10 problems

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

Let $X$ be an affine variety of dimension at least $2$, and let $D_1,\ldots,D_k$ be locally nilpotent derivations on $\mathbb K[X]$. For each $i$, let $H_i=\exp(tD_i)$ be the corre…

Let $X$ be an irreducible homogeneous quasi-affine variety with … and with $\mathbb{K}[X]^{\times}=\mathbb{K}^{\times}$. Here $\theta(X)$ denotes the transitivity degree of the aut…

Infinite transitivity conjecture for affine toric varieties. The group $G$ acts on the open orbit of $X$ infinitely transitively if and only if

Infinite transitivity conjecture for affine toric surfaces. The group $G$ acts on the open orbit of $X$ infinitely transitively if and only if

A family $\mathfrak{F}$ of $\mathbb{G}_a$-subgroups of $\operatorname{Aut}(X)$ is called saturated if it contains every replica of each of its members and is closed under conjugati…

Let $X$ be an irreducible quasiaffine variety of dimension at least $2$. Let conditions (1)–(4) refer to the four conditions in Theorem, whose equivalence is being conjectured. Tra…

Let $X$ be an irreducible quasiaffine variety over an algebraically closed field ${\mathbb K}$ of characteristic zero. Write $\operatorname{Aut}(X)$ for its automorphism group and…

Infinite transitivity conjecture. For each $n\geq 1$, the action of $G$ on $\mathcal{C}_n$ is infinitely transitive.

Let $X$ be a unirational algebraic variety over a field $\mathbf{k}$. The field $\mathbf{k}(X)(y_1,\ldots,y_m)$ is said to admit an infinitely transitive model if it has a model th…