## Second Main Theorem for Algebraic Curves on Compact Riemann Surfaces

Duan Lizhen, Cao Hongzhe,

Department of Mathematics, College of Science, Nanchang University, Nanchang 330031

 基金资助: 国家自然科学基金.  12061041国家自然科学基金.  12061042

 Fund supported: the NSFC.  12061041the NSFC.  12061042

Abstract

In this paper, we first establish some second main theorems for algebraic curves from a compact Riemann surface into a complex projective subvariety of the complex projective space, which is ramified over hypersurfaces in subgeneral position. Then we use it to study the ramification for the generalized Gauss map of complete regular minimal surfaces in $\mathbb{R}^{m}$ with finite total curvature.

Keywords： Second main theorem ; Gauss map of minimal surfaces ; Algebraic curves ; Hypersurfaces

Duan Lizhen, Cao Hongzhe. Second Main Theorem for Algebraic Curves on Compact Riemann Surfaces. Acta Mathematica Scientia[J], 2021, 41(6): 1585-1597 doi:

## 2 预备知识

${\Bbb P}^{n}({\Bbb C})$的齐次坐标为$(\omega_{0}:\cdots:\omega_{m})$, $Q $${\Bbb P}^{n}({\Bbb C}) 中次数是 d 的超曲面. 本文中，在不产生混淆的情况下超曲面由多项式给出, 即 其中 {\cal T}_{d} = \{(i_{0}, \cdots, i_{n})\in {\Bbb Z}^{n+1}_{\geq 0};i_{0}+\cdots+i_{n} = d\} , \omega = (\omega_{0}, \cdots, \omega_{n}) , \omega^{I} = \omega_{0}^{i_{0}}\cdots\omega_{n}^{i_{n}} , I = (i_{0}, \cdots, i_{n})\in {\cal T}_{d}$$ a_{I}(I\in {\cal T}_{d})$是非零常数.

$f^{-1}(H) = \{P_{1}, \cdots, P_{r}\}$, 则

${\Bbb C}^{n+1}$中选择合适的局部坐标系满足: $f(0) = (1, 0, \cdots, 0)$, $f_{1}(0) = \cdots = f_{n}(0) = 0, $$(f_{1}(z), \cdots, f_{n}(z)) = z^{\delta_{1}}(f^{1}_{1}(z), \cdots, f^{1}_{n}(z))$$ (f^{1}_{1}(0) = \cdots = f^{1}_{n}(0))\neq 0$; $(f^{1}_{1}(0) = \cdots = f^{1}_{n}(0)) = (1, 0, \cdots, 0)$, $(f^{1}_{2}(z) = \cdots = f^{1}_{n}(z)) = z^{\delta_{2}-\delta_{1}}(f^{2}_{2}(z) = \cdots = f^{2}_{n}(z))$以及$(f^{2}_{2}(0) = \cdots = f^{2}_{n}(0))\neq 0, \ldots$, 依次变换下去, 我们得到以下形式

$$$f(z) = (z^{\delta_{0}}+\cdots, z^{\delta_{1}}+\cdots, \cdots, z^{\delta_{n}}+\cdots),$$$

$$$\sum\limits^{n}_{i = 0}(n-i)\nu_{i}(P)+\frac{1}{2}n(n+1) = \delta_{0}(P)+\delta_{1}(P) +\cdots+\delta_{n}(P).$$$

