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 $f \otimes_R R/m$ has a section for all maximal ideals $m \subset R$. 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.

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 $\mathfrak{m}$).

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 $R$ be the ring of integers in $K$. Let $X$ and $Y$ be projective flat $R$-schemes, and fix an $R$-map $f:X \to Y$. Assume that $f \otimes_R R/m$ has a section for all maximal ideals $m \subset R$. 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.

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 $f \otimes_R R/m$ has a section for all maximal ideals $m \subset R$. 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.