Weighted projective lines as fine moduli spaces of quiver representations

TL;DR

This paper links weighted projective lines to moduli spaces of quiver representations, studies derived categories of their canonical bundles, and provides a moduli construction for certain total spaces.

Contribution

It introduces a moduli problem characterization of weighted projective lines and explores the properties of generators in their derived categories, including a moduli construction for specific cases.

Findings

Weighted projective lines are described via a moduli problem on canonical algebras.

Generators of derived categories are rarely tilting.

A moduli construction is provided for total spaces with three orbifold points.

Abstract

We describe weighted projective lines in the sense of Geigle and Lenzing by a moduli problem on the canonical algebra of Ringel. We then go on to study generators of the derived categories of coherent sheaves on the total spaces of their canonical bundles, and show that they are rarely tilting. We also give a moduli construction for these total spaces for weighted projective lines with three orbifold points.