Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 1 down vote accepted

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.

share|cite|improve this answer
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.

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.