53 problems

Persistence Conjecture. If $X$ is arithmetically hyperbolic over $k$, then $X$ is absolutely arithmetically hyperbolic.

Let $\mathfrak{X}$ be a “nice” algebraic stack defined over a number field $K$, and let $\mathcal{E}$ be a “nice” vector bundle on $\mathfrak{X}$. Let…

Let $A$ be a geometric $n$-stack, and let $D=\operatorname{Spec}(k[\epsilon]/(\epsilon^2))$. Let $TDef_A$ denote the $(n+1)$-category of deformations of $A$ parametrized by $D$. Le…

Let a geometric characterization theorem be the theorem characterizing toric triples by the properties (i), (ii), and (iii) stated in the introduction. In positive characteristic,…

Let $T$ be a very presentable geometric $n$-stack, and let $X$ be a smooth projective variety. Write $X_{DR}$ for its de Rham formal stack. The de Rham mapping-stack conjecture. Th…

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 $T$ be a geometric $n$-stack whose homotopy sheaves $\pi_i$ are vector sheaves for every $i\geq 3$, and let $X\to S$ be the morphism under consideration. The higher-vector-homo…

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 $\mathcal{M}_{g,n}^r$ denote the moduli stack of $A_r$-stable curves. In the setting of topological moduli spaces, suppose that the relevant properties are expressed by $M$-sep…

Reductive local structure conjecture. Under these hypotheses, such an étale morphism exists.

The stack of all curves conjecture. 1. $\mathcal{M}_g^{\text{all}}$ satisfies Vakil's Murphy's law. 2. $\operatorname{Pic}(\mathcal{M}_g^{\text{all}})=0$. 3. The structure morphism

Very good implies $L^2$. If $\mathcal X$ is very good, then $\mathcal X$ is an $L^2$-stack.

Let $X$ be a quasi-separated Deligne–Mumford stack with generically trivial stabilizer. Let $\mathcal{M}\in \operatorname{QCoh}(X)$ and let $\alpha\in \mathrm{Br}(X)$. Write…

Fix a prime $p$, let $n\in\mathbb{N}$, let $G$ be a smooth affine group scheme over $\mathbb{Z}/p^n\mathbb{Z}$, and let $\mu:\mathbb{G}_m\to G$ be a cocharacter. Let…

Let $k$ be a number field, and let $\mathcal X$, $\mathcal Y$, and $\mathcal Z$ be algebraic $k$-stacks. Let $F$ and $G$ be linear $k$-groups, and let … be torsors. Torsor covering…

Let $\textup{BT}$ be the stack of Barsotti–Tate groups over $S$, let $\textup{Disp}$ be the stack of displays over $S$, and let $\phi\colon \textup{BT}\rightarrow \textup{Disp}$ be…

Smoothness conjecture. The stack $\operatorname{BT}_{n,m}^{G,\mu}$ is smooth over ${\mathbb{Z}}/p^m$.

Coherent completeness and formal functions conjecture. The stack $[X/G]$ is cohomologically proper over $A^G$, and the pair $([X/G],\mathcal{G}_x)$ is coherently complete and satis…

Coherent completeness conjecture. The quotient stack $[\operatorname{Spec} A/G]$ is coherently complete along its unique closed point.

Let $G$ be a smooth affine group scheme over ${\mathbb{Z}}/p^n{\mathbb{Z}}$ and let $\mu:{\mathbb{G}}_m\to G$ be 1-bounded. Let $\operatorname{BT}_n^{G,\mu,?}$ be the quotient 2-st…

Let $G$ be a smooth affine group scheme over ${\mathbb{Z}}/p^n{\mathbb{Z}}$ equipped with a 1-bounded homomorphism $\mu:{\mathbb{G}}_m\to G$. Let $\hat{\mathscr R}_n$ be the stack…

Let $G$) be a smooth affine group scheme over ${\mathbb{Z}}/p^n{\mathbb{Z}}$ equipped with a 1-bounded cocharacter $\mu:{\mathbb{G}}_m\to G$. Let $\operatorname{BT}_n^{G,\mu}$ be t…

Drinfeld's algebraicity conjecture. This restriction is a quasicompact smooth algebraic stack over $\mathbb Z/p^m\mathbb Z$ with affine diagonal.

Drinfeld's gerbe isomorphism conjecture. These two gerbes are isomorphic.