
## 9. Other problems

1. #### Problem 9.05.

Can we find a simply-connected variety with supersingular cohomology which is not rationally-connected?
• #### Problem 9.1.

Find a complete subvariety of $\mathcal{M}_{g}$ of $\text{codim}2g-1$.
• #### 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 9.25.

Find other examples of supersingular hypersurfaces.
• #### Problem 9.3.

Is being rationally connected = unirational in 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 super-singular surfaces?
• #### Problem 9.6.

Let $(X, D)$ snc. How can rational curves in $X$ meet $D$? Are there any restrictions?
• #### Problem 9.65.

Can we get alterations of coprime degrees?
• #### 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.

[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.