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