2

1. Introduction

a modern treatment of this proof in Chapter 8. A study of this proof is

surely the best introduction to the methods and ideas underlying the recent

emphasis on valuation-theoretic methods in resolution problems.

The existence of a resolution of singularities has been completely solved

by Hironaka [52], in all dimensions, when K has characteristic zero. We give

a simplified proof of this theorem, based on the proof of canonical resolution

by Encinas and Villamayor ([40], [41]), in Chapter 6.

When K has positive characteristic, resolution is known for curves, sur-

faces and 3-folds (with char(if) 5). The first proof in positive characteris-

tic of resolution of surfaces and of resolution for 3-folds is due to Abhyankar

[1], [4].

We give several proofs, in Chapters 2, 3 and 4, of resolution of curves

in arbitrary characteristic. In Chapter 5 we give a proof of resolution of

surfaces in arbitrary characteristic.

1.1. Notation

The notation of Hartshorne [47] will be followed, with the following differ-

ences and additions.

By a variety over a field K (or a K-variety), we will mean an open subset

of an equidimensional reduced subscheme of the projective space P^. Thus

an integral variety is a "quasi-projective variety" in the classical sense. A

curve is a one-dimensional variety. A surface is a two-dimensional variety,

and a 3-fold is a three-dimensional variety. A subvariety Y of a variety X

is a closed subscheme of X which is a variety.

An affine ring is a reduced ring which is of finite type over a field K.

If X is a variety, and X is an ideal sheaf on X, we denote V{X) =

spec (Ox /X) C X. If Y is a subscheme of a variety X, we denote the ideal

of Y in X by Xy. If Wi, Wi are subschemes of a variety of X, we will denote

the scheme-theoretic intersection of W\ and W2 by W\ • W2. This is the

subscheme

W1W2 = V{XWl +XW2) C W.

A hypersurface is a codimension one subvariety of a non-singular variety.