In these notes the author investigates noncommutative smooth projective curves of genus zero, also called exceptional curves. As a main result he shows that each such curve \(\mathbb{X}\) admits, up to some weighting, a projective coordinate algebra which is a not necessarily commutative graded factorial domain \(R\) in the sense of Chatters and Jordan. Moreover, there is a natural bijection between the points of \(\mathbb{X}\) and the homogeneous prime ideals of height one in \(R\), and these prime ideals are principal in a strong sense.

Table of Contents

• Introduction
• Background

• Graded factoriality
• Global and local structure of the sheaf category
• Tubular shifts and prime elements
• Commutativity and multiplicity freeness
• Automorphism groups

• Insertion of weights
• Exceptional objects
• Tubular exceptional curves
• Appendix A. Automorphism groups over the real numbers
• Appendix B. The tubular symbols
• Bibliography
• Index