The structure of almost Cohen-Macaulay $3$-generated ideals of codimension $2$ in terms of matrix theory

TL;DR

This paper characterizes almost Cohen-Macaulay ideals generated by three degree-d forms in a polynomial ring of codimension two using matrix theory and minimal free resolutions.

Contribution

It introduces level matrices and provides a complete characterization of such ideals via the existence of related matrices and their maximal minors.

Findings

Complete algebraic and geometric examples illustrating the characterization.

Characterization of ideals in terms of level matrices and their minors.

Structural description based on minimal free resolutions and latent data.

Abstract

Let $R$ be a standard graded polynomial ring over a field $k$. The paper focuses on homogeneous ideals $J \subset R$ of codimension $2$ generated by three forms of the same degree $d \geq 2$ that are almost Cohen--Macaulay, i.e., of homological dimension $2$. Based on the structure of the minimal graded free resolution of $J$ and numerical data encoded in certain \emph{latent data}, one introduces the notion of \emph{level matrices} associated with these data. The main result provides a complete characterization of an almost Cohen--Macaulay $3$-generated ideal $J$ of codimension $2$ in terms of the existence of a related level matrix for which $J$ arises as the ideal of its maximal minors that fix a submatrix. One provides algebraic and geometric examples illustrating the results.