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.



Is the locus of points which have irreducible fibers constructible?
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.


