Geometric model for weighted projective lines of type $(p,q)$

We give a geometric model for the category of coherent sheaves over the weighted projective line of type $(p,q)$ in terms of an annulus with marked points on its boundary. We establish a bijection between indecomposable sheaves over the weighted projective line and certain homotopy classes of oriented curves in the annulus, and prove that the dimension of extension group between indecomposable sheaves equals to the positive intersection number between the corresponding curves. By using the geometric model, we provide a combinatorial description for the titling graph of tilting bundles, which is composed by quadrilaterals (or degenerated to a line). Moreover, we obtain that the automorphism group of the coherent sheaf category is isomorphic to the mapping class group of the marked annulus, and show the compatibility of their actions on the tilting graph of coherent sheaves and on the triangulation of the geometric model respectively. A geometric description of the perpendicular category with respect to an exceptional sheaf is presented at the end of the paper.