A logarithmic view towards semistable reduction
Author:
Jakob Stix
Journal:
J. Algebraic Geom. 14 (2005), 119-136
DOI:
https://doi.org/10.1090/S1056-3911-04-00388-1
Published electronically:
June 24, 2004
MathSciNet review:
2092128
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Abstract |
References |
Additional Information
Abstract: A smooth, proper family of curves creates a monodromy action of the fundamental group of the base on the $\textrm {H}^1$ of a fibre. The geometric condition of T. Saito for the action of the wild inertia of a boundary point to be trivial is transformed to the condition of logarithmic smooth reduction. The proof emphasizes methods and results from logarithmic geometry. It applies to quasi-projective smooth curves with étale boundary divisor.
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[Vi02]Vith Vidal, I., Contributions à la cohomologie étale des schémas et des log schémas, thèse, 2002.
[Ab00]Abbes Abbes, A., Réduction semistable des courbes d’après Artin, Deligne, Grothendieck, Mumford, Saito, Winters, $\ldots$, in: Courbes semi-stables et groupe fondamental en géométrie algébrique (ed. J.-B. Bost, F. Loeser, M. Raynaud), Prog. Math. 187, Birkhäuser (2000), 59–110.
[Ba95]Bauer Bauer, W., On smooth, unramified, étale and flat morphisms of fine logarithmic schemes, Math. Nachr. 176 (1995), 5–16.
[Ha59]Hall Hall, M. jr., The theory of groups, The Macmillan Co., New York (1959).
[Il02]OV Illusie, L., An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology, Astérisque 279 (2002), 271–322.
[Ka89]KaJHU Kato, K., Logarithmic structures of Fontaine–Illusie, in Algebraic Analysis, Geometry and Number Theory, The Johns Hopkins University Press (1989), 191–224.
[Ka94]Kats Kato, K., Toric singularities, Amer. J. Math. 116 (1994), 1073–1099.
[Li68]Lich Lichtenbaum, S., Curves over discrete valuation rings, Amer. J. Math. 90 (1968), 380–405.
[Na98]Nacycles Nakayama, C., Nearby cycles for log smooth families, Comp. Math. 112 (1998), 45–75.
[Sai87]Saito Saito, T., Vanishing cycles and geometry of curves over a discrete valuation ring, Amer. J. Math. 109 (1987), 1043–1085.
[Sai04]Saito2 Saito, T., Log smooth extension of a family of curves and semi-stable reduction, J. Algebraic Geom. 13 (2004), 287–321.
[Sx02]Stix Stix, J., Projective anabelian curves in positive characteristic and descent theory for log étale covers, thesis, Bonner Mathematische Schriften 354 (2002).
[Vi02]Vith Vidal, I., Contributions à la cohomologie étale des schémas et des log schémas, thèse, 2002.
Additional Information
Jakob Stix
Affiliation:
Mathematisches Institut, Universität Bonn, Beringstraße 1, 53115 Bonn, Germany
Email:
stix@math.uni-bonn.de
Received by editor(s):
May 13, 2003
Received by editor(s) in revised form:
February 10, 2004
Published electronically:
June 24, 2004
Additional Notes:
The author acknowledges the financial support provided through the European Community’s Human Potential Program under contract HPRN-CT-2000-00114, GTEM
