1.1. ALGEBROGEOMETRIC TERMINOLOGY
9
The set Pic(X) of isomorphism classes of line bundles on X is an Abelian group
with respect to 0. A line bundle is called trivial if it is isomorphic to the line bundle
O := {Ox, can)
where can is the isomorphism
* can can *
a%Ox ^ Ox ~ alOx.
Of course the isomorphism class of O is the identity element of Pic(X). There is a
natural exact sequence of groups
(1.16)
0{X)X

0{X)X
 Pic{X)  Pic(X)
where the left arrow is defined by
the middle arrow is defined by
and the right arrow is induced by
(0,u
(L,p)^L.
In this book a key role will be played by 2 types of line bundles on corre
spondences: flat line bundles and pluricanonical line bundles; cf. the Definitions
below.
DEFINITION 1.6. Assume R0 c R is a subring, X = (X,X,Ji,a2) is a corre
spondence over R, and A is a locally constant ^—valued function on X. Define
the line bundle
(1.17) F(A):=(0,A)
where
A : a2Ox ^ Ox  O^  ^ i ^ x 
We call A the multiplier of F(A). A line bundle on X is called RQ— flat if it isomorphic
to a line bundle of the form F(A). In particular O = F(l). Clearly F(Ai)®F(A2) ^
F(AiA2).
DEFINITION
1.7. If X is smooth over R and etale then the canonical line bundle
is the line bundle
(1.18) K = K
X
= (#*,/?)
where Kx is the canonical line bundle on X and (3 is the composition
&2KX
— &x —
alKX
The line bundle K  1 is called the anticanonical line bundle. For v G Z the line
bundle K1" is called the v—canonical line bundle (or a pluricanonical line bundle).
If L = (L, (3) is a line bundle on X then one can define the line bundle L o n
on X o n by L o n := (L,f3on) where j3on is the obvious "n—fold product" of various
pullbacks of (3. For an RQ— flat line bundle L ~ F(A) with A G i?£ we have
L
o n
~
F(An).
If K = K x is the canonical bundle on X then it is easy to check
that
(1.19) K ^ K
X
o n .