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On Manin's conjecture for quartic del Pezzo fibrations

  • Sho Tanimoto
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Abstract

We generalize the homological sieve method, developed by Das, Lehmann, Tosteson, and the author, to study certain quartic del Pezzo fibrations, and we prove a version of Manin's conjecture over global function fields in these cases. Our proofs combine the $3$-dimensional positive-characteristic minimal model program, the geometry of the space of sections and the Abel--Jacobi mapping, and the homological sieve method. Our quartic del Pezzo surfaces have Picard rank $2$, and admit two birational morphisms to non-split quadric surfaces. In particular, they do not possess any conic fibrations.

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Cite

BibTeXRIS
BibTeX
@misc{tanimoto2026manins,
  title = {{On Manin's conjecture for quartic del Pezzo fibrations}},
  author = {Sho Tanimoto},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28471},
  eprint = {2608.28471},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28471},
}
RIS
TY  - UNPB
TI  - On Manin's conjecture for quartic del Pezzo fibrations
AU  - Sho Tanimoto
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28471
UR  - https://arxiv.org/abs/2608.28471
AB  - We generalize the homological sieve method, developed by Das, Lehmann, Tosteson, and the author, to study certain quartic del Pezzo fibrations, and we prove a version of Manin's conjecture over global function fields in these cases. Our proofs combine the $3$-dimensional positive-characteristic minimal model program, the geometry of the space of sections and the Abel--Jacobi mapping, and the homological sieve method. Our quartic del Pezzo surfaces have Picard rank $2$, and admit two birational morphisms to non-split quadric surfaces. In particular, they do not possess any conic fibrations.
ER  -