities we show that all subgroups arise in this way. We also show that in any dimension, every normal hypersurface singularity has completion isomorphic to that of a geometric UFD. Our methods are global, applying Noether-Lefschetz theory to linear systems with non-reduced base loci.