TITLE: Use irreducible fibers to show $X$ is irreducible
QUESTION [17 upvotes]: Let $\pi:X\rightarrow Y$ be a proper morphism to an irreducible variety and all fibers of $\pi$ are nonempty, irreducible, and of the same dimension.
Show $X$ must also be irreducible.
Thanks (Any hints would work too)

REPLY [2 votes]: I just wanted to add that M. Mustaţă proves a slightly more general fact here - without assuming that the morphism is proper, but assuming $X$ is equidimensional, which is often useful: http://www-personal.umich.edu/~mmustata/Note1_09.pdf.