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

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Let f:X→ 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)≥κ(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 X=2 and Z=1, we have a stronger inequality κ(X)≥ κ(Z)+κ₁(F) where κ₁(F)=κ(F,ωᵒF) is the Kodaira dimension of the dualizing sheaf ωFᵒ.