A Note on Iitaka's Conjecture $C_{3,1}$ in Positive Characteristics

Abstract

Let $f \colon X \to Y$ be a fibration from a smooth projective $3$-fold to a smooth projective curve, over an algebraically closed field $k$ of characteristic $p \gt 5$. We prove that if the generic fiber $X_{\eta}$ has big canonical divisor $K_{X_{\eta}}$, then\[ \kappa(X) \geq \kappa(Y) + \kappa(X_{\eta}).\]