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On the problem of resolution of singularities in positive characteristic (Or: A proof we are still waiting for)


Author: Herwig Hauser
Journal: Bull. Amer. Math. Soc. 47 (2010), 1-30
MSC (2010): Primary 14B05, 14E15, 12D10
DOI: https://doi.org/10.1090/S0273-0979-09-01274-9
Published electronically: October 28, 2009
MathSciNet review: 2566444
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Abstract: Assume that, in the near future, someone can prove resolution of singularities in arbitrary characteristic and dimension. Then one may want to know why the case of positive characteristic is so much harder than the classical characteristic zero case. Our intention here is to provide this piece of information for people who are not necessarily working in the field. A singularity of an algebraic variety in positive characteristic is called wild if the resolution invariant from characteristic zero, defined suitably without reference to hypersurfaces of maximal contact, increases under blowup when passing to the transformed singularity at a selected point of the exceptional divisor (a so called kangaroo point). This phenomenon represents one of the main obstructions for the still unsolved problem of resolution in positive characteristic. In the present article, we will try to understand it.


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  • [Ab1] Abhyankar, S.: Local uniformization of algebraic surfaces over ground fields of characteristic $ p$. Ann. Math. 63 (1956), 491-526. MR 0078017 (17:1134d)
  • [Ab2] Abhyankar, S.: Good points of a hypersurface. Adv. Math. 68 (1988), 87-256. MR 934366 (89e:14012)
  • [AO] Abramovich, D., Oort, F.: Alterations and resolution of singularities. In: Resolution of Singularities (Obergurgl, 1997, ed. H. Hauser et al.) Progress in Math. 181, Birkhäuser 2000, pp. 39-108. MR 1748617 (2001h:14009)
  • [Be] Berthelot, P.: Altérations de variétés algébriques (d'après A.J. de Jong). Sém. Bourbaki 1995-96, exp. 815. Astérisque 241 (1997), 273-311. MR 1472543 (98m:14021)
  • [BM] Bierstone, E., Milman, P.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128 (1997), 207-302. MR 1440306 (98e:14010)
  • [BV] Bravo, A., Villamayor, O.: Singularities in positive characteristic, stratification and simplification of the singular locus. arXiv:0807.4308
  • [Co] Cossart, V.: Sur le polyèdre caractéristique. Thèse d'État. 424 pages. Univ. Paris-Sud, Orsay 1987.
  • [CJS] Cossart, V., Jannsen, U., Saito, S.: Canonical embedded and non-embedded resolution of singularities of excellent surfaces. arXiv:0905.2191
  • [CP1] Cossart, V., Piltant, O.: Resolution of singularities of three-folds in positive characteristic I. J. Algebra 320 (2008) 1051-1082. MR 2427629 (2009f:14024)
  • [CP2] Cossart, V., Piltant, O.: Resolution of singularities of three-folds in positive characteristic II. J. Algebra (2009), to appear. MR 2494751
  • [Cu] Cutkosky, Resolution of singularities for 3-folds in positive characteristic. Amer. J. Math. 131 (2009), 59-127. MR 2488485
  • [dJ] de Jong, J.: Smoothness, semi-stability and alterations. Publ. Math. IHES 83 (1996), 51-93. MR 1423020 (98e:14011)
  • [EiH] Eisenbud, D., Harris, J.: The geometry of schemes. Graduate Texts in Mathematics. Springer 2000. MR 1730819 (2001d:14002)
  • [EH] Encinas, S., Hauser, H.: Strong resolution of singularities in characteristic zero. Comment. Math. Helv. 77 (2002), 421-445. MR 1949115 (2004c:14021)
