Singularities of General Fibers and the LMMP

Abstract: We use the theory of foliations to study the relative canonical divisor of a normalized inseparable base-change. Our main technical theorem states that it is linearly equivalent to a divisor with positive integer coefficients divisible by $p-1$. We deduce many consequences about the fibrations of the minimal model program: for example the general fibers of terminal $3$-fold Mori fiber spaces are normal in characteristic $p\geq 5$ and smooth in characteristic $p\geq 11$.