The split torsor method for Manin's conjecture

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.