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Normal criteria relating to derived curves and shared hyperplanes

DOI：10.13255/j.cnki.jusst.20210707001

 作者 单位 E-mail 范楚君 上海理工大学 理学院 上海 200093 刘晓俊 上海理工大学 理学院 上海 200093 xiaojunliu2007@hotmail.com

基于值分布和正规族理论以及高等代数相关知识，研究了全纯曲线族及其导曲线分担处于$t$次一般位置的超平面的正规定则。设$\mathcal{F}$是一族从区域$D \subset \mathbb{C}$到${\mathbb{P}}^{N}(\mathbb{C})$的全纯曲线，${H_\ell } = \rhbr \left\{ {{\bm{x}} \in {\mathbb{P}^N}(\mathbb{C}):} \right.\left. {\left\langle {{\bm{x}},{{\bm{\alpha}} _\ell }} \right\rangle = {\text{0}}} \right\}$是${\mathbb{P}^N}(\mathbb{C})$中处于$t$次一般位置的超平面，${{\bm{\alpha}} _\ell } = {\left( {{a_{\ell 0}},{a_{\ell 1}}, \cdots ,{a_{\ell N}}} \right)^{\text{T}}},{\text{ }}\ell = 1,2, \cdots ,3t + 1$，${H_0} = \left\{ {{x_0} = {\text{0}}} \right\}$，$t\geqslant N$。假定对任意的$f \in \mathcal{F}$满足条件：若$f(z) \in {H_\ell }$，则$\nabla f(z) \in {H_\ell }$，$\ell = 1,2, \cdots ,3t + 1$；若$f(z) \in \displaystyle \bigcup\limits_{\ell = 1}^{3t + 1} {{H_\ell }}$，则$\dfrac{\left|\langle f(z),{H}_{0}\rangle \right|}{\Vert f(z)\Vert \cdot \Vert {H}_{0}\Vert }\geqslant\delta$，其中，$\delta \in \left(0，1\right)$且为常数。那么，$\mathcal{F}$在$D$上正规。对于$N = 3$，$t = 3,4,5$的特殊情形，本文有效降低了所分担超平面的个数。

Based on the theories of value distribution, normal family and the knowledge of advanced algebra, the normal criteria for the holomorphic curves and their derived curves sharing hyperplanes located in t-subgeneral position was studied. Let $\mathcal{F}$ be a family of holomorphic curves of a domain $D \subset \mathbb{C}$ to ${\mathbb{P}^N}(\mathbb{C})$, ${H_\ell } =\{ {{\bm{x}} \in {\mathbb{P}^N}(\mathbb{C}):} \left. {\left\langle {{\bm{x}},{{\bm{\alpha}} _\ell }} \right\rangle = 0} \right\}$ be hyperplanes in ${\mathbb{P}^N}(\mathbb{C})$ located in t-subgeneral position, where ${{\bm{\alpha}} _\ell } = {\left( {{a_{\ell 0}},{a_{\ell 1}}, \cdots ,{a_{\ell N}}} \right)^{\text{T}}}$, $\ell = 1,2, \cdots ,3t + 1$, ${H_0} = \left\{ {{x_0} = 0} \right\}$, $t\geqslant N$. Assume the following conditions hold for every $f \in \mathcal{F}$: If $f(z) \in {H_\ell }$, then $\nabla f(z) \in {H_\ell }$, $\ell = 1,2, \cdots ,3t + 1$; If $f(z) \in\displaystyle \bigcup\limits_{\ell = 1}^{3t + 1} {{H_\ell }}$, then $\dfrac{\left|\langle f(z),{H}_{0}\rangle \right|}{\Vert f(z)\Vert \cdot \Vert {H}_{0}\Vert }\geqslant \delta$, where $\delta \in \left(0，1\right)$ is a constant. Then $\mathcal{F}$ is normal on $D$. For the special case of $N = 3$ and $t = 3,4,5$, the number of shared hyperplanes can be effectively reduced through this research.
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