Behavior of canonical divisors under purely inseparable base changes

Abstract

Let $k$ be an imperfect field. Let $X$ be a regular variety over $k$ and set $Y$ to be the normalization of $(X{\times }_k{k}^{1/p^{\infty }}{)}_{\mathrm{red}}$. In this paper, we show that $K_Y+C=f^{\ast }{K}_X$ 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.

33 pages; v2: minor revisions; v3: minor changes, v4: the numberings of the theorems are changed

Cited by in corpus (5)

• Structure of geometrically non-reduced varieties
• Pathologies on Mori fibre spaces in positive characteristic
• Rational points on log Fano threefolds over a finite field
• Iitaka's $C_{n,m}$ conjecture for 3-folds over finite fields
• Smooth rational surfaces violating Kawamata-Viehweg vanishing