# Methods of showing a map has integral or good reduction

### Question

Say we have a map, C->D, of relative curves over a Dedekind scheme, S. What are some of the available methods for showing that this map has good reduction, or integral reduction, at some s∈S? By this I mean: what are some popular conditions that imply this? What are the tricks people usually use?

### Clarification

By a map having good reduction I mean that both Cs and Ds are regular integral curves. By integral reduction I mean that both Cs and Ds are integral curves.

You may assume whatever you want, this is part of the question. Assuming, for example, that C->D is generically Galois; or that D is smooth over S; is legitimate. This is pretty open-ended. Hence, community wiki.

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## 2 Answers

You might have a look here.

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Marvelous! This is exactly what I had in mind. Thank you, and welcome to mathoverflow. –  H. Hasson Jan 23 '10 at 18:20

I don't see why there is a map. Why don't you ask when C/S has good reduction? For this, look at Liu, chapter 10.1.

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That's because I was hoping for some theorems exhibiting the interplay between good/integral reduction on the bottom and on the top; taking into consideration other conditions, like ramification behavior. –  H. Hasson Jan 15 '10 at 14:32