0.3. CANONICAL ALGEBRAS AND EXCEPTIONAL CURVES 17
uniquely determined by X. We call this skew field the function field. We denote it
by k(X) = k(H):
H/H0 mod(k(X)).
We call an exceptional curve X commutative if the function field k(X) is commuta-
tive.
The function field is known to be of finite dimension over its centre and to be
an algebraic function (skew) field of one variable over k (in the sense of [106]),
see [7].
If L H+ is a line bundle, that is, of rank one, then k(X) is isomorphic to
the endomorphism ring of L considered as object in H/H0 (given by fractions of
morphisms of the same degree). Moreover, the rank of an object X H agrees
with the dimension of the vector space over k(X) corresponding to X considered as
object in H/H0.
The function field coincides with the endomorphism ring of the generic module
associated with the separating tubular family mod0(Σ) and was already studied in
detail in [90].
0.3.14 (Special line bundle). From each of the exceptional tubes choose a simple
sheaf Si Uxi . Note that these simple sheaves are exceptional. In the following
let L H+ be a line bundle, and assume additionally that for each i {1, . . . , t}
we have Hom(L, τ j Si) = 0 if and only if j 0 mod pi. Such a line bundle L
exists by [70, Prop. 4.2] and is called special. It follows from [70, 5.2] that L is
exceptional, since EndH(L) is a skew field and a := [L] is a root in K0(X). Recall
from [66, 57] that v K0(X) is a root if v, v 0 and
v,x
v,v
Z for all x K0(X).
For example, the class of an exceptional object is a root. Moreover, an exceptional
object is uniquely determined (up to isomorphism) by its class.
In the sequel, we will always consider H together with a special line bundle L,
also called a structure sheaf . Of course, if X is homogeneous then each line bundle
is special.
0.3.15 (Degree). Let p be the least common multiple of the weights p1, . . . , pt.
Define −,− :=
∑p−1
j=0
τ
j
−,− and define the degree function deg : K0(X) −→ Z
by
deg x :=
1
c
(
a, x− rk x a, a
)
,
where as above a = [L].
0.3.16 (Underlying tame bimodule). Let L be a special line bundle
and S1, . . . , St simple objects from the different exceptional tubes such that
Hom(L, Si) = 0. Let S =
j
Si | 1 i t, j −1 mod pi}. Then the right
perpendicular category
S⊥
is equivalent to mod(Λ), where Λ is a tame hereditary
k-algebra of the form
Λ =
G 0
M F
,
where M =
F
MG is a tame bimodule (also called affine bimodule), that is:
F and G are skew fields, finite dimensional over k;
k lies in the centres of F and G and acts centrally on M .
For the dimensions, [M : F ] · [M : G] = 4;
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