On exceptional sets in Manin’s conjecture

Abstract

In this survey paper, we study Manin’s conjecture from a geometric perspective. The focus of the paper is the recent conjectural description of the exceptional set in Manin’s conjecture due to Lehmann–Sengupta–Tanimoto. After giving an extensive background, we give a precise description of this set and compute it in many examples.

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Author's contributions

Acknowledgements

The authors would like to thank Marta Pieropan and Yuri Tschinkel for a stimulating question leading to this paper and to thank Marta for many helpful comments on an earlier draft. The authors would also like to thank Brendan Hassett, Akash Sengupta, and Yuri Tschinkel for collaborations helping to shape our perspective on the a and b invariants. The first author would like to thank Jian Xiao for a useful conversation about [39]. The authors would like to thank the referee for careful reading of this paper and helpful suggestions. Brian Lehmann is supported by NSF Grant 1600875. Sho Tanimoto is partially supported by MEXT Japan, Leading Initiative for Excellent Young Researchers (LEADER).

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Lehmann, B., Tanimoto, S. On exceptional sets in Manin’s conjecture. Res Math Sci 6, 12 (2019). https://doi.org/10.1007/s40687-018-0174-9

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