# Étale morphism over unirational/uniruled variety

Suppose we have an étale morphism between smooth quasi-projective (complex) varieties $$X \rightarrow Y$$ and assume that $$Y$$ is unirational. I am wondering whether we can somehow deduce that $$X$$ is unirational, or rationally connected, or uniruled, or has negative Kodaira dimension.
And what if $$Y$$ is just uniruled but non-necessarily unirational?

• An étale cover of a rational curve is a disjoint union of rational curves... – Francesco Polizzi May 27 '19 at 14:11
• @FrancescoPolizzi What you write is correct for an etale morphism that is also finite, e.g., proper. However, the OP only indicates that the morphism is etale, not etale and finite. – Jason Starr May 27 '19 at 14:17
• @JasonStarr: I guessed that finite was implict, but of course you are right... – Francesco Polizzi May 27 '19 at 14:18
• Without properness assumptions you cannot deduce anything. For instance take any projective variety $X$ of dimension $n$; a general projection to $Y=\mathbb{P^n}$ is finite. Just take out of $X$ the ramification locus, you get an étale map to $\mathbb{P}^n$. – abx May 27 '19 at 15:34
• "assuming all the other hypotheses": I am lost with which hypotheses you want to make. Anyway there is no reasonable statement of the type you ask for. – abx May 28 '19 at 16:13

In the case where $$Y$$ is unirational and projective, there exists no non-trivial étale cover of $$Y$$. In fact, the fundamental group of a complex, projective, smooth unirational variety is trivial, see