MEMOIRS

0.3. CANONICAL ALGEBRAS AND EXCEPTIONAL CURVES 17uniquely determined by X. We call this skew field the function field. Wedenoteitby k(X) =k(H):H/H 0 ≃ mod(k(X)).We call an exceptional curve X commutative if the function field k(X) iscommutative.The function field is known to be of finite dimension over its centre and to bean 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 tothe endomorphism ring of L considered as object in H/H 0 (given by fractions ofmorphisms of the same degree). Moreover, the rank of an object X ∈Hagreeswith the dimension of the vector space over k(X) corresponding to X considered asobject in H/H 0 .The function field coincides with the endomorphism ring of the generic moduleassociated with the separating tubular family mod 0 (Σ) and was already studied indetail in [90].0.3.14 (Special line bundle). From each of the exceptional tubes choose a simplesheaf S i ∈U xi . Note that these simple sheaves are exceptional. In the followinglet L ∈H + be a line bundle, and assume additionally that for each i ∈{1,...,t}we have Hom(L, τ j S i ) ≠ 0 if and only if j ≡ 0modp i . Such a line bundle Lexists by [70, Prop. 4.2] and is called special. It follows from [70, 5.2]thatL isexceptional, since End H (L) isaskewfieldanda := [L] isarootinK 0 (X). Recallfrom [66, 57] thatv ∈ K 0 (X) isaroot if 〈v, v〉 > 0and 〈v,x〉〈v,v〉 ∈ Z for all x ∈ K 0(X).For example, the class of an exceptional object is a root. Moreover, an exceptionalobject 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 .Ofcourse,ifX is homogeneous then each line bundleis special.0.3.15 (Degree). Let p be the least common multiple of the weights p 1 ,...,p t .Define 〈〈−, −〉〉 := ∑ p−1j=0 〈τ j −, −〉 and define the degree function deg : K 0 (X) −→ Zbywhere as above a =[L].deg x := 1 c(〈〈a, x〉〉 − rk x〈〈a, a〉〉),0.3.16 (Underlying tame bimodule). Let L be a special line bundleand S 1 ,...,S t simple objects from the different exceptional tubes such thatHom(L, S i ) ≠0. LetS = {τ j S i | 1 ≤ i ≤ t, j ≢−1modp i }. Then the rightperpendicular category S ⊥ is equivalent to mod(Λ), where Λ is a tame hereditaryk-algebra of the form( )G 0Λ= ,M Fwhere M = F M G 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;