Ideals generated by diagonal 2-minors

TL;DR

This paper studies binomial ideals generated by diagonal minors associated with simple graphs, proving they are prime complete intersections and analyzing their divisor class groups, with applications to constructing normal domains of arbitrary divisor class rank.

Contribution

It establishes that these ideals are prime complete intersections and determines their divisor class groups, providing tools for constructing normal domains with specified divisor class groups.

Findings

$P_G$ is a prime complete intersection ideal for any graph $G$.

The divisor class group of $K[X]/ P_G$ is explicitly determined.

Applications include constructing normal domains with any given free divisor class group.

Abstract

With a simple graph $G$ on $[n]$, we associate a binomial ideal $P_G$ generated by diagonal minors of an $n \times n$ matrix $X=(x_{ij})$ of variables. We show that for any graph $G$, $P_G$ is a prime complete intersection ideal and determine the divisor class group of $K[X]/ P_G$. By using these ideals, one may find a normal domain with free divisor class group of any given rank.