Notations: As in Hatshorne's book.
Suppose that $f:X\longrightarrow Y$ is a flat morphism between two non-singular projective varieties over an algebraically closed field.
Are the fibers of $f$ also varieties over $k$?
I think the answer is no, indeed the problem is the irreducibility of the fibers. What kind of hypotheses do we need in order to give an affirmative answer? Maybe the smoothness of $f$?
Edit: Are the fibers over closed points always varieties?