The purpose of this article and of" Resolution of singularities of threefolds in positive
characteristic II" is to prove the theorem of resolution: resolution of singularities holds for …

The notion of singularity is basic to mathematics. In algebraic geometry, the resolution of
singularities by simple algebraic mappings is truly a fundamental problem. It has a complete …

A study of the relation between a noetherian local domain with a given valuation and its
associated graded ring with respect to the valuation, which in some cases is an esentially …

We classify all possible extensions of a valuation from a ground field $ K $ to a rational
function field in one or several variables over $ K $. We determine which value groups and …

We study Abhyankar valuations of excellent equicharacteristic local domains with an
algebraically closed residue field. For zero dimensional valuations we prove that whenever …

We classify Artin–Schreier extensions of valued fields with nontrivial defect according to
whether they are connected with purely inseparable extensions with nontrivial defect, or not …

This work was partially supported by a Canadian NSERC grant and by a sabbatical grant
from the University of Saskatchewan. I thank Bernard Teissier, Olivier Piltant, Michael …

A morphism of algebraic varieties (over a field characteristic 0) is monomial if it can locally
be represented in e'tale neighborhoods by a pure monomial mappings. The book gives …

We give a simple algorithm showing that the reduction of the multiplicity of a characteristic
p> 0 p> 0 hypersurface singularity along a valuation is possible if there is a finite linear …

We study in detail the valuation theory of deeply ramified fields and introduce and
investigate several other related classes of valued fields. Further, a classification of defect …