On the Iitaka conjecture for anticanonical divisors in positive characteristic

TL;DR

This paper investigates an Iitaka-type inequality for anticanonical divisors in positive characteristic, establishing conditions under which it holds and providing counterexamples in certain cases.

Contribution

It proves the inequality for specific fibrations in positive characteristic and constructs counterexamples using Tango--Raynaud surfaces.

Findings

Iitaka inequality holds when the source is a threefold or the target is a curve with regular fibers.

Counterexamples are provided in characteristics 2 and 3 for fibrations with non-normal fibers.

Counterexamples are constructed from Tango--Raynaud surfaces.

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.