Skip to main content
SpringerLink
Log in

A Course on Constructive Desingularization and Equivariance

  • Chapter
Book cover Resolution of Singularities

Part of the book series: Progress in Mathematics ((PM,volume 181))

Abstract

We study a constructive proof of desingularization, as the outcome of a process obtained by successively blowing up the maximum stratum of a function f X . We focus on canonical properties of this desingularization such as compatibility with change of base field and that of equivariance, namely the lifting of any group action on X to an action on the desingularization defined by this procedure.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S.S. Abhyankar, Good points of a hypersurface, Advances in Math. 68, (1988), pp. 87–256.

    Article  MathSciNet  MATH  Google Scholar 

  2. S.S. Abhyankar, Resolution of Singularities of Embedded Algebraic Surfaces, Second enlarged edition, Springer-Verlag, 1998.

    Book  MATH  Google Scholar 

  3. J.M. Aroca, H. Hironaka and J.L. Vicente, The theory of maximal contact, de Matemática del Instituto “Jorge Juan” (Madrid), 30, (1977).

    Google Scholar 

  4. J.M. Aroca, H. Hironaka and J.L. Vicente, Desingularization theorems, Memorias Memorias de Matemática del Instituto “Jorge Juan” (Madrid), 29, (1975).

    Google Scholar 

  5. S.S. Abhyankar and T.T. Moh, Newton-Puiseux expansion and generalized Tschirnhausen transformation I-II. J. Reine Angew. Math. 260, (1973), pp. 4783, ibid. 261, (1973), pp. 29–54.

    Google Scholar 

  6. D. Abramovich and A.J. de Jong, Smoothness, semistability and toroidal geometry. Journal of Algebraic Geometry, 6 (1997), pp. 789–801.

    MathSciNet  MATH  Google Scholar 

  7. D. Abramovich and J. Wang, Equivariant resolution of singularities in characterisitic O. Mathematical Research Letters 4 (1997), pp. 427–433.

    MathSciNet  MATH  Google Scholar 

  8. M. Artin, Algebraic approximation of structures over complete local rings, Pub. Math. I.H.E.S., 36, (1969), pp. 23–58.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Berthelot, Altérations des variétés algébriques, Sem. Bourbaki 48, 815, (1995–96).

    Google Scholar 

  11. E. Bierstone and P. Milman, A simple constructive proof of canonical resolution of singularities, Effective Methods in Algebraic Geometry. Prog. in Math, 94, Birkhäuser, Boston, (1991), pp. 11–30.

    Chapter  Google Scholar 

  12. E. Bierstone and P. Milman, Canonical desingularization in characteristic zero by blowing-up the maximal strata of a local invariant, Inv. Math. 128(2), (1997), pp. 207–302.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Bodnár, J. Schicho, Automated resolution of singularities for hypersurfaces. Preprint 1999, available at http://www.risc.uni-linz.ac.at/projects/basic/adjoints/blowup/

    Google Scholar 

  14. F. Bogomolov and T. Pantev, Weak Hironaka Theorem Mathematical Research Letters 3 (1996) pp. 299–307.

    MathSciNet  MATH  Google Scholar 

  15. V. Cossart, J. Giraud and U. Orbanz, Resolution of Surface Singularities. Lectures Notes in Mathematics, Springer Verlag 1101, (1984).

    MATH  Google Scholar 

  16. S. Encinas, Resolución Constructiva de Singularidades de Familias de Esquemas, PhD thesis, Universidad de Valladolid, (1996).

    Google Scholar 

  17. S. Encinas, On constructive desingularization of non-embedded schemes., Pre-print, (1998).

    Google Scholar 

  18. S. Encinas and O. Villamayor, Good points and constructive resolution of singularities, Acta Math. Vol. 181:1 (1998).

    Article  MathSciNet  Google Scholar 

  19. S. Encinas and O. Villamayor, A new theorem of desingularization over fields of characteristic zero. Preprint 1999.

    Google Scholar 

  20. J. Giraud, Analysis Situs, Sem. Bourbaki, 256, (1962–63).

    Google Scholar 

  21. J. Giraud, Etude locale des singularités, Pub. Math. Orsay, France 1972.

    Google Scholar 

  22. J. Giraud, Sur la théorie du contact maximal, Math. Zeit., 137, (1974), pp. 285–310.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Giraud, Contact maximal en caractéristique positive, Ann. Scien. de l’Ec. Norm. Sup., 4 série, 8 (2), (1975), pp. 201–234.

