18 0. BACKGROUND

We say that M is a (tame) bimodule of (dimension) type (2, 2), (1, 4) or (4, 1) if

this pair is ([M : F ], [M : G]). We call the number ε ∈ {1, 2} the numerical type of

M (or of X), which is defined by

ε =

1 if M is of type (2, 2).

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 : K0(X), w ] as above

we have c = κε.

0.3.17 (Automorphism groups). Let X be an exceptional curve with associ-

ated abelian hereditary category H and structure sheaf L. Denote by Aut(H) the

automorphism class group of H, that is, the group of isomorphism classes of au-

toequivalences of H (in the literature sometimes also called the Picard group [8],

which has a different meaning in our presentation). We call this group the auto-

morphism group of H and call the elements automorphisms. (If there is need to

emphasize the base field k, we also write Autk(X) and use a similar notation in

analogue situations.)

By a slight abuse of terminology, we will also call the autoequivalences them-

selves automorphisms, that is, the representatives of such classes; if F is an autoe-

quivalence, then its class in the automorphism group is also denoted by F .

The subgroup of elements of Aut(H) fixing L (up to isomorphism) is denoted

by Aut(X), the automorphism group of X. (We will later see that this group does

not dependent on L.)

Each element φ ∈ Aut(H) induces a bijective map φ on the points of X by

φ(Ux) = Uφ(x) for all x ∈ X. We call φ the shadow of φ. If φ lies in the kernel of the

homomorphism Aut(H) −→ Bij(X), φ → φ, then we call φ point fixing (or invisible

on X). If φ(x) = x we also say (by a slight abuse of terminology) that the point x

is fixed by φ. Similarly, if φ(x) = y we also write φ(x) = y.

Denote by Aut0(H) the (normal) subgroup of Aut(H) given by the point fixing

automorphisms.

Non-trivial elements of Aut(X) which are point fixing are called ghost automor-

phisms, or just ghosts. The subgroup G of Aut(X) formed by the ghosts is called the

ghost group. It is a normal subgroup of Aut(H). We have G = Aut(X) ∩ Aut0(H).

We call the factor group Aut(X)/G the geometric automorphism group of X, its

elements geometric automorphisms. By a slight abuse of terminology, we also call

the elements in Aut(X) which are not ghosts geometric.

Denote by

Aut(Db(X))

the group of isomorphism classes of exact autoequiva-

lences of the triangulated category

Db(X),

called the automorphism group of

Db(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 abelian

group of rank one, which is equipped with a partial order ≤, compatible with the

group structure. Let R =

h∈H

Rh be an H-graded k-algebra, such that each

homogeneous component Rh is finite dimensional over k and such that Rh = 0 for

0 ≤ 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

modH

(R) the category of finitely generated right H-graded R-

modules, and by mod0

H

(R) the full subcategory of graded modules of finite length