# 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.

• You mean "a" generic point? Commented Feb 11, 2015 at 20:07
• Yeah, in my head I'm thinking irreducible. But the general picture is useful to me, so I've fixed that. Thanks! Commented Feb 12, 2015 at 4:33

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 $$f(\eta)=\theta$$.

• What is the definition of dominant? Is it $\overline{f(X)} = Y$? Commented Jul 17, 2020 at 9:00
• yes, it is just that Commented Jul 17, 2020 at 11:41
• I have a follow-up question. This same statement should hold in the case that $f$ is not a morphism $X \to Y$ but a rational map $X \to Y$ (i.e., exercise 7.5.A of Vakil's Foundations of AG: a rational map $\pi: X \to Y$ of irr. schemes is dominant iff the generic point is sent to the generic point). In this case, you only have some morphism $f: U \to Y$, for $U$ a dense open subset. given this, is it valid to say that $f(U) \subseteq f(\overline{\eta}) \subseteq \overline{f(\eta)}$? can we take $\overline{\eta}\subseteq U$ as the closure of $\eta$ in the subspace topology $U \subseteq X$ ? Commented Nov 2, 2022 at 23:10

Only irreducible sets have a unique generic point. Since you mention 'the' generic point I will assume $X,Y$ are irreducible from now on. Let $\eta$ be the generic point of $Y$ and $\mu$ of $X$. Then $\eta$ is contained in each affine open of $Y$ and the same goes for $\mu$ in $X$. So we can restrict our attention to some affines $U\subset X$ and $V \subset Y$. The map becomes

$$f: U = \mbox{Spec } R \rightarrow \mbox{Spec } S = V,$$ $$\varphi: S \rightarrow R$$

and your question reduces to $\varphi^{-1}(\sqrt{(0)}) = \sqrt{(0)}$. Being irreducible gives that this nilradicals are prime. If you further assume that the schemes are reduced, then the nilradicals will be zero. Then we demand injectivity of $\varphi$. This must hold for every open affine $V$ and $U$, which is equivalent to the sheaf component $f^{\#}$ being injective or $f$ being dominant, i.e. having dense image.

• It seems to me that you are assuming $X$ and $Y$ are not merely irreducible but also integral? Commented Feb 11, 2015 at 20:31
• This injectivity has a more geometric interpretation: the morphism $X \to Y$ should be dominant, i.e. have dense image. Commented Feb 12, 2015 at 2:16
• @ZhenLin: You are right. Integrality comes in when I assume that the generic point is the zero ideal in the affine opens. Is this the only place? I only knew of the implication injective -> dominant. That is why I avoided using this, but after giving it a second thought the converse is also true. Thanks! Commented Feb 12, 2015 at 13:45