9. Other problems

Problem 9.05.
Can we find a simplyconnected variety with supersingular cohomology which is not rationallyconnected? 
Problem 9.15.
Fix $q$, let $g\rightarrow \infty$, what subvariety of $\mathcal{M}_{g} / \mathcal{A}_{g}$ contributes the most $\#$ of rational points? 
Problem 9.2.
Is there a supersingular hypersurface of every degree, dimension and char. $p$? 
Problem 9.35.
Can a surface with $c_{1}^{2}=3c_{2}$ contain a rational curve in char. $p$? 
Problem 9.4.
Does an automorphism of a separably rationally connected variety necessarily have a fixed point? 
Problem 9.45.
Is there an irrational smooth proper variety over $\text{Spec}\mathbb{Z}$? (Can we find such a hypersurface?) 
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? 
Problem 9.55.
Can we check Bloch’s conjecture for known examples of supersingular surfaces? 
Problem 9.6.
Let $(X, D)$ snc. How can rational curves in $X$ meet $D$? Are there any restrictions? 
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. 
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? 
Problem 9.8.
[Colliotthélène] For separably rationally connected varieties, is the BrauerManin obstruction the only obstruction for the Hasse principle?
Cite this as: AimPL: Rational subvarieties in positive characteristic, available at http://aimpl.org/ratsubvarpos.