Tilting bundles and the missing part on a weighted projective line of type (2,2,n)(2,2,n)

This paper is devoted to investigate the tilting bundles, i.e. tilting sheaves that are vector bundles, in the category of coherent sheaves on a weighted projective line of type (2,2,n)(2,2,n). We classify all the tilting bundles into two classes, one containing the tilting bundles that are consisting of line bundles, and the other one containing tilting bundles with indecomposable summands that are not line bundles. Moreover, for each tilting bundle T (with endomorphism algebra Λ) we prove that the missing part, from the category of coherent sheaves to the category of finitely generated right Λ-modules, carries the structure of an abelian category.