$$$\sigma_{i} = \sum\limits_{P\in S}\nu_{i}(P).$$$

$$$\sum\limits^{n}_{i = 0}(n-i)\sigma_{i} = (n+1)\deg(f)+n(n+1)(g-1),$$$

2.2  设$X\subset {\Bbb P}^{n}({\Bbb C})$是维数为$k$的射影簇且度为$\Delta$.$X$上的多项式

${\bf u}_{i} = (u_{i0}, \cdots, u_{in})$, $i = 0, \cdots, k$, 满足以下条件: $F_{X} $${\Bbb C}[u_{00}, \cdots, u_{kn}] 中的不可约的齐次多项式, 每个分量 {\bf u}_{i} , i = 0, \cdots, k 的次数为 \Delta , 并且 F_{X}({\bf u}_{0}, \cdots, {\bf u}_{k}) = 0 当且仅当 X\cap H_{{\bf u}_{0}}\cap H_{{\bf u}_{k}}\neq\emptyset, 这里 H_{{\bf u}_{i}} , i = 0, \cdots, k 是由 {\bf u}_{i0}x_{0}+\cdots+{\bf u}_{in}x_{n} = 0 所定义的超平面. 我们称 F_X$$ X$上的Chow形式.

$F_{X} $$X 上的Chow形式, 令 {\bf c} = (c_{0}, \cdots, c_{n}) 为一实数数组. 我们作如下分解 其中 G_{0}, \cdots, G_{r}\in {\Bbb C}[u_{00}, \cdots, u_{0n};\cdots;u_{k0}, \cdots, u_{kn}] , e_{0}>e_{1}>\cdots>e_{r} , 则 X 上的Chow形式关于 c 的权定义为 \{0, \cdots, n\} 的任意子集 J = \{j_{0}, \cdots, j_{k}\} , j_{0}<j_{1}<\cdots<j_{k} , 定义 其中 {\bf u}_{i} = (u_{i0}, \cdots, u_{in})$$ n+1$个变量的块. 令$\beta: = \left( \begin{array}{c} n+1 \\ k+1 \end{array} \right)$, $J_{1}, \cdots, J_{\beta} $$\{0, \cdots, n\} 的子集, 且每个 J_i 中元的个数是 k+1 . X 的Chow形式 F_{X}$$ [J_{1}], \cdots, [J_{\beta}]$中可以写成度为$\Delta$的齐次多项式. 对$J_{1}, \cdots, J_{\beta}$中的任意$J$, ${\bf c} = (c_{0}, \cdots, c_{n})\in {\Bbb R}^{n+1}$,

${\bf a} = (a_{0}, \cdots, a_{n})\in {\Bbb Z}^{n+1}$, 记${\bf x}^{{\bf a}}: = x^{a_{0}}_{0}\cdots x^{a_{n}}_{n}$.$I = I_{X} $${\Bbb C}[x_{0}, \cdots, x_{n}] 中由 X 生成的理想, {\Bbb C}[x_{0}, \cdots, x_{n}]_{m} 表示 {\Bbb C}[x_{0}, \cdots, x_{n}] 中次数为 m (包括0次)的齐次多项式组成的向量空间, I_{m}: = {\bf C}[x_{0}, \cdots, x_{n}]_{m}\cap {I} , m = 1, 2, \cdots . 我们定义 X 中的希尔伯特函数 H_{X}(m) 由希尔伯特多项式性质得 {\bf c} = (c_{0}, \cdots, c_{n})\in {\Bbb R}^{n+1} , 我们定义 X$$ m$次希尔伯特函数权重$S_{X}(m, {\bf c})$如下

$Y = \Phi(V)$. 因为$V\cap \bigcap\limits^{k+1}_{j = 1}P_{1, j} = \emptyset$, 所以$\Phi$是一个$V$中的有限态射, $Y $${\Bbb P}^{l-1}({\Bbb C}) 中次数为 \dim Y = k 的复射影子簇且 \Delta: = \deg Y\leq d^{k}\cdot \deg V . 对于每一个 {\bf y}^{{\bf a}} = y^{a_{1, 1}}_{1, 1}\cdots y^{a_{1, k+1}}_{1, k+1}\cdots y^{a_{n_{0}, 1}}_{n_{0}, 1}\cdots y^{a_{n_{0}, k+1}}_{n_{0}, k+1} . Y_{u} 是有限维的向量空间, 设 \{\upsilon_{0}, \cdots, \upsilon_{n_{u}}\} 是它的一组基. 考虑全纯映射 因为 f 是代数非退化的, 因此 F 是线性非退化的. 由于 \frac{1}{\deg Q}\sum\limits_{z\in S}\nu^{0}_{Q(f)}(z) 不依赖于 Q 的选择. 取 Q = \{w_{0} = 0\}$$ {\Bbb P}^{n_{u}}({\Bbb C})$中的超曲面, 因此

