×

On the problem of resolution of singularities in positive characteristic (Or: a proof we are still waiting for). (English) Zbl 1185.14011

This article is a report on recent work on resolution of singularities of algebraic varieties defined over fields of positive characteristic. It mainly discusses the work of the author on the subject. He is an expert in the field, who for a number of years has been investigating the obstructions to extend the arguments leading to a resolution process in characteristic zero to the general case, and ways to overcome them. Probably the main problem is the lack, in the situation where the base field \(k\) has characteristic \(\mathrm{ch}(k)=p >0\), of hypersurfaces of maximal contact (HMC).
Given a closed subvariety \(X\) of a variety \(W\), smooth over \(k\), a HMC is a hypersurface \(H\) of \(W\) containing the singular locus of \(X\), so that this property is preserved when we blow-up \(W\), using a suitable center, and take strict transforms of \(X\) and \(H\) respectively. They are available (locally, near a singular point \(x\) of \( X\)) if \(\mathrm{ch}(k)=0\) and they allow us, by looking at an induced situation where \(W\) is replaced by \(H\), and using induction on the dimension of the ambient variety, to introduce a useful numerical invariant that “controls” a desingularization process. Indeed, after a blowing-up with an appropriate centre, this invariant decreases. But an HMC may not exist if \(\mathrm{ch}(k)>0\).
The author proposes a “weaker” substitute for these HMC, available in any characteristic, which allows him to introduce a natural analogue of the mentioned invariant in the characterisitc zero case. Unfortunately, this new invariant does not work so well if \(\mathrm{ch}(k) =p >0\). Sometimes, when we blow-up what seems to be a perfectly acceptable centre, there are points \(z\) lying over \(x\), a point of \(X\) of maximum multiplicity, such that the invariant at \(z\) is strictly bigger that that at \(x\). Hauser calls them kangaroo and antelope points respectively (and the singularity \(x \in X\) is called wild).
In the more technical part of the article, he studies necessary conditions for the presence of wild singularities (the Kangaroo Theorem), and ways to modify the basic invariants, trying to control these jumping phenomena. He is able to complete this project if \(X\) is two-dimensional, obtaining a new proof of desingularization for surfaces.
The article is partially expository. The first sections review basic known results, mainly without proofs, and is a good introduction to the subject. Later (more technical) sections include precise definitions and proofs of several results, e.g., the mentioned Kangaroo Theorem. There are numerous interesting examples, and a brief but clear discussion of the ongoing efforts of other geometers in a similar direction (e.g., among others, Hironaka, Kawanoue, Villamayor, who in general use other methods). The concluding comments and the given bibliography should be useful to readers interested in learning more on these topics.

