Combinatorial aspects of exceptional sequences on (rational) surfaces

TL;DR

This paper explores the combinatorial structure of exceptional sequences on algebraic surfaces, linking them to toric surfaces with specific singularities, and demonstrates their transformation into sequences of rank one objects.

Contribution

It establishes a canonical association between exceptional sequences and toric surfaces with particular singularities, and shows how to mutate sequences into rank one objects.

Findings

Association between exceptional sequences and toric surfaces with singularities

Existence of mutations transforming sequences into rank one objects

Characterization of surfaces with smooth or cyclic T-singularities

Abstract

We investigate combinatorial aspects of exceptional sequences in the derived category of coherent sheaves on certain smooth and complete algebraic surfaces. We show that to any such sequence there is canonically associated a complete toric surface whose torus fixpoints are either smooth or cyclic T-singularities (in the sense of Wahl) of type $\frac{1}{r^2}(1, kr - 1)$. We also show that any exceptional sequence can be transformed by mutation into an exceptional sequence which consists only of objects of rank one.