Characterizing almost Cohen-Macaulay $3$-generated ideals of codimension $2$ in terms of prescribed shift

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 shifts}, one introduces the notion of \emph{level matrices} associated with these shifts. Our main result provides a complete characterization of almost Cohen--Macaulay ideals of codimension $2$ in terms of the existence of an associated level matrix for which $J$ arises as the ideal of minors obtained by fixing the lower block. We provide algebraic and geometric examples illustrating the results.