Geometric-combinatorial approaches to tilting theory for weighted projective lines

TL;DR

This paper introduces a geometric-combinatorial model for the category of coherent sheaves on weighted projective lines of type (2,2,n), linking skew-curves and tilting sheaves via pseudo-triangulations on a cylindrical surface.

Contribution

It establishes a novel geometric-combinatorial framework connecting skew-curves, pseudo-triangulations, and tilting theory for weighted projective lines of type (2,2,n).

Findings

Bijective correspondence between indecomposable sheaves and skew-curves.

Pseudo-triangulations correspond to tilting sheaves.

Flip operations match tilting mutations and prove connectivity.

Abstract

We provide a geometric-combinatorial model for the category of coherent sheaves on the weighted projective line of type (2,2,n) via a cylindrical surface with n marked points on each of its upper and lower boundaries, equipped with an order 2 self-homeomorphism. A bijection is established between indecomposable sheaves on the weighted projective line and skew-curves on the surface. Moreover, by defining a skew-arc as a self-compatible skew-curve and a pseudo-triangulation as a maximal set of distinct pairwise compatible skew-arcs, we show that pseudo-triangulations correspond bijectively to tilting sheaves. Under this bijection, the flip of a skew-arc within a pseudo-triangulation coincides with the tilting mutation. As an application, we prove the connectivity of the tilting graph for the category of coherent sheaves.