UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

February 26, 2019

939 Evans Hall

3:45PM: Codepth, complete intersections, and quasi-cyclic modules. (Joint work with Claudia Polini, in progress)

Robin Hartshorne

Codepth is the dual notion to depth, being the greatest length of a coregular sequence for a module, meaning the first element maps the module surjectively, the second is subjective on the kernel of the first, and so on. For a curve in P3, let M be the local cohomology module of the graded coordinate ring with supports in the ideal of the curve. Then the theorem of Hellus says that C is a set theoretic complete intersection if and only if M has codepth 2. This criterion is not directly applicable, so we define the notion of a quasi-cyclic module, which is an increasing limit of cyclic modules. In this talk I will recall the still open problem of whether every irreducible nonsingular curve in P3 is a set theoretic complete intersection, and derive a number of consequences using the concepts introduced above.

5:00PM: Chow rings of matroids, ring of matroid quotients, and beyond

Chris Eur

We introduce a certain nef generating set for the Chow ring of the wonderful compactification of a hyperplane arrangement complement. This presentation yields a monomial basis of the Chow ring that admits a geometric and combinatorial interpretation with several applications. Geometrically, one can recover Poincare duality, compute the volume polynomial and verify its log-concavity, and identify a portion of a polyhedral boundary of the nef cone. Combinatorially, one can generalize Postnikov's result on volumes of generalized permutohedra, prove Mason's conjecture on the log-concavity of independent sets for certain matroids, and define a new valuative invariant of a matroid that measures its closeness to uniform matroids. This is an on-going joint work with Spencer Backman and Connor Simpson.

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