When the fibers of a flat morphism are varieties.

Question

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?

Answer 1

No, Alex gives one counterexample. The main point is that there can be ramification. Say we have a degree $d$ map between curves-- this means that the fibre is usually $d$ distinct points. But sometimes two points come together, and ramify. When this happens the fibre is no longer reduced.

For completeness take $\mathbb P^1 \to \mathbb P^1$ via $[x:y] \mapsto [x^2: y^2]$, and look at the fibre over $[0:1]$.