Does there exist a finite morphism $\pi\colon X\to \mathbb{P}\_{\mathbb{C}}^n$, that does not admit a rational section along any prime divisor $D\subset\mathbb{P}^n$ in the locus where $\pi$ is etale?



Does a finite morphism to $\mathbb{P}^n$ necessarily split at some height one point in the etale locus?
Let $f:X\to Y$ be a finite surjective morphism of smooth (projective) varieties over complex numbers. Let $B\subset Y$ be the branch locus and let $q\in Y-B$. Let $f^{-1}(q)=\{p=p\_1,\ldots, p\_n\}$, where $n=\deg f$. Take $D\subset X$, a smooth divisor passing through $p$, but not passing through $p\_i, i>1$. Let $E=f(D)$. Then, one can check that $E$ is smooth at $q$ and $f:D\to E$ is birational. This gives on an open set of $E$ the section you desire (or not desire, as the question is framed).


