ISSN:
1570-5846
Keywords:
branch locus
;
covers
;
curves
;
discrete valuation ring
;
models
;
semi-stable
;
simultaneous resolution.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let K be a discrete valuation field with ring of integers O K .Letf : X ! 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 ! 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,thenX achieves semi-stable reduction over some explicit tame extension of K.B/.WhenK 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/ 〉 2. We show that if X has a stable model X over O K ,thenY has a stable model Y over O K , and the morphism f extends to a morphism X ! Y. ! 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 O K such that f extends to a finite morphism X ! Y ?As wasshown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situ-ations, 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 order 2 or 3.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1001141725199
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