10 problems

Let $\delta\geq 2$. For a polynomial $g\in\mathbb{Z}[T_1,\ldots,T_\nu]$, let $\mathfrak{h}(g)$ denote its homogeneous part of maximal degree, and let $\mathcal{J}_\delta$ be the se…

Danielewski's conjecture. The surfaces $V_1$ and $V_2$ are isomorphic if and only if $n_1=n_2$ and

Let $F$ be a field of characteristic zero and let $S_F$ be a normal affine surface over $F$. Suppose that $f,g\rbrace\in\operatorname{Aut}(S_F)$ are loxodromic automorphisms. Stron…

Positive geodesic and hyperbolic-direction conjecture. The answers to these three questions should be positive, with greater reserve concerning the assertion that the set of hyperb…

Gizatullin's conjecture. Every smooth quasi-homogeneous surface over $\mathbb K$ is homogeneous.

Gizatullin's transitivity conjecture. The action of $\operatorname{Aut}(V)$ is transitive on $V$; equivalently,

Let $\Bbbk$ be a field of characteristic zero, and let $X$ be a Gizatullin surface, meaning a normal affine surface that admits a completion by a chain of smooth rational curves. W…

Linear plane divisor conjecture. Every smooth $cQ$-homology plane of general type has a linear plane divisor.

Let $C\subset\mathbb{P}^2_{\mathbb{C}}$ be an irreducible curve. A curve is of type I if there exists a point $a\in C$ such that $C\backslash a$ is isomorphic to the affine line; 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…