Ram Abhyankar Title: Simultaneous
Surface Resolution
Abstract: Let K/k be a two dimensional algebraic function field
over an algebraically
closed ground field k. Recall that K/k has a minimal
model if it is not ruled: Among all nonsingular projective models of K/k
one is dominated by all others [Ab2]. (Ruled means K is a
simple transcendental extension of a one dimensional
algebraic function field over k.) A finite algebraic field
extension L/K has a simultaneous resolution if there are
nonsingular
projective models V and W of K/k and L/k,
respectively, with W the normalization of V in L.
Given any prime number q prime to the characteristic of K,
[Ab1] considers cyclic extensions L/K of degree q. It
shows that if q is 2 or 3, then there is simultaneous
resolution, whereas if q>3 and K/k is not ruled,
then there is a cyclic L/K of degree q with no
simultaneous resolution. At the September 2003 Galois Theory Conference
in Banff (Canada), Ted Chinberg
asked whether simultaneous resolution was always possible with L/K
Galois
with group a direct sum of any finite number of copies, say m,
of Z/2. This talk will prove no if m=2 and the
characteristic of K is 2 (and K/k has a minimal
model). This also provides
a negative answer to a question of David Harbater: If a positive answer
for two Galois groups implies a positive
answer for their direct sum.
Also the talk will extend the q>3 result to nonprime q
divisible by the square of some prime p. By taking q=4,
this answers a question raised by Ted Chinburg at the March 2006 AMS
Meeting in New Hampshire: Does a Z_2 extension L/J of a
Z_2 extension J/K, have a
simultaneous resolution. Using a Theorem of David Harbater and Florian
Pop,
we generalize our extended result by replacing Z/q by its
direct sum HxZ_q with any finite group H.
[Ab1] S. S. Abhyankar, Simultaneous resolution for algebraic
surfaces, AJM 78 (1956), 761-790.
[Ab2] S. S. Abhyankar, Resolution of Singularities of Embedded
Algebraic Surfaces, Springer Verlag (1998).
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