Nash multiplicities and resolution invariants

• Autores: A. Bravo, Santiago Encinas Carrión , Beatriz Pascual Escudero
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 68, Fasc. 2, 2017, págs. 175-217
• Idioma: inglés
• DOI: 10.1007/s13348-016-0188-9
• Enlaces
  • Texto completo (pdf)
• Resumen

  • The Nash multiplicity sequence was defined by Lejeune-Jalabert as a non-increasing sequence of integers attached to a germ of a curve inside a germ of a hypersurface. Hickel generalized this notion and described a sequence of blow ups which allows us to compute it and study its behavior. In this paper, we show how this sequence can be used to compute some invariants that appear in algorithmic resolution of singularities. Moreover, this indicates that these invariants from constructive resolution are intrinsic to the variety since they can be read in terms of its space of arcs. This result is a first step connecting explicitly arc spaces and algorithmic resolution of singularities.

• Referencias bibliográficas
  • Benito, A.: The tau-invariant and elimination. J. Algebra 324(8), 1903–1920 (2010)
  • Bennett, B.M.: On the characteristic functions of a local ring. Ann. Math. 91(2), 25–87 (1970)
  • Bierstone, E., Milman, P.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent....
  • Blanco, R., Encinas, S.: Coefficient and elimination algebras in resolution of singularities. Asian J. Math. 15(2), 251–271 (2011)
  • Bravo, A., Villamayor U, O.: Singularities in positive characteristic, stratification and simplification of the singular locus. Adv. Math....
  • Bravo, A., Villamayor U, O.E.: Elimination algebras and inductive arguments in resolution of singularities. Asian J. Math. 15(3), 321–355...
  • Bravo, A., García-Escamilla, M.L., Villamayor U, O.E.: On Rees algebras and invariants for singularities over perfect fields. Indiana Univ....
  • Bravo, A., Villamayor U, O.E.: On the behavior of the multiplicity on schemes: stratification and blow-ups. The resolution of singular algebraic...
  • Dade, E.C.: Multiplicity and monoidal transformations. Thesis (Ph.D.), Princeton University (1960)
  • Ein, L., Mustaţă, M., Yasuda, T.: Jet schemes, log discrepancies, and inversion of adjunction. Invent. Math. 153, 519–535 (2003)
  • Ein, L., Mustaţă, M.: Inversion of adjunction for local complete intersection varieties. Am. J. Math. 126, 1355–1365 (2004)
  • Ein, L., Mustaţă, M.: Jet schemes and singularities. Proc. Symp. Pure Math. 80(2), 505–546 (2009)
  • Encinas, S., Villamayor, O.: A course on constructive desingularization and equivariance. Resolution of singularities. Obergurgl: Progr. Math.,...
  • Encinas, S., Villamayor, O.: A new proof of desingularization over fields of characteristic zero. Proceedings of the international conference...
  • Encinas, S., Villamayor, O.: Rees algebras and resolution of singularities. Proceedings of the XVIth Latin American algebra colloquium (Spanish),...
  • Hickel, M.: Sur quelques aspects de la géométrie de l’espace des arcs tracés sur un espace analytique. Annales de la faculté des sciences...
  • Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero I, II. Ann. Math. 79(2), 109–326 (1964)
  • Hironaka, H.: Idealistic exponents of a singularity. J.J Sylvester Sympos., Baltimore, Md.: Algebraic Geometry. The Johns Hopkins Centennial...
  • Ishii, S.: Geometric properties of jet schemes. Commun. Algebra 39(5), 1872–1882 (2011)
  • Lejeune-Jalabert, M.: Courbes Tracées sur un Germe D’Hypersurface. Am. J. Math. 112(4), 525–568 (1990)
  • Mustaţă, M.: Spaces of arcs in birational geometry. In: Lecture notes (available at the author’s personal web page)
  • Mustaţă, M.: Jet schemes of locally complete intersection canonical singularities, with an appendix by David Eisenbud and Edward Frenkel....
  • Mustaţă, M.: Singularities of pairs via jet schemes. J. Am. Math. Soc. 15, 599–615 (2002)
  • Nobile, A.: Equivalence and resolution of singularities. J. Algebra 420, 161–185 (2014)
  • Villamayor U, O.E.: Constructiveness of Hironaka’s resolution. Ann. Sci. École. Norm. Sup 22(1), 1–32 (1989)
  • Villamayor U, O.E.: Patching local uniformizations. Ann. Sci. École. Norm. Sup. (4) 25(6), 629–677 (1992)
  • Villamayor, O.E.: Tschirnhausen transformations revisited and the multiplicity of the embedded hypersurface. In: Colloquium on Homology and...
  • Villamayor U, O.: Hypersurface singularities in positive characteristic. Adv. Math. 213(2), 687–733 (2007)
  • Villamayor U, O.: Rees algebras on smooth schemes: integral closure and higher differential operator. Rev. Mat. Iberoam. 24(1), 213–242 (2008)
  • Villamayor U, O.E.: Equimultiplicity, algebraic elimination, and blowing-up. Adv. Math. 262, 313–369 (2014)
  • Vojta, P.: Jets via Hasse–Schmidt derivations. In: Diophantine Geometry. CRM Series, 4th ed., pp. 335–361. Norm., Pisa (2007)