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

TL;DR

This paper classifies all tilting bundles on a specific weighted projective line and examines the algebraic properties of the associated categories, enhancing understanding of their structure and relationships.

Contribution

It provides a complete classification of tilting bundles on weighted projective lines of type (2, 2, n) and analyzes the abelianness of the missing part in their associated categories.

Findings

Complete classification of tilting bundles for type (2, 2, n)

Identification of conditions for abelianness of the missing part

Insights into the structure of the associated tilted algebra

Abstract

This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted projective line of weight type $(2, 2, n)$, and investigates the abelianness of the "missing part" from the category of coherent sheaves to the category of finitely generated right modules on the associated tilted algebra for each tilting bundle.