18 problems

Let $G$ be a reductive algebraic group acting on affine space $\mathbb{A}^n$ over a field $k$. Linearization conjecture. The action is linear after a suitable polynomial change of…

Collective infinite transitivity conjecture. The $G$-action on

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 $G$…

Flenner's conjecture. The threefold $X$ does not admit a nontrivial $\mathbb C_+$-action.

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…

Nonexistence conjecture. There is no $k>1$ such that $Hilb_k$ of a two-fold has a regular $B(2)$-variety structure.

Let $X$ be a smooth affine surface. Call $X$ rigid if it admits no nontrivial action of the additive group $\mathbb{G}_{a}$, and call $X$ an affine-line-bundle total space if it is…

Combinatorial description conjecture. ${\rm Rt}(X)$ can be described in a similar combinatorial way for an arbitrary affine spherical variety $X$.

Let $\mathbf{T}=\mathbb{C}^{*d}$ act on affine space $\mathbb{C}^d$. Linearization conjecture. Any $\mathbf{T}$-action on $\mathbb{C}^d$ is linear after a suitable change of coordi…

Biaynicki-Birula's conjecture. Every normal variety $X$ contains finitely many maximal quasi-projective open subvarieties.

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

Algebraic Linearization Conjecture. The group $G$ has a fixed point $p$, and the action of $G$ is linear with respect to a suitable coordinate system of $\mathbb{C}^n$ having $p$ a…

Flenner–Kaliman–Zaidenberg conjecture. Among smooth affine $\mathbb{C}^*$-surfaces, the toric surfaces and the Danilov–Gizatullin surfaces are the only exceptions to uniqueness of…