Simultaneous Surface Resolution in Cyclic Galois Extensions

We show that simultaneous surface resolution is not always possible in a cyclic extension whose degree is greater than three and is not divisible by the characteristic. This answers a recent question of Ted Chinburg. Section 1: Introduction Let K be a two dimensional algebraic function field over an algebraically closed ground field k. Recall that K/k has a minimal model means that amongst all the nonsingular projective models of K/k there is one which is dominated by all others (basic reference [?] or [?]). Also recall that K/k 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 [?]). A finite algebraic field extension L/K is said to have a simultaneous resolution if there exist nonsingular projective models V and W of K/k and L/k, respectively, such that W is the normalization of V in L. Given any positive integer q which is not divisible by the characteristic char(K) of K and letting Zq denote a cyclic group of order q, in [?] it was shown that if q ≤ 3 and L/K is a Zq extension, i.e., a Galois extension whose Galois group is a cyclic group of order q, then it has a simultaneous resolution, whereas if K/k has a minimal model and q > 3 with q being a prime number, then there exists a Zq extension L/K which has no simultaneous resolution. Here we shall extend this second result to those nonprimes q which are 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 to the effect whether every Z2 by Z2 extension L/K, i.e., a Z2 extension L/J of a Z2 extension J/K, has a simultaneous resolution. By using a Theorem of David Harbater and Florian Pop, we generalize our extended result by replacing Zq by its direct sum H ⊕ Zq with any finite group H. For related matter see [?]. In Lemma (2.2) of Section 2 we shall give a consequence of the Harbater-Pop Theorem to be used in proving our generalized extended result. In Lemma (2.1) of Section 2 we shall summarize some technical results form our previous papers [?] and [?]. These technical results deal with the structure of the integral closure of a normal noetherian domain in a cyclic extension. They are used in the proof of Theorem (3.1) of Section 3 which gives a sufficient condition for a two dimensional local domain to be nonregular. Theorem (3.1) is used in proving the special case of Theorem (3.2) of Section 3 which corresponds to our extended result, i.e., the H = 1 case of our generalized extended result. The general case of Theorem (3.2), which corresponds to our generalized extended result, then follows by using Lemma (2.2). 1991 Mathematics Subject Classification. Primary 14A05. 1