TITLE: Why is the number of irreducible components upper semicontinuous in nice situations?
QUESTION [10 upvotes]: Suppose $f: X \rightarrow Y$ is a flat projective morphism of finite type schemes over an algebraically closed field so that the fibers over the closed points of Y are (geometrically) reduced. Why is it true (or do I need some additional assumptions?) that the number of irreducible components of the geometric fibers is upper semicontinuous on Y?
I have asked Joe Harris and Allen Knutson about this statement, and they both believed it was true, but I have not been able to find a reference or a proof.

REPLY [2 votes]: An expanded version of nfdc23's answer in the comments to this question can be found at http://arxiv.org/pdf/1601.05840v1.pdf, Proposition 2.9. An even more expanded version can be found at http://arxiv.org/pdf/1605.01117v1.pdf, Proposition 3.2.5.