  • [EV1] Encinas, S., Villamayor, O.: A course on constructive desingularization and equivariance. In: Resolution of singularities (Obergurgl, 1997, ed. H. Hauser et al.) Progress in Math. 181, Birkhäuser 2000. MR 1748620 (2001g:14018)
  • [EV2] Encinas, S., Villamayor, O.: Good points and constructive resolution of singularities. Acta Math. 181 (1998), 109-158. MR 1654779 (99i:14020)
  • [EV3] Encinas, S., Villamayor, O.: Rees algebras and resolution of singularities. Rev. Mat. Iberoamericana. Proceedings XVI-Coloquio Latinoamericano de Álgebra 2006.
  • [FH] Faber, E., Hauser, H.: Today's menu: Geometry and resolution of singular algebraic surfaces. Bull. Amer. Math. Soc., to appear.
  • [GS] Gonzalez-Sprinberg, G.: Désingularisation des surfaces par des modifications de Nash normalisées. Sém. Bourbaki 1985/86. Astérisque 145-146 (1987), 187-207. MR 880033 (89a:14015)
  • [Ha1] Hauser, H.: Why the characteristic zero proof of resolution fails in positive characteristic. Manuscript 2003, available at www.hh.hauser.cc.
  • [Ha2] Hauser, H.: Wild singularities and kangaroo points in the resolution of algebraic varieties over fields of positive characteristic. Preprint 2009.
  • [Ha3] Hauser, H.: The Hironaka theorem on resolution of singularities. (Or: A proof that we always wanted to understand.) Bull. Amer. Math. Soc. 40 (2003), 323-403. MR 1978567 (2004d:14009)
  • [Ha4] Hauser, H.: Excellent surfaces over a field and their taut resolution. In: Resolution of Singularities (Obergurgl, 1997, ed. H. Hauser et al.) Progress in Math. 181, Birkhäuser 2000. MR 1748627 (2001f:14028)
  • [Ha5] Hauser, H.: Three power series techniques. Proc. London Math. Soc. 88 (2004), 1-24. MR 2063657 (2005f:14032)
  • [Ha6] Hauser, H.: Seven short stories on blowups and resolution. In: Proceedings of Gökova Geometry-Topology Conference 2005, (ed. S. Akbulut et al.) International Press, 2006, pp. 1-48. MR 2282008 (2008a:14023)
  • [Ha7] Hauser, H.: Seventeen obstacles for resolution of singularities. In: The Brieskorn Anniversary volume, Progress in Math. 162, Birkhäuser 1997. MR 1652479 (99g:14016)
  • [HR] Hauser, H., Regensburger, G.: Explizite Auflösung von ebenen Kurvensingularitäten in beliebiger Charakteristik. L'Enseign. Math. 50 (2004), 305-353. MR 2116719 (2006b:14022)
  • [HW] Hauser, H., Wagner, D.: Two new invariants for the embedded resolution of surfaces in positive characteristic. Preprint 2009.
  • [Hi1] Hironaka, H.: Program for resolution of singularities in characteristics $ p > 0$. Notes from lectures at the Clay Mathematics Institute, September 2008.
  • [Hi2] Hironaka, H.: A program for resolution of singularities, in all characteristics $ p > 0$ and in all dimensions. Lecture Notes ICTP Trieste, June 2006.
  • [Hi3] Hironaka, H.: Theory of infinitely near singular points. J. Korean Math. Soc. 40 (2003), 901-920. MR 1996845 (2005a:14003)
  • [Hi4] Hironaka, H.: Desingularization of excellent surfaces. Notes by B. Bennett at the Conference on Algebraic Geometry, Bowdoin 1967. Reprinted in: Cossart, V., Giraud, J., Orbanz, U.: Resolution of surface singularities. Lecture Notes in Math. 1101, Springer 1984. MR 775681 (87e:14032)
  • [Hi5] Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79 (1964), 109-326. MR 0199184 (33:7333)
  • [Hi6] Hironaka, H.: On Nash blowing-up. In: Arithmetic and Geometry II. Birkhäuser, 1983, pp. 103-111. MR 717608 (84k:14012)
