UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

May 03, 2013

939 Evans Hall

2:00PM: The Shimoda conjecture

Justin Chen

Let (R,m) be a Noetherian local ring. In 1978, J. Sally posed the following question: if there is a uniform bound on the number of generators of prime ideals in R, is then dim R at most 2? It remains open to this day, and in 2007, Y. Shimoda asked an even stronger question: if every prime ideal in R except the maximal ideal is a complete intersection, is dim R at most 2? Although this is still unknown in general, I will talk about a paper of Goto, O'Carroll, and Planas-Vilanova, in which they obtain some partial positive results to Shimoda's question, by considering ideals of Herzog-Northcott type.

3:00PM: Closures of Linear Spaces

Adam Boocher

If $L\subset \mathbb{A}^n$ is a linear space then we can take its closure in $(\mathbb{P}^1)^n$ once we fix coordinates. In this talk I'll present joint work in progress with Federico Ardila concerning the defining ideal of the closure. It turns out the combinatorics of this ideal are completely determined by a matroid associated to $L$. We'll compute its degree, universal Gr\"obner basis, and initial ideals - and tell how to read all of this from the matroid!

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