Let $k$ be an arbitrary field. Suppose that $C/k$ is an integral curve which is birationally equivalent to a projective line. Is it true that $C$ is geometrically reduced and irreducible?
This is of course true if $C$ is a normal, since then $C$ is just isomorphic to some open subset of the projective line. So I guess the above question can be generalized to the following one:
Let $X$ and $Y$ be two integral (so both reduced and irreducible) schemes of finite type over $\text{Spec }k$ and let $X\longrightarrow Y$ be a finite, birational $k$-morphism. Suppose that $X$ is geometrically reduced (irreducible). Is then $Y$ also geometrically reduced (irreducible)?