Contemporary Mathematics
Volume 390, 2005
Simultaneous Surface Resolution in Quadratic and
Biquadratic Galois Extensions
Shreeram S. Abhyankar and Manish Kumar
ABSTRACT. We show that simultaneous surface resolution is always possible
in a quadratic extension, and if the characteristic is different from two then
in every compositum of such extensions. We also construct examples to show
that the latter is not always possible if the characteristic is two.
1.
Introduction
Let
K
be a two dimensional algebraic function field over an algebraically closed
ground field k. Recall that Klk has a minimal model means that amongst all
the nonsingular projective models of Klk there is one which is dominated by all
others (basic reference
[A09]).
Also recall that Klk has a minimal model if and
only if it is not a ruled function field, i.e., K is not a simple transcendental field
extension of a one dimensional algebraic function field over
k
(see
[Z02]).
A finite
algebraic field extension
L
I
K
is said to have a simultaneous resolution
if
there exist
nonsingular projective models V and W of Klk and Llk, respectively, such that
W is the normalization of V in L. Given any prime number q
-#
char(K), where
char denotes characteristic, in
[A02]
it was shown that if
q
~
3 and Ll K is a cyclic
Galois extension of degree
q
then it has a simultaneous resolution, whereas if
q
3
and K
I
k has a minimal model then there exists a cyclic Galois extension L
I
K
of degree
q
which has no simultaneous resolution. At the September 2003 Galois
Theory Conference in Banff (Canada), Ted Chinberg asked whether simultaneous
resolution was always possible if Ll K was Galois with Galois group a direct sum of
any finite number of copies, say m, of a cyclic group Z2 of order 2. The purpose of
this note is to prove yes if either char(K)-# 2 or m
=
1, and no if char(K)
=
2
=
m
and K
I
k has a minimal model. This also provides a negative answer to the question
which David Harbater raised at that conference and which asks if a positive answer
for two Galois groups implies a positive answer for their direct sum. It may be
noted that our yes answer remains valid also in the arithmetic case and in fact for
surfaces over any excellent domain.
2000 Mathematics Subject Classification. Primary 12F10, 14H30; Secondary 20006, 20E22.
Abhyankar's work was partly supported by NSF Grant DMS 99-88166 and NSA Grant MDA
904-99-1-0019.
©
2005 American Mathematical Society
1
http://dx.doi.org/10.1090/conm/390/07289
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