Paper detail

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 $κ(X)\geqκ(Z)+κ(F)$, where $F$ is the generic geometric fiber of $f$, and $κ(F)$ is the Kodaira dimension of the normalization of $F$. Moreover, if $\dim X=2$ and $\dim Z=1$, we have a stronger inequality $κ(X)\geq κ(Z)+κ_1(F)$ where $κ_1(F)=κ(F,ω^o_F)$ is the Kodaira dimension of the dualizing sheaf $ω_F^o$.