25 problems

Let $r+s=n$, with signature $(r,s)$, and let $M_I^{\mathrm{loc}}(r,s)$ be the scheme-theoretic closure of the generic fiber of the naive local model $M_I^{\mathrm{naive}}(r,s)$. Le…

Let $f\colon X\to Z$ be a klt-trivial fibration of relative dimension $k$, with $(X,B)$ a klt projective pair whose boundary $B$ has rational coefficients. Let $r$ be the positive…

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 ${\mathcal T}$ be a rational normal scroll of dimension $5$ in ${\mathbf P}^g$, and let ${\mathcal M}({\mathcal T},c)$ denote the set of projective models of $K3$ surfaces $S$…

Let a Kleinian group be called admissible if it has generators $\varphi_i$, $i\in I_+$, satisfying the conditions of the amplification. For an admissible Kleinian group, a fundamen…

Let $T$ be any geometric, locally geometric, or slightly geometric $n$-stack, and let $X\to S$ be a flat projective morphism. The slightly geometric mapping-stack conjecture. The m…

Let $S$ be a base scheme, let $X\to S$ be a flat projective morphism, and let $K({\bf G}_m,2)$ denote the Eilenberg–Mac Lane stack. For any group scheme $G$ of finite type, let…

Let $T$ be a geometric $n$-stack that is very presentable in the sense that its fundamental groups over artinian bases are affine and its higher homotopy groups are vector sheaves.…

Let $S$ be a smooth and irreducible quasi-projective variety over $\mathbb C$, let $N$ be a positive integer, and let $\Phi:S\times\mathbb P^N\to S\times\mathbb P^N$ be an algebrai…

Let $\mathcal{M}^o$ be a finite-type algebraic $\mathbbm{k}$-stack underlying a $\mathbb{Q}$-Gorenstein faithfully flat affine family of $\mathbb{Q}$-Fano fibrations, equipped with…

Consider $T$-equivariant Fano fibrations $\pi\colon X\to Y$, fixing the multi-Hilbert function of $T\curvearrowright Y$ and the Hilbert polynomial of $-K_X$ on the fibers of $\pi$.…

Let $M_I^{\mathrm{loc}}(r,s)$ be the local model and let $M_I(r,s)$ be the closed subscheme defined by the strengthened spin condition. Smithling's strengthened-spin conjecture. On…

Stein-degree boundedness conjecture. The quantity $\operatorname{sdeg}(S/Z)$ is bounded from above depending only on $d$ and $t$.

Odaka's CM minimization conjecture. One has

Finiteness conjecture. There is an open set $U\subseteq Y$ such that for every $y\in U$ the set

Let $A$ be a connective dga and let $S$ be a one-dimensional $A$-module. Let $E\coloneqq {\mathrm{\normalfont\mathbb{R}}}\mathrm{End}_A(S)$ be the derived endomorphism dga of $S$.…

Let $f:X\to Y$ be a smooth projective family of manifolds with ample canonical bundle, so the fibers are canonically polarized. Suppose that the algebraic structure of the family v…

Let $\pi:X\to T$ be a projective family of varieties. Assume that $T$ is normal, every fiber $X_t$ is klt, and $-K_{X/T}$ is $\mathbb{Q}$-Cartier and ample. For a geometric point…

Parameter-scaling conjecture. For all rank vectors $\mathbf{r}$,

Constancy criterion for simultaneous canonical models. The morphism $f:(X,\Delta)\to S$ has a simultaneous canonical model if and only if the function

Let $X_n$ denote the locus of configurations $(P_1,\ldots,P_n)$ for which the canonical ring has exceptional generation behavior, and let the fat diagonal be the locus where at lea…

Let $C$ be the canonically embedded complete-intersection curve of degree $8$ and genus $5$ arising as the base locus of a net of quadrics in $\mathbb{P}^4$, let…

Let $C$ be the canonically embedded complete-intersection curve of degree $8$ and genus $5$ arising as the base locus of a net of quadrics in $\mathbb{P}^4$. Fix ramification data…

Let $Q$ be a conic bundle and let $\operatorname{Cl}_0(Q)$ be its even Clifford algebra. Consider orders Morita equivalent to $\operatorname{Cl}_0(Q)$ and having the same rank and…

Let $f:X\to S$ be a fibre space $X/S$, let $X_{\bar{\eta}}$ be its generic geometric fibre, and let $\omega_{X/S}$ and $\omega_{X_{\bar{\eta}}}$ denote the relative and generic-fib…