ON THE INTEGRAL POINTS ON THE COMPLEMENT OF RAMIFICATION-DIVISORS
ON THE INTEGRAL POINTS ON THE COMPLEMENT OF RAMIFICATION-DIVISORS
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论导数除数的补积分
DOI：
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发表时间：
2005
影响因子：
0.9
通讯作者：
U. Zannier
中科院分区：
文献类型：
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作者：
U. Zannier
继Faltings最近的一篇论文之后，我们研究了$m{P}_2setminusmathcal{D}$上的积分点，其中$mathcal{D}$是曲面$mathcal{X}$的投影的分支轨迹；分析中的关键一点是，在投影的伽罗瓦闭包中$mathcal{D}$的回拉通常分成几个部分。与Faltings的论文一样，在一定的假设下，我们得到了积分点的有限性（定理3.1）；例如，我们将发现，如果投影是足够一般的，并且$mathcal{X}$有Kodaira数$ge0$（推论4.1），则它是充分的。我们从法尔廷斯的论文中随意借用了整个几何背景。至于算法，我们的方法部分不同，依赖于Corvaja和Zannier最近的论文，并导致明显的新条件。我们还将使用一种更基本的方法来研究任意维的类似情况，其中投影取自超曲面（定理2.1）。具体来说，这些结果处理某些丢番图方程F(x_0,dots,x_n)=c$。AMS 2000数学学科分类：小学11D72；11 g35;11可以
Following a recent paper by Faltings, we study the integral points on $m{P}_2setminusmathcal{D}$, where $mathcal{D}$ is the branch locus of a projection from a surface $mathcal{X}$; a crucial point in the analysis is that the pull-back of $mathcal{D}$ in the Galois closure of the projection often splits into several components. As in the paper by Faltings, under certain assumptions we obtain finiteness of the integral points (Theorem 3.1); for instance, we shall find that it suffices if the projection is sufficiently general and if $mathcal{X}$ has Kodaira number $ge0$ (Corollary 4.1). We have borrowed freely from Faltings’s paper, for the whole geometrical setting. As to the arithmetic, our method is in part different, relying on the recent paper by Corvaja and Zannier and leading to apparently new conditions. We shall also use a more elementary approach to study a similar situation in arbitrary dimension, where the projection is taken from a hypersurface (Theorem 2.1). In concrete terms, these results deal with certain diophantine equations $F(x_0,dots,x_n)=c$. AMS 2000 Mathematics subject classification: Primary 11D72; 11G35; 11G99