This work deals with weighted projective lines, a class of non-commutative curves modelled by Geigle and Lenzing on a graded commutative sheaf theory. They play an important role in representation theory of finite-dimensional algebras; the complexity of the classification of coherent sheaves largely depends on the genus of these curves. We study exceptional vector bundles on weighted projective lines and show in particular that the braid group acts transitively on the set of complete exceptional sequences of such bundles. We further investigate tilting sheaves on weighted projective lines and determine the Auslander-Reiten components of modules over their endomorphism rings. Finally we study tilting complexes in the derived category and present detailed classification results in the case of weighted projective lines of hyperelliptic type.

Readership

Graduate students and research mathematicians interested in algebraic geometry and representations of finite-dimensional algebras.

Table of Contents

• Background
• Summary
• Weighted projective lines
• Mutations of exceptional sequences
• Tubular mutations
• Twisted mutations
• On the number of exceptional vector bundles
• Tilting sheaves
• Tilting complexes
• Hyperelliptic weighted projective lines
• Bibliography
• Index