Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Log smooth extension of a family of curves and semi-stable reduction

Author: Takeshi Saito
Translated by:
Journal: J. Algebraic Geom. 13 (2004), 287-321
Published electronically: December 3, 2003
MathSciNet review: 2047700
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Abstract | References | Additional Information

Abstract: We show that a family of smooth stable curves defined on the interior of a log regular scheme is extended to a log smooth scheme over the whole log regular scheme, if it is so at each generic point of the boundary, under a very mild assumption. We also include a proof of the fact that a log smooth scheme over a discrete valuation ring has potentially a semi-stable model. As a consequence, we show that a hyperbolic polycurve in the sense of Mochizuk over a discrete valuation field has potentially a proper semi-stable model if the characteristic of the residue field is sufficiently large.

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

  • 1. D.Abramovich and Jong, Smoothness, semistability, and toroidal geometry, J. of Algebraic Geometry, 6 (1997) 789-801.
  • 2. P.Deligne, La formule de Picard-Lefschetz, Exp. XV, SGA 7 II, LNM 340, Springer, (1973) 165-196.
  • 3. P.Deligne and D.Mumford, The irreducibility of the space of curves of given genus, Publ. Math. IHES, 36 (1970) 75-109.
  • 4. A. de Jong and F. Oort, On extending families of curves, J. of Algebraic Geometry, 6 (1997) 545-562.
  • 5. A.Grothendieck with J.Dieudonné, Eléments de Géométrie Algébrique II, Publ. Math. IHES 8 (1961).
  • 6. L.Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology, Astérisque, 279, 2002, pp. 271-322.
  • 7. K.Kato, Logarithmic structure of Fontaine-Illusie, Algebraic Analysis, Geometry and Number Theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191-224.
  • 8. - Toric singularities, American J. of Math., 116 (1994) 1073-1099.
  • 9. G.Kempf, F.Knudsen, D.Mumford, and B.Saint-Donat, Toroidal Embeddings I, LNM 339, 1973, Springer.
  • 10. S.Mochizuki, Extending families of curves over log regular schemes, J. Reine Angew. Math., 511 (1999), 43-71.
  • 11. C.Nakayama, Nearby cycles for log smooth families, Compositio Math., 112 (1998), 45-75.
  • 12. W.Niziol, Toric singularities: Log-blow-ups and global resolutions, J. Alg. Geom., to appear.
  • 13. T.Saito, Vanishing cycles and geometry of cuves over a discrete valuation ring, Amer. J. of Math. 109, (1987), 1043-1085.
  • 14. T.Tsuji, Saturated morphisms of log schemes, preprint, (1997).
  • 15. H.Yoshioka, Semistable reduction theorem for logarithmically smooth varieties, Master thesis at Univ. of Tokyo, (1995).

Additional Information

Takeshi Saito
Affiliation: Department of Mathematics, University of Tokyo, Tokyo 153-8914 Japan

Received by editor(s): October 3, 2001
Published electronically: December 3, 2003

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
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Online ISSN 1534-7486; Print ISSN 1056-3911
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