[Paper Review] Regularity and multiplicity of toric rings of three-dimensional Ferrers diagrams

Kuei-Nuan Lin, Yi-Huang Shen|arXiv (Cornell University)|Sep 22, 2018

Algebraic structures and combinatorial models15 references3 citations

TL;DR

This paper establishes explicit formulas for the Castelnuovo–Mumford regularity and multiplicity of toric rings associated with three-dimensional Ferrers diagrams, particularly in the rectangular case, and provides bounds for general cases using algebraic and combinatorial techniques on Stanley–Reisner complexes and lexicographic initial ideals.

ABSTRACT

We investigate the Castelnuovo-Mumford regularity and the multiplicity of the toric ring associated with a three-dimensional Ferrers diagram. In particular, in the rectangular case, we provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanying pair of Ferrers diagrams under some mild conditions and bound the Castelnuovo-Mumford regularity for more general cases.

Motivation & Objective

• To determine the Castelnuovo–Mumford regularity and multiplicity of the special fiber ring associated with a three-dimensional Ferrers diagram.
• To extend previous results on two-dimensional Ferrers diagrams to the three-dimensional case.
• To provide direct formulas for these invariants in the rectangular case and bounds for general cases.
• To analyze the structure of the associated Stanley–Reisner complex and its homological properties.
• To establish bounds on regularity via induction on diagram size and projection properties.

Proposed method

• Utilizes the lexicographic order to show that the special fiber ideal of a 3D Ferrers diagram has a squarefree quadratic initial ideal.
• Applies algebraic techniques to transfer the computation of regularity and multiplicity to the Stanley–Reisner complex of the diagram.
• Employs induction on the size of the diagram, reducing to smaller subdiagrams that preserve the projection property.
• Defines a concise subdiagram $\overline{\mathcal{D}}$ to simplify the ambient space and bound the regularity via structural decomposition.
• Uses Hochster’s formula and Reisner’s criterion to analyze the homological properties of the Stanley–Reisner ring.
• Leverages the pure vertex-decomposability of the Stanley–Reisner complex to ensure acyclicity and support inductive arguments.

Experimental results

Research questions

• RQ1 What are the explicit formulas for the Castelnuovo–Mumford regularity and multiplicity of the toric ring associated with a 3D Ferrers diagram in the rectangular case?
• RQ2 How do the regularity and multiplicity of an accompanying pair of Ferrers diagrams compare under mild conditions?
• RQ3 What bounds can be established for the Castelnuovo–Mumford regularity in the general case of 3D Ferrers diagrams?
• RQ4 How does the structure of the Stanley–Reisner complex influence the regularity and multiplicity of the associated toric ring?
• RQ5 Can the regularity be bounded via inductive reduction on subdiagrams satisfying the projection property?

Key findings

• For rectangular 3D Ferrers diagrams, the Castelnuovo–Mumford regularity and multiplicity are given by explicit closed-form formulas.
• The regularity of the Stanley–Reisner complex associated with a 3D Ferrers diagram satisfying the projection property is bounded above by $\mu_{\mathcal{D}} - 4$.
• The Stanley–Reisner complex $\Delta(\mathcal{D})$ of a 3D Ferrers diagram is Cohen–Macaulay and hence acyclic over any field.
• The multiplicity of the toric ring is related to the volume of the associated edge polytope, generalizing known results from 2D cases.
• The subdiagram $\overline{\mathcal{D}}$ constructed via projection and truncation preserves the essential structure and allows inductive bounds on regularity.
• The regularity bound $\operatorname{reg}(\Delta(\widetilde{\mathcal{H}})) \leq \mu_{\widetilde{\mathcal{D}}} - 4$ is established through careful decomposition and induction on the diagram size.