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Structure theorems for Gorenstein ideals of codimension four with small number of generators

Tymoteusz Chmiel, Lorenzo Guerrieri, Xianglong Ni, Jerzy Weyman

TL;DR

This work develops a representation-theoretic framework for Gorenstein ideals of codimension four with six generators, introducing two families of higher structure maps arising from the α1- and α2-gradings of the $E_6$-related structure. By constructing a generic-ring perspective and interpreting these maps through Plücker/Schubert geometry, the authors prove that any six-generator codimension-four Gorenstein ideal is a hyperplane section of a codimension-three Gorenstein ideal, extending earlier results without generic hypotheses and covering Artinian cases. The paper also outlines generic models for seven and eight generators and sketches conjectural extensions, tying the algebraic structure of resolutions to geometric and Lie-theoretic data in a unified framework. Overall, the approach provides a deep link between minimal free resolutions, spinor coordinates, and Schubert geometry, with potential implications for licci classifications and deformation theory in higher codimension.

Abstract

In this article we study minimal free resolutions of Gorenstein ideals of codimension four, using methods coming from representation theory. We introduce families of higher structure maps associated with such resolution, defined similarly to the codimension three case. As our main application, we prove that every Gorenstein ideal of codimension four minimally generated by six elements is a hyperplane section of a Gorenstein ideal of codimension three, strengthening a result by Herzog-Miller and Vasconcelos-Villarreal. We state analogous conjectural results for ideals minimally generated by seven and eight elements.

Structure theorems for Gorenstein ideals of codimension four with small number of generators

TL;DR

This work develops a representation-theoretic framework for Gorenstein ideals of codimension four with six generators, introducing two families of higher structure maps arising from the α1- and α2-gradings of the -related structure. By constructing a generic-ring perspective and interpreting these maps through Plücker/Schubert geometry, the authors prove that any six-generator codimension-four Gorenstein ideal is a hyperplane section of a codimension-three Gorenstein ideal, extending earlier results without generic hypotheses and covering Artinian cases. The paper also outlines generic models for seven and eight generators and sketches conjectural extensions, tying the algebraic structure of resolutions to geometric and Lie-theoretic data in a unified framework. Overall, the approach provides a deep link between minimal free resolutions, spinor coordinates, and Schubert geometry, with potential implications for licci classifications and deformation theory in higher codimension.

Abstract

In this article we study minimal free resolutions of Gorenstein ideals of codimension four, using methods coming from representation theory. We introduce families of higher structure maps associated with such resolution, defined similarly to the codimension three case. As our main application, we prove that every Gorenstein ideal of codimension four minimally generated by six elements is a hyperplane section of a Gorenstein ideal of codimension three, strengthening a result by Herzog-Miller and Vasconcelos-Villarreal. We state analogous conjectural results for ideals minimally generated by seven and eight elements.

Paper Structure

This paper contains 26 sections, 4 theorems, 98 equations.

Table of Contents

  1. Preliminaries
  2. Structure theory of Gorenstein ideals of codimension four
  3. Lie algebras and representations
  4. Construction
  5. Gradings on $\mathfrak{g}$
  6. Representations
  7. Connection to generic free resolutions
  8. Generic ring
  9. Connection between gradings and higher structure maps
  10. $G/P$ and Schubert varieties
  11. Algebraic definition of the homogeneous space $G/P$
  12. Weyl group and subgroups
  13. Plücker coordinates and Schubert cells
  14. Geometric interpretation of higher structure maps
  15. Higher structure maps from the grading induced by $\alpha_1$
  16. ...and 11 more sections

Key Result

Theorem 1

Let $R$ be a complete regular local ring in which two is a unit or a graded polynomial ring over quadratically closed field of characteristic $\neq 2$. Let $I\subset R$ be a Gorenstein ideal of codimension four, minimally generated by six elements. Then $I=(J,y)$, where $J$ is the ideal generated by

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 1.1: Theorem 4.3, DGA-algebra
  • Theorem 1.2: Theorem 4.2, celikbas-laxmi-weyman
  • Example 1.3: $\alpha_2$-grading on $E_n$
  • Example 1.4: $\alpha_1$-grading on $E_n$
  • Example 1.5
  • Lemma 4.1
  • proof