Log smooth extension of a family of curves and semi-stable reduction
Author:
Takeshi Saito
Journal:
J. Algebraic Geom. 13 (2004), 287-321
DOI:
https://doi.org/10.1090/S1056-3911-03-00338-2
Published electronically:
December 3, 2003
MathSciNet review:
2047700
Full-text PDF
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.
a-dj D.Abramovich and A.de Jong, Smoothness, semistability, and toroidal geometry, J. of Algebraic Geometry, 6 (1997) 789-801.
sga7 P.Deligne, La formule de Picard-Lefschetz, Exp. XV, SGA 7 II, LNM 340, Springer, (1973) 165-196.
d-m P.Deligne and D.Mumford, The irreducibility of the space of curves of given genus, Publ. Math. IHES, 36 (1970) 75-109.
dj-o A. de Jong and F. Oort, On extending families of curves, J. of Algebraic Geometry, 6 (1997) 545-562.
ega2 A.Grothendieck with J.Dieudonné, Eléments de Géométrie Algébrique II, Publ. Math. IHES 8 (1961).
illusie 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.
kk-log 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.
kk-toric ---Toric singularities, American J. of Math., 116 (1994) 1073-1099.
kkms G.Kempf, F.Knudsen, D.Mumford, and B.Saint-Donat, Toroidal Embeddings I, LNM 339, 1973, Springer.
mochi S.Mochizuki, Extending families of curves over log regular schemes, J. Reine Angew. Math., 511 (1999), 43-71.
naka C.Nakayama, Nearby cycles for log smooth families, Compositio Math., 112 (1998), 45-75.
niziol W.Niziol, Toric singularities: Log-blow-ups and global resolutions, J. Alg. Geom., to appear.
t-m T.Saito, Vanishing cycles and geometry of cuves over a discrete valuation ring, Amer. J. of Math. 109, (1987), 1043-1085.
tsuji T.Tsuji, Saturated morphisms of log schemes, preprint, (1997).
yoshi H.Yoshioka, Semistable reduction theorem for logarithmically smooth varieties, Master thesis at Univ. of Tokyo, (1995).
a-dj D.Abramovich and A.de Jong, Smoothness, semistability, and toroidal geometry, J. of Algebraic Geometry, 6 (1997) 789-801.
sga7 P.Deligne, La formule de Picard-Lefschetz, Exp. XV, SGA 7 II, LNM 340, Springer, (1973) 165-196.
d-m P.Deligne and D.Mumford, The irreducibility of the space of curves of given genus, Publ. Math. IHES, 36 (1970) 75-109.
dj-o A. de Jong and F. Oort, On extending families of curves, J. of Algebraic Geometry, 6 (1997) 545-562.
ega2 A.Grothendieck with J.Dieudonné, Eléments de Géométrie Algébrique II, Publ. Math. IHES 8 (1961).
illusie 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.
kk-log 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.
kk-toric ---Toric singularities, American J. of Math., 116 (1994) 1073-1099.
kkms G.Kempf, F.Knudsen, D.Mumford, and B.Saint-Donat, Toroidal Embeddings I, LNM 339, 1973, Springer.
mochi S.Mochizuki, Extending families of curves over log regular schemes, J. Reine Angew. Math., 511 (1999), 43-71.
naka C.Nakayama, Nearby cycles for log smooth families, Compositio Math., 112 (1998), 45-75.
niziol W.Niziol, Toric singularities: Log-blow-ups and global resolutions, J. Alg. Geom., to appear.
t-m T.Saito, Vanishing cycles and geometry of cuves over a discrete valuation ring, Amer. J. of Math. 109, (1987), 1043-1085.
tsuji T.Tsuji, Saturated morphisms of log schemes, preprint, (1997).
yoshi 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
MR Author ID:
236565
Email:
t-saito@ms.u-tokyo.ac.jp
Received by editor(s):
October 3, 2001
Published electronically:
December 3, 2003