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Desingularization: Invariants and Strategy

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2270))

Abstract

Let X be an irreducible and reduced excellent noetherian scheme.

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References

  1. S.S. Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristic p≠0. Ann. Math. (2) 63, 491–526 (1956)

    Google Scholar 

  2. S.S. Abhyankhar, Resolution of singularities of arithmetical surfaces, in Arithmetical Algebraic Geometry (Proceedings of the Conference on Purdue University, 1963) (Harper and Row, New York, 1965), pp. 111–152

    Google Scholar 

  3. S.S. Abhyankar, An algorithm on polynomials in one indeterminate with coefficients in a two dimensional regular local domain. Ann. Mat. Pura Appl. (4) 71, 25–59 (1966)

    Google Scholar 

  4. S.S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, in Pure and Applied Mathematics, vol. 24 (Academic, New York 1966), ix+291 pp.

    Google Scholar 

  5. S.S. Abhyankar, Nonsplitting of valuations in extensions of two dimensional regular local domains. Math. Ann. 170, 87–144 (1967)

    Article  MathSciNet  Google Scholar 

  6. S.S. Abhyankar, Good points of a hypersurface. Adv. Math. 68(2), 87–256 (1988)

    Article  MathSciNet  Google Scholar 

  7. G. Albanese, Trasformazione birazionale di una superficie algebrica in un’altra priva di punti multipli. Rend. Circ. Mat. Palermo 48(3), 321–332 (1924)

    Article  Google Scholar 

  8. J.M. Aroca, H. Hironaka, J.L.J. Vicente, The Theory of the Maximal Contact. Memorias de Matemática del Instituto “Jorge Juan”, No. 29 (Consejo Superior de Investigaciones Científicas, Madrid, 1975), 135 pp.

    Google Scholar 

  9. B. Bennett, On the characteristic functions of a local ring. Ann. Math. 91, 25–87 (1970)

    Article  MathSciNet  Google Scholar 

  10. E. Bierstone, P.D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997). (English Summary)

    Google Scholar 

  11. V. Cossart, Desingularization of embedded excellent surfaces. Tohoku Math. J. 33, 25–33 (1981)

    Article  MathSciNet  Google Scholar 

  12. V. Cossart, Desingularization: a few bad examples in dim. 3, characteristic p > 0, (English summary), in Topics in Algebraic and Noncommutative Geometry (Luminy/Annapolis, MD, 2001), Contemporary Mathematics, vol. 324 (American Mathematical Society, Providence, 2003), pp. 103–108

    Google Scholar 

  13. V. Cossart, O. Piltant, 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). (English Summary)

    Google Scholar 

  14. V. Cossart, O. Piltant, Resolution of singularities of threefolds in positive characteristic. II. J. Algebra 321(7), 1836–1976 (2009)

    Article  MathSciNet  Google Scholar 

  15. V. Cossart, O. Piltant, Resolution of singularities of arithmetical threefolds. J. Algebra 529, 268–535 (2019)

    Article  MathSciNet  Google Scholar 

  16. V. Cossart, B. Schober, A strictly decreasing invariant for resolution of singularities in dimension two. Publ. Res. Inst. Math. Sci. 56(2), 217–280 (2020)

    Article  MathSciNet  Google Scholar 

  17. V. Cossart, O. Piltant, B. Schober, Faîte du cône tangent à une singularité : un théorème oublié. Compt. Rendus Math. 355(4), 455–459 (2017)

    Article  Google Scholar 

  18. S.D. Cutkosky, Resolution of Singularities. Graduate Studies in Mathematics, vol. 63 (American Mathematical Society, Providence, 2004), viii+186 pp.

    Google Scholar 

  19. S.D. Cutkosky, Resolution of singularities for 3-folds in positive characteristic. Amer. J. Math. 131(1), 59–127 (2009)

    Article  MathSciNet  Google Scholar 

  20. A.J. de Jong, Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. No. 83, 51–93 (1996)

    Article  MathSciNet  Google Scholar 

  21. S. Encinas, H. Hauser, Strong resolution of singularities in characteristic zero. Comment. Math. Helv. 77(4), 821–845 (2002). (English summary)

    Google Scholar 

  22. J. Giraud, Étude Locale des Singularités. Cours de 3e cycle, Orsay, Publication No. 26 (1971–1972). http://portail.mathdoc.fr/PMO/PDF/G_GIRAUD-50.pdf

  23. J. Giraud, Sur la théorie du contact maximal. Math. Z. 137, 285–310 (1974). (French)

    Google Scholar 

  24. J. Giraud, Contact maximal en caractéristique positive. Ann. Sci. École Norm. Sup. (4) 8(2), 201–234 (1975). (French)

    Google Scholar 

  25. A. Grothendieck, Travaux de Heisuke Hironaka sur la résolution des singularités. Actes C.I.M. (Nice, 1970), vol. I (Gauthier-Villars, Paris, 1971), pp. 7–9

    Google Scholar 

  26. A. Grothendieck, J. Dieudonné, Éléments de géométrie algébrique IV. Publ. Math. I.H.É.S. 1: No. 20 (1964), 2: No. 24 (1965), 3: No. 28 (1966), 4: No. 32 (1967).

