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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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##### References:
  Cossart, V.: Sur le polyèdre caractéristique d’une singularité. Bull. Soc. Math. F 103, 13–19 (1975) · Zbl 0333.32008  Cossart, V.: Desingularization of embedded excellent surfaces. Tohoku Math. J. II. Ser 33, 25–33 (1981) · Zbl 0472.14019  Cossart, V.: Resolution of surface singularities. In: Lecture Notes in Mathematics, vol. 1101, pp. 79–98. Springer, Berlin (1984)  Cossart, V.: Forme normale d’une fonction sur un k-schéma de dimension 3 and de caractéristique positive. Géométrie algébrique et applications, C. R. 2ieme Conf. int., La Rabida/Espagne, 1984: I: Géométrie and calcul algébrique. Trav. Cours 22, 1–21 (1987)  Cossart, V.: Sur le polyèdre caractéristique. Thèse d’État Orsay, pp. 1–424 (1987)  Cossart, V.: Polyèdre caractéristique et éclatements combinatoires. Rev. Mat. Iberoam. 5(1/2), 67–95 (1989) · Zbl 0708.14010  Cossart, V.: Contact maximal en caractéristique positive and petite multiplicité. Duke Math. J. 63(1), 57–64 (1991) · Zbl 0752.14011  Cossart, V.: Modèle projectif régulier et désingularisation. Math. Ann. 293(1), 115–122 (1992) · Zbl 0735.14011  Cossart, V.: Désingularisation en dimension 3 et caractéristique it p. Proceedings de La Rabida. Progress in mathematics, vol. 134, pp. 1–7. Birkhauser, Boston (1996)  Cossart, V.: Uniformisation et désingularisation des surfaces. dédié à O. Zariski (Hauser, Lipman, Oort, Quiros Éd.). Progress in mathematics, vol 181, pp. 239–258. Birkhauser, Boston (2000) · Zbl 0973.14006  Cossart, V., Janssen, U., Saito, S.: Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes. arXiv:0905.2191  Cossart, V., Piltant, O.: Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings. J. Algebra 320(3), 1051–1082 (2008) · Zbl 1159.14009  Cossart, V., Piltant, O.: Resolution of singularities of threefolds in positive characteristic. II. J. Algebra 321(7), 1836–1976 (2009) · Zbl 1173.14012  Giraud, J.: Étude locale des singularités Cours de 3’eme cycle, Pub. no 26. Univ. d’Orsay (1972)  Giraud, J.: Contact maximal en caractéristique positive. Ann. Scient. Ec Norm. Sup. 4ème série, t.8, pp. 201–234 (1975)  Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic 0, I–II. Ann Math. 109–326 (1964) · Zbl 0122.38603  Hironaka, H.: Characteristic polyhedra of singularities. J. Math. Kyoto Univ. 7(3), 251–293 (1967) · Zbl 0159.50502  Hironaka, H.: Additive groups associated with points of a projective space. Ann. Math 92, 327–334 (1970) · Zbl 0228.14007  Hironaka, H.: Idealistic exponents of singularity. (J.J. Sylvester symposium, John Hopkins University, Baltimore, 1976). John Hopkins University Press, pp. 52–125 (1977) · Zbl 0496.14011  Hironaka, H.: Theory of infinitely near singular points. J. Korean Math. Soc. 40(5), 901–920 (2003) · Zbl 1055.14013  Matsumura, H.: Commutative ring theory. In: Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press (1986) · Zbl 0603.13001
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