Exceptional collections of line bundles on projective homogeneous varieties

TL;DR

This paper constructs and analyzes exceptional collections of line bundles on projective homogeneous varieties associated with split semisimple algebraic groups of rank 2, establishing existence for certain types and non-existence for others.

Contribution

It provides explicit constructions of exceptional collections for types A_2 and B_2=C_2 and proves non-existence for G_2, settling the question for rank 2 groups.

Findings

Exceptional collections exist for types A_2 and B_2=C_2.

No exceptional collection exists for type G_2.

The existence question is settled for split groups of rank at most 2.

Abstract

We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G of rank 2 over a field. We exhibit exceptional collections of the expected length for types A_2 and B_2=C_2 and prove that no such collection exists for type G_2. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G-varieties for split linear algebraic groups G of rank at most 2.