Introduction to coherent sheaves on weighted projective lines

2009/11/23 by Xiao‐Wu Chen, Chen, Xiao-Wu, Henning Krause +1 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.0911.4473

openalex publication_date 2009/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Quiver algebras, weighted projective lines, and the Deligne-Simpson problem
• Noncommutative curves of genus zero: related to finite dimensional algebras

Cited by

• The tangent space to the space of 0-cycles
• Fabric idempotents and homological dimensions

Related

• Expansions of abelian categories
• A recollement approach to Geigle-Lenzing weighted projective varieties
• Connections for weighted projective lines
• Tilting bundles and the "missing part" on the weighted projective line of type (2, 2, n)
• General sheaves over weighted projective lines