When does a generic point map to a generic point? This question may be too vague, so feel free to specialize to particular examples.
Given a morphism of schemes $f:X\to Y$, I want to know what conditions one can impose on $f,X$ or $Y$ such that a generic point of $X$ will map to a generic point of $Y$. For example, if $X$ and $Y$ are irreducible etc.

Let $f:X\rightarrow Y$ be a map between irreducible schemes with generic points $\eta$ and $\theta$ respectively. It's possible to prove that $f(\eta)=\theta$ is equivalent to $f$ being dominant without assuming that the schemes are reduced. Here is the proof:
If $f(\eta)=\theta$ we have $\overline{f(X)}\supseteq \overline{f(\eta)}=\bar{\theta}=Y$. On the other hand, if $f$ is dominant then, as $f(\overline{A})\subseteq \overline {f(A)}$ for every continuous function and every set $A$, we have $$f(X)=f(\bar{\eta})\subseteq \overline{f(\eta)}.$$
So $f(\eta)$ is a point whose closure is $Y$, hence $f(\eta)=\theta$.
