From: Brenda Sue Cook (brenda@math.missouri.edu)
Date: Fri Apr 11 2008 - 08:10:32 CDT
Ph.D. DEFENSE
Department of Mathematics
Samar El Hitti
(University of Missouri)
Algebraic Resolution of Formal Ideals along a Valuation
Abstract: Let X be a possibly singular complete algebraic variety,
defined over a field k of characteristic zero. X is nonsingular at p
\in X if O_{X,p} is a regular local ring. The problem of resolution
of singularities is to show that there exists a nonsingular complete
variety X, which birationally dominates X. Resolution of
singularities (in characteristic zero) was proven by Hironaka in
1964. Let v be a valuation of the function field of X, v dominates a
unique point p, on any complete variety Y, which birationally
dominates X. The problem of local uniformization is to show that,
given a valuation v of the function field of X, there exists a
complete variety Y, which birationally dominates X such that the
center of v on Y, is a regular local ring. Zariski proved local
uniformization (in characteristic zero) in 1944. His proof gives a
very detailed analysis of rank 1 valuations, and produces a
resolution which reflects invariants of the valuation.
We extend these methods to higher rank in our thesis to give a proof
of local uniformization which reflects important properties of the
valuation. We simultaneously resolve the centers of all the composite
valuations, and resolve certain formal ideals associated to the
valuation.
Dissertation Advisor: S. Dale Cutkosky
Friday, April 18
2:30 p.m.
312 Math Sciences
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