# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [1] Cossart, V.: Sur le polyèdre caractéristique d’une singularité. Bull. Soc. Math. F 103, 13–19 (1975) · Zbl 0333.32008 [2] Cossart, V.: Desingularization of embedded excellent surfaces. Tohoku Math. J. II. Ser 33, 25–33 (1981) · Zbl 0472.14019 [3] Cossart, V.: Resolution of surface singularities. In: Lecture Notes in Mathematics, vol. 1101, pp. 79–98. Springer, Berlin (1984) [4] 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) [5] Cossart, V.: Sur le polyèdre caractéristique. Thèse d’État Orsay, pp. 1–424 (1987) [6] Cossart, V.: Polyèdre caractéristique et éclatements combinatoires. Rev. Mat. Iberoam. 5(1/2), 67–95 (1989) · Zbl 0708.14010 [7] Cossart, V.: Contact maximal en caractéristique positive and petite multiplicité. Duke Math. J. 63(1), 57–64 (1991) · Zbl 0752.14011 [8] Cossart, V.: Modèle projectif régulier et désingularisation. Math. Ann. 293(1), 115–122 (1992) · Zbl 0735.14011 [9] 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) [10] 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 [11] Cossart, V., Janssen, U., Saito, S.: Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes. arXiv:0905.2191 [12] 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 [13] Cossart, V., Piltant, O.: Resolution of singularities of threefolds in positive characteristic. II. J. Algebra 321(7), 1836–1976 (2009) · Zbl 1173.14012 [14] Giraud, J.: Étude locale des singularités Cours de 3’eme cycle, Pub. no 26. Univ. d’Orsay (1972) [15] Giraud, J.: Contact maximal en caractéristique positive. Ann. Scient. Ec Norm. Sup. 4ème série, t.8, pp. 201–234 (1975) [16] 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 [17] Hironaka, H.: Characteristic polyhedra of singularities. J. Math. Kyoto Univ. 7(3), 251–293 (1967) · Zbl 0159.50502 [18] Hironaka, H.: Additive groups associated with points of a projective space. Ann. Math 92, 327–334 (1970) · Zbl 0228.14007 [19] 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 [20] Hironaka, H.: Theory of infinitely near singular points. J. Korean Math. Soc. 40(5), 901–920 (2003) · Zbl 1055.14013 [21] Matsumura, H.: Commutative ring theory. In: Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press (1986) · Zbl 0603.13001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.