Models of Curves and Finite Covers

Abstract

Let K be a discrete valuation field with ring of integers OK .L etf :X! Y be afi nite morphism of curves overK. In this article, we study some possible relationships between the models over OK ofX and ofY . Three such relationships are listed below. Consider a Galois coverf :X!Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semi-stable reduction over K ,t henX achieves semi-stable reduction over some explicit tame extension of K.B/ .W henK is strictly henselian, we determine the minimal extensionL=K with the property thatXL has semi-stable reduction. Let f :X ! Y be a finite morphism, with g.Y/> 2. We show that if X has a stable model X over OK ,t henY has a stable model Y over OK , and the morphism f extends to a morphism X! Y. Finally, given any finite morphism f :X! Y , is it possible to choose suitable regular models X and Y of X and Y over OK such that f extends to a finite morphism X! Y ?A s was shown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situ- ations, withf a cyclic cover of any order> 4. On the other hand, we prove, without any hypotheses on the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3. Mathematics Subject Classifications ( 1991): 14G20, 11G20, 14H30, 14H25. Let OK be a Dedekind domain with field of fractions K.L etf :X ! Y be a finite 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 first 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. Definitions and standard facts about models are reviewed in the first section. Let B Y denote the branch locus of f ,a nd letK.B/ be the compositum, in an algebraic closure of K, of the residue fields of points of B. In the second section, we consider a Galois cover X! Y of degree prime top and show that, ifY has semi-stable reduction over K ,t henX achieves semi-stable reduction over an explicit tame extension of the field K.B/ (Theorem 2.3). When K is strictly Henselian, there exist extensions LX=K and