Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Toric singularities: Log-blow-ups and global resolutions

Author: Wiesława Nizioł
Journal: J. Algebraic Geom. 15 (2006), 1-29
Published electronically: June 27, 2005
MathSciNet review: 2177194
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Abstract | References | Additional Information

Abstract: We prove that singularities of Kato’s log-regular schemes (a base-free analogue of toroidal embeddings) can be resolved globally by a log-blow-up. This is done by showing that the classical desingularization algorithms can be globalized and extended to log-schemes.

References [Enhancements On Off] (What's this?)

  • Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302. MR 1440306, DOI
  • Siegfried Bosch and Werner Lütkebohmert, Formal and rigid geometry. I. Rigid spaces, Math. Ann. 295 (1993), no. 2, 291–317. MR 1202394, DOI
  • Br J.-L. Brylinski, Eventails et variétés torique, Séminaire sur les singularités des surfaces, Centre de Math. de l’École Polytechnique, Palaiseau 1976-1977, Lect. Notes Math. 777, Springer-Verlag, Berlin, Heidelberg and New York, 1980, 247–288.
  • David A. Cox, Toric varieties and toric resolutions, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 259–284. MR 1748623
  • V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247 (Russian). MR 495499
  • Günter Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, vol. 168, Springer-Verlag, New York, 1996. MR 1418400
  • FK K. Fujiwara, K. Kato, Logarithmic étale topology theory, preprint, 1995.
  • William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037
  • I L. Illusie, Logarithmic smoothness and vanishing cycles, preprint, 1996.
  • Luc Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology, Astérisque 279 (2002), 271–322. Cohomologies $p$-adiques et applications arithmétiques, II. MR 1922832
  • Takeshi Kajiwara, Logarithmic compactifications of the generalized Jacobian variety, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40 (1993), no. 2, 473–502. MR 1255052
  • Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191–224. MR 1463703
  • Kazuya Kato, Toric singularities, Amer. J. Math. 116 (1994), no. 5, 1073–1099. MR 1296725, DOI
  • Ka2 K. Kato, Logarithmic degeneration and Dieudonné theory, preprint.
  • G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518
  • O A. Ogus, Logarithmic de Rham cohomology, preprint, 1997. Ts T. Tsuji, Saturated morphisms of logarithmic schemes, preprint, 1997.

Additional Information

Wiesława Nizioł
Affiliation: Department of Mathematics, College of Science, University of Utah, Salt Lake City, Utah 84112-0090

Received by editor(s): January 6, 2003
Received by editor(s) in revised form: February 10, 2005
Published electronically: June 27, 2005