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.