Tilting objects on tubular weighted projective lines: a cluster tilting approach

TL;DR

This paper employs cluster tilting theory to construct and classify tilting objects and their endomorphism algebras in the stable category of vector bundles on a specific weighted projective line, advancing understanding in algebraic geometry and representation theory.

Contribution

It introduces a cluster tilting mutation method to explicitly construct tilting objects of rank two and classifies their endomorphism algebras in the context of weighted projective lines.

Findings

Constructed tilting objects of rank two via mutation.

Classified endomorphism algebras of tilting objects.

Provided a new approach to study coherent sheaves and derived categories.

Abstract

Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type $(2, 2, 2, 2)$. More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.