环面簇上的广义 Campana 点与 adelic 逼近
Generalized Campana points and adelic approximation on toric varieties
AI总结：
本文提出 $\mathcal{M}$-点框架统一推广整点、Campana 点和 Darmon 点，引入 $M$-逼近概念，并刻画了分裂环面簇满足 $M$-逼近及 $\mathcal{M}$-点集非薄的条件。
摘要：
我们引入一个研究代数簇上有理点特殊子集的一般框架，称之为 $\mathcal{M}$-点。$\mathcal{M}$-点的概念推广了整点、Campana 点和 Darmon 点的概念。我们引入并研究数域和函数域上的 $M$-逼近，这是一个推广弱逼近和强逼近的概念。我们证明该性质蕴含 $\mathcal{M}$-点集不是薄的。随后我们给出分裂环面簇何时满足 $M$-逼近的简单刻画，推广了 Nakahara 和 Streeter 的工作。进一步，我们确定了分裂环面簇上 $\mathcal{M}$-点集何时是薄的。
英文摘要：
We introduce a general framework for studying special subsets of rational points on an algebraic variety, termed $\mathcal{M}$-points. The notion of $\mathcal{M}$-points generalizes the concepts of integral points, Campana points and Darmon points. We introduce and study $M$-approximation over number fields and function fields, which is a notion that generalizes weak and strong approximation. We show that this property implies that the set of $\mathcal{M}$-points is not thin. We then give a simple characterisation of when a split toric variety satisfies $M$-approximation, generalizing work of Nakahara and Streeter. Further, we determine when the set of $\mathcal{M}$-points on a split toric variety is thin.
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