Exceptional sequences on rational ℂ* -surfaces

Inspired by Bondal's conjecture, we study the behavior of exceptional sequences of line bundles on rational ℂ* -surfaces under homogeneous degenerations. In particular, we provide a sufficient criterion for such a sequence to remain exceptional under a given degeneration. We apply our results to show that, for toric surfaces of Picard rank 3 or 4, all full exceptional sequences of line bundles may be constructed via augmentation. We also discuss how our techniques may be used to construct noncommutative deformations of derived categories. © 2012 Springer-Verlag Berlin Heidelberg.