  We associate to a regular system of weights a weighted projective line over
an algebraically closed field of characteristic zero in two different ways. One
is defined as a quotient stack via a hypersurface singularity for a regular
system of weights and the other is defined via the signature of the same
regular system of weights.
  The main result in this paper is that if a regular system of weights is of
dual type then these two weighted projective lines have equivalent abelian
categories of coherent sheaves. As a corollary, we can show that the
triangulated categories of the graded singularity associated to a regular
system of weights has a full exceptional collection, which is expected from
homological mirror symmetries.
  Main theorem of this paper will be generalized to more general one, to the
case when a regular system of weights is of genus zero, which will be given in
the joint paper with Kajiura and Saito. Since we need more detailed study of
regular systems of weights and some knowledge of algebraic geometry of
Deligne--Mumford stacks there, the author write a part of the result in this
paper to which another simple proof based on the idea by Geigle--Lenzing can be
applied.
