Let X be an algebraic variety over a field k. The objective of the program of resolution of singularities is to find a birational projective morphism Y → X where Y is smooth. A much sharper version of the following conjecture was proved by Hironaka in his celebrated paper [10] for the case k is algebraically closed of characteristic zero. There have been many attempts to prove (at least) the following weaker statement in other characteristics.
Conjecture 9.1 (Weak Hironaka). Let X be an algebraic variety and Z a closed subvariety. There is a projective birational morphism f : Y → X so that Y is smooth and f1(Z) is a divisor with normal crossings.
Recent work by de Jong [11]has led to some results in this direction. The author has given a much simplified and more geometric proof of the above result in characteristic zero based on fundamental ideas of Bogomolov and Pantev [3]. Moreover, their work has been extended to non-zero characteristic so that the resolution of singularities even in that case has been reduced to the problem of resolution in the following context:
Conjecture 9.2 (Resolution of “Toric” singularities). The Weak Hironaka result is true when X is a branched covering over a smooth variety W, with branch locus D a divisor with normal crossings in W.
In characteristic zero one can show that this case is identical to the toroidal embeddings studied and resolved by Mumford, Kempf and others [12]. In non-zero characteristic this case is known to be difficult due the the fact that the fundamental group of the affine line being non-trivial.