Combinatorial Approaches to Exceptional Sequences for Weighted Projective Lines of Type $(p,q)$

TL;DR

This paper introduces a combinatorial and geometric framework for understanding exceptional sequences and tilting bundles in the category of coherent sheaves over weighted projective lines of type (p,q), providing classification, enumeration, and new proofs of known properties.

Contribution

It offers a novel combinatorial description of morphisms and exceptional sequences, along with a geometric realization and enumeration methods for tilting sheaves in this setting.

Findings

Classification of complete exceptional sequences

Effective method for enlarging exceptional sequences

Counting of tilting sheaves up to Auslander-Reiten translation

Abstract

We provide a combinatorial description of morphisms in the coherent sheaf category ${\rm coh}\mbox{-}\mathbb{X}(p,q)$ over weighted projective line of type $(p,q)$ via a marked annulus. This leads to a geometric realization of exceptional sequences in ${\rm coh}\mbox{-}\mathbb{X}(p,q)$. As applications, we present a classification of complete exceptional sequences, an effective method for enlarging exceptional sequences, and a new proof of the transitivity of the braid group action on complete exceptional sequences. Besides, we offer a combinatorial description of tilting bundles via lattice paths and count the number of tilting sheaves in ${\rm coh}\mbox{-}\mathbb{X}(p,q)$, up to the Auslander-Reiten translation.