On the Problem of Resolution of Singularities in Positive Characteristic

Introduction. The embedded resolution of singular algebraic varieties of dimension > 3 defined over fields of characteristic p > 0 is still an open problem. The inductive argument which works in characteristic zero fails for positive characteristic. The main obstruction is the failure of maximal contact, which, in turn, manifests in the occurence of wild singularities and kangaroo points at certain stages of a sequence of blowups. At these points the standard characteristic zero resolution invariant increases instead of decreasing. The induction breaks down. No remedy has been found yet.