Fibers of a morphism Let $f:X\to Y$ be a birational morphism of projective varieties, with $Y$ non-singular. 

Consider a fiber $X_y=f^{-1}(y)$ for a closed point $y\in Y$. Is $X_y$ also a variety, or at least a finite union of projective varieties?
If $X$ is projective, it is obvious that the fiber $X_y$ is a closed subset of $X$, therefore it is a projective variety, obviously not necessarily irreducible.
I do not know if under your hypotheses ($f$ birational, $Y$ projective and smooth) it is true that the fibers are irreducible.
The fibre needs not be irreducible: consider a singular curve $C$ and its normalization $C^\prime\rightarrow C$ within the function field $K(C)$. Then the fibre over a double or ordinary multiple point of $C$ consists of finitely many closed points.