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

1. #### 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 non-vanishing differential forms?
• #### Problem 6.4.

Look for new applications of Kollár / Totaro’s techique.
• #### 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 non-trivial Brawer classes.

Cite this as: AimPL: Rational subvarieties in positive characteristic, available at http://aimpl.org/ratsubvarpos.