6. New applications of Kollar’s and Totaro’s techniques

Problem 6.1.
Is the very general hyperusrfae of degree $d\geq 2 \lceil(n+3)/3\rceil$ not rational / stably rational / ruled in char $p>0$ for $p>>0$? 
Problem 6.2.
Can we improve those techniques to find differential form for lower degree hypersurfaces? 
Problem 6.3.
Can we find new exmaples of unirational varieties that have nonvanishing differential forms? 
Problem 6.5.
Find an example of varieties which are degenerations of Fano hypersurfaces with $H^{i}(\mathcal{O}_{X})\neq 0$ for some $i$. Try to construct varieties with nontrivial Brawer classes.
Cite this as: AimPL: Rational subvarieties in positive characteristic, available at http://aimpl.org/ratsubvarpos.