Introduction to coherent sheaves on weighted projective lines

Abstract

These notes provide a description of the abelian categories that arise as categories of coherent sheaves on weighted projective lines. Two different approaches are presented: one is based on a list of axioms and the other yields a description in terms of expansions of abelian categories. A weighted projective line is obtained from a projective line by inserting finitely many weights. So we describe the category of coherent sheaves on a projective line in some detail, and the insertion of weights amounts to adding simple objects. We call this process `expansion' and treat it axiomatically. Thus most of these notes are devoted to studying abelian categories, including a brief discussion of tilting theory. We provide many details and have tried to keep the exposition as self-contained as possible.

62 pages, including an index; minor changes

References in corpus (2)

• Quiver algebras, weighted projective lines, and the Deligne-Simpson problem
• Expansions of abelian categories

Cited by in corpus (12)

• Weighted noncommutative regular projective curves
• Higher S-dualities and Shephard-Todd groups
• Tilting bundles on orders on $P^d$
• Fabric idempotents and homological dimensions
• Expansions of abelian categories
• Graded mutation in cluster categories coming from hereditary categories with a tilting object
• Composition algebra of a weighted projective line
• Extended Weyl groups, Hurwitz transitivity and weighted projective lines I: Generalities and the tubular case
• Bounded t-structures on the bounded derived category of coherent sheaves over a weighted projective line
• Hereditary uniserial categories with Serre duality
• The tangent space to the space of 0-cycles
• Piecewise Hereditary algebras of Dynkin and extended Dynkin type