8. Fano varieties, characterization of projective spaces

Problem 8.14.
Does every Fano variety in char. $p$ contain a rational, unirational and rationallyconnected surface? 
Problem 8.44.
Is every smooth Fano hypersurface / variety separably rationally connected? 
Problem 8.46.
Does every smooth projective Fano lift to char. $0$? Are they all simplyconnected? Does boundedness hold in char. $p$? 
Problem 8.5.
What extra hypothesis need to be added to make characterizations of $\mathbb{P}^{n}$ in terms of ampleness of a subbundle of the tangent bundle hold in char. $p$? 
Problem 8.52.
If there exist $\phi\colon E\longrightarrow T_{X}$ with $E$ ample, $X$ smooth projective irreducible, does it follow $X\cong\mathbb{P}^{n}$? 
Problem 8.6.
Can we find a separably rationally connected Fano hypersurface of every degree and dimension over $\mathbb{F}_{p}$?
Cite this as: AimPL: Rational subvarieties in positive characteristic, available at http://aimpl.org/ratsubvarpos.