Koszul blowup algebras associated to three-dimensional Ferrers diagrams

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.

Regularity and multiplicity of toric rings of three-dimensional Ferrers diagrams

“…Then, we can actually prove that the minimal generating set of the special fiber ideal J D contains a degree p binomial, generalizing the phenomenon observed in [22,Example 2.4] for degree three.…”

“…An identical estimate can be achieved for the regularity under the same (weaker) condition as assumed in [22]. No doubt, the proof, given in Section 6, invites more intricate combinatorial maneuver.…”

“…As the main result of [22], we proved that the special fiber ring F pI D q is a Koszul Cohen-Macaulay normal domain, provided that D is a three-dimensional Ferrers diagram which satisfies the projection property. Its proof is quite involved and depends on the following two key ingredients.…”

“…On the other hand, to give rise to the toric variety, the squarefree monomial ideal I that we are particularly interested in here, is the I D associated to the three-dimensional Ferrers diagram D. Ferrers diagram is an important object studied in combinatorics and it has applications in permutation statistics [7] and inverse rook problems [15]. This paper is the continuation of our previous work in [22]. There, we have shown that, under some mild conditions, the special fiber ideal J D associated to a three-dimensional Ferrers diagram D has a squarefree quadratic initial ideal with respect to the lexicographic order, and the accompanying Stanley-Reisner complex ∆pDq is pure vertex-decomposable.…”

On the Conjecture of Vasconcelos for Artinian Almost Complete Intersection Monomial Ideals

“…The next question will be if one can at least find a bound on the degree of generators of the defining ideal without narrowing down to special cases. It is unfortunate that even for a degree square-free monomial ideal, one can get a generator of any degree if no other restriction is placed; see, for example, [12, 13]. Finding the degree bound is actually a classical question in the commutative algebra community.…”