MEMOIRS

18 0. BACKGROUNDWe say that M is a (tame) bimodule of (dimension) type (2, 2), (1, 4) or (4, 1) ifthis pair is ([M : F ], [M : G]). We call the number ε ∈{1, 2} the numerical type ofM (or of X), which is defined by{1 if M is of type (2, 2).ε =2 if M is of type (1, 4) or (4, 1).The numerical type is an invariant of the curve X.With κ := 〈[L], [L]〉, for the normalization factor c =[Z : 〈K 0 (X), w〉] asabovewe have c = κε.0.3.17 (Automorphism groups). Let X be an exceptional curve with associatedabelian hereditary category H and structure sheaf L. DenotebyAut(H) theautomorphism class group of H, that is, the group of isomorphism classes of autoequivalencesof H (in the literature sometimes also called the Picard group [8],which has a different meaning in our presentation). We call this group the automorphismgroup of H and call the elements automorphisms. (If there is need toemphasize the base field k, wealsowriteAut k (X) and use a similar notation inanalogue situations.)By a slight abuse of terminology, we will also call the autoequivalences themselvesautomorphisms, that is, the representatives of such classes; if F is an autoequivalence,then its class in the automorphism group is also denoted by F .The subgroup of elements of Aut(H) fixingL (up to isomorphism) is denotedby Aut(X), the automorphism group of X. (We will later see that this group doesnot dependent on L.)Each element φ ∈ Aut(H) induces a bijective map φ on the points of X byφ(U x )=U φ(x)for all x ∈ X. Wecallφ the shadow of φ. Ifφ lies in the kernel of thehomomorphism Aut(H) −→ Bij(X), φ ↦→ φ, thenwecallφ point fixing (or invisibleon X). If φ(x) =x we also say (by a slight abuse of terminology) that the point xis fixed by φ. Similarly, if φ(x) =y we also write φ(x) =y.Denote by Aut 0 (H) the (normal) subgroup of Aut(H) given by the point fixingautomorphisms.Non-trivial elements of Aut(X) which are point fixing are called ghost automorphisms,orjustghosts. The subgroup G of Aut(X) formed by the ghosts is called theghost group. It is a normal subgroup of Aut(H). We have G =Aut(X) ∩ Aut 0 (H).We call the factor group Aut(X)/G the geometric automorphism group of X, itselements geometric automorphisms. By a slight abuse of terminology, we also callthe elements in Aut(X) which are not ghosts geometric.Denote by Aut(D b (X)) the group of isomorphism classes of exact autoequivalencesof the triangulated category D b (X), called the automorphism group of D b (X).(Compare also [9]. There is also the related notion of the derived Picard group [82].)0.3.18 (Projective coordinate algebras). Let H be a finitely generated abeliangroup of rank one, which is equipped with a partial order ≤, compatible with thegroup structure. Let R = ⊕ h∈H R h be an H-graded k-algebra, such that eachhomogeneous component R h is finite dimensional over k and such that R h =0for0 ≰ h. Assume moreover that R is a finitely generated k-algebra and noetherian.Note that we do not require that R is commutative.Denote by mod H (R) the category of finitely generated right H-graded R-modules, and by mod H 0 (R) the full subcategory of graded modules of finite length