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

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 $\kappa (X)\ge \kappa (Z)+\kappa (F)$, 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 $\kappa (X)\ge \kappa (Z)+{\kappa }_1(F)$ where ${\kappa }_1(F)=\kappa (F,{\omega }_F^o)$ is the Kodaira dimension of the dualizing sheaf ${\omega }_F^o$.

14 pages, welcome comments

References in corpus (1)

• Semi-positivity in positive characteristics

Cited by in corpus (3)

• Behavior of canonical divisors under purely inseparable base changes
• Projectivity of the moduli space of stable log-varieties and subadditvity of log-Kodaira dimension
• On the characterization of abelian varieties in characteristic $p>0$