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Resolution of singularities for 3-folds in positive characteristic. (English) Zbl 1170.14011

The author gives a concise, complete proof of resolution of singularities of 3-folds in characteristic \(> 5\) using the same main steps as Abhyankar’s original (508 pages long) proof of this theorem from the mid-1960s. More precisely, Cutkosky breaks the proof down into the following 5 steps to emphasize the overall structure:
1. proof of embedded resolution of surface singularities and principalization of ideals
2. proof of birational equivalence of a projective variety of dimension \(n\) to an appropriate normal one which does not contain points of multiplicity \(>n!\)
3. proof of local resolution of points of multiplicity not exceeding the characteristic of the ground field
4. patching of local resolutions to produce a nonsingular projective variety which is birationally equivalent to the original one
5. change of the resolution obtained in steps 1-4 to also satisfy the condition that the resolution should be an isomorphism outside the singular locus.
To allow the readers to become familiar with the main constructions and ideas before entering too deep into the technical details, very clear and compact outlines of the proofs of embedded resolution and of the constructions of steps 2-5 precede the complete proofs thereof, which makes the article accessible also to algebraic geometers who are not specialists in the desingularization.

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14B05 Singularities in algebraic geometry
32S45 Modifications; resolution of singularities (complex-analytic aspects)
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