Models of Curves and Finite Covers

Let K be a discrete valuation field with ring of integers OK . Let f :X → Y be a finite morphism of curves overK . In this article, we study some possible relationships between the models over OK of X and of Y . Three such relationships are listed below. Consider a Galois cover f :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 , then X achieves semi-stable reduction over some explicit tame extension of K(B). When K is strictly henselian, we determine the minimal extension L/K with the property that XL 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 , then Y 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 ? As was shown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situations, with f 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