Inseparable Local Uniformization

The aim of this paper is to prove that an algebraic variety over a field can be desingularized locally along a valuation after a purely inseparable alteration. Zariski was first to study the problem of desingularizing algebraic varieties along valuations. He called this problem local uniformization of valuations and observed that it should be considered as the local part of the desingularization problem. Zariski established in [Zar1] local uniformization in characteristic zero and deduced in [Zar2] desingularization of threefolds. In positive characteristic, local uniformization of threefolds was proved very recently by Cossart and Piltant, see [CP1] and [CP2], and the case of larger dimensions is open in general (the case of char(k) > dim(X)! is due to Abhyankar). Since the general local uniformization and desingularization problems look very difficult, it is natural to seek for a reasonable weakening which can be proved with concurrent methods. A very successful weakening of the global desingularization problem was discovered by de Jong in [dJ1]: the problem of desingularizing an algebraic variety X by an alteration f : Y → X (i.e. f is a proper generically finite morphism) is much simpler than the classical desingularization, in particular, it can be solved with reasonable efforts. However, this weaker desingularization can replace the classical desingularization in many applications. In some sense, the difference between de Jong’s theorem and classical desingularization is in the control on the extension of the fields of rational functions k(Y )/k(X). de Jong proves that it can be chosen to be separable, while the desingularization conjecture predicts that it can be made trivial. It seems that the only known intermediate result is announced by Gabber and states that one can make n = [k(Y ) : k(X)] to be prime to a prime number l invertible on X , see a survey on Gabber’s work by Illusie, [Ill, Cor. 1,2]. In particular, the following conjecture, which appeared in [AO, 2.9], is absolutely open.