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# 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

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n3.a3

#### Authors

#### Abstract

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 $\omega^o_F$.