Exceptional Sequences of Line Bundles and Spherical Twists - a Toric Example

TL;DR

This paper explores the construction of full exceptional sequences of line bundles on toric surfaces, demonstrating that some can be obtained via augmentation while others require spherical twists, expanding understanding of their structure.

Contribution

It introduces new examples of full exceptional sequences that cannot be constructed through augmentation but are achieved using spherical twists.

Findings

Some exceptional sequences are not constructible via augmentation.

Spherical twists can produce full exceptional sequences beyond augmentation methods.

The paper provides explicit examples on toric surfaces.

Abstract

Exceptional sequences of line bundles on a smooth projective toric surface are automatically full when they can be constructed via augmentation. By using spherical twists, we give examples that there are also exceptional sequences which can not be constructed this way but are nevertheless full.