Models of curves and finite covers, by Qing Liu and Dino Lorenzini

This paper has appeared in Compositio Math 118, 61-102 (1999) and so the preprint has been removed. Let K be a discrete valuation field with ring of integers O_K. Let f : X \to Y be a finite morphism of curves over K. In this article, we study some possible relationships between the models over O_K of X and of Y. Three such relationships are listed below.

Consider a Galois cover f : X \to 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 X_L has semi-stable reduction.

Let f : X \to Y be a finite morphism, with g(Y) \ge 2. We show that if X has a stable model {\cal X} over O_K, then $Y$ has a stable model {\cal Y} over O_K, and the morphism f extends to a morphism {\cal X} \to {\cal Y}.

Finally, given any finite morphism f : X \to Y, is it possible to choose suitable regular models {\cal X} and {\cal Y} of X and Y over O_K such that f extends to a finite morphism {\cal X} \to {\cal 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 \ge 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.

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Qing Liu and Dino Lorenzini <,>