We introduce the split torsor method to count rational points of bounded height on Fano varieties. As an application, we prove Manin's conjecture for all nonsplit quartic del Pezzo surfaces of type $\mathbf A_3+\mathbf A_1$ over arbitrary number fields. The counting problem on the split torsor is solved in the framework of o-minimal structures.

中文翻译：

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Manin猜想的分裂torsor方法

我们引入分裂torsor方法来计算Fano品种上有界高度的有理点。作为一个应用，我们在任意数域上证明了所有类型为 $\mathbf A_3+\mathbf A_1$ 的非分裂四次 del Pezzo 曲面的 Manin 猜想。在o-极小结构的框架内解决分裂torsor上的计数问题。