17 problems

Let $n$ be a positive integer, let $X$ be an $n$-dimensional rational variety, and let $G$ be a finite group acting on $X$. Geometric Jordan conjecture. There exists a constant…

Let $\mathrm{Cr}_3(\mathbb{R})$ denote the Cremona group of birational self-maps of real projective 3-space, and let $\mathfrak{A}_6$ denote the alternating group on six letters. U…

Let $\mathcal{A}_3$ and $\mathcal{B}_3$ be the classes of finite abelian groups arising in the paper, and let … while … where $G_1=(\mathbb{Z}/4)^4$,…

Let $G$ be a finite group over the field of complex numbers. Its essential dimension is the minimal dimension of a faithful $G$-variety admitting a compression from a faithful line…

Let $k$ be a field of characteristic $p>0$, and let $\mathrm{Cr}_n(k)=\mathrm{Bir}_k(\mathbb{P}^n_k)$ be the Cremona group of birational automorphisms of projective $n$-space. An e…

Let $n\ge1$, let $k\in\mathbb{R}$, and let $g(k)$ be the polynomial map obtained from the coefficients $j_{k,i}(\mathbf{l},\mathbf{r})$ in the expansion of the two-sided orthogonal…

Let $n\ge 1$, let $k\in\mathbb{R}$, and let $f(k)$ be the polynomial map obtained from the coefficients $j_{k,i}(\mathbf{l})$ in the expansion of the one-sided orthogonal Jacobi po…

Let $n\geq 2$. The Cremona group ${\rm Bir}({\mathbb P}^n)$ consists of birational transformations of complex projective $n$-space, and a Borel subgroup is a maximal closed connect…

Let $n\geq 5$. The Cremona group ${\rm Bir}({\mathbb P}^n)$ is the group of birational transformations of complex projective $n$-space, and a Borel subgroup is a maximal closed con…

Discreteness conjecture. The embeddings into $\operatorname{Cr}_3$ of $\mathcal{A}_6$ and $\operatorname{PSL}_2(7)$ up to conjugation form discrete families.

Klein's generation conjecture. The group of contact maps is generated by $\mathscr{K}$ and the Legendre involution. This concerns the structure of the group of birational self-maps…

Let $H$ be an irreducible hypersurface in $\mathbb{P}^s\times\mathbb{P}^s$, and let $X,Y$ be nonempty Zariski-open subsets of $\mathbb{P}^s$. A hypersurface restriction is $(s,t)$-…

Let $H$ be an irreducible hypersurface in $\mathbb{P}^s\times\mathbb{P}^s$. A set $V\subset\mathbb{P}^s\times\mathbb{P}^s$ is almost-$(s,t)$-grid-free if there are nonempty Zariski…

Let $G_K\cong \mathrm{PSL}_2(\mathbb{F}_7)$ be the Klein simple group, and let $X_n/\mathbb{P}^1$ be the $G_K$-Mori fiber spaces described above. A $G_K$-Mori fiber space is a thre…

Cheltsov–Shramov conjecture. The varieties

Irrationality conjecture. The varieties $\mathcal{X}_n$ are all irrational except for those two particular members; consequently, only two embeddings of…

Dolgachev's inequality.