Quasi-\'etale covers of Du Val del Pezzo surfaces and Zariski dense exceptional sets in Manin's conjecture

Abstract

We construct first examples of singular del Pezzo surfaces with Zariski dense exceptional sets in Manin's conjecture, varying in degrees $1, 2$ and $3$. The obstructions arise from accumulating quasi-\'etale covers. We classify all quasi-\'etale covers of Du Val del Pezzo surfaces, extending earlier works of Miyanishi-Zhang. Then, we identify all potential examples by studying group actions on the pseudo-effective cones, and show that no such example exists in degree more than $3$. Relevant results on the geometry and descent problem of quasi-\'etale covers are also established, providing a systematic method to construct other examples.