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

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 \(f:Y\rightarrow X\) $f:Y\to X$ 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.