Memoirs of the American Mathematical Society 2004; 139 pp; softcover Volume: 171 ISBN10: 082183519X ISBN13: 9780821835197 List Price: US$64 Individual Members: US$38.40 Institutional Members: US$51.20 Order Code: MEMO/171/808
 This work deals with weighted projective lines, a class of noncommutative curves modelled by Geigle and Lenzing on a graded commutative sheaf theory. They play an important role in representation theory of finitedimensional 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 AuslanderReiten 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 finitedimensional 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
