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

TL;DR

This paper demonstrates that specific relative flag varieties lack full exceptional collections and explores their categorical representability, linking it to rational points and rationality over the base field.

Contribution

It establishes the non-existence of full exceptional collections on certain relative flags and connects categorical representability to rationality properties.

Findings

Certain relative flags cannot have full exceptional collections.

Categorical representability in dimension zero is equivalent to having a $k$-rational point.

Calculated categorical representability dimension for the studied 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.