Commutative Algebra Seminar

Title

Classifying extremal pairs in equal characteristic

Abstract

Let R be a polynomial ring, J ⊆ R an ideal, and P 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 P: 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.

In 1972, Skoda showed that lct_P(R, J) >= 1/ord_P(J), where ord_P denotes the order of vanishing at P. Skoda’s bound has been generalized and refined many times since, most recently by Demailly and Pham using mixed multiplicities of J and P. We extend Demailly and Pham’s lower bound to positive characteristic and study the pairs (R, J) for which lct_P(R, J) (or fpt_P(R, J)) equals the lower bound. We conjecture a classification of these “extremal pairs,” which we confirm in codimension 1, when P, J are homogeneous, and when char(R) = 0 and dim(R) = 2.