8
MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN
with
f.
These
N
form a basis for the topology. They are connected. The sets
A(N) form a basis for a topology on
1-{ab,
such that A:
1-(in - 1-{ab
becomes an
( unramified) covering.
(2. 7) r-tuples of unordered branch points. Let
Ur
denote the space of
all subsets of cardinality r of the Riemann sphere JP1
.
It has a natural structure
of algebraic variety defined over Q
[FVl, §1.1];
it is isomorphic to
wr,D,
where
D,
the discriminant locus, is a hypersurface in JPr. In particular,
Ur
is an affine
variety. FUrthermore, if K is a subfield of C and a = { a
1
, ... ,
ar} E
Ur
with
a
1
, ... ,
ar
=I
oo, then a is K-rational if and only if
[1~= 1
(X- ai)
E
K[X]. As
a complex manifold, the topology of
Ur
has a basis consisting of sets D of the
following form: Given pairwise disjoint open discs D
1
, ... ,
Dr
on JP1
,
let
D
be
the set of all a
E
Ur
with jan
Dil
= 1 fori= 1, ... ,
r.
(2.8) Maps to
Ur.
Define 111:
1-lin -
Ur
and
q,:
1-lab -
Ur
by sending
[X,
h]
and [x], respectively, to the set of branch points of These maps are (unrami-
fied) coverings and
q,
o A= 111. Through these coverings the spaces
1-{ab
and
1-{in
inherit a structure of complex manifold from
Ur·
(2.9) The algebraic structure on covers. Each cover
x:
X
-
JP1 as above
is an algebraic morphism of algebraic varieties over C, compatible with its ana-
lytic structure (Riemann's existence theorem). An automorphism
f3
of C defines
an automorphism
/3*
of JP1 by (
x0
:
xt)
t---t
(/3-
1
(
x0
) :
(3-
1
(
xt)).
Consider the
cover
f3(x): f3(X) _,
JP1 obtained from
x: X
-
JP1 through base change with
f3*.
FUrthermore, for each o:
E
Aut(X/JP1) let f3*(o:) = f3(o:)
E
Aut(j3(X)/JP1) be the
morphism obtained by the same base change.
(2.10) The algebraic structure on
1-{in.
The spaces
1-{ab
and
1-(in
have a
unique structure of (the set of complex points of) a (reducible) algebraic variety
defined over Q
[FVl,
Theorem 1]. This variety structure is compatible with
the analytic structure of
1-{ab
and
1-{in,
and it makes the maps 111,
q,
and A into
algebraic morphisms defined over Q. Also, each automorphism
f3
of C-in its
natural action (x
1
, ... ,
Xn)
t---t
(f3(x
1
), ... ,
f3(xn))
on the complex points of (the
affine pieces of) a variety defined over Q-sends the point [x]
E
1-{ab
to [f3(x)]
and the point [x, h]
E 1-{in
to
[f3(x), h
o (3; 1].
(2.11) Complex conjugation acting on
1-{in.
We can describe the action
of complex conjugation c on the triples of (2.4) that compose
1-{in.
Namely,
c naturally acts on paths in JP1
.
Thus, it induces a map
II
1
(JP1 '-.a, a0
)
-
II
1
(JP1.,c(a),c(a0
)).
Denote this map by'"'(
t---t
C'"'f.
LEMMA. Ifp = [a, a0
,
f]
E 1-{in,
then c(p) = [c(a), c(a0
),
cf]. Here (cf)(q) =
fh)
for each '"'(
E II1
(JP
1
'-.a, ao).
PROOF.
Write p as p = [x, h]. Then c(p) = [c(x), h o c:;-
1].
It remains
to show that this point is represented by the triple (c(a), c(a0
),
cf). This is a
straightforward consequence of the definitions
(cf. [DFl,
Lemma 2.1]). o
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