6 problems

Let $k$ be a field of characteristic $p>0$, and let a cyclic extension mean a finite Galois extension with cyclic Galois group. Oort's conjecture. Every cyclic extension lifts to c…

Strong Oort conjecture. A cyclic-by-$p$ group $G$ is an Oort group for $k$ if and only if $G$ is of the form $C_m$, $D_{2p^n}$, or $A_4$, with the last case occurring only if $p=2$…

Let $k$ be an algebraically closed field of characteristic $p$, and let $k[[t]]$ be the formal power series ring. Local Oort conjecture. Every cyclic extension of $k[[t]]$ lifts to…

Let $X$ be a smooth cyclic curve over a field $k$ of characteristic $p$. Let $N$ be a cyclic normal subgroup of $G=\operatorname{Aut}_{k}(X)$ such that the quotient $X/N$ has genus…

Arithmetic ring-specific Oort conjecture. There exist $l>0$ and a normal, $C$-Galois, regular branched cover $Y\rightarrow D$ such that $Y_k$ is reduced, $L_m=M$ for $m\gg0$ and…

Ring-specific Oort conjecture. The cover $f$ lifts over $R$. This is a stronger, ring-specific form of Oort's conjecture because it specifies the mixed-characteristic ring over whi…