Rational subvarieties in positive characteristic

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

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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 $n\geq 11$ ?
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Problem 4.2.
Find rational curves in $\mathcal{A}_{g}$ in char. $p$ , that do not lift to char. $0$ .
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Problem 4.3.
What are the rationality properties of $\mathcal{A}_{g,s.s.}$ ? You can ask the same question for Newton polygon strata.
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Problem 4.4.
Are complete rational curves in $\mathcal{A}_{g}$ dense in char. $p$ ?

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