On the Iitaka conjecture for anticanonical divisors in positive characteristic

Abstract

Given a fibration over a perfect field of positive characteristic, we study an Iitaka-type inequality for the anticanonical divisors. We conclude that it holds when the source of the fibration is a threefold or when the target is a curve, the general fibre is regular and the pair induced on it from the ambient space is strongly F-regular. We then give counterexamples in characteristics 2 and 3 for fibrations with non-normal fibres, constructed from Tango--Raynaud surfaces.

23 pages, revised version, updated with suggestions from referee, to appear in Michigan Mathematical Journal