Mathematical Research Letters

Volume 22 (2015)

Number 3

The subadditivity of the Kodaira dimension for fibrations of relative dimension one in positive characteristics

Pages: 675 – 696



Yifei Chen (Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China)

Lei Zhang (College of Mathematics and Information Sciences, Shaanxi Normal University, Xi’an, China)


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 $\omega^o_F$.

Accepted 8 July 2014

Published 20 May 2015