Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic

TL;DR

This paper proves that for certain morphisms between smooth projective varieties over a field of positive characteristic, the relative canonical divisor is pseudo-effective, linking uniruledness properties with cohomological conditions.

Contribution

It establishes new criteria connecting uniruledness, pseudo-effectivity, and cohomological invariants in positive characteristic algebraic geometry.

Findings

If the generic fiber is not uniruled, then the relative canonical divisor is pseudo-effective.

Cyclic covers of degree not divisible by the characteristic are not uniruled under ample line bundles.

A cohomological condition involving semi-stable parts of cohomology groups characterizes non-uniruled varieties.

Abstract

We show that if $f\colon X \to T$ is a surjective morphism between smooth projective varieties over an algebraically closed field $k$ of characteristic $p>0$ with geometrically integral and non-uniruled generic fiber, then $K_{X/T}$ is pseudo-effective. The proof is based on covering $X$ with rational curves, which gives a contradiction as soon as both the base and the generic fiber are not uniruled. However, we assume only that the generic fiber is not uniruled. Hence, the hardest part of the proof is to show that there is a finite smooth non-uniruled cover of the base for which we show the following: If $T$ is a smooth projective variety over $k$ and $\mathcal{A}$ is an ample enough line bundle, then a cyclic cover of degree $p \nmid d$ given by a general element of $\left|\mathcal{A}^d\right|$ is not uniruled. For this we show the following cohomological uniruledness condition, which might be of independent interest: A smooth projective variety $T$ of dimenion $n$ is not uniruled whenever the dimension of the semi-stable part of $H^n(T, \mathcal{O}_T)$ is greater than that of $H^{n-1}(T, \mathcal{O}_T)$. Additionally, we also show singular versions of all the above statements.