Franz-Viktor Kuhlmann
Title: Local Uniformization and the Defect
Abstract:
The problem of local uniformization (the local form of resolution of
singularities) can be reformulated as a problem about the structure of
valued function fields. I will quickly sketch this reformulation.
Local uniformization was proved by Zariski in 1940 for all algebraic
varieties over ground fields of characteristic 0.
One part of the problem is elimination of ramification. Zariski
implicitly eliminated tame ramification. The task is harder in
positive characteristic because there, you also have to deal with wild
ramification, and the main obstacle turns out to be the defect of
valued field extensions. I will give examples of non-trivial defect.
Then I will show why certain valuations (called "Abhyankar
valuations") always admit local uniformization, and describe the
valuation theoretical theorems used for the solution. The crucial
theorem used here is the so-called Generalized Stability Theorem.
A second theorem, the Henselian Rationaity Theorem, was then used to
extend the results in order to prove Local Uniformization by
alteration for all valuations on arbitrary algebraic varieties. This
follows in principle from de Jong's Resolution by Alteration, but our
proof is purely valuation theoretical, and it provides more precise
information on the alteration we have to take into account.
The Henselian Rationaity Theorem has been proven over tame ground
fields, which I will define in my presentation.
In order to generalize these theorems, we have to learn more about the
defect. I will describe a classification of defects. It has been
observed that one of the two types of defects seems to be more
harmless than the other (one indication being Temkin's "Inseparable
Local Uniformization"). Recently, it has been shown that deeply
ramified fields only admit these more harmless defects, so the hope is
that results proven for tame fields can be generalized to deeply
ramified fields.