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

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.

26 pages, v2: minor revisions, v3: the proofs of the main theorems have been improved

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