    MathSciNet  MATH  Google Scholar 

  24. J. Giraud, Remarks on desingularization problems, Nova Acta Leopoldina, 52, (1981), pp. 103–107.

    MathSciNet  MATH  Google Scholar 

  25. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer Verlag, New York, 1983.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. H. Hironaka, Idealistic exponent of a singularity, Algebraic Geometry. The John Hopkins centennial lectures, Baltimore, Johns Hopkins University Press (1977), pp. 52–125.

    Google Scholar 

  28. A.J. de Jong, Smoothness, semi-stability and alterations, Pub. Math. I.H.E.S. (1996) no. 83, pp. 51–93.

    MATH  Google Scholar 

  29. J. Lipman, Introduction to resolution of singularities, Proc. Symp. in Pure Math., 29, (1975), pp. 187–230.

    MathSciNet  Google Scholar 

  30. M. Lejeune and B. Teissier, Quelques calculs utiles pour la résolution des singularités, Centre de Mathématique de l’Ecole Polytechnique, (1972).

    Google Scholar 

  31. H. Matsumura. Commutative Algebra. W. A. Benjamin Co., New York, 1970.

    MATH  Google Scholar 

  32. H. Matsumura. Commutative Ring Theory. Cambridge University Press, Londres, 1986.

    MATH  Google Scholar 

  33. T. T. Moh. Quasi-canonical uniformization of hypersurface singularities of characteristic zero. Communications in Algebra, 20(11), (1992), pp. 3207–3251.

    MathSciNet  Google Scholar 

  34. M. Raynaud. Anneaux locaux henséliens, volume 169 of Lecture Notes in Math-ematics. Springer-Verlag, 1970.

    Google Scholar 

  35. T. Oda, Infinitely Very Near-Singular Points, Complex Analytic Singularities, Advanced Studies in Pure Mathematics, 8, (1986), pp. 363–404.

    MathSciNet  Google Scholar 

  36. O.E. Villamayor, Constructiveness of Hironaka’s resolution, Ann. Scient. Ec. Norm. Sup., 4e serie, 22, (1989) pp. 1–32.

    MathSciNet  MATH  Google Scholar 

  37. O.E. Villamayor, Patching local uniformizations, Ann. Scient. Ec. Norm. Sup., 25, (1992), pp. 629–677.

    MATH  Google Scholar 

  38. O.E. Villamayor, On good points and a new canonical algorithm of resolution of singularities (Announcement), Real Analytic and Algebraic Geometry, Walter de Gruyter Berlin, New York, 1995.

    Google Scholar 

  39. O.E. Villamayor. Introduction to the algorithm of resolution. Proceedings of the 1991 La Rábida Meeting. Progress in Mathematics, Vol 134, (1996), Birkhäuser Verlag Basel/Switzerland.

    Google Scholar 

  40. O. Zariski. Local uniformization of algebraic varieties. Ann. of Math., 41(4), (1940), pp. 852–860.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dpto. Matemática Aplicada Fundamental E.T.S. Arquitectura, Universidad de Valladolid, Valladolid, 47014, Spain

    Santiago Encinas

  2. Dpto. Matemáticas, Universidad Autónoma de Madrid, Madrid, 28049, Spain

    Orlando Villamayor

Authors
  1. Santiago Encinas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Orlando Villamayor
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Mathematik, Universität Innsbruck, Innsbruck, A-6020, Österreich

    Herwig Hauser

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Joseph Lipman

  3. Department of Mathematics, Universiteit Utrecht, Utrecht, 3508 TA, The Netherlands

    Frans Oort

  4. Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, 28049, Spain

    Adolfo Quirós

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Basel AG

About this chapter

Cite this chapter

Encinas, S., Villamayor, O. (2000). A Course on Constructive Desingularization and Equivariance. In: Hauser, H., Lipman, J., Oort, F., Quirós, A. (eds) Resolution of Singularities. Progress in Mathematics, vol 181. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8399-3_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8399-3_6

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9550-7

  • Online ISBN: 978-3-0348-8399-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation