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arXiv · 2311.04138

The geometric exceptional set in Manin's conjecture for Batyrev and Tschinkel's example

Runxuan Gao
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Abstract

Batyrev and Tschinkel's example is a Fermat cubic surface bundle $X$ which is a Fano $5$-fold. It is the first example for which Manin's conjecture can never hold for a proper closed exceptional set. Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural geometric description of the exceptional set in Manin's conjecture and showed that it is always contained in a thin set. Over a field of characteristic $0$, we explicitly construct finitely many thin maps such that any thin map $f:Y\rightarrow X$ with equal or larger $a$- and $b$-values in lexicographical order factors rationally through one of them. In particular, this defines a thin set which coincides with Lehmann-Sengupta-Tanimoto's conjectural exceptional set.

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Runxuan Gao. 2023-11-07. The geometric exceptional set in Manin's conjecture for Batyrev and Tschinkel's example. https://doi.org/10.1007/s40879-024-00778-6

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