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Blowup Algebras of $n$--dimensional Ferrers Diagrams

Kuei-Nuan Lin, Yi-Huang Shen

TL;DR

The paper addresses implicit equations and homological properties of blowup algebras arising from collections of ideals, with a focus on those tied to $n$-dimensional Ferrers diagrams. It extends the strong $\ell$-exchange property to the multi-Rees setting to establish a fiber-type decomposition and develops a tableau-based ordering and standardization framework to obtain $G$-quadratic presentations. The main results show that for standardizable Ferrers diagrams, the special fiber and multi-Rees algebras are Koszul, Cohen–Macaulay, and normal, with rational singularities in characteristic zero and $F$-rationality in positive characteristic; a critical role is played by a Gröbner-basis description combining $\mathbf{I}(\widetilde{\mathcal{D}}_r)$ with additional binomial relations. The work highlights the necessity of the standardizable condition and demonstrates boundary cases where the method does not directly extend to non-identical diagrams, thereby guiding future developments of monomial-order frameworks in combinatorial settings.

Abstract

We demonstrate that the direct sum of ideals satisfying the strong $\ell$-exchange property is of fiber type. Furthermore, we provide Gröbner bases of the presentation ideals of multi-Rees algebras and the corresponding special fibers, when they are associated with an $n$-dimensional Ferrers diagram that is standardizable. In particular, we show that these blowup algebras are Koszul Cohen--Macaulay normal domains and classify their singularities.

Blowup Algebras of $n$--dimensional Ferrers Diagrams

TL;DR

The paper addresses implicit equations and homological properties of blowup algebras arising from collections of ideals, with a focus on those tied to -dimensional Ferrers diagrams. It extends the strong -exchange property to the multi-Rees setting to establish a fiber-type decomposition and develops a tableau-based ordering and standardization framework to obtain -quadratic presentations. The main results show that for standardizable Ferrers diagrams, the special fiber and multi-Rees algebras are Koszul, Cohen–Macaulay, and normal, with rational singularities in characteristic zero and -rationality in positive characteristic; a critical role is played by a Gröbner-basis description combining with additional binomial relations. The work highlights the necessity of the standardizable condition and demonstrates boundary cases where the method does not directly extend to non-identical diagrams, thereby guiding future developments of monomial-order frameworks in combinatorial settings.

Abstract

We demonstrate that the direct sum of ideals satisfying the strong -exchange property is of fiber type. Furthermore, we provide Gröbner bases of the presentation ideals of multi-Rees algebras and the corresponding special fibers, when they are associated with an -dimensional Ferrers diagram that is standardizable. In particular, we show that these blowup algebras are Koszul Cohen--Macaulay normal domains and classify their singularities.

Paper Structure

This paper contains 5 sections, 12 theorems, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Multi-Rees Algebras and its special fibers
  3. Ferrers diagrams
  4. Ordering of tableau
  5. Multi-Rees algebras of standardizable Ferrers diagrams

Key Result

Theorem 2.5

Assume that $I_1,\dots,I_r$ are ideals as in notation:collection which satisfy the strong $\ell$-exchange property. Let $\mathcal{H}$ be the collection of binomials $x_{k_1}T_{i,j}-x_{k_2}T_{i,j'}$ where $x_{k_1}f_{i,j}=x_{k_2}f_{i,j'}$, $k_1<k_2$, and $k_2$ is the largest for which $x_{k_1}f_{i,j}/

Figures (2)

  • Figure 1: A three-dimensional standardizable Ferrers diagram
  • Figure 2: Two $12 \times 5$ semi-standard tableaux with the same support

Theorems & Definitions (39)

  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3: Lin-Shen3D
  • Example 3.4
  • ...and 29 more