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Resolution of singularities of threefolds in positive characteristic. II. (English) Zbl 1173.14012

This article completes the proof of the following theorem. If \(k\) is a field of positive characteristic, which is an extension of a perfect field \(k_0\), so that \(\Omega ^1 _{k/{k_0}}\) is a finite dimensional \(k\)-vector space, then for any quasi-projective algebraic variety \(Z\) of dimension three (a 3-fold) over \(k\) there is a projective morphism \(\pi :{\tilde Z} \to Z\), with \(\tilde Z\) regular, inducing an isomorphism off \(S\) (the singular locus of \(X\)) such that \(\pi ^{-1}(S)\) is a divisor in \({\tilde Z}\) with strict normal crossings.
In a previous article [J. Algebra 320, 1051–1082 (2008; Zbl 1159.14009)], the authors showed that the given theorem follows from the following: result: if \(k\) is as above, \(S\) is a three-dimensional local \(k\)-algebra essentially of finite type (with maximal ideal \(M\)), \(R'\) either an Artin-Schreier or purely inseparable singularity over \(S\), and \(L\) the fraction field of \(R'\), then any valuation \(\mu\) of \(L\) dominating \(R'\) (satisfying certain conditions) admits a local uniformization. Here, \(R'\) being an Artin-Scheier (resp. purely inseparable) singularity over \(S\) means: \(S\) is a ring of the form \(S= \text{Spec} ((S[X]/(h))_{(X,M)}\), \(X\) an indeterminate, and \(h=X^p - g ^{p-1}+f\), with \(f\) and \(g\) in \(M\), \(g \not=0\) (resp. \(f \in M\) and \(g=0\)). The present paper proves this result on uniformization.
With \(K_0\) the function field of \(X_0:= {\text{Spec}} (R)\), after some preliminary simplifications, one is reduced (using results of Hironaka and Giraud) to showing the existence of a local hypersurface model \((X',x')\) of \(K_0\) such that \(\mu\) is centred on it and \(x'\) is not in the locus of multiplicity \(p\) (the “local uniformization problem for \(\mu\)”). The authors construct these models by means of successive blowing-ups of \(X_0\) with suitable centres. The main invariant used to control the process is (for \(x \in X_0\)) \(\iota (x) = (\omega (x), \omega ' (x))\), where \(\omega (x)\) is a positive integer whose definition involves certain “jacobian ideals”, obtained by using suitable associated characteristic polyhedra (introduced by Hironaka); the values of \(\omega ' (x)\) are the numbers 1,2 or 3.
This leads to the consideration of several cases. The cases where \(\omega (x) =0\), or \(\omega (x) > 0\) but \(\omega '(x)\) takes the value 1 or 3 are relatively simple. The case where \(\omega '(x) = 2\) is harder, and again breaks down into several cases depending on the value of an auxiliary invariant \(\kappa (x)\).
The whole proof is very involved, but the reader’s task is facilitated by the inclusion of a preliminary informal discussion which gives an idea about the general strategy.

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
14J30 \(3\)-folds
14E05 Rational and birational maps
13A18 Valuations and their generalizations for commutative rings

Citations:

Zbl 1159.14009
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References:

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