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A Zariski dense exceptional set in Manin’s Conjecture: dimension 2

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Abstract

Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural construction of the exceptional set in Manin’s Conjecture, which we call the geometric exceptional set. We construct a del Pezzo surface of degree 1 whose geometric exceptional set is Zariski dense. In particular, this provides the first counterexample to the original version of Manin’s Conjecture in dimension 2 in characteristic 0. Assuming the finiteness of Tate-Shafarevich groups of elliptic curves over \({\mathbb Q}\) with j-invariant 0, we show that there are infinitely many such counterexamples.

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Acknowledgements

The author would like to thank Sho Tanimoto for suggesting the problem and for many stimulating conversations. The author also wishes to thank Brian Lehmann and Anthony Várilly-Alvarado for useful comments and Julian Lyczak for pointing out an error in Proposition 3.1 of a previous version of this paper. This paper is based on the author’s master’s thesis at Nagoya University. The author was partially supported by JST FOREST program Grant Number JPMJFR212Z and JSPS Bilateral Joint Research Projects Grant Number JPJSBP120219935.

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  1. Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan

    Runxuan Gao

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Gao, R. A Zariski dense exceptional set in Manin’s Conjecture: dimension 2. Res. number theory 9, 42 (2023). https://doi.org/10.1007/s40993-023-00450-0

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