On non-existence of full exceptional collections on some relative flags

Abstract

In this short note we show that certain relative flags cannot have full exceptional collections. We also prove that some of these flags are categorically representable in dimension zero if and only if they admit a full exceptional collection. As a consequence, these flags are representable in dimension zero if and only if they have k-rational points.

Amitsur subgroup and noncommutative motives

“…In the last decades, the bounded derived category D b (X) of coherent sheaves on a smooth projective variety X has been recognized as an interesting invariant, encoding a lot of geometric information. For instance, there are links between the semiorthogonal decomposition of D b (X) and the birational geometry of X (see for instance [16], [2], [3], [24], [26] and references therein). From a motivic point of view, it is quite natural to ask how birational geometry of a given variety X is detected by its noncommutative motive.…”

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.…”

Rational points on symmetric powers and categorical representability

“…This question is now central for arithmetic aspects of the theory of derived categories, see [8], [1], [2], [5], [6], [7], [18], [23] [29] and [30]. In the present work we use a potential measure for rationality to study Question 1.1 in some special cases.…”

“…Quite recently, it has been shown that certain varieties X admit k-rational points if and only if rdim(X) = 0, see [8], [29] and [30]. Among these varieties are certain Fano varieties having full weak exceptional collections.…”