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 = {τ

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;