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

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 $d\ge 2$ 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.

In this version, we've made several changes to the article's introduction and exposition. We've corrected some typos found in the previous version