Existence of local sections

Question

I would like to know when the property of "having a section" for a morphism of varieties in characteristic $0$ can be detected by spreading out to characteristic $p$.

Take a number field $K$, and let $R := \mathcal{O}_K[1/N]$ for some integer $N$. Let $X$ and $Y$ be projective flat $R$-schemes, and fix an $R$-map $f:X \to Y$. Assume that $f \otimes_R R/\mathfrak{m}$ has a section for all maximal ideals $\mathfrak{m} \subset R$. Does $f \otimes_R \overline{K}$ have a section for some algebraic closure $\overline{K}$ of $K$?

Note that taking the algebraic closure of $K$ is necessary (by the consideration of Brauer classes).

I suspect the answer is 'no' but cannot find a counterexample.

Edit: As Piotr's answer shows, the answer is 'yes' if one can bound the degrees of the sections of $f \otimes_R R/\mathfrak{m}$ independently of $\mathfrak{m}$. However, in practice, this could be hard to arrange, especially if the sections modulo $\mathfrak{m}$ are constructed using Frobenius (whose degree depends on $\mathfrak{m}$).

Answer 1

The answer is yes if you can bound the Hilbert polynomials of the graphs of the sections over different $\mathfrak{m}$. In such a situation, sections $Y\to X$ are points in finitely many components of the Hilbert scheme of $X\times Y$. Hence you have a scheme of finite type over $R$ (the scheme of sections with bounded Hilbert polynomial), which has non-empty fibers over all closed points of the base. It follows that the generic fiber is non-empty, hence has a $\bar K$-point.

Therefore in a hypothetical counterexample, the degrees of the sections over different $\mathfrak{m}$ have to go to infinity. Wouldn't that be weird?

Answer 2

Since nobody else has mentioned this, let me point out the article of Bogomolov and Tschinkel.

MR2154371 (2006e:14024) Reviewed
Bogomolov, Fedor(1-NY-X); Tschinkel, Yuri(D-GTN)
Rational curves and points on K3 surfaces. (English summary)
Amer. J. Math. 127 (2005), no. 4, 825–835.
14G05 (11G35 14J28)
http://arxiv.org/pdf/math/0310254.pdf

Let $Z$ be a Kummer K3 surface with a specified very ample divisor class $D$, let $Y$ be $\mathbb{P}^1$ with one marked point $0$, and let $X$ be $Y\times Z$ with its obvious projection $\text{pr}_Y$ to $Y$, with the $Y$-ample divisor class $\text{pr}_Z^*D$, and with an arbitrary $K$-rational point $x_0$ in the fiber over $0$ marked. Then Bogomolov and Tschinkel prove that for every finite field reduction $R/\mathfrak{m}$, there exists a section $s_{\mathfrak{m}}$ sending $0$ to $x_0$ and having positive degree with respect to $\text{pr}_Z^* D$, i.e., the section is not "horizontal". They point out that it may well be that the same holds over $\overline{K}$, but nobody really knows this, and some people are skeptical. So this is an example worth bearing in mind.