The subadditivity of the Kodaira Dimension for Fibrations of Relative Dimension One in Positive Characteristics
The subadditivity of the Kodaira Dimension for Fibrations of Relative Dimension One in Positive Characteristics
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DOI：
10.4310/mrl.2015.v22.n3.a3
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发表时间：
2013-05
期刊：
影响因子：
--
通讯作者：
Yifei Chen;Lei Zhang
中科院分区：
文献类型：
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作者：
Yifei Chen;Lei Zhang
设$f：x\right行Z$是具有正特征的代数闭域$k$上光滑射影簇之间相对维度1的可分纤维.证明了Kodaira维度$\kappa(X)\geq\kappa(Z)+\kappa(F)$的次可加性，其中$F$是$f$的一般几何纤维，$kappa(F)$是$F$的正规化的Kodaira维度。此外，如果$\dim X=2$且$\dim Z=1$，则我们有一个更强的不等式$\kappa(X)\geq\kappa(Z)+\kappa_1(F)$其中$\kappa_1(F)=\kappa(F，omega^o_F)$是对偶层$\omega_F^o$的Kodaira维度。
Let $f:X\rightarrow Z$ 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 $\omega_F^o$.