Generalized binomial edge ideals are Cartwright-Sturmfels

TL;DR

This paper addresses whether generalized binomial edge ideals are Cartwright–Sturmfels (CS) and develops a versatile toolkit to analyze CS properties in multigraded settings. It introduces a base-change framework and several CS-preserving constructions, enabling the combination and extension of CS ideals while controlling their multidegree interactions. The main result proves that generalized binomial edge ideals $I_G(m)$ are CS by decomposing them into sums indexed by connected induced subgraphs and applying the established preservation principles, with precise descriptions of their generic initial ideals. It also extends the discussion to higher-order minors, identifying cases where CS holds and highlighting obstructions, thereby enriching the landscape of CS ideals and their geometric/combinatorial implications.

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.

Figures (3)

• Figure 1: Example of CS-minor ideal
• Figure 2: Example of a $5$-uniform hypergraph satisfying the conditions of Lemma \ref{['complete']} for $j_1 = 1$, $j_2 = 3$ and $j_3 = 5$
• Figure 3: Non CS Configurations from Proposition \ref{['prop:obstruct']} and Example \ref{['ex:strange']}

Theorems & Definitions (29)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• Proposition 3.1
• proof
• Corollary 3.2