Let $X$ and $Y$ be varieties over some base field (i.e. integral schemes of finite type over that field), and let $f:X \to Y$ be a morphism of schemes. Then $Y$ (being irreducible) has a unique generic point, call it $\eta$, and one defines the generic fiber as $f^{-1}(\eta)$.
- What's the geometric meaning of the generic fiber?
- What differentiates the generic fiber from other inverse images? In particular, If $Z$ is a subvariety of $Y$ then is the generic fiber $f^{-1}(\theta)$, with $\theta$ the generic point of $Z$, equal to $f^{-1}(Z)$?
I think the way to think about it is the fiber of the generic point of an irreducible subscheme $Z$ of $Y$ is the generic points of subschemes $Z'$ of $X$ such that the restriction $f|_{Z'}$ is contained in $Z$ and is not contained in any proper subvariety, which I think implies it's surjective onto an open subset of $Z$ (dominant?). I think this basically the definition, but is a slightly more geometric way of saying it.
e.g. take the projection of the plane onto the x-axis. The fiber of some closed point consists of the closed points in the vertical line above it and the generic point of that vertical line. All other 1 dim subvarieties are "horizontal" in the sense that the projection is surjective, and the generic point of the plane is also in the fiber of the generic point.