  Let $k$ be an imperfect field. Let $X$ be a regular variety over $k$ and set
$Y$ to be the normalization of $(X \times_k k^{1/p^{\infty}})_{{\rm red}}$. In
this paper, we show that $K_Y+C=f^*K_X$ for some effective divisor $C$ on $Y$.
We obtain the following three applications. First, we show that a $K_X$-trivial
fiber space with non-normal fibers is uniruled. Second, we prove that general
fibers of Mori fiber spaces are rationally chain connected. Third, we obtain a
weakening of the cone theorem for surfaces and threefolds defined over an
imperfect field.
