Local Picard Groups

TL;DR

This paper extends the Noether-Lefschetz theorem to describe local class groups of hypersurface singularities, providing new insights into their structure and applications to rational double points and unique factorization domains.

Contribution

It introduces an extension of the Noether-Lefschetz theorem to characterize generators of local class groups of hypersurface singularities, with several key applications.

Findings

Every subgroup of the class group of a rational double point arises from a hypersurface in complex projective 3-space.

Every complete local ring from a normal hypersurface singularity over complex numbers is a completion of a UFD of finite type.

The paper provides explicit descriptions of local class groups for various singularities.

Abstract

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.