Rational points on symmetric powers and categorical representability

Abstract

In this paper we observe that for geometrically integral projective varieties $X$, admitting a full weak exceptional collection consisting of pure vector bundles, the existence of a $k$-rational point implies $\mathrm{rdim}(X)=0$. We also study the symmetric power $S^n(X)$ of Brauer--Severi and involution varieties over $\mathbb{R}$ and prove that the equivariant derived category $D^b_{S_n}(X^n)$ admits a full weak exceptional collection. As a consequence, we find $\mathrm{rdim}(X)=0$ if and only if $\mathrm{rdim}(D^b_{S_n}(X^n))=0$ for $1\leq n\leq 3$. If $X$ is Brauer--Severi, the existence of a $\mathbb{R}$-rational point on $X$ or $S^3(X)$ is equivalent to $\mathrm{rdim}(D^b_{S_3}(X^3))=0$.