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This is vaguely similar to, but quite different from, this question.

In the above linked question the focus is on fields of large cardinality per se. Here we fix a base field $k$ (say algebraically closed and of characteristic zero), and we consider "objects" $\xi$ over a base $k$-variety or scheme $S$ (for example, $\xi$ could be a morphism $X\to S$, a family of curves, a coherent sheaf on $S$, a flat family of coherent sheaves on some $X$ parametrized by $S$, etc).

This is potentially a "big list" sort of question (if it's not even too broad to be legit).

I would like to ask about interesting cases in which facts and phenomena about $\xi$ can be deduced from more classical algebraic geometry about the base change $\xi_\eta$ of $\xi$ to a/the generic point $\eta$ of $S$. So $\xi_\eta$ will be an "object" over a field $K=\kappa(\eta)$ larger than the base field.

For example, it could be a nontrivial fact about elliptic surfaces over $k$ that are deduced from the geometry of elliptic curves over $K$

Also: are there cases in which it is "natural" to consider (objects over) the algebraic closure of $K$?

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    $\begingroup$ I think it may indeed be too broad. What percentage of algebraic geometry papers do you think use an argument of this type at least once? It's got to be at least 20% and may be more like 50%. $\endgroup$ – Will Sawin Jun 21 at 13:07
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    $\begingroup$ I agree with the previous comment (qualitatively at least). For the last question, searching for "geometric generic fibre" (and the same with last two letters transposed) returns many results. $\endgroup$ – Bort Jun 21 at 14:37

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