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

Ricardo Burity, Thiago Fiel, Zaqueu Ramos, Aron Simis

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.

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

Abstract

Let be a standard graded polynomial ring over a field . The paper focuses on homogeneous ideals of codimension generated by three forms of the same degree that are almost Cohen--Macaulay, i.e., of homological dimension . Based on the structure of the minimal graded free resolution of 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 in terms of the existence of an associated level matrix for which arises as the ideal of minors obtained by fixing the lower block. We provide algebraic and geometric examples illustrating the results.
Paper Structure (10 sections, 7 theorems, 83 equations)

This paper contains 10 sections, 7 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. On the shifts of the minimal free resolution
  3. Extending a result of Dimca--Sticlaru
  4. Ideals with prescribed shifts
  5. Ideals of minors fixing a submatrix, I
  6. Skew-symmetric matrices and an idea of Vasconcelos
  7. Theory guiding examples
  8. Ideals of minors fixing a submatrix, II
  9. Plane curves whose Jacobian ideal admits only three generating syzygies
  10. Plane curves with $4$-generated Jacobian syzygies

Key Result

Proposition 2.1

Let $R=k[x_1,\ldots,x_n]$ be a standard graded polynomial ring in $n\geq 3$ variables over a field $k$. If $J$ is a homogeneous ideal of $R$ of height $2$ generated by three forms of degree $d$, then the following conditions are equivalent$:$

Theorems & Definitions (23)

  • Proposition 2.1
  • Theorem 2.2
  • proof
  • Definition 3.3
  • Example 3.4
  • Theorem 3.5
  • proof
  • Lemma 3.6
  • proof
  • Example 3.7
  • ...and 13 more