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.