40 problems

Let $X$ be an affine algebraic variety defined over an algebraically closed field, and let $G$ be a subgroup of $\operatorname{Aut}(X)$ generated by a finite collection of…

Let $\mathcal{M}_g$ be the coarse moduli space of genus $g$ Riemann surfaces, viewed as a quasiprojective complex variety. Looijenga's conjecture. For every $g\geq 2$, the variety…

Let $(M,g)$ be a real-analytic Riemannian manifold, and let $T^\infty M:=TM$ be an entire Grauert tube, meaning that the tangent bundle admits a Stein structure with strictly pluri…

Knop's conjecture. The weight monoid $\Gamma(X)$ determines $X$ up to equivariant automorphism.

Pham–Brieskorn rigidity conjecture. The coordinate ring $\mathcal O_{\bar Y}(\bar Y)$ is rigid if and only if $a_1\geq 3$.

Let $k$ be an algebraically closed field, and let $X$ be a smooth affine $k$-scheme of dimension $d\geq2$. Suslin's cancellation conjecture. If $\mathcal{E}$ and $\mathcal{E}'$ are…

Effective-bound conjecture. These three quantities equal $2^k-1$ for $k\geq 1$. This conjecture gives the expected sharp value of the embedding dimension bound for the incidence st…

Moduli conjecture. For every $m\geq 3$ and every $n\geq 0$, there exists such an $S$ and $f$ for which the fibers over the $k$-points of $S$ are all non-isomorphic.

Let $U$ be a smooth affine scheme of dimension $d$ over the finite field ${{\mathbb F}}_q$, and let $H_i^{c}(U,{{\mathbb Q}}(n))$ be rational motivic Borel–Moore homology. Affine P…

Let $Y$ be an algebraic manifold of dimension $n$, let $X$ be a completion of $Y$, and let $D$ be an effective boundary divisor with support $X-Y$. For integers $i,j\geq 0$, write…

Let $A$ be the coordinate ring of the affine curve under consideration, and let $pi_1(A)$ denote its algebraic fundamental group. Harbater's conjecture. The algebraic fundamental g…

Let $\Bbbk$ be an algebraically closed field. An algebraic monoid is an algebraic variety $M$ with an associative product $M\times M\rightarrow M$ that is a morphism and has a neut…

Let ${\mathcal{O}}$ be a non-simplicial MaxDeg planar order ideal, and let $\mathbb{B}_{\mathcal{O}}$ denote its border basis scheme. A $Z$-separating re-embedding means a re-embed…

Finite-generation conjecture. The Lie algebra $\operatorname{Vect}(X)$ is finitely generated as a Lie algebra.

Let $X$ be an affine variety admitting no effective $\mathbb G_{\mathrm a}$-action and no effective $\mathbb G_{\mathrm m}$-action. The rigid-affine trivial-automorphism conjecture…

Let $X$ be a rigid affine variety, meaning that it admits no effective $\mathbb G_{\mathrm a}$-action. The rigid-affine automorphism-torus conjecture. The identity component…

Let $Y$ be a variety, let ${\mathbb K}[Y]$ be its coordinate ring, let $I$ be the ideal generated by the images of all locally nilpotent derivations on $Y$, and let…

Let $Y$ be a variety, let ${\mathbb K}[Y]$ be its coordinate ring, and let $I$ be the ideal of ${\mathbb K}[Y]$ generated by the images of all locally nilpotent derivations on $Y$.…

Flexible affine variety conjecture. Every smooth flexible complex affine variety is an $RR$-space.

Let $X$ be a topologically contractible smooth affine complex variety, and let $x$ be a chosen closed basepoint. Asok–Østvær's conjecture. The pointed variety $(X,x)$ is stably…

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 $\mathbb K$ be an algebraically closed field of characteristic zero. Let $Y$ be an affine algebraic variety. Call $Y$ rigid if it admits no effective $\mathbb G_{\rm a}$-action…

Let $X$ be a compact algebraic manifold and let $Y\subset X$ be a prime divisor. The normal bundle $N_{Y/X}$ is the bundle of normal directions to $Y$ in $X$. Goodman's conjecture.…

Let $M$ be a smooth affine variety with log Kodaira dimension $-\infty$. Cyclic dilation conjecture. The existence of a cyclic dilation whose image under the marking map equals the…

Let $M$ be an affine conic bundle of the form described in the paper, and suppose that the polynomial $p(z_3,\ldots,z_{n+1})$ is weighted homogeneous and invertible. Dilation conje…