8 problems

Let $G$ be an algebraic group, and let $V$ and $W$ be generically free linear representations of $G$. Katsylo's conjecture. If … then $V$ and $W$ are birationally isomorphic as $G$…

Work over an algebraically closed field $\mathbb K$ of characteristic zero. Let $Y$ be an affine variety, and let $\mathbb{G}_{\rm a}$ and $\mathbb{G}_{\rm m}$ denote the additive…

Flexibility conjecture. $X$ is flexible.

Let $X$ be a variety. The non-torsion automorphism conjecture. If $\operatorname{Aut}(X)$ is infinite dimensional (respectively, has a nontrivial connected component), then…

Conjectural approach. There exist such $N$, hypersurfaces $W_i$, $\mathbb{G}_a$-subgroups $G_i$, a point $p$, and a subvariety $H$ for which the restricted map $\psi|_H$ is surject…

Let $Y$ be an affine variety over an algebraically closed field of characteristic zero. Assume that $Y$ admits no effective action of the additive group $\mathbb G_{\rm a}$ and no…

Let $F_n$ be the free associative algebra on $n$ generators, and let $\mathbb{T}_{n-1}$ be the $(n-1)$-dimensional algebraic torus. An action is linearizable when it is conjugate t…

Sylow-versality conjecture. If $X$ is $G_p$-versal for every prime $p$, then $X$ is $G$-versal.