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

TL;DR

This paper constructs the first known examples of singular del Pezzo surfaces with dense exceptional sets in Manin's conjecture, using quasi-étale covers, and classifies all such covers for Du Val del Pezzo surfaces.

Contribution

It extends classification of quasi-étale covers to all Du Val del Pezzo surfaces and identifies potential examples related to group actions, revealing no such examples exist beyond degree 3.

Findings

Constructed examples of singular del Pezzo surfaces with dense exceptional sets in degrees 1, 2, and 3.

Classified all quasi-étale covers of Du Val del Pezzo surfaces.

Proved non-existence of such examples in degrees higher than 3.

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.