    Google Scholar 

  27. R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics (Springer, Berlin, 1977)

    Google Scholar 

  28. H. Hauser, Excellent surfaces and their taut resolution, in Resolution of singularities (Obergurgl, 1997). Progress in Mathematics, vol. 181 (Birkhhäuser, Basel, 2000), pp. 341–373

    Google Scholar 

  29. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero: I–II. Ann. Math. 79, 109–326 (1964)

    Article  MathSciNet  Google Scholar 

  30. H. Hironaka, On the characters ν and τ of singularities. J. Math. Kyoto Univ. 7, 19–43 (1967)

    Article  MathSciNet  Google Scholar 

  31. H. Hironaka, Characteristic polyhedra of singularities. J. Math. Kyoto Univ. 7, 251–293 (1967)

    Article  MathSciNet  Google Scholar 

  32. H. Hironaka, Desingularization of excellent surfaces, in Advanced Science Seminar in Algebraic Geometry, (Summer 1967 at Bowdoin College), Mimeographed notes by B. Bennet, Lecture Notes in Mathematics, 1101 (Springer, Berlin, 1984), pp. 99–132.

    Google Scholar 

  33. H. Hironaka, Additive groups associated with ponts of a projective space. Ann. Math. 92, 327–334 (1970)

    Article  MathSciNet  Google Scholar 

  34. H. Hironaka, Idealistic exponents of singularity, in Algebraic Geometry (J. J. Sylvester Symposium, Johns Hopkins University, Baltimore, 1976) (Johns Hopkins University Press, Baltimore, 1977), pp. 52–125

    Google Scholar 

  35. U. Jannsen, S. Saito, Kato conjecture and motivic cohomology over finite fields (2007, preprint, arxiv.org/abs/0910.2815)

    Google Scholar 

  36. U. Jannsen, Resolution of singularities for embedded curves, Appendix of: A finiteness theorem for zero-cycles over p-adic fields. Ann. Math. 172(3), 1593–1639 (2010).

    Article  MathSciNet  Google Scholar 

  37. J. Kollár, Resolution of Singularities. Annals of Mathematics Studies, vol. 166 (Princeton University Press, Princeton, 2007)

    Google Scholar 

  38. J. Lipman, Desingularization of two-dimensional schemes. Ann. Math. (2) 107(1), 151–207 (1978)

    Google Scholar 

  39. R. Narasimhan, Hyperplanarity of the equimultiple locus. Proc. Amer. Math. Soc. 87(3), 403–408 (1983)

    Article  MathSciNet  Google Scholar 

  40. R. Narasimhan, Monomial equimultiple curves in positive characteristic. Proc. Amer. Math. Soc. 89(3), 402–406 (1983)

    Article  MathSciNet  Google Scholar 

  41. V. Puiseux, Recherches sur les fonctions algébriques. J. de Math. Pures et Appl. 15, 207 (1850)

    Google Scholar 

  42. H. Reitberger The turbulent fifties in resolution of singularities, in Resolution of Singularities, ed. by H. Hauser, J. Lipman, F. Oort, A. Quirós. Progress in Mathematics, vol 181 (Birkhäuser, Basel, 2000)

    Google Scholar 

  43. J-P. Serre, Algèbre Locale Multiplicitès. Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel. Lecture Notes in Mathematics, vol. 11, 2nd edn. (Springer, Berlin, 1965), vii+188 pp.

    Google Scholar 

  44. B. Singh, Effect of a permissible blowing-up on the local Hilbert functions. Invent. Math. 26, 201–212 (1974)

    Article  MathSciNet  Google Scholar 

  45. M. Temkin, Desingularization of quasi-excellent schemes of characteristic zero. Adv. Math. 219, 488–522 (2008)

    Article  MathSciNet  Google Scholar 

  46. M. Temkin, Functorial desingularization of quasi-excellent schemes in characteristic zero: the non-embedded case. Duke J. Math. 161, 2208–2254 (2012)

    Article  Google Scholar 

  47. U.Villamayor, Patching local uniformizations. Ann. Sci. École Norm. Sup. (4) 25(6), 629–677 (1992)

    Google Scholar 

  48. J. Walker, Resolution of singularities of an algebraic surface. Ann. Math. (2) 36, 336-365 (1935)

    Google Scholar 

  49. O. Zariski, The reduction of the singularities of an algebraic surface. Ann. Math. (2) 40, 639–689 (1939)

    Google Scholar 

  50. O. Zariski, A simplified proof for the resolution of singularities of an algebraic surface. Ann. Math. (2) 43, 583–593 (1942)

    Google Scholar 

  51. O. Zariski, Reduction of the singularities of algebraic three dimensional varieties. Ann. Math. (2) 45, 472–542 (1944)

    Google Scholar 

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Acknowledgements

We would like to thank our colleagues D. Cutkosky, O. Piltant, B. Schober, B. Teissier, O. Villamayor and some others whom we forgot, for long conversations on this topic and who encourage us to put this monograph in a suitable form to be published. Special thanks to B. Schober who sent us a long list of misprints and who helped us for many corrections, B. Tiefenbach who made the pictures and G. Moreno-Socias who helped us for the formatting and the readability of this book.

Many thanks to Dan Abramovich who encouraged us to publish this book and who, at the very end of the process, asked us 5 questions on the characteristic polyhedra. Great many thanks to Bernd Schober who accepted to write down, on short notice, the Appendix answering Dan Abramovich’s questions.

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Cossart, V., Jannsen, U., Saito, S. (2020). Introduction. In: Desingularization: Invariants and Strategy. Lecture Notes in Mathematics, vol 2270. Springer, Cham. https://doi.org/10.1007/978-3-030-52640-5_1

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