Blowup Algebras of $n$--dimensional Ferrers Diagrams

TL;DR

This paper studies the algebraic properties of blowup algebras associated with n-dimensional Ferrers diagrams, providing explicit Gr"obner bases and demonstrating their favorable properties such as being Koszul Cohen–Macaulay normal domains.

Contribution

It introduces a method to obtain Gr"obner bases for multi-Rees algebras of ideals with the strong ll-exchange property related to Ferrers diagrams, establishing their algebraic structure.

Findings

Blowup algebras are Koszul Cohen–Macaulay normal domains.

Explicit Grf6bner bases are provided for these algebras.

Classification of singularities of these blowup algebras.

Abstract

We demonstrate that the direct sum of ideals satisfying the strong $\ell$-exchange property is of fiber type. Furthermore, we provide Gr\"obner 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.