Rational subvarieties in positive characteristic

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8. Fano varieties, characterization of projective spaces

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Problem 8.14.
Does every Fano variety in char. $p$ contain a rational, unirational and rationally-connected surface?
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Problem 8.44.
Is every smooth Fano hypersurface / variety separably rationally connected?
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Problem 8.46.
Does every smooth projective Fano lift to char. $0$ ? Are they all simply-connected? Does boundedness hold in char. $p$ ?
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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. $p$ ?
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Problem 8.52.
If there exist $\phi\colon E\longrightarrow T_{X}$ with $E$ ample, $X$ smooth projective irreducible, does it follow $X\cong\mathbb{P}^{n}$ ?
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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.