Stable reductions of algebraic curves. (English) Zbl 0588.14021

From the author’s introduction: Let X be a projective, absolutely non- singular and irreducible curve of genus \(\geq 2\) over a field k which is complete with respect to a discrete valuation. A theorem of P. Deligne and D. Mumford [Publ. Math., Inst. Haut. Étud. Sci. 36 (1969), 75-109 (1970; Zbl 0181.488)] states: (*) there exists a finite field extension \(\ell\) of k and a projective, flat \(\ell^ 0\)-scheme \(\chi\) of finite presentation \((\ell^ 0\) denotes the valuation ring of \(\ell\) and \({\bar \ell}\) is the residue field of \(\ell)\) such that \((i)\quad \chi \times \ell \cong X\times_ k\ell\) and \((ii)\quad \chi \times {\bar \ell}\) is a stable curve over \({\bar \ell}\). In this paper the result (*) is generalized to the case of arbitrary complete fields k (w.r.t. some non-archimedean valuation. The proof is necessarily very far from the known algebraic proofs of stable reduction since Néron minimal models and desingularization of surfaces lose their meaning or validity for non-discrete valuation rings.
Reviewer: H.Yanagihara


14H25 Arithmetic ground fields for curves
14G20 Local ground fields in algebraic geometry
12J25 Non-Archimedean valued fields


Zbl 0181.488