Equivariant approach to weighted projective curves

TL;DR

This paper explores how group actions on weighted projective lines relate to coherent sheaves on weighted projective curves, providing explicit formulas and classifications that connect to various types of algebraic curves and singularities.

Contribution

It introduces an equivariant framework linking weighted projective lines and curves, including a classification and applications to hyperelliptic and genus 2 curves.

Findings

Equivariant categories are equivalent to coherent sheaves over weighted projective curves.

Explicit formulas relate genus and weights of curves to group actions.

Applications include descriptions of sheaves on hyperelliptic and genus 2 curves.

Abstract

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 category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations. This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a description for the category of coherent sheaves over a hyperelliptic curve of arbitrary genus. Links to smooth projective curves of genus 2 and also to Arnold's exceptional unimodal singularities are also established.