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9. Other problems


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      Problem 9.05.

      Can we find a simply-connected variety with supersingular cohomology which is not rationally-connected?


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          Problem 9.1.

          Find a complete subvariety of \mathcal{M}_{g} of \text{codim}2g-1.


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              Problem 9.15.

              Fix q, let g\rightarrow \infty, what subvariety of \mathcal{M}_{g} / \mathcal{A}_{g} contributes the most \# of rational points?


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                  Problem 9.2.

                  Is there a supersingular hypersurface of every degree, dimension and char. p?


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                      Problem 9.25.

                      Find other examples of supersingular hypersurfaces.


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                          Problem 9.3.

                          Is being rationally connected = unirational in char p?


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                              Problem 9.35.

                              Can a surface with c_{1}^{2}=3c_{2} contain a rational curve in char. p?


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                                  Problem 9.4.

                                  Does an automorphism of a separably rationally connected variety necessarily have a fixed point?


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                                      Problem 9.45.

                                      Is there an irrational smooth proper variety over \text{Spec}\mathbb{Z}? (Can we find such a hypersurface?)


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                                          Problem 9.5.

                                          For every p, d with d\geq n+1, can we find hypersurface over \mathbb{F}_{p} of degree d in \mathbb{P}^{n} such that either \# X \not\equiv 1 \text{mod} p or X is not rationally connected?


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                                              Problem 9.55.

                                              Can we check Bloch’s conjecture for known examples of super-singular surfaces?


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                                                  Problem 9.6.

                                                  Let (X, D) snc. How can rational curves in X meet D? Are there any restrictions?


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                                                      Problem 9.65.

                                                      Can we get alterations of coprime degrees?


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                                                          Problem 9.7.

                                                          Characterize a variety X over \text{Spec}\mathbb{Z} such that the indecomposable summands of F_{*}\mathcal{O}_{X} have universally bounded rank.


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                                                              Problem 9.75.

                                                              If X is separably rationally connected, there exists N(\text{dim} X, \text{deg} X, m) such that, for any p_{1},\cdots, p_{m}\in X(\mathbb{F}_{p}), there exists a rational curve C defined over \mathbb{F}_{p^{N}} containing all p_{i}. (due to Kollar) Can you do this without lifting the cardinality of field tend to \infty? Let X be a cubic surface over \mathbb{F}_{q}. Can you find a rational curve defined over \mathbb{F}_{q} through each point?


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                                                                  Problem 9.8.

                                                                  [Colliot-thélène] For separably rationally connected varieties, is the Brauer-Manin obstruction the only obstruction for the Hasse principle?

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