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

TL;DR

This paper introduces a geometric model for the category of coherent sheaves over weighted projective lines of type (p,q), using an annulus with marked points to establish bijections, compute extension groups, and describe tilting structures.

Contribution

It provides a novel geometric framework linking coherent sheaves to curves in an annulus, enabling combinatorial and group-theoretic analysis of the category.

Findings

Bijection between indecomposable sheaves and homotopy classes of curves

Extension group dimensions equal positive intersection numbers

Automorphism group is isomorphic to the mapping class group

Abstract

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.