Stable reduction and uniformization of abelian varieties. I. (English) Zbl 0554.14012

Let R be a discrete valuation ring and denote by K its field of fractions. The ”stable reduction theorem” of P. Deligne and D. Mumford [cf. Publ. Math., Inst. Haut. Étud. Sci. 36(1969), 75-109 (1970; Zbl 0181.488)] asserts that each smooth projective curve C over K of genus \(\geq 2\) has potential stable reduction. This means that (up to finite separable extension of the ground field K) C can be extended to a curve \({\mathcal C}\) over R with the following properties: (a) The singularities of the special fibre \({\mathcal C}_ s\) of \({\mathcal C}\) are at most ordinary double points. (b) Each smooth rational component of \({\mathcal C}_ s\) meets other components of \({\mathcal C}_ s\) in at least 3 different points. In the first part of their work, the authors discuss stable reductions of curves in terms of rigid analysis. They prove an analytic version of the stable reduction theorem, valid also for non- discrete valuations, and show how to deduce the algebraic version of Deligne-Mumford from it. The stable reduction theorem is an indispensible tool for attacking the uniformization of curves and also for many questions arising from number theory. See part II of this work for applications to abelian varieties (Invent. Math. 78, 257-297 (1984; Zbl 0554.14015)].


14H25 Arithmetic ground fields for curves
13F30 Valuation rings
14G20 Local ground fields in algebraic geometry
Full Text: DOI EuDML


[1] Bosch, S., Lütkebohmert, W.: Stable reduction and uniformization of abelian varieties II. Invent. Math.78, 257-297 (1984) · Zbl 0554.14015
[2] Abhyankar, S.: Resolution of singularities of arithmetical surfaces. Proc. Conf. on Arithm. Alg. Geometry, Purdue, 1963 · Zbl 0147.20503
[3] Artin, M., Winters, G.: Degenerate fibres and stable reduction of curves. Topology,10, 373-383 (1971) · Zbl 0221.14018
[4] Bosch, S.: Orthonormalbasen in der nichtarchimedischen Funktionentheorie. Manuscripta Math.1, 35-57 (1969) · Zbl 0164.21202
[5] Bosch, S.:k-affinoide Tori. Math. Ann.192, 1-16 (1971) · Zbl 0214.19703
[6] Bosch, S.: Zur Kohomologietheorie rigid analytischer Räume. Manuscripta Math.20, 1-27 (1977) · Zbl 0343.14004
[7] Bosch, S.: Eine bemrkenswerte Eigenschaft der formellen Fasern affinoider Räume. Math. Ann.229, 25-45 (1977) · Zbl 0385.32008
[8] Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean analysis. A systematic approach to rigid analytic geometry. Grundlehren der Mathematischen Wissenschaften, Bd. 261. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0539.14017
[9] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math. IHES36 (1969) · Zbl 0181.48803
[10] Gerritzen, L., Grauert, H.: Die Azyklizität der affinoiden Überdeckungen. Global Analysis, Papers in Honor of K. Kodaira 159-184. University of Tokyo Press, Princeton University Press 1969 · Zbl 0197.17303
[11] Grauert, H., Remmert, R.: Über die Methode der diskret bewerteten Ringe in der nichtarchimedischen Funktionentheorie. Invent. Math.2, 87-133 (1966) · Zbl 0148.32401
[12] Kiehl, R.: Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie. Invent. Math.2, 256-273 (1967) · Zbl 0202.20201
[13] Kiehl, R.: Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie. Invent. Math.2, 191-214 (1967) · Zbl 0202.20101
[14] Köpf, U.: Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen. Schr. Math. Inst. Univ. Münster, 2. Serie, Heft 7 (1974) · Zbl 0275.14006
[15] Lütkebohmert, W.: Ein globaler Starrheitssatz für Mumfordkurven. J. Reine Angew. Math.340, 118-139 (1983) · Zbl 0505.14023
[16] Lipman, J.: Desingularization of two-dimensional schemes. Ann. Math.107, 151-207 (1978) · Zbl 0369.14005
[17] Mumford, D.: An analytic construction of degenerating curves over complete local fields. Compositio Math.24, 129-174 (1972) · Zbl 0228.14011
[18] Mumford, D.: An analytic construction of degenerating abelian varieties over complete rings. Compositio Math.24, 239-272 (1972) · Zbl 0241.14020
[19] Néron, A.: Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Publ. Math. IHES21 (1964)
[20] van der Put, M.: Stable reductions of algebraic curves. Report Rijksuniversiteit Groningen (1981) · Zbl 0509.14024
[21] Raynaud, M.: Modèles de Néron. C.R. Acad. Sci. Paris262, Série A, 345-347 (1966) · Zbl 0141.18203
[22] Raynaud, M.: Variétés abéliennes et géométrie rigide. Actes, Congrès intern. math.1, 473-477 (1970)
[23] Raynaud, M.: Spécialisation du functeur de Picard. C.R. Acad. Sci. Paris264, Série A. 941-943, 1001-1004 (1966)
[24] Grothendieck, A.: Groupes de monodromie en géométrie algébrique (SGA7I). Lecture Notes in Mathematics, Vol. 288. Berlin, Heidelberg, New York: Springer 1972
[25] Shafarevitch, J.R.: Lectures on minimal models and birational transformations of two-dimensional schemes. TIFR, No. 37 (1966)
[26] Tate, J.: Rigid analytic spaces. Private Notes (1962). Reprinted in Invent. Math.12, 257-289 (1971)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.