6
MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN
Furthermore, assume that
E
is a totally real extension of a field
K,
and let
P
E
X(K). Denote the involutions
E
E
G(FIE) for which
P
extends to an
ordering of F(E) by Ip(FIE). For
X~
X(K), let Ix(FIE)
=
UPEX
Ip(FIE).
IfF is the algebraic closure of E, write Ip(E) for Ip(F I E), etc.
REMARK 1.8. (a) If E
=
K, then Ip(FIE) is a conjugacy class in G(FIE).
In the general case Ip(FIE) is a conjugacy domain in G(FIE); in fact,
Ip(FIE) = nQEX(E)IQ(FIE).
Q::JP
(b) If MIN is a finitely generated extension of fields, then the restriction
map of orderings X(M)
----+
X(N) is closed and open [ELW, Theorem 4.1 and
4.bis]. In particular, let I be a set of involutions in G(FIE), and assume that
FIKis finitely generated. Then so is F(E)IK, for every involution
E
E
G(FIE).
Hence the set
{P'
E
X(K)I Ip(FIE)
=I}
is closed and open in X(K).
LEMMA 1.9. Let (K, P)
~
(K', P') be an extension of ordered fields. Let x
be transcendental over K', and put E = K(x) and E' = K'(x). Furthermore,
let FIE and F' IE' be Galois extensions with F'
=
F · E. Assume that F, and
hence also F', is not formally real. Then Ip(FI E)= Conc(F/E) resFIP'(F' IE').
PROOF. We have Ip(FIE)
=
nQEX(EliQ(FIE). Also,
Qd_P
Ip,(F'IE')
=
nQ'EX(E')IQ'(F'IE').
Q':JP'
As
E
and
K'
are linearly disjoint over
K,
each extension
Q
of
P
to
E
extends
to an ordering
Q'
of
E'
that extends
P'.
Thus it suffices to show
IQ(FI E)= Conc(F/E) resFIQ'(F' IE')
for each ordering Q of
E
and for each extension Q' of Q to
E'.
Let E1
E
I Q' ( F' IE') and
E
= res FE
1

There is an ordering R' of F' ( E
1
)
that
extends
Q'.
Its restriction to F
(E)
is an extension of
Q,
and hence
E
E
I Q (FIE).
Since IQ(F I E) is a conjugacy class in G(F I E), the assertion follows. o
2. Moduli spaces for covers of the Riemann sphere
In this section we add remarks to the notation and results from [FVl
J
in the
form to be used later. Let G be a finite group, and r
~
3 an integer.
(2.1) Covers of the sphere. Let 1P'1 = CU{oo} denote the Riemann sphere.
We consider covers x: X
----+
1P'1 of compact (connected) Riemann surfaces. Call
two such covers x: X
----+
1P'1 and x': X'
----+
1P'1 equivalent if there exists an
isomorphism o:: X
---
X' with
x'
o o:
=
Let Aut(XI1P'1
)
denote the group of
automorphisms o: of X with
x
o o:
=
We say that xis Galois if Aut(XI1P'
1
)
is
transitive on the fibers of From now on
x
will always denote a Galois cover.
All but finitely many points of 1P'1 have the same number of inverse images under
These finitely many exceptional points are called the branch points of
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