  • [Ka] Kawanoue, H.: Toward resolution of singularities over a field of positive characteristic. I. Publ. Res. Inst. Math. Sci. 43 (2007), 819-909. MR 2361797 (2008m:14028)
  • [Ko] Kollár, J.: Lectures on resolution of singularities. Annals Math. Studies, vol. 166. Princeton 2007. MR 2289519 (2008f:14026)
  • [Lp1] Lipman, J.: Desingularization of 2-dimensional schemes. Ann. Math. 107 (1978), 151-207. MR 0491722 (58:10924)
  • [Lp2] Lipman, J.: Introduction to resolution of singularities. Proc. Symp. Pure Appl. Math. 29, Amer. Math. Soc. 1975, 187-230. MR 0389901 (52:10730)
  • [MK] 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. arXiv:math/0612008
  • [Mo] Moh, T.-T.: On a stability theorem for local uniformization in characteristic $ p$. Publ. Res. Inst. Math. Sci. 23 (1987), 965-973. MR 935710 (89d:14018)
  • [Mu] Mulay, S.: Equimultiplicity and hyperplanarity. Proc. Amer. Math. Soc. 87 (1983), 407-413. MR 715854 (85d:14005)
  • [Na1] Narasimhan, R.: Monomial equimultiple curves in positive characteristic. Proc. Amer. Math. Soc. 89 (1983), 402-413. MR 715853 (85d:14006)
  • [Na2] Narasimhan, R.: Hyperplanarity of the equimultiple locus. Proc. Amer. Math. Soc. 87 (1983), 403-406. MR 684627 (84f:14001)
  • [Pa] Panazzolo, D.: Resolution of singularities of real analytic vector fields in dimension three. Acta Math. 197 (2006) 167-289. MR 2296055 (2007m:37044)
  • [Se] Seidenberg, A.: Reduction of singularities of the differential equation $ A dy=B dx$. Amer. J. Math. 90 (1968), 248-269. MR 0220710 (36:3762)
  • [Sp] Spivakovsky, M.: Sandwiched singularities and desingularization of surfaces by normalized Nash transformations. Ann. Math. 131 (1990), 411-491. MR 1053487 (91e:14013)
  • [Vi1] Villamayor, O.: Constructiveness of Hironaka's resolution. Ann. Sci. École Norm. Sup. 22 (1989), 1-32. MR 985852 (90b:14014)
  • [Vi2] Villamayor, O.: Hypersurface singularities in positive characteristic. Adv. Math. 213 (2007), 687-733. MR 2332606 (2008g:14021)
  • [Vi3] Villamayor, O.: Elimination with applications to singularities in positive characteristic. Publ. Res. Inst. Math. Sci. 44 (2008), 661-697. MR 2426361 (2009h:14028)
  • [Wl] Włodarczyk, J.: Simple Hironaka resolution in characteristic zero. J. Amer. Math. Soc. 18 (2005), 779-822. MR 2163383 (2006f:14014)
  • [Ya] Yasuda, T.: Higher Nash blowups. Compos. Math. 143 (2007), 1493-1510. MR 2371378 (2008j:14029)
  • [Za] Zariski, O.: Reduction of the singularities of algebraic three dimensional varieties. Ann. Math. 45 (1944), 472-542. MR 0011006 (6:102f)
  • [Ze] Zeillinger, D.: Polyederspiele und Auflösen von Singularitäten. PhD Thesis, Universität Innsbruck, 2005.

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Additional Information

Herwig Hauser
Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090, Wien, Austria
Email: herwig.hauser@univie.ac.at

DOI: https://doi.org/10.1090/S0273-0979-09-01274-9
Received by editor(s): November 17, 2008
Received by editor(s) in revised form: December 27, 2008, February 12, 2009, and July 10, 2009
Published electronically: October 28, 2009
Additional Notes: Supported within the project P-18992 of the Austrian Science Fund FWF. The author thanks the members of the Clay Institute for Mathematics at Cambridge and the Research Institute for Mathematical Science at Kyoto for their kind hospitality.
Article copyright: © Copyright 2009 American Mathematical Society

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