#### Vol. 13, No. 9, 2019

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

### Sho Ejiri

Vol. 13 (2019), No. 9, 2057–2080
##### 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.

##### Keywords
Fano variety, weak Fano variety, anticanonical divisor, restricted base locus, augmented base locus
Primary: 14D06
Secondary: 14J45
##### Milestones
Received: 22 May 2018
Revised: 9 May 2019
Accepted: 13 June 2019
Published: 7 December 2019
##### Authors
 Sho Ejiri Department of Mathematics Graduate School of Science Osaka University Japan