5 problems

Let $\overline{M}_{g,n}$ be the moduli space of genus $g$ curves with $n$ marked points over $\operatorname{Spec} \mathbb{Z}$, and let $L$ be a nef line bundle on it. Mixed-charact…

Weak hyperkähler SYZ conjecture. If $q(M)=0$ and $h^0(X,\mathcal{O}_X(M))\geq n+1$, then $M$ is semiample; that is, some positive multiple of $M$ is globally generated.

Assume the base field has positive characteristic. Let $\overline{\operatorname{M}}_{g,n}$ be the moduli space of stable $n$-pointed curves of genus $g$, and let $D$ be a nef divis…

Hyper-Kähler SYZ conjecture. The line bundle $L$ is semi-ample.

Let $\overline{M}_{g,n}$ be the moduli space of genus $g$ curves with $n$ marked points in positive characteristic, and let $L$ be a nef line bundle on $\overline{M}_{g,n}$. Keel's…