6 problems

Let $F$ be a field, and let $A$ and $B$ be central simple algebras over $F$. Write $SB(A)$ and $SB(B)$ for their associated Severi–Brauer varieties, and let $[A]$ and $[B]$ denote…

Let $k$ be a field, let $d\geq 1$, and let $D$ be a central division algebra of degree $d$ over $k$. For a positive rational number $r$, write $\operatorname{Mod}_{D,r}$ for the fu…

Product Severi–Brauer conjecture. The following are equivalent:

Stable rationality conjecture. This quotient is always stably rational.

Let $D$ be a central division $F$-algebra, and let $M_{m,D}$ denote its upper motive associated with the generalized Severi–Brauer variety $X(p^m,D)$. Let $K/F$ be a field extensio…

Let $D$ be a division algebra over $F$ and let $E/F$ be a field extension such that $D_E$ remains a division algebra. Consider the Severi–Brauer varieties of $D$ and their motivic…