26 problems

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 $Y$ 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 ${\mathcal{O}}$ 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…

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 $Y$ 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 a rigid affine algebraic variety over ${\mathbb K}$. The connected component $\operatorname{Aut}^0(Y)$ is an algebraic torus of rank at most $\dim Y$. Perepechko–Zaidenb…

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 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 $X$ 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 $k$ 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…

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…

Let $M$ be a smooth affine variety with $\kappa(M)=-\infty$, and let $c_1^{\mathit{GH}}(M)$ denote its first Gutt–Hutchings capacity. Gutt–Hutchings capacity conjecture. … The sour…

Let $M$ be an $n$-dimensional smooth affine variety over a field $\mathbb{K}$ of characteristic zero. Let $\kappa(M)$ denote its relevant Kodaira dimension. A cyclic dilation is un…

Let $X$ be an affine variety over an algebraically closed field ${\mathbb K}$ of characteristic zero. Let ${\mathcal U}(X)\subset \operatorname{Aut}(X)$ be the subgroup generated b…

Let $X$ be an affine variety over an algebraically closed field ${\mathbb K}$ of characteristic zero. Let ${\mathbb G}_a$ denote the additive group of ${\mathbb K}$, and let…

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…

Let $X$ and $Y$ be smooth affine varieties over $\mathbb{C}$, and let $T^*(X)$ and $T^*(Y)$ denote their cotangent bundles. The cotangent-bundle isomorphism conjecture. If … then ……

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 $\mathcal{M}_g$ 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 $k$ be the ground field, let $T$ be the coordinate ring considered in the paper, let $T^\ast$ be its group of units, and let $n$ be the relevant degree or group order. Let $G$…

Let $k$ be the ground field, let $f$ be the polynomial defining the affine variety in the paper, and let $T$ be its coordinate ring. Write $T^\ast$ for the group of units of $T$. U…