The primary components of positive critical binomial ideals

TL;DR

This paper investigates the structure of positive critical binomial ideals, revealing their unmixedness properties, primary components, and conditions for primality, especially in relation to monomial curves and Herzog-Northcott ideals.

Contribution

It provides a complete description of primary components of critical binomial ideals for all dimensions and characterizes when their hulls are prime, advancing understanding of binomial ideal structures.

Findings

Critical binomial ideals are not unmixed for n≥4.

Complete description of primary components for n≤3.

Explicit characterization of embedded components and prime hulls.

Abstract

A natural candidate for a generating set of the (necessarily prime) defining ideal of an $n$-dimensional monomial curve, when the ideal is an almost complete intersection, is a full set of $n$ critical binomials. In a somewhat modified and more tractable context, we prove that, when the exponents are all positive, critical binomial ideals in our sense are not even unmixed for $n\geq 4$, whereas for $n\leq 3$ they are unmixed. We further give a complete description of their isolated primary components as the defining ideals of monomial curves with coefficients. This answers an open question on the number of primary components of Herzog-Northcott ideals, which comprise the case $n=3$. Moreover, we find an explicit, concrete description of the irredundant embedded component (for $n\geq 4$) and characterize when the hull of the ideal, i.e., the intersection of its isolated primary components, is prime. Note that these last results are independent of the characteristic of the ground field. Our techniques involve the Eisenbud-Sturmfels theory of binomial ideals and Laurent polynomial rings, together with theory of Smith Normal Form and of Fitting ideals. This gives a more transparent and completely general approach, replacing the theory of multiplicities used previously to treat the particular case $n=3$.