Exceptional Sequences on Rational C*-Surfaces

TL;DR

This paper investigates how exceptional sequences of line bundles on rational C*-surfaces behave under degenerations, providing criteria for their preservation and applications to toric surfaces and noncommutative deformations.

Contribution

It offers a sufficient criterion for exceptional sequences to remain under degenerations and shows all full sequences on certain toric surfaces can be constructed via augmentation.

Findings

Exceptional sequences are preserved under specific degenerations.

All full exceptional sequences on certain toric surfaces are constructible via augmentation.

Techniques enable construction of noncommutative deformations of derived categories.

Abstract

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.