Positivity of anticanonical divisors and $F$-purity of fibers

Abstract

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 nef anticanonical divisor, respectively, then so is the target space. We also show that, in arbitrary characteristic, a generically smooth surjective morphism between smooth projective varieties cannot have nef and big relative anticanonical divisor, if the target space has positive dimension.