5 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…

Novaković's conjecture. $D^b(X)$ does not admit a full strongly exceptional collection.

Period-index conjecture. The index should divide the period raised to the power one less than the dimension of $B_0$:

Let $X$ be the Fargues–Fontaine curve, and let $\mathcal{C}$ be the division algebra introduced by Colmez. The Brauer–Severi variety of a central division algebra is the variety pa…

Let $X$ be the Brauer–Severi variety of an order of type $F^2_3$, $F^2_4$, or $F^{2-}_5$, as described in the table in the source. Let $Y$ be another variety birational to $X$ over…