$$$\deg F = \sum\limits^{r}_{j = 1} \nu_{\upsilon_{0}(\Phi\circ\widetilde{f})}(z_{j}) = ud\cdot\deg(f).$$$

$\begin{eqnarray} \sum^{H_{Y}(u)}_{i = 1}\nu^{0}_{L_{i}(\tilde{F})}(z)-\sum^{H_{Y}(u)}_{i = 1} \min\{\nu_{L_{i}(\tilde{F})}(z), n_{u}\} & = &\sum^{H_{Y}(u)}_{i = 1}\max\{\nu_{L_{i}(\tilde{F})}(z)-n_{u}, 0\}{}\\ & \leq&\sum^{H_{Y}(u)-1}_{i = 0}(\delta_{i}(z)-i). \end{eqnarray}$

$\begin{eqnarray} \sum^{H_{Y}(u)-1}_{i = 0}(H_{Y}(u)-i)\nu_{i}(z) = \sum^{H_{Y}(u)-1}_{i = 0}(\delta_{i}(z)-i) \geq \sum^{H_{Y}(u)}_{i = 1}{\bf a}_{i}\cdot{{\bf c}} = S_{Y}(u, {\bf c}). \end{eqnarray}$

$\begin{eqnarray} S_{Y}(u, {\bf c})\geq \frac{uH_{Y}(u)}{(k+1)\Delta}e_{Y}({\bf c})-(2k+1)\Delta H_{Y}(u) \max\limits_{1\leq i\leq n_{0}\atop 1\leq j\leq k+1}{\bf c}_{ij}. \end{eqnarray}$

$P_{1, 1}, \cdots, P_{1, k+1}$处于一般位置, 根据引理2.2, 可得

$\begin{eqnarray} e_{Y}({\bf c)}\geq\Delta\sum\limits_{1\leq j\leq k+1}{\bf c}_{1, j} = \Delta\sum\limits_{1\leq j\leq k+1}\max\{\nu_{P_{1, j}(f)}(z)-n_{u}, 0\}. \end{eqnarray}$

$$$2(g-1)+|E|\leq \deg(\overline{G})-r<\deg(\overline{G}).$$$

$\begin{eqnarray} & &(q-p(n+1)-\varepsilon)\deg(\overline{G}){}\\ &\leq&\frac{1}{d_{j}}\sum\limits_{p\not\in E}\sum^{r_{0}}_{j = 1}\min\{\nu_{Q_{j}(\overline{G}})(z), M_{0}-1\} +\frac{1}{d_{j}}\sum\limits_{p\not\in E}\sum^{q}_{j = r_{0}+1}\min\{\nu_{Q_{j}(\overline{G}})(z), M_{0}-1\}{}\\ & &+\frac{1}{2u}p(n+1)(M_{0}-1)\frac{\deg(\overline{G})}{d}. \end{eqnarray}$

$P\not\in E$, 则$Q_{r_{0}+1}, \cdots, Q_{q}$是例外超曲面, 故$\nu^{0}_{Q_{j}(\overline{G})}(P) = 0$, $r_{0}+1\leq j\leq q$.$P\in S$, $1\leq j\leq r_{0}$,

$$$\min\{\nu_{Q_{j}(\overline{G})}(P), M_{0}-1\}\leq (M_{0}-1)\cdot \min \{\nu_{Q_{j}(\overline{G})}(P), 1\} \leq \frac{M_{0}-1}{m_{j}}\nu^{0}_{Q_{j}(\overline{G})}(P).$$$

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