Rational points on symmetric powers and categorical representability

TL;DR

This paper links the existence of rational points on certain varieties to their categorical dimension, showing that for varieties with specific collections, rational points imply minimal categorical complexity, especially in symmetric powers.

Contribution

It establishes a connection between rational points and categorical dimension for varieties with full weak exceptional collections, and analyzes symmetric powers of Brauer--Severi and involution varieties over .

Findings

Rational points imply im(X)=0 for varieties with full weak exceptional collections.

The equivariant derived category of symmetric powers admits a full weak exceptional collection.

Existence of -rational points on X or S^3(X) is equivalent to im of the equivariant derived category being zero.

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 $\mathrm{rdim}(D^b_{S_3}(X^3))=0$.