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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