Geometric Divisors in Normal Local Domains

TL;DR

This paper investigates the possible class groups of local rings at points on normal complex varieties, demonstrating that all finitely generated abelian groups can occur as class groups in certain geometric contexts.

Contribution

It proves that all finitely generated subgroups of class groups are realizable in specific geometric cases, extending understanding of class group structures in algebraic geometry.

Findings

All finitely generated subgroups of class groups are realizable.

Every finitely generated abelian group appears as a class group of a local ring at a cone vertex.

Uses Noether-Lefschetz theory to establish these results.

Abstract

Let A be the local ring at a point of a normal complex variety with completion R. Srinivas has asked about the possible images of the induced map from Cl A to Cl R over all geometric normal domains A with fixed completion R. We use Noether-Lefschetz theory to prove that all finitely generated subgroups are possible in some familiar cases. As a byproduct we show that every finitely generated abelian group appears as the class group of the local ring at the vertex of a cone over some smooth complex variety of each positive dimension.