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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)].

MSC:
14H25 Arithmetic ground fields for curves
13F30 Valuation rings
14G20 Local ground fields in algebraic geometry
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