Models of Curves and Nite Covers

Let O K be a Dedekind domain with eld of fractions K. Let f : X ! Y be a nite morphism of projective, smooth, and geometrically connected curves over Spec(K). In this paper, we study some possible relationships between the models of X and of Y. In the rst part of the paper, we look at semi-stable and stable models, while in the second part we investigate regular models. Let us describe the content of this paper. Deenitions and standard facts about models are reviewed in the rst section. Let B Y denote the branch locus of f, and let K(B) be the compositum, in an algebraic closure of K, of the residue elds of points of B. In the second section, we consider a Galois cover X ! Y of degree prime to p and show that, if Y has semi-stable reduction over K, then X achieves semi-stable reduction over an explicit tame extension of the eld K(B) (Theorem 2.3). When K is strictly henselian, there exist extensions L X =K and L Y =K minimal with the property that X L X and Y L Y have semi-stable reduction. In the third section, we assume that K is strictly henselian and strengthen the result obtained in the second section to show that for a Galois cover of degree prime to p, the p-part of L X =K is equal to the compositum of the p-part of L Y =K and the p-part of K(B)=K (Corollary 3.2). We later completely describe L X in terms of L Y K(B) and some vertical ramiication data (Theorem 3.9). Let f : X ! Y be a nite morphism, with g(Y) 2. In the fourth section, we show that if X has a stable model X=O K , then Y has a stable model Y=O K , and f extends to a (not necessarily nite) morphism ' : X ! Y (Proposition 4.4). As a corollary, we give a new proof a theorem of Lange which states that if X has good reduction, then Y has good reduction. Given any nite morphism f : X ! Y as above, it is interesting in some situations to be able to compare the regular models of X and Y over O K. In particular, it is natural to wonder whether it is possible to choose suitable regular models X=O K and Y=O K …