Models of Curves and Finite Covers

Abstract

Let K be a discrete valuation field with ring of integers ${\mathcal O}$$_K$. Let f:X→Y be a finite morphism of curves over K. In this article, we study some possible relationships between the models over ${\mathcal O}$$_K$ 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 X$_L$ has semi-stable reduction.

Let f:X→Y be a finite morphism, with g(Y)[ges ]2. We show that if X has a stable model ${\mathcal X}$ over ${\mathcal O}$$_K$, then Y has a stable model ${\mathcal Y}$ over ${\mathcal O}$$_K$, and the morphism f extends to a morphism ${\mathcal X}$→${\mathcal Y}$.

Finally, given any finite morphism f:X→Y, is it possible to choose suitable regular models ${\mathcal X}$ and ${\mathcal Y}$ of X and Y over ${\mathcal O}$$_K$ such that f extends to a finite morphism ${\mathcal X}$→${\mathcal Y}$? 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 [ges ] 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.