  We use our extension of the Noether-Lefschetz theorem to describe generators
of the class groups at the local rings of singularities of very general
hypersurfaces containing a fixed base locus. We give several applications,
including (1) every subgroup of the class group of the completed local ring of
a rational double point arises as the class group of such a singularity on a
surface in complex projective 3-space and (2) every complete local ring arising
from a normal hypersurface singularity over the complex numbers is the
completion of a unique factorization domain of essentially finite type over the
complex numbers.
