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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.