Exceptional Sequences of Invertible Sheaves on Rational Surfaces

TL;DR

This paper studies exceptional sequences of invertible sheaves on rational surfaces, establishing a canonical association with toric surfaces and classifying such sequences, especially on toric surfaces.

Contribution

It introduces a canonical link between exceptional sequences on rational surfaces and toric surfaces, and classifies full strongly exceptional sequences on toric surfaces.

Findings

Associates each exceptional sequence with a toric surface

Constructs full strongly exceptional sequences for many rational surfaces

Provides a complete classification for toric surfaces

Abstract

In this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.