4. Rational subvarieties of the moduli space of rational curves and of abelian varieties

Problem 4.1.
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 4.2.
Find rational curves in $\mathcal{A}_{g}$ in char. $p$, that do not lift to char. $0$. 
Problem 4.3.
What are the rationality properties of $\mathcal{A}_{g,s.s.}$? You can ask the same question for Newton polygon strata.
Cite this as: AimPL: Rational subvarieties in positive characteristic, available at http://aimpl.org/ratsubvarpos.