Isolated singularities with large class group

TL;DR

This paper constructs high-dimensional local normal domains over complex numbers with isolated singularities whose class groups contain large, continuous subgroups, revealing new insights into the structure of singularities.

Contribution

It introduces a geometric construction using symmetric products of curves to produce examples of isolated singularities with large, complex class groups.

Findings

Constructs explicit examples of isolated singularities with large class groups.

Shows the class group can contain a subgroup isomorphic to $( eals/z)^ {2g}$.

Provides a method to realize complex class group structures in singularities.

Abstract

Let $d\geq3$ and $g\geq1$ be integers. Using a geometric construction involving the symmetric product of a projective curve, we exhibit a $d$-dimensional complete local normal domain over $\mathbb{C}$ with an isolated singularity such that its class group contains a copy of $(\mathbb{R}/\mathbb{Z})^{\oplus 2g}$ as subgroup.