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