# The split torsor method for Manin's conjecture

## TL;DR

This paper introduces the split torsor method to count rational points on Fano varieties and proves Manin's conjecture for specific nonsplit quartic del Pezzo surfaces using o-minimal structures.

## Contribution

The paper develops the split torsor method and applies it to establish Manin's conjecture for certain nonsplit quartic del Pezzo surfaces over any number field.

## Key findings

- Proves Manin's conjecture for all nonsplit quartic del Pezzo surfaces of type A3+A1
- Introduces the split torsor method for counting rational points
- Utilizes o-minimal structures to solve the counting problem

## Abstract

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.