The subadditivity of the Kodaira Dimension for Fibrations of Relative Dimension One in Positive Characteristics

Let $f:X\rightarrow Z$ be a separable fibration of relative dimension 1 between smooth projective varieties over an algebraically closed field $k$ of positive characteristic. We prove the subadditivity of Kodaira dimension $\kappa(X)\geq\kappa(Z)+\kappa(F)$, where $F$ is the generic geometric fiber of $f$, and $\kappa(F)$ is the Kodaira dimension of the normalization of $F$. Moreover, if $\dim X=2$ and $\dim Z=1$, we have a stronger inequality $\kappa(X)\geq \kappa(Z)+\kappa_1(F)$ where $\kappa_1(F)=\kappa(F,\omega^o_F)$ is the Kodaira dimension of the dualizing sheaf $\omega_F^o$.