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PreprintCC BY 4.0arXiv

Hasse Obstructions to Rationality for Special Fourfolds

  • Aideen Fay
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Abstract

For every nonempty Hassett divisor 𝒞d parameterizing special cubic fourfolds, we consider the quaternion class βd=(d/2,−3) in the two-torsion Brauer group. We prove that if a Hodge-general member of Cd is rational then βd=0 -- or, equivalently, that Huybrechts' twisted-K3 condition (**') holds. Consequently, a very general member of 𝒞d is irrational if βd≠0. Thus, a very general cubic fourfold containing a smooth cubic scroll or a Veronese surface is irrational, and hence so is a very general Küchle fourfold of type (c7). We also obtain an analogous obstruction for Hodge--special Gushel--Mukai fourfolds: a very general member in the discriminant--d locus can be rational only if d is a sum of two squares. Consequently, a very general Gushel--Mukai fourfold containing a cubic scroll is irrational.

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Cite

BibTeXRIS
BibTeX
@misc{fay2026hasse,
  title = {{Hasse Obstructions to Rationality for Special Fourfolds}},
  author = {Aideen Fay},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06759},
  eprint = {2609.06759},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06759},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Hasse Obstructions to Rationality for Special Fourfolds
AU  - Aideen Fay
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06759
UR  - https://arxiv.org/abs/2609.06759
AB  - For every nonempty Hassett divisor 𝒞_d parameterizing special cubic fourfolds, we consider the quaternion class β_d=(d/2,−3) in the two-torsion Brauer group. We prove that if a Hodge-general member of C_d is rational then β_d=0 -- or, equivalently, that Huybrechts' twisted-K3 condition (**') holds. Consequently, a very general member of 𝒞_d is irrational if β_d≠0. Thus, a very general cubic fourfold containing a smooth cubic scroll or a Veronese surface is irrational, and hence so is a very general Küchle fourfold of type (c7). We also obtain an analogous obstruction for Hodge--special Gushel--Mukai fourfolds: a very general member in the discriminant--d locus can be rational only if d is a sum of two squares. Consequently, a very general Gushel--Mukai fourfold containing a cubic scroll is irrational.
ER  -