10 problems

Makar-Limanov invariant stability conjecture.

Let $V(p)$ be a hypersurface in affine $n$-space, and let $R$ be its coordinate $K$-algebra. A hyperplane in affine $n$-space is isomorphic to affine $(n-1)$-space. Zariski's cance…

Let $K$ be a field, let $A:=K[T]=K[t_1,\dots,t_n]$ be the polynomial algebra, and let $B$ be a commutative $K$-algebra. Zariski cancellation problem. Does … imply … This asks wheth…

Cancellation conjecture.

Let $R$ be an algebra over a field $K$, let $z$ be a free generator, and let $x_1,\dots,x_n$ be noncommuting free generators. Write $K\langle x_1,\dots,x_n\rangle$ for the free ass…

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

Let $K$ be the ground field, let $V(p)$ be a hypersurface in affine $n$-space, and call two subvarieties equivalent if an automorphism of the ambient affine space takes one onto th…

Let $R$ be a $\mathbb{K}$-algebra, and let $\mathbb{K}\langle x_1,\ldots,x_n\rangle$ and $\mathbb{K}\langle y\rangle$ denote free associative algebras. Free associative cancellatio…

Let $X$ be a smooth affine surface. Say that $X$ has negative logarithmic Kodaira dimension when its logarithmic Kodaira dimension is negative. Say that $X$ fails the…

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…