Commutative Algebra Seminar

Regular sequences are a fundamental tool in commutative algebra. In this talk, we introduce a notion of regular sequences in $R$-linear triangulated categories, where $R$ is a graded-commutative ring. As an application of this definition, we show that the length of regular sequences provides lower bounds on levels. This is joint work with Janina C. Letz and Marc Stephan.

A weakening of Frobenius splitting, Quasi-F-Splittings have proven to be a vital invariant in the study of varieties in positive characteristic, with numerous applications to arithmetic and birational geometry. This weaker condition extends the application of Frobenius to study singularities of arithmetically supersingular varieties, encompassing a much broader class of examples.  In this talk I'll provide an overview of Quasi-F-Splittings and introduce a local analogue, Quasi-F-Purity. I will also discuss how quasi-F-pure hypersurfaces are "as close to being F-pure as possible" by computing the F-pure threshold of an arbitrary quasi-F-pure hypersurface. This talk includes joint work with Jack J Garzella.

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.

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