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arXiv · 2610.10509

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 $\geq 8$) 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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James Hotchkiss. 2026-10-07. The rationality problem for hypersurfaces of degree at least five. https://arxiv.org/abs/2610.10509

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