The structure of almost Cohen-Macaulay $3$-generated ideals of codimension $2$ in terms of matrix theory
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912994409054208 |
|---|---|
| author | Burity, Ricardo Fiel, Thiago Ramos, Zaqueu Simis, Aron |
| author_facet | Burity, Ricardo Fiel, Thiago Ramos, Zaqueu Simis, Aron |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19175 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The structure of almost Cohen-Macaulay $3$-generated ideals of codimension $2$ in terms of matrix theory Burity, Ricardo Fiel, Thiago Ramos, Zaqueu Simis, Aron Commutative Algebra Algebraic Geometry 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. |
| title | The structure of almost Cohen-Macaulay $3$-generated ideals of codimension $2$ in terms of matrix theory |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2603.19175 |