MEMOIRS

96 6. INSERTION OF WEIGHTSh ∈ H and p ≥ 2 be an integer. Then denote by H[ h p] the abelian group given by(H ⊕ Z)/Z(−h, p). Similarly, H[ h 1p 1,..., h tp t] is defined inductively.The next proposition follows immediately with 6.1.5.Proposition 6.3.3. With the same notations as in Proposition 6.3.1, we havePic(X) =Pic(X)[σx1p 1,..., σ x tp t]. □The following is the extension of Proposition 5.2.3 to the weighted case.Proposition 6.3.4. Let X be an exceptional curve, such that for the underlyinghomogeneous situation there is an exhaustive automorphism in the Picard group,and such that the underlying bimodule is not of orbit case IIIb. Let G be the ghostgroup. Then the group Pic(X)/ Pic(X)∩G acts simply transitive on the Aut(H)-orbitof the structure sheaf L, and there is a split exact sequence of groups1 −→ Pic(X)/ Pic(X) ∩G−→Aut(H)/G −→Aut(X)/G −→1.Proof. Let L ′ be lying in the same Aut(H)-orbit X as L. After applyingsuitable shifts associated to exceptional points we can assume that L and L ′ arespecial with respect to the same set of exceptional simple objects. By perpendicularcalculus, L and L ′ are line bundles over the associated homogeneous curve. Byassumption, there is a Picard element mapping L onto L ′ . Hence Pic(X) actstransitively on X. Each ghost fixes L, hence also any other member of X. Wegetan induced action of Pic(X)/ Pic(X) ∩G on X, which is obviously simply transitive.Define Aut(H)/G −→Aut(X)/G by [φ] ↦→ [σ ◦ φ], where σ ∈ Pic(X) such thatσ(φ(L)) ≃ L. This induces the split exact sequence.□In the special situation k = R Proposition 6.3.4 can be formulated as follows.Proposition 6.3.5. Let X be an exceptional curve over the real numbers. LetG be the ghost group. Then the group Pic(X)/G acts simply transitive on the set ofall (isomorphism classes of ) special line bundles and there is a split exact sequenceof groups1 −→ Pic(X)/G −→Aut(H)/G −→Aut(X)/G −→1. □Note that [58, Lem. 4+Thm. 5] is not quite correct in the twisted case C ⊕ C,where G is non-trivial; moreover, we have to restrict to special line bundles as inthe preceding proposition. We will give an example, where there are line bundleswhich are not special in 8.5.1.For the domestic curves and the tubular curves over the real numbers theautomorphism groups are listed in Appendix A.1.