Mathematical Research Letters
Volume 22 (2015)
The subadditivity of the Kodaira dimension for fibrations of relative dimension one in positive characteristics
Pages: 675 – 696
Let $f : X \to Z$ 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 $\omega^o_F$.