Multi-Rees Algebras of Strongly Stable Ideals

TL;DR

This paper investigates the algebraic structure of multi-Rees algebras formed from strongly stable ideals, proving they are of fiber type, providing explicit Gr"obner bases, and establishing conditions for Koszulness, including a quadratic Gr"obner basis case.

Contribution

It proves multi-Rees algebras of strongly stable ideals are of fiber type and constructs explicit Gr"obner bases, advancing understanding of their algebraic properties.

Findings

Multi-Rees algebra of strongly stable ideals is of fiber type.

Provided a Gr"obner basis for the defining ideal.

Established Koszulness under certain conditions.

Abstract

We prove that the multi-Rees algebra $\mathcal{R}(I_1 \oplus \cdots \oplus I_r)$ of a collection of strongly stable ideals $I_1, \ldots, I_r$ is of fiber type. In particular, we provide a Gr\"obner basis for its defining ideal as a union of a Gr\"obner basis for its special fiber and binomial syzygies. We also study the Koszulness of $\mathcal{R}(I_1 \oplus \cdots \oplus I_r)$ based on parameters associated to the collection. Furthermore, we establish a quadratic Gr\"obner basis of the defining ideal of $\mathcal{R}(I_1 \oplus I_2)$ where each of the strongly stable ideals has two quadric Borel generators. As a consequence, we conclude that this multi-Rees algebra is Koszul.