Adam LaClair : F-purity of binomial edge ideals

Posted by Wenliang Zhang , part of the Commutative Algebra Seminar.

At

Jan. 15, 2025, 11 a.m.

In

1227 SEO

Abstract

Binomial edge ideals provide a way to associate to any graph a binomial ideal. Many researchers have investigated the algebraic properties of binomial edge ideals in terms of the combinatorics of the graph. One such question is: When are binomial edge ideals F-pure? F-purity describes a class of rings in positive characteristic that exhibit a "mild" singularity, such rings have assumed a prominent position amongst the study of F-singularities. In 2012, K. Matsuda introduced the class of weakly closed graphs (i.e., incomparability graphs), and he proved that the binomial edge ideal associated to such graphs is F-pure in any positive characteristic. He conjectured that the converse should hold in characteristic 2. In this talk, we will review binomial edge ideals, F-purity, and other relevant background, and then we will present the main ideas going into the proof of Matsuda's conjecture.