  Let k be an uncountable algebraically closed field and let Y be a smooth
projective k-variety which does not admit a decomposition of the diagonal. We
prove that Y is not stably birational to a very general hypersurface of any
given degree and dimension. We use this to study the variation of the stable
birational types of Fano hypersurfaces over fields of arbitrary characteristic.
This had been initiated by Shinder, whose method works in characteristic zero.
