Weakly- closed graphs and F-purity of binomial edge ideals

Abstract: Herzog et al. characterized closed graphs as the graphs whose binomial edge ideals have a quadratic Groebner basis. In this talk, we focus on a generalization of closed graphs, namely weakly-closed graphs (or co-comparability graphs). Building on known results about Knutson ideals of generic matrices, we characterize weakly-closed graphs as the only graphs whose binomial edge ideals are Knutson ideals (associated with a certain polynomial f). In doing so, we re-prove Matsuda's theorem about the F-purity of binomial edge ideals of weakly-closed graphs in prime characteristic and we extend it to generalized binomial edge ideals. Lastly, we will discuss some open conjectures on the F-purity of binomial edge ideals and on the relation between Knutson ideals and compatible ideals.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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