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.

In 1972, Skoda showed that lct_m(R, J) >= 1/ord_m(J), where ord_m denotes the order of vanishing at m. Skoda's bound has been generalized and refined many times since; among these improvements is a 2014 result by Demailly and Pham 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, the codimension 1 case, and the dimension 2 case, confirming special cases of a conjecture by Bivià-Ausina.