On the rationality problem for low degree hypersurfaces

Source abstract

Abstract 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].