TITLE: What are the most important instances of the "yoga of generic points"?
QUESTION [19 upvotes]: In algebraic geometry, an irreducible scheme has a point called "the generic point." The justification for this terminology is that under reasonable finiteness hypotheses, a property that is true at the generic point is actually generically true (i.e. is true on a dense open subset).
For example, there is a result called "generic flatness" (EGA IV (2), Theorem 6.9.1). Suppose Y is locally noetherian and integral, with f:X→Y a morphism of finite type and F is a coherent OX-module. If F is flat over the generic point of Y (a condition which is always satisfied, since anything is flat over a point), then there is a dense open subscheme U⊆Y such that F is flat over U.
I'm sure that there are lots of instances of this "yoga of generic points", but whenever I try to come up with one, it's kind of lame (in my example above, the condition at the generic point is vacuous). What are the main examples of the yoga of generic points?

REPLY [2 votes]: There is a spectacular proof of the ACC for log canonical thresholds in $\mathbb C ^N$ due to Ein, de Fernex and Mustata https://arxiv.org/abs/0905.3775 that relies on taking generic limits of polynomials / power series (see also the beautiful paper by Koll\'ar especially \S 4 of https://arxiv.org/pdf/0805.0756.pdf). The idea is that the use of generic points allows us to take the "correct" limit of a sequence of polynomials.