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Resolution of singularities of threefolds in mixed characteristic: case of small multiplicity. (English) Zbl 1308.13004
From the authors’ introduction: “This article is part of the authors’ program whose purpose is to prove the following conjecture on Resolution of Singularities of threefolds in mixed characteristic. The conjecture is a special case of Grothendieck’s Resolution conjecture for quasi-excellent schemes.
Conjecture 1.1 Let \(C\) be an integral regular excellent curve with function field \(F\). Let \(S/F\) be a reduced algebraic projective surface and \(\mathcal X\) be a flat projective \(C\)-scheme with generic fiber \({\mathcal X}_F =S\). There exists a birational projective \(C\)-morphism \(\pi :{\mathcal Y} \to \mathcal X\) such that
(i)
\({\mathcal Y}\) is everywhere regular.
(ii)
\(\pi^{-1}(\text{Reg} {\mathcal X}) \to \text{Reg} {\mathcal X} \) is an isomorphism.”
In a previous paper [J. Algebra 320, No. 3, 1051–1082 (2008; Zbl 1159.14009)], the authors have developed equicharacteristic techniques which extend to that situation. Using classical invariants introduced by Hironaka, they present a proof of the following:
Main Theorem 1.3. Let \((R, {\mathcal M}, k=k(x):= R/{\mathcal M} )\) be an excellent regular local ring of dimension four, \((Z,x):= (\text{Spec} R,{\mathcal M})\) and \((X,x):= (\text{Spec} R/(h),x)\) be a reduced hypersurface. Assume that the multiplicity \(m(x)\) of \((X,x)\) satisfies \(m(x) < p:=\text{char} k(x)\). Let \(v\) be a valuation of \(K(X)\) centered at \(x\). Then there exists a finite sequence of local blowing ups \[ (X,x)=:(X_0,x_0) \leftarrow (X_1,x_1) \leftarrow \dots \leftarrow (X_n,x_n) , \] where \(x_i\in X_i\), \(0\leq i \leq n\) is the center of \(v\), each blowing up center \(Y_i\subset X_i\) is permissible at \(x_i\) (in Hironaka’s sense), such that \(x_n\) is regular.
The authors point out that the methods applied are global in nature and thus an extension of the main theorem to a global version should be possible.

MSC:
13A18 Valuations and their generalizations for commutative rings
14B05 Singularities in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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