Generalized binomial edge ideals are Cartwright-Sturmfels

TL;DR

This paper proves that generalized binomial edge ideals, associated with matrices of multiple rows, are Cartwright-Sturmfels, extending known properties of binomial edge ideals and exploring ideal constructions that preserve this property.

Contribution

It establishes that generalized binomial edge ideals are Cartwright-Sturmfels, broadening understanding of their algebraic properties and providing new results on ideal constructions.

Findings

Generalized binomial edge ideals are Cartwright-Sturmfels.

Ideal constructions can preserve the Cartwright-Sturmfels property.

Examples and counterexamples for higher minors are provided.

Abstract

Binomial edge ideals associated to a simple graph G were introduced by Herzog and collaborators and, independently, by Ohtani. They became an ``instant classic" in combinatorial commutative algebra with more than 100 papers devoted to their investigation over the past 15 years. They exhibit many striking properties, including being radical and, moreover, Cartwright-Sturmfels. Using the fact that binomial edge ideals can be seen as ideals of 2-minors of a matrix of variables with two rows, generalized binomial edge ideals of 2-minors of matrices of m rows were introduced by Rauh and proved to be radical. The goal of this paper is to prove that generalized binomial edge ideals are Cartwright-Sturmfels. On the way we provide results on ideal constructions preserving the Cartwright-Sturmfels property. We also give examples and counterexamples to the Cartwright-Sturmfels property for higher minors.