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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?

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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.

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At least, $f$ being birational, it has connected fibers. –  Henri Aug 21 '11 at 21:46

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.

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Dear Hagen, It was assumed in the question that the target was non-singular. Regards, –  Matt E Aug 8 '11 at 13:16

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