Ben Baily of University of Michigan : Classifying extremal pairs in equal characteristic

Posted by Kevin Tucker , part of the Algebraic Geometry Seminar.

At

Aug. 31, 2026, 3 p.m.

In

427 SEO

Abstract

Let R be a polynomial ring, J ⊆ R an ideal, and m a maximal ideal containing J. We consider invariants of the pair (R, J) which measure the singularities of the embedding Spec(R/J) ⊆ Spec(R) at m: the log canonical threshold (lct) in characteristic zero and the F-pure threshold (fpt) in positive characteristic. A smaller value of the lct/fpt means that the embedding is "more singular;" we seek to classify pairs which are as singular as possible. Skoda (1972) showed that lct_m(R, J) >= 1/ord_m(J); Demailly and Pham (2014) sharpened Skoda's bound using mixed multiplicities of J and m. We extend Demailly and Pham's lower bound to positive characteristic and study the pairs (R, J) for which lct_m(R, J) (or fpt_m(R, J)) equals the lower bound. We classify these "extremal pairs" in the standard graded case and the height-1 case, answering a special case of a conjecture by Bivià-Ausina.