Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\mathcal{O}_Y$ (...so fibres connected by Stein).

I have a question about some points in Jason Starr's answer here: Firstly as his example shows it may happen that the (geometric) generic fibre of $f$ is nowhere reduced.

But in first paragraph he mentioned a criterion by means of intersection numbers that the geometric generic fibre is generically reduced, namely that for any class of an ample divisor $A$ on $X$, and for any ample divisor class $B$ on $Y$, if the intersection number $A^{\text{dim}(X)-\text{dim}(Y)}.(f^*B)^{\text{dim}(Y)}$ is prime to $p$, then the geometric generic fiber of $f$ is generically reduced.

Question: Could somebody give a reference discussing this result or sketch the involved ideas?