TITLE: Is the locus of points which have irreducible fibers constructible?
QUESTION [5 upvotes]: Suppose $X \rightarrow Y$ is a map of projective schemes over a field $k$. Is $\{y \in Y: \pi^{-1}(y) \text{ is irreducible}\}$ a constructible subset of $Y$?
Note: One cannot hope to do "better" than constructible. That is, if we let H be the hilbert scheme of conics in $\mathbb P^2$ and let $\mathscr C$ be the universal curve over $H$, then $\mathscr C \rightarrow H$ has irreducible fibers consisting of smooth conics and double lines, which is constructible but not locally closed.

REPLY [4 votes]: I double-checked.  This is Théorème 4.10, p. 36 of the following.
MR0725671 (86b:13007) Reviewed 
Jouanolou, Jean-Pierre 
Théorèmes de Bertini et applications. (French) 
Progress in Mathematics, 42. Birkhäuser Boston, Inc., Boston, MA, 1983. ii+127 pp. 
ISBN: 0-8176-3164-X 
As mentioned above, you need to replace "irreducible" by "geometrically irreducible".  You can also prove this from Lemma 3.2 as I sketched above.  But Lemma 3.2 is really designed for a different purpose.