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:

=

X· Let Aut(XI1P'1

)

denote the group of

automorphisms o: of X with

x

o o:

=

X· We say that xis Galois if Aut(XI1P'

1

)

is

transitive on the fibers of X· 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

X· These finitely many exceptional points are called the branch points of X·