We introduce the split torsor method to count rational points of bounded height on Fano varieties. As an application, we prove Manins 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.
We give a definition of Cox rings and Cox sheaves for varieties over nonclosed fields that is compatible with torsors under quasitori, including universal torsors. We study their existence and classification, we make the relation to torsors precise, and we present arithmetic applications.
