8 problems

Let $\mathfrak X$ be a hyperelliptic curve with extra automorphisms, meaning automorphisms beyond the hyperelliptic involution. Shaska's conjecture. Every such curve can be defined…

Wittenberg's descent conjecture. Let $X$ be a smooth $k$-variety and let $G$ be a linear algebraic $k$-group. Let $f:Y\to X$ be a $G$-torsor with $Y$ rationally connected. Assume t…

Let $({\mathcal{T}},{\mathcal{V}})$ be the context of $({\mathcal{T}},{\mathcal{V}})$-categories, let $(X,a)$ and $(Y,b)$ be such categories, and let $f\colon (X,a)\to(Y,b)$ be a…

Let $A,A_1,A_2,A_{12}$ be noetherian $\mathbb{E}_1$-Banach rings in the indicated simplicial ind-Banach categories. Suppose that their homotopy rings fit into a short strictly exac…

Atlas descent conjecture. The following conditions on $F$ are equivalent: (1) $F$ satisfies descent along atlases; (2) $F$ satisfies hyperdescent. In particular, the full subcatego…

Let $(X,\mathcal{A})$ be a compact ringed space with soft structure sheaf $\mathcal{A}$. Let $U_{\bullet}$ be a hypercover of $X$, and let $\mathfrak{L}(X)$ denote the dg-category…

Affine Demazure descent conjecture. The descent category is equivalent to

Persistence conjecture. Any effective descent data which is not destroyed by a given deformation remains effective in the deformation.