6 problems

Support-space conjecture. The morphism $\Sigma\rightarrow S$ is a disjoint union of locally closed immersions.

Smooth algebraic-space general-position conjecture. Every smooth algebraic space has the curve-free-point property; equivalently, there is no smooth algebraic space with a unique p…

Algebraicity conjecture for deformations of varieties of general type. The space $\operatorname{Def}(X_0)$ has a representative ${\mathbf X}\to S$ where $S$ is a scheme of finite t…

Let $K'/K$ be a finite field extension, let $X'$ be a separated $K'$-scheme locally of finite type over $K'$, and let $(X')^\mathrm{an}$ be the associated $K'$-analytic Berkovich s…

Grothendieck's conjecture. The morphism $X\to S$ is integral.

Let a groupoid action have a geometric quotient, and suppose the additional weak assumptions referred to in the source hold. A geometric quotient is categorical among locally separ…