So far in this introduction we have concentrated on the homogeneous case.
These notes also deal with the weighted case. The following results show that the
problem of determining the geometry of an exceptional curve can often be reduced
to the homogeneous case.
We show that insertion of weights into a central prime element in a graded
factorial coordinate algebra preserves the graded factoriality; the resulting
graded algebra is a projective coordinate algebra of a (weighted) excep-
tional curve (6.2.4).
The automorphism group of a (weighted) exceptional curve is given by
the automorphisms of the underlying homogeneous curve preserving the
weights (6.3.1). In particular, both curves have the same ghost group.
The insertion of weights is particularly important for our treatment of the
tubular case in Chapter 8. The tubular exceptional curves have a strong relationship
to elliptic curves. They are defined by the condition that the so-called virtual
(orbifold) genus is one. The main feature of the tubular case is that, very similar to
Atiyah’s classification of vector bundles over an elliptic curve, H consists entirely
of tubular families. In fact, there is a linear form deg, called the degree, which
together with the rank rk defines the slope µ(X) =
deg X
rk X
of (non-zero) objects X
in H. Denote for q Q = Q ∪{∞} by
the additive closure of indecomposable
objects in H of slope q. Then H is the additive closure of all
where q Q.
In case the base field is algebraically closed all the tubular families
isomorphic to each other as categories, and moreover each is parametrized by the
curve X. The reason for this is that in this case the natural action of the automor-
phism group
on the set Q is transitive. This is not true in general over
an arbitrary base field. We show in Chapter 8 that in general this action may have
up to three orbits [53, 59]. Accordingly, there are up to three different tubular
exceptional curves which are Fourier-Mukai partners.
Another interesting effect treated in the same chapter is the occurrence of line
bundles which are not exceptional. Over an algebraically closed field each line bun-
dle L over an exceptional curve X is exceptional, that is, satisfies
L) = 0.
But this does not extend to arbitrary base fields, the simplest counterexamples ex-
isting in the tubular case. We characterize the tubular cases where non-exceptional
line bundles exist and show how they can be determined explicitly (Section 8.5).
Applications to finite dimensional algebras. The study of noncommuta-
tive curves of genus zero has strong applications in the representation theory of finite
dimensional algebras. Conjecturally these curves yield the natural parametrizing
sets for one-parameter families of indecomposable modules over finite dimensional
tame algebras. This is reflected by the definition of tame algebras over an alge-
braically closed field k, using as parametrizing curves (affine subsets of) the pro-
jective line
and in a certain sense this “explains” that in Drozd’s Tame and
Wild Theorem [32, 17] only rational one-parameter families occur. Note that in
the algebraically closed case
is the only homogeneous curve of genus zero.
For the class of tame hereditary algebras and the class of tame canonical al-
gebras [92] over an arbitrary field it is well-known that the parametrizing sets are
precisely the (affine) curves of genus zero. For a tame algebra, in general more
than one exceptional curve is needed to parametrize the indecomposables: there is
a tubular (canonical) algebra which requires three such curves (Section 8.3).
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