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

Dayane Lira, Geisa Oliveira, Zaqueu Ramos, Aron Simis

TL;DR

This work studies Cohen–Macaulay ideals of codimension two in standard graded rings by examining Hilbert–Burch matrices without assuming equigeneration, linear presentation, or $G_d$. It blends algebraic and geometric viewpoints, revisiting Vasconcelos’ $3\times2$ matrix problem and Geramita’s plane-point geometry, and develops a framework using minors fixing submatrices and Sylvester forms to control Rees algebras and reductions. Key contributions include criteria for when the Rees algebra is Cohen–Macaulay, a reduction/linear-type analysis via $G_d$ in two-degree generators, and a final application to plane reduced points in tight generic position with birationality results and explicit degree computations for the associated rational maps. The results also yield a combinatorial proof of the Jacobian ideal conjecture for hyperplane arrangements in the generic (BuSiTo2022) setting, integrating homological methods with geometric configurations and providing detailed resolutions in several regimes.

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.

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

TL;DR

This work studies Cohen–Macaulay ideals of codimension two in standard graded rings by examining Hilbert–Burch matrices without assuming equigeneration, linear presentation, or . It blends algebraic and geometric viewpoints, revisiting Vasconcelos’ matrix problem and Geramita’s plane-point geometry, and develops a framework using minors fixing submatrices and Sylvester forms to control Rees algebras and reductions. Key contributions include criteria for when the Rees algebra is Cohen–Macaulay, a reduction/linear-type analysis via in two-degree generators, and a final application to plane reduced points in tight generic position with birationality results and explicit degree computations for the associated rational maps. The results also yield a combinatorial proof of the Jacobian ideal conjecture for hyperplane arrangements in the generic (BuSiTo2022) setting, integrating homological methods with geometric configurations and providing detailed resolutions in several regimes.

Abstract

We consider classes of codimension two Cohen--Macaulay ideals over a standard graded polynomial ring over a field. We revisit Vasconcelos' problem on 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 condition of Artin--Nagata.

Paper Structure

This paper contains 16 sections, 23 theorems, 96 equations.

Table of Contents

  1. Notation
  2. A primer of $3\times 2$ matrices over $k[x,y,z]$
  3. The ideal generated by the entries
  4. A column of degree one
  5. The $(x,y)$-primary component of the ideal of entries
  6. A role of the Sylvester forms
  7. Codimension two perfect ideals generated in two degrees
  8. Basic players
  9. Minors fixing a submatrix
  10. A role of the Buchsbaum--Rim complex
  11. A main theorem
  12. A player in the theory of plane reduced points
  13. Preliminaries
  14. Points in tight generic position
  15. Tight generic position: range $0\leq h\leq s/2$
  16. ...and 1 more sections

Key Result

Lemma 2.3

(De_Jonq) Let $\{g_1,\ldots, g_n \}\subset k[x_1,\ldots,x_{n}]$ be forms of the same degree with no proper common factor and let $\mathfrak{J}$ and $\mathfrak{F}$ be confluent rational maps as above. Then $\mathfrak{F}=(g_1:\cdots : g_n)$ if and only if $\mathfrak{J}=(fg_1 :\cdots : fg_n:g)$ for sui

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Example 2.8
  • ...and 42 more