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

TL;DR

This paper investigates the distribution of rational points on singular weak Fano surfaces in the context of Manin's conjecture, focusing on the emergence of Zariski-dense exceptional sets generated by accumulating quasi-étale covers. It develops a complete framework to classify quasi-étale covers of Du Val del Pezzo surfaces, connecting fundamental groups, extremal-curve correspondences, and Cremona isometries via a lattice- and graph-theoretic approach. The authors construct explicit accumulating examples in degrees $3$, $2$, and $1$ (with four concrete types) and show nonexistence of such phenomena for degrees $ ext{d} exists 4$, providing a systematic descent methodology to realize other examples over non-closed fields. These results yield first instances where Manin's conjecture fails for a thin-set version in dimension two and establish a robust toolkit for generating additional accumulating coverings and analyzing their geometric exceptional sets.

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-étale covers. We classify all quasi-étale 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-étale covers are also established, providing a systematic method to construct other examples.

Figures (1)

• Figure 1:

Theorems & Definitions (75)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Theorem \ref{['theo:classification']}
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6: $\subset$ Theorem \ref{['theo-corr']}
• Proposition 1.7: Proposition \ref{['prop-descend']}
• Definition 2.1
• Definition 2.2
• Remark 2.3