Characteristic polyhedra of singularities without completion: part II

• Cossart, Vincent ^[1] ; Schober, Bernd ^[2]
  1. [1] Laboratoire de Mathémathiques de Versailles CNRS UMR 8100, Université Paris-Saclay, 45 avenue des États-Unis, 78035, Versaills Cedex, France
  2. [2] Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 72, Fasc. 2, 2021, págs. 351-392
• Idioma: inglés
• DOI: 10.1007/s13348-020-00291-5
• Enlaces
  • Texto completo
• Resumen

  • Hironaka’s characteristic polyhedron is an important combinatorial object reflecting the local nature of a singularity. We prove that it can be determined without passing to the completion if the local ring is a G-ring and if additionally either it is Henselian, or a certain polynomiality condition (Pol) holds, or a mild condition (*) on the singularity holds. For example, the latter is fulfilled if the residue field is perfect.

• Referencias bibliográficas
  • Berthomieu, J., Hivert, P., Mourtada, H.: Computing Hironaka’s invariants: ridge and directrix. In: Arithmetic, Geometry, Cryptography and...
  • Cossart, V., Giraud, J., Orbanz, U.: Resolution of Surface Singularities, with an Appendix by H. Hironaka, Lecture Notes in Mathematics, vol....
  • Cossart, V., Jannsen, U., Saito, S.: Desingularization: Invariants and Strategy. Application to Dimension 2. Lecture Notes in Mathematics,...
  • Cossart, V., Jannsen, U., Schober, B.: Invariance of Hironaka’s characteristic polyhedron. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat....
  • Cossart, V., Piltant, O.: Resolution of singularities of threefolds in positive characteristic I. J. Algebra 320, 1051–1081 (2008)
  • Cossart, V., Piltant, O.: Resolution of singularities of threefolds in positive characteristic II. J. Algebra 321, 1836–1976 (2009)
  • Cossart, V., Piltant, O.: Characteristic polyhedra of singularities without completion. Math. Ann. 361, 157–167 (2015)
  • Cossart, V., Piltant, O.: Resolution of Singularities of Arithmetical Threefolds. J. Algebra 529, 268–535 (2019)
  • Cossart, V., Piltant, O., Schober, B.: Faîte du cône tangent à une singularité: un théorème oublié. C. R. Acad. Sci. Paris, Ser. I 355, 455–459...
  • Cossart, V., Schober, B.: A strictly decreasing invariant for resolution of singularities in dimension two. Publ. Res. Inst. Math. Sci. 56(2),...
  • Dietel, B.: A refinement of Hironaka’s additive group schemes for an extended invariant, Ph.D. thesis, University of Regensburg (2015). http://epub.uni-regensburg.de/31359/
  • Giraud, J.: Étude locale des singularités, Cours de 3^{\acute{\text{e}}\text{ me }} cycle, 1971–1972. Publ. Math. d’Orsay 26, (1972)
  • Grothendieck, A., Dieudonné, J.: Eléments de géométrie algébrique IV, Publ. Math. I.H.E.S. 24 (1965)
  • Grothendieck, A., Dieudonné, J.: Eléments de géométrie algébrique IV, Publ. Math. I.H.E.S. 32 (1967)
  • Hironaka, H.: Characteristic polyhedra of singularities. J. Math. Kyoto Univ. 7, 251–293 (1967)
  • Hironaka, H.: Additive groups associated with points of a projective space. Ann. of Math. (2) 92, 327–334 (1970)
  • Hironaka, H.: Desingularization of Excellent Surfaces. Adv. Sci. Sem. in Alg. Geo, Bowdoin College, Summer 1967. Notes by Bruce Bennett, appendix...
  • Matsumura, H.: Commutative Ring Theory, Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics, 8. Cambridge University...
  • Mizutani, H.: Hironaka’s additive group schemes. Nagoya Math. J. 52, 85–95 (1973)
  • Mourtada, H., Schober, B.: A polyhedral characterization of quasi-ordinary singularities. Moscow Math. J. 18, 755–785 (2018)
  • Oda, T.: Hironaka’s additive group scheme, Number theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Kinokuniya,...
  • Schober, B.: Characteristic polyhedra of idealistic exponents with history, PhD thesis, University of Regensburg (2013). available on http://epub.uni-regensburg.de/28877/