  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.
