Koszul blowup algebras associated to three-dimensional Ferrers diagrams

TL;DR

This paper studies the algebraic structures of monomial ideals from three-dimensional Ferrers diagrams, showing they have desirable properties like being Koszul and Cohen–Macaulay, with explicit descriptions under certain conditions.

Contribution

It provides explicit presentations of Rees algebras and toric rings for these ideals and proves their Koszul and Cohen–Macaulay properties under the projection property.

Findings

Toric ring is a Koszul Cohen–Macaulay normal domain.

Rees algebra is Koszul with an ideal of fiber type.

Explicit presentation ideals are described under the projection property.

Abstract

We investigate the Rees algebra and the toric ring of the squarefree monomial ideal associated to the three-dimensional Ferrers diagram. Under the projection property condition, we describe explicitly the presentation ideals of the Rees algebra and the toric ring. We show that the toric ring is a Koszul Cohen--Macaulay normal domain, while the Rees algebra is Koszul and the defining ideal is of fiber type.