# Mutation of tilting bundles of tubular type

## TL;DR

This paper proves the connectedness of the tilting graph for coherent sheaves on tubular weighted projective lines, using mutations and slope changes of tilting sheaves, providing new insights into their structure.

## Contribution

It demonstrates the connectedness of the tilting graph for coherent sheaves on tubular weighted projective lines and explores slope changes under mutations.

## Key findings

- Connectedness of the tilting graph is established.
- Mutation affects slopes of tilting sheaves.
- Provides an alternative proof for tilting graph connectedness.

## Abstract

Let $\mathbb{X}$ be a weighted projective line of tubular type and $\operatorname{coh}\mathbb{X}$ the category of coherent sheaves on $\mathbb{X}$. The main purpose of this note is to show that the subgraph of the tilting graph consisting of all basic tilting bundles in $\operatorname{coh}\mathbb{X}$ is connected. This yields an alternative proof for the connectedness of the tilting graph of $\operatorname{coh}\mathbb{X}$. Our approach leads to the investigation of the change of slopes of a tilting sheaf in $\operatorname{coh}\mathbb{X}$ under (co-)APR mutations, which may be of independent interest.