From: Brenda Sue Frazier (brenda@math.missouri.edu)
Date: Mon Mar 19 2007 - 07:13:23 CDT
Special Algebra Seminar
Professor Yasuyuki Kachi, University of Kansas
Structure theory of birational transforms.
I focus on one open problem in the theory of global
meromorphic functions - the rationality (= the generic
parametrizability) criterion of compact manifolds/domains.
After a quick review on 1-dimensional theory (Riemann
surfaces), I explain a partial solution
to the problem by Kobayashi-Ochiai (in any dimension)
(and my own humble generalization to char = p > 0).
The difficulty of the original problem seems to have to do
with our incomplete knowledge on the key underlying concept
- birational/bimeromorphic equivalence. I introduce a new
group functor SG_n with an action on the space S_n of
uniformizing parameters (roughly the set of regular local
subrings of the universal coefficient ring HVR_n (k)) which
yields local birational transforms.
I emphasize that this last object is worthy enough to study
in its own right. Indeed, as it turns out, it enables us
to view the following two outstanding theories in algebraic
geometry in one context, at least locally:
1. Hironaka's theory on resolution of singularities,
2. Cutkosky's theory on factoring birational mappings.
312 Mathematical Sciences
4:00-4:50 p.m.
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