Positivity of anticanonical divisors and F-purity of fibers

When is the Albanese morphism an algebraic fiber space in positive characteristic?

“…The above theorem is a special case of[7, Theorem 4.5] (see[7, Remark 4.7]). (2) The F -purity of the geometric generic fiber is equivalent to the F -purity of the geometric general fiber (see[28, Corollary 3.31] or [7, Lemma 2.3]).…”

Iitaka’s $C_{n,m}$ conjecture for $3$-folds in positive characteristic

“…It is proved as a consequence of Theorems 3.1 and 3.2. 20]). Let f : X → Z be a surjective morphism from a normal projective variety X over an algebraically closed field of characteristic p > 0 to a smooth projective variety Z, and let ∆ be an effective Q-divisor on X such that a∆ is integral for some a > 0 not divisible by p. Assume that (X η , ∆ η ) is F -pure, where η is the geometric generic point of Z.…”

Nef anti-canonical divisors and rationally connected fibrations

“…This theorem answers another question posed in [10] that asks if the inequality in the statement of the theorem holds. Theorem 1.2 is actually proved in the case where f is an almost holomorphic map, that is, a rational map with proper general fibres (Theorem 4.2).…”