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In this paper, we study a version of the Iitaka conjectures for anticanonicaldivisors over perfect fields of positive characteristic. That is, we prove theinequality $\kappa(X,-K_X) \leq \kappa(X_y,-K_{X_y})+ \kappa(Y,-K_Y)$, for acontraction $f: X \rightarrow Y$ with general fibre $X_y$ having goodarithmetic properties.

Superadditivity of anticanonical Iitaka dimension for contractions with F-split fibres