Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes

TL;DR

This paper establishes a canonical, functorial resolution of singularities for excellent two-dimensional schemes, including embedded and non-embedded cases, using stratifications and characteristic polyhedra, extending to non-noetherian schemes, algebraic spaces, and stacks.

Contribution

It provides the first canonical resolution method for all excellent two-dimensional schemes, including non-noetherian cases, and relates embedded and non-embedded resolutions.

Findings

Resolution exists via permissible blow-ups

The method is functorial and canonical

Standard characteristic zero techniques fail in positive characteristic

Abstract

We prove the existence of resolution of singularities for arbitrary (not necessarily reduced or irreducible) excellent two-dimensional schemes, via permissible blow-ups. The resolution is canonical, and functorial with respect to automorphisms or etale or Zariski localizations. We treat the embedded case as well as the non-embedded case, with or without a boundary, and we relate the diferent versions. In the non-embedded case, a boundary is a collection of locally principal closed subschemes. Our main tools are the stratifications by Hilbert-Samuel functions and the characteristic polyhedra introduced by H. Hironaka. In an appendix we show that the standard method used in characteristic zero - the theory of maximal contact - does not work for surfaces in positive characteristic (the counterexamples are hypersurfaces in affine threespace and work over any field of positive characteristic). In this new version, we treat the case of locally noetherian but not necessarily noetherian schemes in an appropriate way. Here one does not have a finite resolution sequence, but still a canonical resolution morphism by glueing. The same techniques allow to treat algebraic spaces and stacks.