Title: Valued function fields and the defect
I will give a survey on the role that the valuation theretical phenomenon
of "defect" (or "ramification deficiency") plays in the structure theory
of valued function fields in positive characteristic. The defect is the
main enemy that we meet when we deal with the following two deep problems
in positive (or mixed) characteristic:
- the model theory of valued fields, in particular that of Laurent series
fields over finite fields,
- local uniformization (the local form of resolution of singularities.
Even if resolution of singularities in positive characteristic is proved
soon (as several experts now believe), the model theoretic problem is not
solved. We will need to know much more about the defect and the structure
of valued function fields in several variables.
A question connected with both problems is: under which conditions is a
ground field existentially closed in a function field? Is the existence of
a rational place sufficient? It is if the ground field is perfect and
large. We do not know whether the "perfect" condition can be dropped. I
will discuss some background of this question.