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.

There are some figures in separated eps files. You need them in order to view papier.dvi. But you can also get the single file papier.ps which contains all the figures.

[Note: Several figures have been prepared separately as postscript files, so download them also if you prefer to use the dvi file, but it might be better to simply download the postscript file, which already incorporates the figures.]



Qing Liu and Dino Lorenzini <liu@math.u-bordeaux.fr, lorenz@math.uga.edu>