Ideals of Herzog-Northcott type

TL;DR

This paper explores a broad class of ideals generated by 2x2 minors of matrices with entries as powers of three elements, analyzing their algebraic properties, prime and radical nature, and liaison characteristics, extending prior work on monomial curves.

Contribution

It generalizes the study of ideals related to monomial curves by identifying them as Northcott ideals, analyzing their set-theoretic complete intersection property, and characterizing their prime and radical structures.

Findings

Ideals are set-theoretically complete intersections.

Characterization of when these ideals are prime.

Examples of characteristic-dependent prime and primary structures.

Abstract

This paper takes a new look at ideals generated by 2x2 minors of 2x3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this are the ideals determining monomial curves in three dimensional space, which were already studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.