Wild Singularities and Kangaroo Points for the Resolution of Singularities in Positive Characteristic

Wild singularities are the main obstacle for applying the characteristic zero proof of resolution of singularities to the case of characteristic p > 0. They cause at certain points of the exceptional divisor of blowups, so called kangaroo points, the increase of the resolution invariant. This increase destroys the induction argument. In the article, we determine the structure of wild singularities and the occurrence of kangaroo points. A complete characterization of both phenomena is given. This will explain, in particular, why the resolution of curves has no characteristic p obstruction, and how, for surfaces, the obstruction can be handled elegantly via a new resolution invariant. Also, we indicate strategies based on the understanding of wild singularities for attacking the general resolution problem.