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

Hauser, Herwig

doi:10.1090/S0273-0979-09-01274-9

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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
Posted: 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.

[Ab1] Shreeram Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic 𝑝≠0, Ann. of Math. (2) 63 (1956), 491–526. MR 0078017 (17,1134d)
[Ab2] Shreeram S. Abhyankar, Good points of a hypersurface, Adv. in Math. 68 (1988), no. 2, 87–256. MR 934366 (89e:14012), http://dx.doi.org/10.1016/0001-8708(88)90015-1
[AO] 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. MR 1748617 (2001h:14009), http://dx.doi.org/10.1007/978-3-0348-8399-3_3
[Be] 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. MR 1472543 (98m:14021)
[BM] 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. MR 1440306 (98e:14010), http://dx.doi.org/10.1007/s002220050141
[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] 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. MR 2427629 (2009f:14024), http://dx.doi.org/10.1016/j.jalgebra.2008.03.032
[CP2] Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. II, J. Algebra 321 (2009), no. 7, 1836–1976. MR 2494751 (2010c:14009), http://dx.doi.org/10.1016/j.jalgebra.2008.11.030
[Cu] Steven Dale Cutkosky, Resolution of singularities for 3-folds in positive characteristic, Amer. J. Math. 131 (2009), no. 1, 59–127. MR 2488485 (2010e:14010), http://dx.doi.org/10.1353/ajm.0.0036
[dJ] A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020 (98e:14011)
[EiH] David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. MR 1730819 (2001d:14002)
[EH] Santiago Encinas and Herwig Hauser, Strong resolution of singularities in characteristic zero, Comment. Math. Helv. 77 (2002), no. 4, 821–845. MR 1949115 (2004c:14021), http://dx.doi.org/10.1007/PL00012443
[EV1] 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. MR 1748620 (2001g:14018)
[EV2] S. Encinas and O. Villamayor, Good points and constructive resolution of singularities, Acta Math. 181 (1998), no. 1, 109–158. MR 1654779 (99i:14020), http://dx.doi.org/10.1007/BF02392749
[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] 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. 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] 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 (electronic). MR 1978567 (2004d:14009), http://dx.doi.org/10.1090/S0273-0979-03-00982-0
[Ha4] Herwig Hauser, Excellent surfaces and their taut resolution, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 341–373. MR 1748627 (2001f:14028)
[Ha5] Herwig Hauser, Three power series techniques, Proc. London Math. Soc. (3) 89 (2004), no. 1, 1–24. MR 2063657 (2005f:14032), http://dx.doi.org/10.1112/S0024611503014588
[Ha6] 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. MR 2282008 (2008a:14023)
[Ha7] Herwig Hauser, Seventeen obstacles for resolution of singularities, Singularities (Oberwolfach, 1996) Progr. Math., vol. 162, Birkhäuser, Basel, 1998, pp. 289–313. MR 1652479 (99g:14016)
[HR] Herwig Hauser and Georg Regensburger, Explizite Auflösung von ebenen Kurvensingularitäten in beliebiger Charakteristik, Enseign. Math. (2) 50 (2004), no. 3-4, 305–353 (German). 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 . Notes from lectures at the Clay Mathematics Institute, September 2008.
[Hi2] Hironaka, H.: A program for resolution of singularities, in all characteristics and in all dimensions. Lecture Notes ICTP Trieste, June 2006.
[Hi3] Heisuke Hironaka, Theory of infinitely near singular points, J. Korean Math. Soc. 40 (2003), no. 5, 901–920. MR 1996845 (2005a:14003), http://dx.doi.org/10.4134/JKMS.2003.40.5.901
[Hi4] 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. MR 775681 (87e:14032)
[Hi5] 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. MR 0199184 (33 #7333)
[Hi6] Heisuke Hironaka, On Nash blowing-up, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Mass., 1983, pp. 103–111. MR 717608 (84k:14012)
[Ka] 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. MR 2361797 (2008m:14028)
[Ko] János Kollár, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR 2289519 (2008f:14026)
[Lp1] Joseph Lipman, Desingularization of two-dimensional schemes, Ann. Math. (2) 107 (1978), no. 1, 151–207. MR 0491722 (58 #10924)
[Lp2] 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. 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] T. T. Moh, On a stability theorem for local uniformization in characteristic 𝑝, Publ. Res. Inst. Math. Sci. 23 (1987), no. 6, 965–973. MR 935710 (89d:14018), http://dx.doi.org/10.2977/prims/1195175867
[Mu] S. B. Mulay, Equimultiplicity and hyperplanarity, Proc. Amer. Math. Soc. 89 (1983), no. 3, 407–413. MR 715854 (85d:14005), http://dx.doi.org/10.1090/S0002-9939-1983-0715854-9
[Na1] R. Narasimhan, Monomial equimultiple curves in positive characteristic, Proc. Amer. Math. Soc. 89 (1983), no. 3, 402–406. MR 715853 (85d:14006), http://dx.doi.org/10.1090/S0002-9939-1983-0715853-7
[Na2] R. Narasimhan, Hyperplanarity of the equimultiple locus, Proc. Amer. Math. Soc. 87 (1983), no. 3, 403–408. MR 684627 (84f:14001), http://dx.doi.org/10.1090/S0002-9939-1983-0684627-8
[Pa] Daniel Panazzolo, Resolution of singularities of real-analytic vector fields in dimension three, Acta Math. 197 (2006), no. 2, 167–289. MR 2296055 (2007m:37044), http://dx.doi.org/10.1007/s11511-006-0011-7
[Se] A. Seidenberg, Reduction of singularities of the differential equation 𝐴𝑑𝑦=𝐵𝑑𝑥, Amer. J. Math. 90 (1968), 248–269. MR 0220710 (36 #3762)
[Sp] Mark Spivakovsky, Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Ann. of Math. (2) 131 (1990), no. 3, 411–491. MR 1053487 (91e:14013), http://dx.doi.org/10.2307/1971467
[Vi1] Orlando Villamayor, Constructiveness of Hironaka’s resolution, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1–32. MR 985852 (90b:14014)
[Vi2] Orlando Villamayor U., Hypersurface singularities in positive characteristic, Adv. Math. 213 (2007), no. 2, 687–733. MR 2332606 (2008g:14021), http://dx.doi.org/10.1016/j.aim.2007.01.006
[Vi3] Orlando Villamayor U., Elimination with applications to singularities in positive characteristic, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 661–697. MR 2426361 (2009h:14028), http://dx.doi.org/10.2977/prims/1210167340
[Wl] Jarosław Włodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005), no. 4, 779–822 (electronic). MR 2163383 (2006f:14014), http://dx.doi.org/10.1090/S0894-0347-05-00493-5
[Ya] Takehiko Yasuda, Higher Nash blowups, Compos. Math. 143 (2007), no. 6, 1493–1510. MR 2371378 (2008j:14029), http://dx.doi.org/10.1112/S0010437X0700276X
[Za] Oscar Zariski, Reduction of the singularities of algebraic three dimensional varieties, Ann. of Math. (2) 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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