Binomial ideals attached to finite collections of cells

TL;DR

This paper studies binomial ideals generated by inner 2-minors of finite cell collections, providing insights into their height and characterizing the coordinate rings for cases with isolated singularities.

Contribution

It introduces the concept of cell ideals, interprets their height combinatorially, and characterizes the coordinate rings with isolated singularities.

Findings

Height of the ideal relates to the number of cells in the collection.

Coordinate rings with isolated singularities are explicitly characterized.

Provides a new interpretation of the algebraic properties of cell ideals.

Abstract

We consider the ideal of inner $2$-minors $I_{\mathcal{P}}$ of a finite set of cells $\mathcal{P}$, which we call the cell ideal of $\mathcal{P}$. A nice interpretation for the height of an unmixed ideal $I_{\mathcal{P}}$, in terms of the number of cells of $\mathcal{P}$ is given. Moreover, the coordinate rings of cell ideals with isolated singularities are determined.