This question stems from Oliver Debarre's Higher-Dimensional Algebraic Geometry, proposition 5.7.
Let $X$ be a normal quasi-projective variety over an algebraically closed field of characteristic $p > 0$. In the second step of the proof of his statement, he constructs a constructible set $V \subset X \times X$ and says that general fibers of the first projection map $\pi_1 : \overline{V} \to X$ may not be reduced.
I suppose I can abstractly say that $A \otimes_B k$ may have nilpotent elements, where $A$ and $B$ are domains and $k$ is a field of positive characteristic. Can anyone give an example of a projection map $X \times X \to X$ with non-reduced general fibers ($X$ satisfying the conditions above)?