Take $X$ and $Y$ to be $k$-varieties, where $k$ is a field of characteristic 0. Assume also that $X$ is geometrically integral, and let $f: Y \to X$ be an etale a morphism of $k$-varieties.
Question: by the addition of which conditions on $f$ do we have that $X$ geometrically integral implies that $Y$ is geometrically integral?
Some thoughts: when I think of etale I think of local properties, and geometrically connectedness is not. Also geometrically connectedness is not preserved by base change (if I recall correctly). So I doubt that etalness alone can allow us to lift geometrically integralness (even though I can't think of a counterexample, but I'm sure there are easy ones). Maybe $f$ finite (and etale) will suffice?