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

Abstract

In this paper we show that certain relative flags cannot have full exceptional collections. We also prove that some of these flags are categorical representable in dimension zero if and only if they admit a full exceptional collection. As a consequence, these flags are categorical representable in dimension zero if and only if they have $k$-rational points if and only if they are $k$-rational. Moreover, we calculate the categorical representability dimension for the flags under consideration.