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 rdim(X) = 0. We also study the symmetric power S n (X) of a Brauer-Severi variety over R and prove that the equivariant derived category D b Sn (X n ) admits a full weak exceptional collection. As a consequence, we find rdim(X) = 0 if and only if rdim (D b Sn (X n )) = 0 for 1 ≤ n ≤ 3 and that the existence of a R-rational point on X or S 3 (X) is equivalent to rdim(D b S 3 (X 3 )) = 0.

Rationality of inner twisted flags of type $A_n$

“…There are several results suggesting that this question has a positive answer (see [1], [2], [7], [15], [16], [17] and references therein). In this context, we prove: Theorem 1.1.…”

No phantoms in the derived category of curves over arbitrary fields, and derived characterizations of Brauer-Severi varieties

“…For instance, there are links between the semiorthogonal decomposition of D b (X) to the birational geometry of X (see for instance [16], [19] and references therein). There are also links between the existence of special types of semiorthogonal decompositions of D b (X) and the existence of k-rational points (see [3], [22], [21] and references therein). Recently, special examples of semiorthogonal decompositions have been constructed.…”

“…By a theorem of Bondal and Orlov, a Brauer-Severi variety can be recovered from its derived category. So it is natural to study how birationality of two Brauer-Severi varieties is detected in their respective derived categories (see [21], [22]). In [22] the author gives a derived characterization for a Brauer-Severi variety to be split, i.e.…”