We generalize the gcd results of Corvaja and Zannier and of Levin on ${\mathbb{𝔾}}_m^n$ to more general settings. More specifically, we analyze the height of a closed subscheme of codimension at least $2$ inside an $n$-dimensional Cohen–Macaulay projective variety, and show that this height is small when evaluated at integral points with respect to a divisor $D$ when $D$ is a sum of $n+1$ effective divisors which are all numerically equivalent to some multiples of a fixed ample divisor. Our method is inspired by Silverman’s gcd estimate, but instead of his usage of Vojta’s conjecture, we use the recent result of Ru and Vojta.

中文翻译：

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数值等效除数的积分点的最大公约数

我们将Corvaja和Zannier以及Levin的gcd结果归纳为 ${\mathbb{𝔾}}_{米}^{ñ}$进行更一般的设置。更具体地说，我们至少分析一个余维封闭子方案的高度$\mathrm{2个}$ 里面 $ñ$维科恩-马考莱投影射影，并且证明了当相对于除数在积分点上进行计算时，该高度很小 $d$ 什么时候 $d$ 是一个总和 $ñ+\mathrm{1个}$有效除数在数值上都等于固定的充足除数的某些倍数。我们的方法受到Silverman的gcd估计的启发，但是我们使用Ru和Vojta的最新结果代替了他对Vojta猜想的使用。