Hi everyone,


let X be a variety over a field k, S an integral scheme such that the function field K of S is contained in k. An S-scheme **X** is called model of X/k if **X** x\_S k = X, i.e. if the generic fiber of **X** over S is isomorphic to X.


* Are there general conditions on X, S, k, such that **X** exists?
* If X is smooth and projective, what are the conditions, such that there is a smooth model **X**?
* Any good references that go into models and reduction in general, and not only in the case of curves?



Existence of (smooth) models
I would also be very interested if someone could point to good references for this question, but I am not yet allowed to vote for the question.


Nekovar's survey article on the Beilinson conjectures from the early 90s mentions some results for varieties over Q. He says in section 5.3 that given a smooth projective variety over Q, there always exists a proper flat model over Z, but that a regular such model is rarely known to exist. However, in the published version of the same survey, there is an added note at the very end of the article saying that "Spivakovsky recently announced a general result on resolution of singularities, which implies that a regular proper flat model of X mentioned in 5.3 always exist". However, I have never seen this result of Spivakovsky mentioned anywhere else, so I doubt that it is true. Does anyone else know more about this?


The survey is available here:
<http://people.math.jussieu.fr/~nekovar/pu/mot.pdf>


For the published version, google "Serre Jannsen Motives", click at the Google Books link, and then search for "Spivakovsky" within the book.


In general for the case when k is a number field and S is the ring of integers in k, it seems unreasonable to ask for general theorems about the existence of smooth models, although one could maybe hope for something about regular models. For example, I think there are no nontrivial smooth and proper curves over Spec(Z) at all (although there are smooth and proper such schemes of higher dimension, and there are also smooth and proper such schemes in any dimension, including 1, over rings of integers in other number fields).


