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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