Resolving Surface Singularities in Positive Characteristic

Abstract

The paper contains a systematic proof for the embedded resolution of two-dimensional hypersurface singularities over a field of arbitrary characteristic. It relies on the construction of a local upper semicontinuous invariant whose top locus defines at each stage of the resolution process the center of a permissible blowup under which the invariant improves. As the invariant belongs to a well-ordered set, resolution is achieved in finitely many steps. To define the invariant we analyze in detail the obstructions which occur when transcribing the inductive characteristic zero proof à la Hironaka. These difficulties are then overcome by introducing finer invariants than in zero characteristic: They reflect and control typical characteristic $p$ phenomena. This may give new ideas for approaching the characteristic $p$ resolution in higher dimensions.