The geometric exceptional set for Fano threefolds

@misc{gao2026geometric,
  title = {{The geometric exceptional set for Fano threefolds}},
  author = {Runxuan Gao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23024},
  eprint = {2610.23024},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23024},
}

TY  - UNPB
TI  - The geometric exceptional set for Fano threefolds
AU  - Runxuan Gao
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23024
UR  - https://arxiv.org/abs/2610.23024
AB  - We study the geometric exceptional set in Manin's conjecture for smooth Fano threefolds. We prove that every adjoint rigid surface with a-invariant one on a non-product smooth Fano threefold is normal. This yields a description of the geometric exceptional set in terms of the higher-a locus, anticanonical conics, and del Pezzo fibrations. Zariski density arise only from the latter two contributions. For del Pezzo fibrations, we construct a finite collection of parameter covers which controlling the associated breaking maps. We classify their low-genus components for general smooth Fano threefolds yielding examples with potentially Zariski dense geometric exceptional sets in $22$ of the $105$ deformation families. Extending Batyrev-Tschinkel's example, we construct Fano fourfolds with quartic del Pezzo fibrations and an explicit Fano threefold with two degree-six del Pezzo fibrations, all with Zariski dense geometric exceptional sets. For this threefold, the original form of Manin's conjecture fails on every non-empty open subset for every anticanonical height.
ER  -