Cohen--Macaulay ideals of codimension two and the geometry of plane points
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arXiv
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| Format: | Preprint |
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2025
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| author | Lira, Dayane Oliveira, Geisa Ramos, Zaqueu Simis, Aron |
| author_facet | Lira, Dayane Oliveira, Geisa Ramos, Zaqueu Simis, Aron |
| contents | We consider classes of codimension two Cohen--Macaulay ideals over a standard graded polynomial ring over a field. We revisit Vasconcelos' problem on $3\times 2$ matrices with homogeneous entries and describe the homological details of Geramita's work on plane points. An additional topic is the homological discussion of minors fixing a submatrix in the context of a perfect codimension two ideal. A combinatorial outcome of the results is a proof of the conjecture on the Jacobian ideal of a hyperplane arrangement stated by Burity, Simis and Tohǎneanu. The basic drive behind the present landscapes is a thorough analysis of the related Hilbert--Burch matrix, often without assuming equigeneration, linear presentation or even the popular $G_d$ condition of Artin--Nagata. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03728 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohen--Macaulay ideals of codimension two and the geometry of plane points Lira, Dayane Oliveira, Geisa Ramos, Zaqueu Simis, Aron Commutative Algebra Algebraic Geometry We consider classes of codimension two Cohen--Macaulay ideals over a standard graded polynomial ring over a field. We revisit Vasconcelos' problem on $3\times 2$ matrices with homogeneous entries and describe the homological details of Geramita's work on plane points. An additional topic is the homological discussion of minors fixing a submatrix in the context of a perfect codimension two ideal. A combinatorial outcome of the results is a proof of the conjecture on the Jacobian ideal of a hyperplane arrangement stated by Burity, Simis and Tohǎneanu. The basic drive behind the present landscapes is a thorough analysis of the related Hilbert--Burch matrix, often without assuming equigeneration, linear presentation or even the popular $G_d$ condition of Artin--Nagata. |
| title | Cohen--Macaulay ideals of codimension two and the geometry of plane points |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2503.03728 |