Positivity of anti-canonical divisors and $F$-purity of fibers

TL;DR

This paper investigates how the positivity properties of anti-canonical divisors are preserved under certain morphisms between algebraic varieties, especially in relation to $F$-purity and nefness, across different characteristics.

Contribution

It establishes conditions under which the anti-canonical divisor positivity is preserved in morphisms with $F$-pure fibers and explores the nefness of relative anti-canonical divisors in various settings.

Findings

Anti-canonical positivity is preserved under specific morphisms with $F$-pure fibers.

Relative anti-canonical divisors are not nef and big in general.

Conditions for the preservation of Fano and weak Fano properties in morphisms.

Abstract

In this paper, we prove that given a flat generically smooth morphism between smooth projective varieties with $F$-pure closed fibers, if the source space is Fano, weak Fano or a variety with the nef anti-canonical divisor, then so is the target space. We also show that relative anti-canonical divisors of generically smooth surjective nonconstant morphisms are not nef and big in arbitrary characteristic.