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Introduction to coherent sheaves on weighted projective lines

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.