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I am confused about the notion of good reduction. Let $R$ be a DVR, let $K$ be its fraction field. If we have a smooth proper $K$-scheme $V$, then I believe $V$ is said to have good reduction at the unique non-zero prime ideal if there exists a smooth proper $R$-scheme whose generic fiber is $V$.

I tend to dislike the word "exists". I think for abelian varieties, a condition equivalent to good reduction can be formulated in terms of the 1st cohomology. Can a condition not involving the existential quantifier and equivalent to good reduction be given for general smooth proper $K$-schemes?

A second question, if $F$ is a number field, then what is the right notion of good reduction modulo a non-zero prime ideal of the ring of integers of $F$ for smooth proper $F$-schemes? The issue for me is the compatibility between different prime ideals: if a smooth proper $F$-scheme has "good reduction everywhere", does it mean that there is a single integral model that has smooth fibers over every prime ideal, or just that for any prime ideal you can find a (proper flat, or I don't know what should be required really) model that has smooth fiber over that prime ideal?

Third question: given a smooth proper scheme, is there some functorially constructed "best" integral model so that all questions of reduction can be just answered using that particular model? I have heard something about Neron models but I think they only work for abelian varieties.

I apologize for these naive questions but all references I found so far refer to good reduction without giving a definition. If there is a reference addressing the above questions I will gladly study it.

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(1) Can a condition not involving the existential quantifier and equivalent to good reduction be given for general smooth proper $K$-schemes?

Not in terms of their "topology" in general. For abelian varieties, there's the Neron-Ogg-Shafarevich criterion: $X$ has good reduction iff $H^1(X_{\bar K}, \mathbf{Q}_\ell)$ is unramified (there is also a $p$-adic Hodge theory variant if $R$ has mixed characteristic $(0,p)$, due to Coleman-Iovita). This has been extended to K3 surfaces recently by Liedtke-Matsumoto and Chiarellotto-Lazda-Liedtke, but even there the answer is quite subtle if the residue field is not algebraically closed (roughly speaking, one can only detect good reduction after an unramified extension using $H^2$, and to detect good reduction one has to do some hard work). Already curves give an example where cohomology is insufficient: there exist "curves of compact type" i.e. with bad reduction but whose Jacobian has good reduction (e.g. a curve whose model has a special fiber whose dual graph has no loops). Andreatta-Iovita-Kim provide a criterion in terms of the Galois action on the geometric fundamental group.

One could wonder whether looking at the Galois action on the etale homotopy type can see good reduction. I don't know the answer, but it might be not too difficult to find a counterexample.

EDIT. I recalled the example due to Cynk and van Straten to the effect that already for Calabi-Yau threefolds, trivial monodromy does not imply (potential) good reduction. Moreover, Chiarellotto-Lazda-Liedtke (see Theorem 4.5) show that for Enriques surfaces one cannot detect good reduction using cohomology of local systems (equivalently, the cohomology of the K3 double cover).

(2) If $F$ is a number field, then what is the right notion of good reduction modulo a non-zero prime ideal of the ring of integers of $F$ for smooth proper $F$-schemes?

To me, this would mean that for every maximal ideal $\mathfrak{p} \subseteq\mathcal{O}_K$, the base change of $X$ to the henselian (or complete, shouldn't matter) local ring of $\operatorname{Spec}\mathcal{O}_K$ at $\mathfrak{p}$ has good reduction in the sense of the definition over a dvr you gave.

An alternative definition would be that $X$ has a smooth proper model over $\mathcal{O}_K$.

For abelian varieties the two notions coincide because of Neron models.

(3) Given a smooth proper scheme, is there some functorially constructed "best" integral model so that all questions of reduction can be just answered using that particular model?

I think the answer is no. Again the most studied case beyond abelian varieties is curves (where one has the Deligne-Mumford compactification) and K3 surfaces (where one has so-called Kulikov models, whose existence is conjectural in general, and which are not unique).

If, however, you are interested with smooth and proper models, and you allow yourself to fix a polarization, then the theorem of Mumford and Matsusaka might be useful. See the answers to this question.

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