Behavior of canonical divisors under purely inseparable base changes

Abstract

Abstract Let k be an imperfect field. Let X be a regular variety over k and set Y to be the normalization of ( X ⊗ k k 1 / p ∞ ) red {(X\otimes_{k}k^{1/p^{\infty}})_{{\rm red}}} . In this paper, we show that K Y + C = f * ⁢ K X {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 {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.