Commutative Algebra Seminar
Spring 2026
Regular Day: Friday
Regular Time: 2:00PM - 3:00PM
Regular Location: LCB 222
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February 6 |
Antonia Kekkou
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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.
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February 13 |
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.
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February 20 |
Ben Baily
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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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April 3 Joint with AG Seminar |
Vasudevan Srinivas
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This talk will discuss some results on etale fundamental groups of varieties over an algebraically closed field of characteristic p > 0, based on joint work with Hélène Esnault and other coauthors. One result, along with Mark Schusterman, is that the tame fundamental group is finitely presented for such a variety which is the complement of an SNC divisor in a smooth projective variety. A second, along with Jakob Stix, is to give an obstruction for a smooth projective variety to admit a lifting to characteristic 0, in terms of the structure of its etale fundamental group as a profinite group. We will finally touch on some open questions.
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April 10 |
Vaibhav Pandey
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We give a characterization of ideals which are monomial under some system of coordinates. Our motivation is to understand when does a homogeneous ideal I, in a polynomial ring S over an infinite field, admit a monomializable Artinian reduction. That is, when does there exist a choice of general hyperplanes which cut down the affine cone over Proj(S/I) to a zero-dimensional scheme which is defined, in some coordinates, by a monomial ideal. This question is difficult in general; an affirmative answer for any class of ideals is extremely useful. For example, if S/I is Cohen-Macaulay, its graded betti table is preserved on passing to an Artinian reduction. If any such Artinian reduction is monomializable, then many invariants of the betti table of S/I (type, multiplicity, regularity, etc.) become readily computable and have strong constraints.
We discuss the subtleties and obstructions involved in the choice of these general hyperplanes and some surprising applications to linkage theory. This is joint work with Alessandro De Stefani and Matteo Varbaro.
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April 17 |
TBA
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Abstract
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