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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.

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    $\begingroup$ 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. $\endgroup$
    – Sasha Anan'in
    Commented Feb 5, 2014 at 16:55
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    $\begingroup$ ... which is quite difficult to characterize by any other property. $\endgroup$
    – abx
    Commented Feb 5, 2014 at 17:02
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    $\begingroup$ 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. $\endgroup$
    – Simon Pepin Lehalleur
    Commented Feb 5, 2014 at 17:39
  • $\begingroup$ @Simon: Actually, it is enough if the general fiber of $f$ is rationally connected (but $Y$ still needs to be a curve). $\endgroup$
    – Sándor Kovács
    Commented Feb 6, 2014 at 14:31

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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).

So, the remaining question is whether having a pseudosection is enough for admitting a generic section. Clearly it is enough over a curve by [GHS]. I don't think it is enough in general and I think it is possible that the exact condition that would be needed here is not known. Again, I defer to Jason to possibly comment on this.

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    $\begingroup$ In general, existence of a pseudosection is strictly weaker than existence of a section, e.g., this is the case with Severi-Brauer varieties. However, you might at least hope to "explain" this via cohomological obstructions (e.g., the Brauer group). $\endgroup$
    – Jason Starr
    Commented Feb 6, 2014 at 12:52

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