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

Runxuan Gao

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arXiv · 2406.16240

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

Runxuan Gao

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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.

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Runxuan Gao. 2024-06-23. Quasi-\'etale covers of Du Val del Pezzo surfaces and Zariski dense exceptional sets in Manin's conjecture. https://arxiv.org/abs/2406.16240

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