14 problems

Let $X$ be a simply connected complex surface, let $f_k:X\to\mathbb{CP}^2$ be a generic projection defined by sections of $L^{\otimes k}$, let $\tilde{X}_k$ be the associated Galoi…

Abhyankar's inertia conjecture. The pair $(G,I)$ is realizable if and only if

Free abelianity conjecture. The group $\pi_1(X_{\operatorname{Gal}})$ is a free abelian group.

Generalized Purely Wild Inertia Conjecture. There is a connected $G$-Galois cover of $\mathbb{P}^1$ étale away from $B$ such that $P_i$ occurs as an inertia group above $x_i$ for…

Generalized Inertia Conjecture. There is a connected $G$-Galois cover of $\mathbb{P}^1$ étale away from $B$ such that $I_i$ occurs as an inertia group above $x_i$ for…

Purely Wild Inertia Conjecture. The subgroup $P$ occurs as the inertia group at a point above $\infty$ in a connected $G$-Galois cover of $\mathbb{P}^1$ branched only at $\infty$ if…

Let $V$ be a divisorially canonical Fano variety satisfying the hypotheses of Theorem 1, and let $X\dashrightarrow V$ be a rationally connected Galois rational cover. A Galois grou…

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 $p=2$…

Oort's conjecture. Every cyclic Galois cover of $k$-curves lifts to characteristic zero.

In the same quotient setting, let $\fie GK$ denote the set of isomorphism classes of large $G$-fields over $K$, and let $\delta_{F,m}$ be the residue invariant associated with $F$…

Let $K$ be the ground field, let $X=\mathbb{P}^1_K$, and let $\bar{x}_0$ be the distinguished branch point with residue field $\kappa_0$. For a character…

Local Oort conjecture. The local lifting problem can always be solved when $G$ is cyclic.

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…