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14B05 Singularities in algebraic geometry
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
14J17 Singularities of surfaces or higher-dimensional varieties
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Shreeram Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic \?\ne 0, Ann. of Math. (2) 63 (1956), 491 – 526. · Zbl 0108.16803
[2] Shreeram S. Abhyankar, Good points of a hypersurface, Adv. in Math. 68 (1988), no. 2, 87 – 256. · Zbl 0657.14008
[3] Dan Abramovich and Frans Oort, Alterations and resolution of singularities, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 39 – 108. · Zbl 0996.14007
[4] Pierre Berthelot, Altérations de variétés algébriques (d’après A. J. de Jong), Astérisque 241 (1997), Exp. No. 815, 5, 273 – 311 (French, with French summary). Séminaire Bourbaki, Vol. 1995/96.
[5] Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207 – 302. · Zbl 0896.14006
[6] Bravo, A., Villamayor, O.: Singularities in positive characteristic, stratification and simplification of the singular locus. · Zbl 1193.14019
[7] Cossart, V.: Sur le polyèdre caractéristique. Thèse d’État. 424 pages. Univ. Paris-Sud, Orsay 1987.
[8] Cossart, V., Jannsen, U., Saito, S.: Canonical embedded and non-embedded resolution of singularities of excellent surfaces.
[9] Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings, J. Algebra 320 (2008), no. 3, 1051 – 1082. · Zbl 1159.14009
[10] Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. II, J. Algebra 321 (2009), no. 7, 1836 – 1976. · Zbl 1173.14012
[11] Steven Dale Cutkosky, Resolution of singularities for 3-folds in positive characteristic, Amer. J. Math. 131 (2009), no. 1, 59 – 127. · Zbl 1170.14011
[12] A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51 – 93. · Zbl 0916.14005
[13] David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. · Zbl 0960.14002
[14] Santiago Encinas and Herwig Hauser, Strong resolution of singularities in characteristic zero, Comment. Math. Helv. 77 (2002), no. 4, 821 – 845. · Zbl 1059.14022
[15] Santiago Encinas and Orlando Villamayor, A course on constructive desingularization and equivariance, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 147 – 227. · Zbl 0969.14007
[16] S. Encinas and O. Villamayor, Good points and constructive resolution of singularities, Acta Math. 181 (1998), no. 1, 109 – 158. · Zbl 0930.14038
[17] Encinas, S., Villamayor, O.: Rees algebras and resolution of singularities. Rev. Mat. Iberoamericana. Proceedings XVI-Coloquio Latinoamericano de Álgebra 2006. · Zbl 1222.14030
[18] Faber, E., Hauser, H.: Today’s menu: Geometry and resolution of singular algebraic surfaces. Bull. Amer. Math. Soc., to appear. · Zbl 1235.14016
[19] Gérardo Gonzalez-Sprinberg, Désingularisation des surfaces par des modifications de Nash normalisées, Astérisque 145-146 (1987), 4, 187 – 207 (French). Séminaire Bourbaki, Vol. 1985/86.
[20] Hauser, H.: Why the characteristic zero proof of resolution fails in positive characteristic. Manuscript 2003, available at www.hh.hauser.cc.
[21] Hauser, H.: Wild singularities and kangaroo points in the resolution of algebraic varieties over fields of positive characteristic. Preprint 2009.
[22] Herwig Hauser, The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand), Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 323 – 403. · Zbl 1030.14007
[23] Herwig Hauser, Excellent surfaces and their taut resolution, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 341 – 373. · Zbl 0979.14007
[24] Herwig Hauser, Three power series techniques, Proc. London Math. Soc. (3) 89 (2004), no. 1, 1 – 24. · Zbl 1065.14019
[25] Herwig Hauser, Seven short stories on blowups and resolutions, Proceedings of Gökova Geometry-Topology Conference 2005, Gökova Geometry/Topology Conference (GGT), Gökova, 2006, pp. 1 – 48. · Zbl 1105.14014
[26] Herwig Hauser, Seventeen obstacles for resolution of singularities, Singularities (Oberwolfach, 1996) Progr. Math., vol. 162, Birkhäuser, Basel, 1998, pp. 289 – 313. · Zbl 0978.14010
[27] Herwig Hauser and Georg Regensburger, Explizite Auflösung von ebenen Kurvensingularitäten in beliebiger Charakteristik, Enseign. Math. (2) 50 (2004), no. 3-4, 305 – 353 (German). · Zbl 1081.14041
[28] Hauser, H., Wagner, D.: Two new invariants for the embedded resolution of surfaces in positive characteristic. Preprint 2009.
[29] Hironaka, H.: Program for resolution of singularities in characteristics \( p > 0\). Notes from lectures at the Clay Mathematics Institute, September 2008.
[30] Hironaka, H.: A program for resolution of singularities, in all characteristics \( p > 0\) and in all dimensions. Lecture Notes ICTP Trieste, June 2006.
[31] Heisuke Hironaka, Theory of infinitely near singular points, J. Korean Math. Soc. 40 (2003), no. 5, 901 – 920. · Zbl 1055.14013
[32] Vincent Cossart, Jean Giraud, and Ulrich Orbanz, Resolution of surface singularities, Lecture Notes in Mathematics, vol. 1101, Springer-Verlag, Berlin, 1984. With an appendix by H. Hironaka. · Zbl 0553.14003
[33] Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109 – 203; ibid. (2) 79 (1964), 205 – 326. · Zbl 0122.38603
[34] Heisuke Hironaka, On Nash blowing-up, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser, Boston, Mass., 1983, pp. 103 – 111. · Zbl 0595.14006
[35] Hiraku Kawanoue, Toward resolution of singularities over a field of positive characteristic. I. Foundation; the language of the idealistic filtration, Publ. Res. Inst. Math. Sci. 43 (2007), no. 3, 819 – 909. · Zbl 1170.14012
[36] János Kollár, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. · Zbl 1113.14013
[37] Joseph Lipman, Desingularization of two-dimensional schemes, Ann. Math. (2) 107 (1978), no. 1, 151 – 207. · Zbl 0349.14004
[38] Joseph Lipman, Introduction to resolution of singularities, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 187 – 230.
[39] Matsuki, K., Kawanoue, H.: Toward resolution of singularities over a field of positive characteristic (The Idealistic Filtration Program) Part II. Basic invariants associated to the idealistic filtration and their properties. · Zbl 1235.14017
[40] T. T. Moh, On a stability theorem for local uniformization in characteristic \?, Publ. Res. Inst. Math. Sci. 23 (1987), no. 6, 965 – 973. · Zbl 0657.14002
[41] S. B. Mulay, Equimultiplicity and hyperplanarity, Proc. Amer. Math. Soc. 89 (1983), no. 3, 407 – 413. · Zbl 0554.14011
[42] R. Narasimhan, Monomial equimultiple curves in positive characteristic, Proc. Amer. Math. Soc. 89 (1983), no. 3, 402 – 406. · Zbl 0554.14010
[43] R. Narasimhan, Hyperplanarity of the equimultiple locus, Proc. Amer. Math. Soc. 87 (1983), no. 3, 403 – 408. · Zbl 0521.13014
[44] Daniel Panazzolo, Resolution of singularities of real-analytic vector fields in dimension three, Acta Math. 197 (2006), no. 2, 167 – 289. · Zbl 1112.37016
[45] A. Seidenberg, Reduction of singularities of the differential equation \?\?\?=\?\?\?, Amer. J. Math. 90 (1968), 248 – 269. · Zbl 0159.33303
[46] Mark Spivakovsky, Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Ann. of Math. (2) 131 (1990), no. 3, 411 – 491. · Zbl 0719.14005
[47] Orlando Villamayor, Constructiveness of Hironaka’s resolution, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1 – 32. · Zbl 0675.14003
[48] Orlando Villamayor U., Hypersurface singularities in positive characteristic, Adv. Math. 213 (2007), no. 2, 687 – 733. · Zbl 1118.14016
[49] Orlando Villamayor U., Elimination with applications to singularities in positive characteristic, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 661 – 697. · Zbl 1210.14017
[50] Jarosław Włodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005), no. 4, 779 – 822. · Zbl 1084.14018
[51] Takehiko Yasuda, Higher Nash blowups, Compos. Math. 143 (2007), no. 6, 1493 – 1510. · Zbl 1135.14011
[52] Oscar Zariski, Reduction of the singularities of algebraic three dimensional varieties, Ann. of Math. (2) 45 (1944), 472 – 542. · Zbl 0063.08361
[53] Zeillinger, D.: Polyederspiele und Auflösen von Singularitäten. PhD Thesis, Universität Innsbruck, 2005.
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.