Minimal primes and radicality of ideals generated by adjacent 2-minors

TL;DR

This paper characterizes the minimal primes and radicality of ideals generated by adjacent 2-minors, revealing their algebraic properties are equivalent and providing combinatorial criteria for these properties.

Contribution

It offers a complete description of minimal primes and radicality for ideals generated by adjacent 2-minors, linking algebraic properties to combinatorial structures.

Findings

Properties like being Cohen-Macaulay, Gorenstein, and complete intersection are equivalent for these ideals.

Provides a combinatorial characterization of convex collections of cells with these properties.

Identifies necessary conditions for the radicality of these ideals based on minimal non-radical configurations.

Abstract

In this paper, we provide a complete description of the minimal primes of ideals generated by adjacent $2$-minors, in terms of the so-called admissible sets and associated lattice ideals. We prove that for these ideals, the properties of being unmixed, Cohen-Macaulay, level, Gorenstein, and complete intersection are equivalent. Moreover, we give a combinatorial characterization of all convex collections of cells satisfying any of these equivalent properties. Finally, we study the radicality of these ideals and derive necessary combinatorial conditions based on minimal non-radical configurations.