A Course on Constructive Desingularization and Equivariance

  1. Encinas, Santiago 1
  2. Villamayor, Orlando 1
  1. 1 Universidad de Valladolid
    info

    Universidad de Valladolid

    Valladolid, España

    ROR https://ror.org/01fvbaw18

Libro:
Resolution of Singularities

ISBN: 9783034895507,  9783034883993

Año de publicación: 2000

Páginas: 147-227

Tipo: Capítulo de Libro

DOI: 10.1007/978-3-0348-8399-3_6 GOOGLE SCHOLAR lock_openAcceso abierto editor

Objetivos de desarrollo sostenible

Resumen

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.

Referencias bibliográficas

  • S.S. Abhyankar, Good points of a hypersurface, Advances in Math. 68, (1988), pp. 87–256.
  • S.S. Abhyankar, Resolution of Singularities of Embedded Algebraic Surfaces, Second enlarged edition, Springer-Verlag, 1998.
  • J.M. Aroca, H. Hironaka and J.L. Vicente, The theory of maximal contact, de Matemática del Instituto “Jorge Juan” (Madrid), 30, (1977).
  • J.M. Aroca, H. Hironaka and J.L. Vicente, Desingularization theorems, Memorias Memorias de Matemática del Instituto “Jorge Juan” (Madrid), 29, (1975).
  • 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.
  • D. Abramovich and A.J. de Jong, Smoothness, semistability and toroidal geometry. Journal of Algebraic Geometry, 6 (1997), pp. 789–801.
  • D. Abramovich and J. Wang, Equivariant resolution of singularities in characterisitic O. Mathematical Research Letters 4 (1997), pp. 427–433.
  • M. Artin, Algebraic approximation of structures over complete local rings, Pub. Math. I.H.E.S., 36, (1969), pp. 23–58.
  • B.M. Bennett, On the characteristic function of a local ring, Ann. of Math., 91, (1970), pp. 25–87.
  • P. Berthelot, Altérations des variétés algébriques, Sem. Bourbaki 48, 815, (1995–96).
  • 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.
  • 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.
  • 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/
  • F. Bogomolov and T. Pantev, Weak Hironaka Theorem Mathematical Research Letters 3 (1996) pp. 299–307.
  • V. Cossart, J. Giraud and U. Orbanz, Resolution of Surface Singularities. Lectures Notes in Mathematics, Springer Verlag 1101, (1984).
  • S. Encinas, Resolución Constructiva de Singularidades de Familias de Esquemas, PhD thesis, Universidad de Valladolid, (1996).
  • S. Encinas, On constructive desingularization of non-embedded schemes., Pre-print, (1998).
  • S. Encinas and O. Villamayor, Good points and constructive resolution of singularities, Acta Math. Vol. 181:1 (1998).
  • S. Encinas and O. Villamayor, A new theorem of desingularization over fields of characteristic zero. Preprint 1999.
  • J. Giraud, Analysis Situs, Sem. Bourbaki, 256, (1962–63).
  • J. Giraud, Etude locale des singularités, Pub. Math. Orsay, France 1972.
  • J. Giraud, Sur la théorie du contact maximal, Math. Zeit., 137, (1974), pp. 285–310.
  • J. Giraud, Contact maximal en caractéristique positive, Ann. Scien. de l’Ec. Norm. Sup., 4 série, 8 (2), (1975), pp. 201–234.
  • J. Giraud, Remarks on desingularization problems, Nova Acta Leopoldina, 52, (1981), pp. 103–107.
  • R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer Verlag, New York, 1983.
  • H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I—II,Ann. Math., 79, (1964), pp. 109–326.
  • H. Hironaka, Idealistic exponent of a singularity, Algebraic Geometry. The John Hopkins centennial lectures, Baltimore, Johns Hopkins University Press (1977), pp. 52–125.
  • A.J. de Jong, Smoothness, semi-stability and alterations, Pub. Math. I.H.E.S. (1996) no. 83, pp. 51–93.
  • J. Lipman, Introduction to resolution of singularities, Proc. Symp. in Pure Math., 29, (1975), pp. 187–230.
  • M. Lejeune and B. Teissier, Quelques calculs utiles pour la résolution des singularités, Centre de Mathématique de l’Ecole Polytechnique, (1972).
  • H. Matsumura. Commutative Algebra. W. A. Benjamin Co., New York, 1970.
  • H. Matsumura. Commutative Ring Theory. Cambridge University Press, Londres, 1986.
  • T. T. Moh. Quasi-canonical uniformization of hypersurface singularities of characteristic zero. Communications in Algebra, 20(11), (1992), pp. 3207–3251.
  • M. Raynaud. Anneaux locaux henséliens, volume 169 of Lecture Notes in Math-ematics. Springer-Verlag, 1970.
  • T. Oda, Infinitely Very Near-Singular Points, Complex Analytic Singularities, Advanced Studies in Pure Mathematics, 8, (1986), pp. 363–404.
  • O.E. Villamayor, Constructiveness of Hironaka’s resolution, Ann. Scient. Ec. Norm. Sup., 4e serie, 22, (1989) pp. 1–32.
  • O.E. Villamayor, Patching local uniformizations, Ann. Scient. Ec. Norm. Sup., 25, (1992), pp. 629–677.
  • 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.
  • 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.
  • O. Zariski. Local uniformization of algebraic varieties. Ann. of Math., 41(4), (1940), pp. 852–860.