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H. Hasson
H. Hasson
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Question

Say we have a map, C->D, of relative curves over a Dedekind scheme, S. What methods are availablesome 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.

Question

Say we have a map, C->D, of relative curves over a Dedekind scheme, S. What methods are available of 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.

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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H. Hasson
H. Hasson
  • 1.5k
  • 10
  • 25

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 methods are available of 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.