4 problems

Let $X$ be a Brauer–Severi variety over a field, with period equal to index. Denote its index by $\operatorname{ind}(X)$ and its categorical representability dimension by…

Let $X\to\mathbb{P}^1$ be a fibration in complete intersections of two four-dimensional quadrics. Let $\operatorname{rcodim}(X)$ denote its categorical representability codimension…

Let $S$ be a surface over a field $k$, and let $\operatorname{rdim}(S)$ denote its categorical representability dimension. Orlov's conjecture. The surface $S$ is $k$-rational if an…

Let $\mathsf{T}$ be a category, and let $\operatorname{rdim}\mathsf{T}$ denote its representability dimension. Finite-dimensionality conjecture. If…