# When does a dominant map generically admit a section?

Let $f:X\to Y$ be a dominant map of algebraic varieties. Say that $f$ generically admits sections if there exists a Zariski (not etale) open set $U \subseteq Y$ over which $f$ has a section $\sigma:Y\to X$. For example, $f:\mathbb C\to \mathbb C$ taking $z\mapsto z^2$ does not generically admit sections.

Is there a characterization of such maps in terms of more standard notions?

I don't mind assuming characteristic $0$ so then we can use the generic smoothness theorem to assume $f$ is a smooth morphism (through shrinking $X$ and $Y$). Quasiprojective is fine too.

Feel free to add tags as seems appropriate.

• Just obvious remark: You can think of $X$ as of a variety over the field $K(Y)$ of rational functions. A section in question exists exactly when that variety has a $K(Y)$-rational point. – Sasha Anan'in Feb 5 '14 at 16:55
• ... which is quite difficult to characterize by any other property. – abx Feb 5 '14 at 17:02
• A nice result on this question (in a very special but already challenging case) is the Graber-Harris-Starr theorem: if Y is a curve over \mathbb{C} and X is rationally connected, then f generically admit sections. – Simon Pepin Lehalleur Feb 5 '14 at 17:39
• @Simon: Actually, it is enough if the general fiber of $f$ is rationally connected (but $Y$ still needs to be a curve). – Sándor Kovács Feb 6 '14 at 14:31

So, as you allow, let's assume characteristic zero and perhaps even working over $\mathbb C$? It seems to me that Graber-Harris-Mazur-Starr gives you at least a necessary condition. Jason might correct me if this is wrong, or add something in the other direction.
Anyway, GHMS introduces the notion of a pseudosection which is a subvariety $Z\subseteq X$ that dominates the base with a rationally connected general fiber and they prove (essentially) that the existence of a pseudosection is equivalent to $f$ admitting a section after a base change to the normalization of any curve in the base. As mentioned in the comments, over a smooth curve the existence of a section follows from the general fiber being rationally connected by an earlier result of Graber-Harris-Starr.
So, I would proceed as follows: Suppose there exists a section over $U$, a Zariski open subset of $Y$. Restrict $f$ to $U$ and observe that the section of $f\big|_U$ implies that any further base change to a smooth curve admits a section. Then by GHMS this $f\big|_U$ admits a pseudosection. Take the closure of this pseudosection in the original $X$. This closure will be a pseudosection of the original $f$.
In other words, if $f$ admits a generic section as you require, then it admits a pseudosection. This shows for example why your example does not admit a generic section (not that you need this condition, I'm just saying that it does imply that).