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.
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