Stable modification of relative curves

Temkin, Michael

doi:10.1090/S1056-3911-2010-00560-7

Author: Michael Temkin
Journal: J. Algebraic Geom. 19 (2010), 603-677
DOI: https://doi.org/10.1090/S1056-3911-2010-00560-7
Published electronically: June 9, 2010
MathSciNet review: 2669727
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Abstract | References | Additional Information

Abstract: We generalize theorems of Deligne-Mumford and de Jong on semi-stable modifications of families of proper curves. The main result states that after a generically étale alteration of the base any (not necessarily proper) family of multipointed curves with semi-stable generic fiber admits a minimal semi-stable modification. The latter can also be characterized by the property that its geometric fibers have no certain exceptional components. The main step of our proof is uniformization of one-dimensional extensions of valued fields. Riemann-Zariski spaces are then used to obtain the result over any integral base.

References

D. Abramovich and A. J. de Jong, Smoothness, semistability, and toroidal geometry, J. Algebraic Geom. 6 (1997), no. 4, 789–801. MR 1487237
D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000), no. 2, 241–273. MR 1738451, DOI https://doi.org/10.1007/s002229900024
Dan Abramovich and Frans Oort, Alterations and resolution of singularities, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 39–108. MR 1748617, DOI https://doi.org/10.1007/978-3-0348-8399-3_3
M. Artin and G. Winters, Degenerate fibres and stable reduction of curves, Topology 10 (1971), 373–383. MR 476756, DOI https://doi.org/10.1016/0040-9383%2871%2990028-0
Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709
Vladimir G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 5–161 (1994). MR 1259429
Vladimir G. Berkovich, Smooth $p$-adic analytic spaces are locally contractible, Invent. Math. 137 (1999), no. 1, 1–84. MR 1702143, DOI https://doi.org/10.1007/s002220050323
S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961
Siegfried Bosch and Werner Lütkebohmert, Stable reduction and uniformization of abelian varieties. I, Math. Ann. 270 (1985), no. 3, 349–379. MR 774362, DOI https://doi.org/10.1007/BF01473432
Siegfried Bosch and Werner Lütkebohmert, Formal and rigid geometry. II. Flattening techniques, Math. Ann. 296 (1993), no. 3, 403–429. MR 1225983, DOI https://doi.org/10.1007/BF01445112
Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Formal and rigid geometry. IV. The reduced fibre theorem, Invent. Math. 119 (1995), no. 2, 361–398. MR 1312505, DOI https://doi.org/10.1007/BF01245187

Bourbaki, N.: Algèbre commutative, Hermann, Paris, 1961.

Brian Conrad, Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc. 22 (2007), no. 3, 205–257. MR 2356346

Brian Conrad, Grothendieck duality and base change, Lecture Notes in Mathematics, vol. 1750, Springer-Verlag, Berlin, 2000. MR 1804902

Pierre Deligne, Le lemme de Gabber, Astérisque 127 (1985), 131–150 (French). Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84). MR 801921

P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240

Ducros, A: Toute forme modérément ramiée d’un polydisque ouvert est triviale, preprint.

A. Johan de Jong, Families of curves and alterations, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 2, 599–621. MR 1450427

Dieudonné, J.; Grothendieck, A.: Éléments de géométrie algébrique, Publ. Math. IHES, 4, 8, 11, 17, 20, 24, 28, 32, (1960-7).

Dieudonné, J.; Grothendieck, A.: Éléments de géométrie algébrique, I: Le langage des schemas, second edition, Springer, Berlin, 1971.

Helmut P. Epp, Eliminating wild ramification, Invent. Math. 19 (1973), 235–249. MR 321929, DOI https://doi.org/10.1007/BF01390208

Kazuhiro Fujiwara and Fumiharu Kato, Rigid geometry and applications, Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math., vol. 45, Math. Soc. Japan, Tokyo, 2006, pp. 327–386. MR 2310255, DOI https://doi.org/10.2969/aspm/04510327

D. Gieseker, Lectures on moduli of curves, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 69, Published for the Tata Institute of Fundamental Research, Bombay; Springer-Verlag, Berlin-New York, 1982. MR 691308

Ofer Gabber and Lorenzo Ramero, Almost ring theory, Lecture Notes in Mathematics, vol. 1800, Springer-Verlag, Berlin, 2003. MR 2004652

Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157

Roland Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30, Friedr. Vieweg & Sohn, Braunschweig, 1996. MR 1734903

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

Kuhlmann, F.-V.: Elimination of Ramification I: The Generalized Stability Theorem, to appear in Trans. Amer. Math. Soc.

Kuhlmann, F.-V.: Elimination of Ramification II: Henselian Rationality of Valued Function Fields, in preparation.

Joseph Lipman, Desingularization of two-dimensional schemes, Ann. of Math. (2) 107 (1978), no. 1, 151–207. MR 491722, DOI https://doi.org/10.2307/1971141

Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461

Marius van der Put, Stable reductions of algebraic curves, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), no. 4, 461–478. MR 770734

Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1–89 (French). MR 308104, DOI https://doi.org/10.1007/BF01390094

Takeshi Saito, Vanishing cycles and geometry of curves over a discrete valuation ring, Amer. J. Math. 109 (1987), no. 6, 1043–1085. MR 919003, DOI https://doi.org/10.2307/2374585

M. Temkin, On local properties of non-Archimedean analytic spaces, Math. Ann. 318 (2000), no. 3, 585–607. MR 1800770, DOI https://doi.org/10.1007/s002080000123

M. Temkin, On local properties of non-Archimedean analytic spaces. II, Israel J. Math. 140 (2004), 1–27. MR 2054837, DOI https://doi.org/10.1007/BF02786625

Temkin, M.: Relative Riemann-Zariski spaces, preprint, arXiv:[0804.2843], to appear in Isr. J. of Math.

Temkin, M.: Inseparable local uniformization, preprint, arXiv:[0804.1554].

R. W. Thomason and Thomas Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, DOI https://doi.org/10.1007/978-0-8176-4576-2_10

Oscar Zariski, Local uniformization on algebraic varieties, Ann. of Math. (2) 41 (1940), 852–896. MR 2864, DOI https://doi.org/10.2307/1968864

Additional Information

Michael Temkin
Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
Address at time of publication: Institute of Mathematics, Hebrew University, Giv$’$at-Ram, 91904 Jerusalem, Israel
MR Author ID: 332870
Email: temkin@math.ias.edu, temkin@math.huji.ac.il

Received by editor(s): July 26, 2007
Received by editor(s) in revised form: February 24, 2010
Published electronically: June 9, 2010
Additional Notes: This article is based on a portion of my Ph.D. thesis; I want to thank my advisor Professor V. Berkovich. I am absolutely indebted to B. Conrad and I owe a lot to A. Ducros for pointing out various mistakes and inaccuracies, and for many suggestions that led to two revisions of the paper that improved the exposition. I express my deep gratitude to the Israel Clore Foundation for supporting my doctoral studies and to the Max Planck Institute for Mathematics, where a portion of this paper was written. A final revision was made when the author was staying at the IAS; the author was supported by NFS grant DMS-0635607.