Geodesic
On the rationality problem for low degree hypersurfaces
Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e25

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We show that a very general hypersurface of degree $d \geq 4$ and dimension $N \leq (d+1)2^{d-4}$ over a field of characteristic $\neq 2$ does not admit a decomposition of the diagonal; hence, it is neither stably nor retract rational, nor $\mathbb {A}^1$-connected. Similar results hold in characteristic $2$ under a slightly weaker degree bound. This improves earlier results in [44] and [33]. 
Lange, Jan; Schreieder, Stefan. On the rationality problem for low degree hypersurfaces. Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e25. doi: 10.1017/fmp.2025.10016
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