Let $f: X \to S$ be a proper morphism (not necessarily smooth!, but perhaps flat) and $S$ be an excellent, regular, affine, integral variety over a finite field (I do not know which of these properties are really necessary).


Does $\{s \in S : X\_s := f^{-1}(s) \text{ is (geometrically) integral}\}$ contain an open subset [edit: this was: "non-empty open subset"]?


(The same question for $\{s \in S : X\_s := f^{-1}(s) \text{ is (geometrically) connected}\}$ seems to have a positive answer by Stein factorisation.)



locus of integral fibres open
If $f$ is proper and flat, the "geometrically integral locus" is open by EGA IV, (12.2.1)(x). This works for any morphism of schemes which is proper, flat and finitely presented.  

The geometrically connected locus is not open in general: think of a ramified double cover.


