1. Problems

Problem 1.02.
When is $\mathcal{M}_{g,n}$, $\mathcal{A}_{g}[d]$ or $\mathcal{K3}_{d}$ unirational over char. $p$? When is $\mathcal{M}_{1,n}$ unirational for $n\geq 11$? 
Problem 1.04.
Find rational curves in $\mathcal{A}_{g}$ in char. $p$, that do not lift to char. $0$. 
Problem 1.06.
Can we find a simplyconnected variety with supersingular cohomology which is not rationallyconnected? 
Problem 1.08.
Let $X$ be a separably rationally connected variety. Does $H^{i}(X,\mathcal{O}_{X})=0$ for $i>0$? 
Problem 1.1.
Can we give lower bounds on the degree of irrationality / covering gonality.... of hypersurfaces in char $p$? How do we make sense of these notions over countable / finite fields? 
Problem 1.12.
How does rationality behave in a family over $\text{Spec} \mathbb{Z}$ or $\text{Spec} \mathbb{Z}[\frac{1}{n}]$. If $X$ is rational / unirational for almost all $p$, does it follow $X$ is rational / unirational in char $0$? 
Problem 1.14.
Can we find a cubic fourfold which is conjectually irratioal, but (most of ) its reductions are rational? 
Problem 1.16.
What are the rationality properties of $\mathcal{A}_{g,s.s.}$? You can ask the same question for Newton polygon strata. 
Problem 1.2.
Fix $q$, let $g\rightarrow \infty$, what subvariety of $\mathcal{M}_{g} / \mathcal{A}_{g}$ contributes the most $\#$ of rational points? 
Problem 1.22.
If the general member of a family of smooth cubic is rational, is the special fiber also rational? 
Problem 1.24.
Let $\mathbb{P}$ be the parametrizing space of higher surfaces of degree $d$ in $\mathbb{P}^{n}$. What is the largest dimensional family of unirational / supersinglar hypersurfaces? 
Problem 1.26.
Is there a supersingular hypersurface of every degree, dimension and char. $p$? 
Problem 1.3.
Does every Fano variety in char. $p$ contain a rational, unirational and rationallyconnected surface? 
Problem 1.34.
$\mathcal{X}\longrightarrow \mathcal{B}$ be a family of smooth hypersurfaces of degree $d=n$ in $\mathbb{P}^{n}$. Is there more than one rational section? Does weak approximation hold? 
Problem 1.36.
If $d^{2}\leq n$, does weak approximation hold in char. $p$for hypersurfaces? (it does in char. $0$) 
Problem 1.4.
Can a surface with $c_{1}^{2}=3c_{2}$ contain a rational curve in char. $p$? 
Problem 1.42.
Does an automorphism of a separably rationally connected variety necessarily have a fixed point? 
Problem 1.44.
In char. $p$ if $n\geq d+2$, are the moduli spaces of rational curves on a general hypersurface of degree $d$ in $\mathbb{P}^{n}$ irreducible of the expected dimension? 
Problem 1.46.
Can we find new char. $p$ cohomological obstruction to an integral decomposition of the diagonal? 
Problem 1.48.
Is the very general hyperusrfae of degree $d\geq 2 \lceil(n+3)/3\rceil$ not rational / stably rational / ruled in char $p>0$ for $p>>0$? 
Problem 1.5.
Is there an irrational smooth proper variety over $\text{Spec}\mathbb{Z}$? (Can we find such a hypersurface?) 
Problem 1.52.
Are there more examples where we can apply the Kollar’s / Totaro’s technique? 
Problem 1.54.
Can we improve those techniques to find differential form for lower degree hypersurfaces? 
Problem 1.56.
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 1.58.
Can we check Bloch’s conjecture for known eamples of supersingular surfaces? 
Problem 1.6.
Can we find new exmaples of unirational varieties that have nonvanishing differential forms? 
Problem 1.62.
Let $(X, D)$ snc. How can rational curves in $X$ meet $D$? Are there any restrictions? 
Problem 1.64.
Is every smooth Fano hypersurface / variety separably rationally connected? 
Problem 1.66.
Does every smooth projective Fano lift to char. $0$? Are they all simplyconnected? Does boundedness hold in char. $p$? 
Problem 1.7.
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 1.72.
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 1.74.
Characterize a variety $X$ over $\text{Spec}\mathbb{Z}$ such that the indecomposable summands of $F_{*}\mathcal{O}_{X}$ have universally bounded rank. 
Problem 1.76.
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 1.78.
[Colliotthélène] For separably rationally connected varieties, is the BrauerManin obstruction the only obstruction for the Hasse principle? 
Problem 1.8.
Can we find a separably rationally connected Fano hypersurface of every degree and dimension over $\mathbb{F}_{p}$? 
Problem 1.84.
Can we give lower bounds on the degree of irrationality / covering gonality of hypersurfaces in char. $p$? 
Problem 1.86.
What is the dimension of rational curves in hypersurfaces of degree $d$ in $\mathbb{P}^{n}$ in char. $p$? Does char. $0$ proof go through? 
4th day
Problem 1.92.
Do there exist a cubic $4$fold defined over number fields which is conjecured to be irrational but many reductions are conjectured to be ratioal? 
Problem 1.94.
Find an example of varieties which are degenerations of Fano hypersurfaces with $H^{i}(\mathcal{O}_{X})\neq 0$ for some $i$. Try to construct varieties with nontrivial Brawer classes. 
Problem 1.96.
Do cubic $3$folds have odd degree unirational prametrizations in char $p$? 
Problem 1.98.
Can irrational varieties over $\#$ fields specialize to rational varieties in char. $p>0$? (ColliotthélèneOjanguren’s examples)
Cite this as: AimPL: Rational subvarieties in positive characteristic, available at http://aimpl.org/ratsubvarpos.