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
Previous Page Next Page