10 1. PRELIMINARIES FROM ALGEBRAIC GEOMETR Y

It will be important later to have a criterion for a line bundle L = (L, /3) on

a correspondence X over R to be flat. The following (trivial) Lemma gives the

answer:

LEMMA 1.8. For A G Rx the following are equivalent:

1)L~ F(A),

2) There exists an invertible section rj G H°(X, L) such that fla^r] — A • air].

PROOF. Let us check that 2 = 1; the converse is proved by reversing the

argument. If 2 holds then we have an isomorphism u : Ox —* L, 1 i— rj inducing a

commutative diagram

Ox=a*Ox -^ o\Ox = Ox l = o-%l i- A = A - ( j J l

a^u[ [a{u , 1 1

j2*L ^ o\L

a2V

^ A-a*7y

which shows that L ~ F(A). D

In what follows fix A:, an algebraically closed field.

DEFINITION 1.9. An irreducible reduced correspondence X

finite type over k is dense if the smallest equivalence relation

(1.20) (a) C X(k) x X(k)

on X(k) containing the image of

a1 x a2 : X{k) - X{k) x X(k)

is Zariski dense in X(k) x X(k).

Density of correspondences is, of course, a common occurrence. For instance,

in the simplest case when X and X are curves and a has an infinite orbit, (a) is

dense in X x X; cf. the Introduction.

DEFINITION 1.10. Two irreducible reduced correspondences X and X' of finite

type over k are called commensurable if there exists a sequence of irreducible reduced

correspondences Xi,X2, ...,Xn of finite type over k such that X = Xi, X7 — X

n

,

and for each i there exists either a morphism (71^,7^) : X^ — X^+i or a morphism

(7Ti, TTi) : Xi+i —• Xi with both 7^ and 7T; dominant and quasi-finite. We say X and

X' are equivalent if if there exists a sequence of irreducible reduced correspondences

Xi, X2,..., X

n

of finite type over k such that X = Xi, X' = X

n

, and for each i there

exists either a morphism (7^,7^) : X^ — X^+i or a morphism (71^,7^) : X^+i — • X$

with TTi an isomorphism and 7^ dominant and quasi-finite.

DEFINITION 1.11. A morphism of irreducible reduced correspondences of finite

type over k

(TT,TT) : (Y,Y,n,T2) ^ (X,X,aua2)

is a Galois cover if both TT and n are Galois covers.

If L = (L, (3) is a line bundle on a correspondence X

we may consider the isomorphism

(3 : H°{X, a%L) ~ H°(X, a*L).

Then we set L — o\L and we define the R—linear map

(1.21) A = A L : H°(X, L) - H°(X, L)

(X,X,T1,a2)of

(X,X,(7i,(T2) over k