# Weighted projective lines and Riemann surfaces

@article{Lenzing2016WeightedPL, title={Weighted projective lines and Riemann surfaces}, author={Helmut Lenzing}, journal={arXiv: Representation Theory}, year={2016} }

For the base field of complex numbers we discuss the relationship between categories of coherent sheaves on compact Riemann surfaces and categories of coherent sheaves on weighted smooth projective curves. This is done by bringing back to life an old theorem of Bundgaard-Nielsen-Fox proving Fenchel's conjecture for fuchsian groups.

## 6 Citations

On the K-theory of weighted projective curves

- Mathematics
- 2017

We present a largely self contained account on the K-theory of a weighted smooth projective curve over an algebraically closed field. In particular, we discuss the weighted version of divisor theory,…

Admissible homomorphisms and equivariant relations between weighted projective lines.

- Mathematics
- 2020

The string group acts on the category of coherent sheaves over a weighted projective line by degree-shift actions. We study the equivariant equivalence relations induced by degree-shift actions…

Equivariant approach to weighted projective curves

- Mathematics
- 2021

We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the…

The algebraic theory of fuchsian
singularties

- Mathematics
- 2019

This article has the following aims:
(1) Extend the notion of fuchsian singularities (of first kind) to base fields of arbitrary characteristic.
(2) Discuss their relationship to mathematical…

The dual actions, equivariant autoequivalences and stable tilting objects

- MathematicsAnnales de l'Institut Fourier
- 2020

For a finite abelian group action on a linear category, we study the dual action given by the character group acting on the category of equivariant objects. We prove that the groups of equivariant…

The dual actions, autoequivalences and stable tilting objects

- Mathematics
- 2017

For a finite abelian group action on a linear category, we study the dual action given by the character group acting on the category of equivariant objects. We prove that the groups of equivariant…

## References

SHOWING 1-10 OF 37 REFERENCES

Weighted Projective Lines as Fine Moduli Spaces of Quiver Representations

- Mathematics
- 2014

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…

MCKAY CORRESPONDENCE AND EQUIVARIANT

- 2006

Let G be a finite subgroup in SU(2), and Q the corresponding affine Dynkin diagram. In this paper, we review the relation between the categories of G-equivariant sheaves on P and Rep Qh, where h is…

Kac's Theorem for weighted projective lines

- Mathematics
- 2005

We prove an analogue of Kac’s Theorem, describing the dimension types of indecomposable coherent sheaves (or parabolic bundles) over weighted projective lines in terms of root systems for loop…

Characters and Automorphism Groups of Compact Riemann Surfaces

- Mathematics
- 2000

Preface Notation 1. Compact Riemann surfaces 2. Group characters 3. Automorphisms of compact Riemann surfaces 4. Generation of groups 5. Classification for small genus 6. Classification for fixed…

Tame Algebras and Integral Quadratic Forms

- Mathematics
- 1985

Integral quadratic forms.- Quivers, module categories, subspace categories (Notation, results, some proofs).- Construction of stable separating tubular families.- Tilting functors and tubular…

Surfaces and Planar Discontinuous Groups

- Physics
- 1980

Free groups and graphs.- 2-Dimensional complexes and combinatorial presentations of groups.- Surfaces.- Planar discontinuous groups.- Automorphisms of planar groups.- On the complex analytic theory…

Uniformization of Deligne-Mumford curves

- Mathematics
- 2005

Abstract We compute the fundamental groups of non-singular analytic Deligne-Mumford curves, classify the simply connected ones, and classify analytic Deligne-Mumford curves by their uniformization…

Monadicity theorem and weighted projective lines of tubular type

- Mathematics
- 2014

We formulate a version of Beck's monadicity theorem for abelian categories, which is applied to the equivariantization of abelian categories with respect to a finite group action. We prove that the…

The automorphism group of the derived category for a weighted projective line

- Mathematics
- 2000

We show that up to a translation each automorphism of the derived category D b X of coherent sheaves on a weighted projective line X, equiv-alently of the derived category D b A of finite dimensional…