Blow-Up Algebras of Strongly Stable Ideals

Abstract: Let $S$ be a polynomial ring and $I_1,\ldots, I_r$ be a collection of ideals in $S$. The multi-Rees algebra $\mathcal{R} (I_1,\ldots, I_r)$ of this collection of ideals encode many algebraic properties of these ideals, their products, and powers. Additionally, the multi-Rees algebra $\mathcal{R} (I_1,\ldots, I_r)$ arise in successive blowing up of $\textrm{Spec } S$ at the subschemes defined by $I_1,\ldots, I_r$. Due to this connection, Rees and multi-Rees algebras are also called blow-up algebras in the literature.

In this talk, we will focus on Rees and multi-Rees algebras of strongly stable ideals. In particular, we will discuss the Koszulness of these algebras through a systematic study of these objects via three parameters: the number of ideals in the collection, the number of Borel generators of each ideal, and the degrees of Borel generators. In our study, we utilize combinatorial objects such as fiber graphs to detect Gröbner bases and Koszulness of these algebras. This talk is based on a joint work with Kuei-Nuan Lin and Gabriel Sosa.

algebraic geometrynumber theory

Audience: researchers in the topic

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca

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