A guiding problem in algebraic geometry is the classification of varieties. In dimension 1, the main invariant for their classification is the genus. Similarly, in higher dimension we study positivity properties of the canonical divisor and a first measure of these is its Iitaka dimension.

A long-standing problem is how we can relate Iitaka dimensions in fibrations: the Iitaka conjecture. Recently, Chang proved that Iitaka dimensions of the anticanonical divisors in fibrations over fields of characteristic 0 satisfy superadditivity, provided that we control the singularities of the anticanonical linear series. Both the Iitaka conjecture and Chang’s theorem are known to fail in positive characteristic. However, in a joint work with Brivio and Chang, we prove that the latter holds when also the “arithmetic properties” of the anticanonical linear series are sufficiently good.