Does anyone know an example of an integral scheme $X$ over a field $k$ such that $X\_{\overline{k}}$ is connected but reducible? Does it make a difference if $k$ is perfect, or if we ask for $X\_{\overline{k}}$ to be reduced as well? 



Example of an integral scheme which is geometrically connected but not geometrically irreducible
I am just posting my comment as an answer. For the field $\mathbb{R}$, the affine $\mathbb{R}$-scheme $\text{Spec}\ \mathbb{R}[x,y]/\langle x^2+y^2\rangle$ is integral and geometrically connected, but it is not geometrically irreducible. If $X\_k$ is an integral, locally finite type $k$-scheme that is *normal* and geometrically connected, then $X\_{\overline{k}}$ is irreducible. For geometric irreducibility, it is irrelevant whether $k$ is perfect: the field extension $\overline{k}/k^{\text{sep}}$ is a universal homeomorphism. However, perfectness is relevant for reducedness.


