## A simpler proof of toroidalization of morphisms from 3-folds to surfaces.(English. French summary)Zbl 1282.14029

Let $$k$$ be an algebraically closed field of characteristic zero. Let $$\Phi :X\to Y$$ be a dominant morphism of non-singular algebraic varieties over $$k$$. Roughly speaking $$\Phi$$ is called toroidal if and only if for each closed point $$x\in X$$ the formal neighborhood of $$X$$ at $$x$$ and the formal neighborhood of $$Y$$ at $$\Phi(x)$$ are isomorphic to formal neighborhoods of toric varieties and the morphism induced by $$\Phi$$ between these toric varieties is a toric morphism (i.e. this morphism is given by monomials in toric coordinates).
In this paper the author gives a new proof of the following result (proved previously in [S. D. Cutkosky, Monomialization of morphisms from 3-folds to surfaces. Berlin: Springer (2002; Zbl 1057.14009)]):
Let $$\Phi : X\to Y$$ be a dominant morphism from a threefold to a surface. Then there exist $$\pi : X_1\to X$$ and $$\rho : Y_1\to Y$$, compositions of blow-ups with non-singular centers, and a toroidal morphism $$f :X_1\to Y_1$$ such that $$\Phi\circ\pi=\rho\circ f$$.
The proof of Cutkosky given in [loc. cit.] consists of two steps: the first one consists to replace $$\Phi$$ by a “strongly prepared” morphism, the second one consists to prove that strongly prepared morphism can be toroidalized. The aim of the paper under review is to give a shorter proof of this first step (it takes 170 pages in [loc. cit.]) with the hope that this one may be generalized to higher dimension (the second step is generalized to higher dimension in [S. D. Cutkosky and O. Kashcheyeva, J. Algebra 275, No. 1, 275–320 (2004; Zbl 1057.14008)]).

### MSC:

 1.4e+100 Birational geometry 1.4e+16 Global theory and resolution of singularities (algebro-geometric aspects)

### Keywords:

morphism; toroidalization; monomialization

### Citations:

Zbl 1057.14009; Zbl 1057.14008
Full Text:

