# Mathematical Sciences Research Institute

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# Seminar

Eisenbud Seminar: Algebraic Geometry and Commutative Algebra March 11, 2014 (03:45 PM PDT - 06:00 PM PDT)
Location: Evans 939
Description No Description

Video
Abstract/Media

Grobner bases for twisted commutative algebras

Speaker: Steven Sam

3:45 PM

Recent work of Draisma-Kuttler, Snowden, and Church-Ellenberg-Farb in commutative algebra, algebraic geometry, and homological algebra, seeks to understand stability results (say, of some invariant of a sequence of algebraic objects) as the finite generation of some other algebraic object.

Examples include the Delta-modules in the study of free resolutions of Segre embeddings and FI-modules in the study of cohomology of configuration spaces. This finite generation often reduces to establishing the Noetherian property: subobjects of finitely generated objects are again finitely generated. I will discuss the situation of modules over twisted commutative algebras, which realize some of these topics as special cases. I will introduce a Grobner basis theory and show how it proves these Noetherian results. The talk will be mostly combinatorial and I will suggest some open problems. This is based on joint work with Andrew Snowden.

Smoothing of limit linear series on metrized complexes of algebraic curves.

1986 and this theory has several applications to algebraic curves. This theory has recently been generalized  to objects called metrized complexes of curves" by Amini and Baker. A metrized complex of curves is essentially a metric graph with algebraic curves plugged into the vertices of this metric graph. Eisenbud and Harris showed that any limit $g^1_d$ on a curve of compact type can be smoothed. We study the question of smoothing a limit $g^1_d$ on a metrized complex.  We provide an effective characterization of a smoothable limit $g^1_d$ on a metrized complex and the talk will include examples demonstrating this characterization.  This is ongoing work with Matthew Baker and Luo Ye.