Noncommutative curves of genus zero: related to finite dimensional algebras

2009/01/01 by Dirk Kussin · 24 citations
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Bijection #Combinatorics #Commutative property #Domain (mathematical analysis) #Genus #Mathematical analysis #Mathematics #Noncommutative geometry #Prime (order theory) #Projective test #Pure mathematics #Zero (linguistics)

paper · doi:10.1090/memo/0942

published in Memoirs of the American Mathematical Society 201(942), 0 (American Mathematical Society)

openalex publication_date 2009/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

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 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 X and the homogeneous prime ideals of height one in R, and these prime ideals are principal in a strong sense.

Citations

• Finite-Dimensional Division Algebras over Fields
• Vector Bundles Over an Elliptic Curve
• On triangulated orbit categories

Cited by

• Tilting Objects via Recollements and p-Cycles on Weighted Projective Lines
• Coxeter-Dynkin algebras of canonical type
• Introduction to coherent sheaves on weighted projective lines
• Witt's theorem for noncommutative conics

Related

• Noncommutative curves and noncommutative surfaces
• Noncommutative projective geometry
• Noncommutative plane curves
• Totally reflexive modules constructed from smooth projective curves of genus g ≥ 2
• Stability conditions on non-commutative curves