In these notes we investigate noncommutative smooth projective curves of
genus zero, also called exceptional curves. As a main result we show 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.
Curves of genus zero have strong applications in the representation theory of
finite dimensional algebras being natural index sets for one-parameter families of
indecomposable modules. They play a key role for an understanding of the notion
of tameness and conjecturally for an extension of Drozd’s Tame and Wild Theorem
to arbitrary base fields. The function field of X agrees with the endomorphism
ring of the unique generic module over the associated tame hereditary algebra.
This skew field is of finite dimension over its centre which is an algebraic function
field in one variable. As another main result we show that the function field is
commutative if and only if the multiplicities determined by the homomorphism
spaces from line bundles to simples sheaves (originally defined by Ringel for tame
hereditary algebras) are equal to one for every point.
The study provides major insights into the nature of arithmetic complications
in the representation theory of finite dimensional algebras that arise if the base field
is not algebraically closed.
Received by the editor March 19, 2006 and in revised form March 18, 2007.
2000 Mathematics Subject Classification. Primary 14H45, 16G10; Secondary 14H60, 14A22,
Key words and phrases. noncommutative curve, genus zero, exceptional curve, one-parameter
family, separating tubular family, tame bimodule, canonical algebra, tubular algebra, orbit algebra,
graded factorial domain, eﬃcient automorphism, ghost automorphism.