2 INTRODUCTION
of a complete exceptional sequence of objects in H.) In the case of genus zero the
request that there are infinitely many points is automatic.
In this setting noncommutativity occurs in two different styles:
(1) The curves are allowed to be “weighted” which gives a parabolic structure
on H. This means that there are some points x in which more than
one simple object is concentrated. Such a point x is called exceptional;
the other points are called homogeneous. We emphasize that for the
weighted curves additionally a genus in the orbifold sense (called virtual
genus in [66]) is of importance.
(2) There is a another kind of noncommutativity of an arithmetic nature,
determined by the function field k(H). This skew field is commutative
only in very special cases.
The first kind of noncommutativity arising by weights is well-known and the
phenomenon is described in its pure form by the weighted projective
lines1
(over
an algebraically closed field) defined by Geigle-Lenzing [34]. Each weighted curve
of genus zero admits only finitely many exceptional points and has an underlying
homogeneous curve of genus zero (where all points are homogeneous) from which
it arises by so-called insertion of weights. Since this homogeneous curve has the
same function field, the homogeneous case and the associated arithmetic effects of
noncommutativity are the main topic of these notes.
In the following we assume this homogeneous case, which can be also expressed
in the following way.
For all simple objects S H we have ExtH(S,
1
S) = 0 (equivalently, τS
S).
Such a homogeneous curve H has genus zero if and only if ExtH(L,
1
L) = 0
for one, equivalently for all line bundles L (which follows from [74]). In this case
the function field k(H) is of finite dimension over its centre which is an algebraic
function field in one variable [7]. Moreover, there is a tilting object T which con-
sists of two indecomposable summands, a line bundle L and a further indecom-
posable bundle L so that HomH(L, L) = 0. The endomorphism ring EndH(T )
is a tame hereditary bimodule k-algebra. This underlying bimodule is given as
End(L)
HomH(L, L)End(L).
We always consider H together with a fixed line bundle L which we consider
as a structure sheaf. This yields a projective coordinate algebra for H, depending
on the choice of a suitable endofunctor σ on H, and given as the orbit algebra with
respect to L and σ defined as
Π(L, σ) =
n≥0
HomH(L,
σnL),
with multiplication given by the rule
g f
def
=
σm(g)
f,
where f Hom(L, σmL) and g Hom(L, σnL). Formation of orbit algebras is
a standard tool for obtaining projective coordinate algebras in algebraic geometry
1Even
though in all of these cases we have graded coordinate rings and function fields
which are commutative, these curves are nonetheless noncommutative since the coherent sheaves
over an affine part correspond to (finitely generated) modules over a ring that is in general not
commutative.
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