3 problems

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

Amitsur's conjecture. If $A$ and $B$ generate the same cyclic subgroup of $\mathrm{Br}(k)$, then $X$ and $Y$ are birational.

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…