Introduction to coherent sheaves on weighted projective lines

TL;DR

This paper explores the structure of coherent sheaves on weighted projective lines, presenting two approaches—axiomatic and expansion-based—to describe their abelian categories, with detailed analysis and tilting theory insights.

Contribution

It introduces a new axiomatic framework and an expansion approach for understanding the abelian categories of coherent sheaves on weighted projective lines, enriching the theoretical foundation.

Findings

Descriptions of abelian categories for weighted projective lines

Development of an axiomatic approach to expansions

Discussion of tilting theory in this context

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.