### References:

 [1] Abhyankar, S., Local uniformization on algebraic surfaces over ground fields of characteristic $$p\ne 0$$, Annals of Math, 63, 491-526 (1956) · Zbl 0108.16803 · doi:10.2307/1970014 [2] Abhyankar, S., Resolution of singularities of embedded algebraic surfaces (1998) · Zbl 0914.14006 [3] Benito, A.; Villamayor, O., Monoidal transforms and invariants of singularities in positive characteristic · Zbl 1278.14019 [4] Bierstone, E.; Millman, P., Canonical desingularization in characteristic zero by blowing up the maximal strata of a local invariant, Inv. Math, 128, 207-302 (1997) · Zbl 0896.14006 · doi:10.1007/s002220050141 [5] Bravo, A.; Encinas, S.; Villamayor, O., A simplified proof of desingularization and applications · Zbl 1086.14012 [6] Cano, F., Reduction of the singularities of codimension one singular foliations in dimension three, Ann. of Math., 160, 907-1011 (2004) · Zbl 1088.32019 · doi:10.4007/annals.2004.160.907 [7] Cossart, V., Desingularization of Embedded Excellent Surfaces, Tohoku Math. Journ., 33, 25-33 (1981) · Zbl 0472.14019 · doi:10.2748/tmj/1178229492 [8] Cossart, V.; Piltant, O., Resolution of singularities of threefolds in positive characteristic I, Journal of Algebra, 320, 1051-1082 (2008) · Zbl 1159.14009 · doi:10.1016/j.jalgebra.2008.03.032 [9] Cossart, V.; Piltant, O., Resolution of singularities of threefolds in positive characteristic II, Journal of Algebra, 321, 1336-1976 (2009) · Zbl 1173.14012 · doi:10.1016/j.jalgebra.2008.11.030 [10] Cutkosky, S. D., Local monomialization and factorization of Morphisms, 260 (1999) · Zbl 0941.14001 [11] Cutkosky, S. D., Monomialization of morphisms from 3-folds to surfaces, 1786 (2002) · Zbl 1057.14009 [12] Cutkosky, S. D., Resolution of Singularities (2004) · Zbl 1076.14005 [13] Cutkosky, S. D., Local monomialization of trancendental extensions, Annales de L’Institut Fourier, 55, 1517-1586 (2005) · Zbl 1081.14020 · doi:10.5802/aif.2132 [14] Cutkosky, S. D., Toroidalization of birational morphisms of 3-folds, 190, 890 (2007) · Zbl 1133.14013 [15] Cutkosky, S. D., Resolution of Singularities for 3-Folds in Positive Characteristic, American Journal of Math., 131, 59-128 (2009) · Zbl 1170.14011 · doi:10.1353/ajm.0.0036 [16] Cutkosky, S. D.; Kascheyeva, O., Monomialization of strongly prepared morphisms from nonsingular $$n$$-folds to surfaces, J. Algebra, 275, 275-320 (2004) · Zbl 1057.14008 · doi:10.1016/S0021-8693(03)00366-1 [17] Cutkosky, S. D.; Piltant, O., Monomial resolutions of morphisms of algebraic surfaces, Communications in Algebra, 28, 5935-5960 (2000) · Zbl 1003.14004 · doi:10.1080/00927870008827198 [18] de Jong, A. J., Smoothness, semistability and Alterations, Publ. Math. I.H.E.S., 83, 51-93 (1996) · Zbl 0916.14005 · doi:10.1007/BF02698644 [19] Encinas, S.; Hauser, H., Strong resolution of singularities in characteristic zero, Comment Math. Helv., 77, 821-845 (2002) · Zbl 1059.14022 · doi:10.1007/PL00012443 [20] Hauser, H., Kangaroo Points and Oblique Polynomials in Resolution of Positive Characteristic [21] Hauser, H., Resolution of Singularities, (Obergurgl, 1997), 181, 341-373 (2000) · Zbl 0979.14007 [22] Hauser, H., The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand), Bull. Amer. Math. Soc., 40, 323-348 (2003) · Zbl 1030.14007 · doi:10.1090/S0273-0979-03-00982-0 [23] Hauser, H., On the problem of resolution of singularites in positive characteristic (Or: a proof we are still waiting for), Bull. Amer. Math. Soc., 47, 1-30 (2010) · Zbl 1185.14011 · doi:10.1090/S0273-0979-09-01274-9 [24] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, Annals of Math, 79, 109-326 (1964) · Zbl 0122.38603 · doi:10.2307/1970486 [25] Hironaka, H., Cossart V., Giraud J. and Orbanz U., Resolution of singularities, 1101 (1980) · Zbl 0122.38602 [26] Hironaka, H., A program for resolution of singularities, in all characteristics $$p>0$$ and in all dimensions (200620082008) [27] Knaf, H.; Kuhlmann, F.-V., Every place admits local uniformization in a finite extension of the function field, Advances in Math., 221, 428-453 (2009) · Zbl 1221.14016 · doi:10.1016/j.aim.2008.12.009 [28] Panazzolo, D., Resolution of singularities of real-analytic vector fields in dimension three, Acta Math., 197, 167-289 (2006) · Zbl 1112.37016 · doi:10.1007/s11511-006-0011-7 [29] Rond, G., Homomorphisms of local algebras in positive characteristic, J. Algebra, 322, 4382-4407 (2009) · Zbl 1189.13022 · doi:10.1016/j.jalgebra.2009.09.011 [30] Seidenberg, A., Reduction of the singlarities of the differential equation $$Ady=Bdx$$, Amer. J. Math., 90, 248-269 (1968) · Zbl 0159.33303 · doi:10.2307/2373435 [31] Teissier, B.; Franz-Viktor Kuhlmann, Salma Kuhlmann; Marshall, Murray, Valuation Theory and its Applications II · Zbl 1061.14016
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.