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In this paper we prove that a finite product of Brauer--Severi varieties iscategorical representable in dimension zero if and only if it admits a$k$-rational point if and only if it is rational over $k$. The same is true forcertain isotropic involution varieties over a field $k$ of characteristicdifferent from two. For finite products of generalized Brauer--Severivarieties, categorical representability in dimension zero is equivalent to theexistence of a full exceptional collection. In this case however categoricalrepresentability in dimension zero is not equivalent to the existence of arational point. We also show that non-trivial twisted flags of classical type$A_n$ and $C_n$ cannot have full exceptional collections, enlarging in this waythe set of previous known examples. Finally, we determine the categoricalrepresentability dimension $\mathrm{rdim}(X)$ for generalized Brauer--Severivarieties of index $\leq 3$ and for certain twisted forms of smooth quadrics(involution varieties).

Non-existence of exceptional collections on twisted flags and categorical representability via noncommutative motives