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Resolution of singularities of threefolds in positive characteristic. I: Reduction to local uniformization on Artin-Schreier and purely inseparable coverings. (English) Zbl 1159.14009

A field \(k\) (of positive characteristic \(p\)) is differentiably finite over a perfect subfield \(k_0\), or a DF-field, if \(\Omega ^1 _{k/{k_0}}\) is a finite dimensional \(k\)-vector space. For instance, a function field over a perfect field is a DF-field.
The authors recently proved resolution of singularities for a quasi-projective algebraic \(X\) variety of dimension three (a 3-fold) over a DF-field \(k\) (in the sense that there is a projective morphism \(\pi : {\tilde X} \to X\), inducing an isomorphism off \(S\), the singular locus of \(X\), such that \(\pi ^{-1}(S)\) is a divisor in \({\tilde X}\) with strict normal crossings). This theorem improves previously known results on resolution of 3-folds in characteristic \(p\), which required stronger assumptions on the base field.
In their proof the authors use, following a strategy proposed by Zariski, and successfully followed by himself in characteristic zero, valuation-theoretic methods: prove local uniformization first and then by patching deduce the theorem. This proof is presented in two parts. In the present paper they show how, for a 3-fold over any field of characteristic \(p > 0\), desingularization follows from a special uniformization result. Namely, essentially it says that if one can prove local uniformization for valuations \(W \) which dominate the local ring of an Artin-Schreier (A-S) or a purely inseparable (p.i.) singularity, then desingularization is available. An A-S (resp. p.i) singularity is one of the form \(S={\text{Spec}} ((R[X]/(h))_{(X,M)})\), where \((R,M)\) is a local, three dimensional regular local ring, essentially of finite type over a field \(k\) of characteristic \(p > 0\), \(X\) an indeterminate, \(h=X^p - g ^{p-1}+f\), with \(f\) and \(g\) in \(M\), \(g \not=0\) (resp. \(f \in M\) and \(g=0\)). In another paper they show that when \(k\) is a DF-field, this type of local uniformization is always possible, concluding the proof of their Main Theorem (see http://hal.archives-ouvertes.fr/hal-00139445).
In the article being reviewed, the authors use refinements of the original valuation-theoretic approach of Zariski and ramification methods pioneered by Abhyankar, as well as a variety of other algebraic and geometric techniques. As usually in this kind of work, parts of the article are highly technical. However, it is very well written and relatively self-contained.

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
14J30 \(3\)-folds
14E22 Ramification problems in algebraic geometry
14E05 Rational and birational maps
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References:

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