
## 8. Fano varieties, characterization of projective spaces

1. #### Problem 8.14.

Does every Fano variety in char. $p$ contain a rational, unirational and rationally-connected 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 simply-connected? 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.