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The rationality problem for hypersurfaces of degree at least five

  • James Hotchkiss
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Abstract

We prove that a very general hypersurface of degree at least five and any positive dimension does not have a decomposition of the diagonal, over an uncountable algebraically closed field of characteristic not $2$. In particular, such hypersurfaces are not stably or retract rational. For the proof, we construct (in each dimension ≥ 8) a certain quintic hypersurface with a nonzero unramified cohomology class, and which is rationally fibered by quadrics over a lower-dimensional projective space. From our construction, the main result follows using methods developed by Schreieder.

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Cite

BibTeXRIS
BibTeX
@misc{hotchkiss2026rationality,
  title = {{The rationality problem for hypersurfaces of degree at least five}},
  author = {James Hotchkiss},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10509},
  eprint = {2610.10509},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10509},
}
RIS
TY  - UNPB
TI  - The rationality problem for hypersurfaces of degree at least five
AU  - James Hotchkiss
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10509
UR  - https://arxiv.org/abs/2610.10509
AB  - We prove that a very general hypersurface of degree at least five and any positive dimension does not have a decomposition of the diagonal, over an uncountable algebraically closed field of characteristic not $2$. In particular, such hypersurfaces are not stably or retract rational. For the proof, we construct (in each dimension ≥ 8) a certain quintic hypersurface with a nonzero unramified cohomology class, and which is rationally fibered by quadrics over a lower-dimensional projective space. From our construction, the main result follows using methods developed by Schreieder.
ER  -