Stable Reduction of Finite Covers of Curves

Let K be the function field of a connected regular scheme S of dimension 1, and let f : X → Y be a finite cover of projective smooth and geometrically connected curves over K with g(X) ≥ 2. Suppose that f can be extended to a finite cover X → Y of semi-stable models over S (it is known that this is always possible up to finite separable extension of K). Then there exists a unique minimal such cover. This gives a canonical way to extend X → Y to a finite cover of semi-stable models over S. Let S be a Dedekind scheme (i.e a connected Noetherian regular scheme of dimension 1), with field of functions K := K(S). Let f : X → Y be a finite morphism of smooth geometrically connected projective curves overK. We can ask how to extend, in some canonical way, the morphism f to a morphism of models of X and Y over S. It is proved in [15],4.4 that if X and Y have stable models X ,Y over S, then f extends uniquely to a morphism X st → Y. However we will in general lose the finiteness of f . On the other hand, after a finite separable extension of K, f extends to a finite morphism of semi-stable models X → Y over S ([4]; [15], Remark 4.6; and Corollary 3.10 below). Following Coleman [4], such a pair X → Y is called a semi-stable model of f , and it is called stable if moreover it is minimal among the semi-stable models of f (cf. 3.1). The stable model of f (if it exists) is unique up to isomorphism. Theorem 0.1. (Corollary 4.6) Suppose that either g(X) ≥ 2, or g(X) = 1 and X has potentially good reduction. Then there exists a finite separable extension K ′ of K such that XK′ → YK′ admits a stable model X ′ → Y ′ over S, where S is the integral closure of S in K . Moreover, for any Dedekind scheme T dominating S, X ′ T → Y ′ T is the stable model of XK(T ) → YK(T ) over T . This gives a canonical way to extend a finite cover of projective smooth curves overK to a finite cover of semi-stable models over (some finite cover of) S. The last part of the theorem says that the stable model of f commutes with flat base change. The stable model of f should be seen as an analogue of the stable model of a curve. In a forthcoming work, we will apply this theorem to study a compactification of Hurwitz moduli spaces of finite covers of curves. We also prove the theorem for smooth marked curves X,Y . Note that in general X ,Y ′ are not the respective stable models of the curves XK′ , YK′ . The proof of 0.1 is based on a more general result: Theorem 0.2. (Theorem 4.5) Let f : X → Y be a finite morphism of smooth geometrically connected projective curves over K. Let X ,Y be respective models of X and Y over S. Then there exists a finite separable extension K ′ of K, and models X ,Y ′ of XK′ and YK′ over S ′ (integral closure of S in K ) such that 2000 Mathematics Subject Classification. 14G20, 14H30, 14D10, 11G20.