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arXiv · 2503.03728

Cohen--Macaulay ideals of codimension two and the geometry of plane points

Dayane Lira·Geisa Oliveira·Zaqueu Ramos·Aron Simis
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Abstract

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.

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Dayane Lira, Geisa Oliveira, Zaqueu Ramos, Aron Simis. 2025-03-19. Cohen--Macaulay ideals of codimension two and the geometry of plane points. https://arxiv.org/abs/2503.03728

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