REAL HILBERTIANITY-TOTALLY REAL NUMBERS

9

(2.12) r-tuples of all conjugacy classes of G. Let b = {bb ... , br}

E

Ur

such that 0

tt

b. We can choose generators 1'1, ... , l'r for the fundamental group

II1(JP1'-

b, 0) so that /'l · · ·!'r = 1 is the only relation among them. Indeed,

assume that b1 = (1, for j = 1, ... , r, where ( = e

2

;;.

Otherwise apply a

homeomorphism

JP1

---*

JP1

that maps 0 onto itself and

b1

onto (j. Let ;:y1 be a

path starting at 0, going up on a straight line to a neighborhood of b1, traversing

a small disk around b1 in the counterclockwise direction, and following the same

straight line back to 0. Then

i

1

, ... ,

ir

do not intersect except at 0. Let

/'j

be the

homotopy class of

i1.

Then 1'1, ... , l'r generate II1

(JP1'.b,

0) and 1'1 · · ·!'r = 1.

Represent a point p

E

w-

1 (b) by a triple (b, 0,

!).

The r-tuple

(0"1 , ... ,

O"r)

=

(!(1'1), ... ,f(!'r))

determines the epimorphism

f:II1(JP1'-..b,O)---*

G.

It has the

following properties:

0"1

· · ·

O"r

= 1,

0"1

, ... ,

O"r

generate G, and

O"j

=/:-

1 for all j

[FV1, §1.3]. Let

Er

denote the set of such r-tuples

(0"1,···

,O"r)·

Clearly, each

(0"1, ...

,O"r)

E

£r

arises in the above way from some p E

w-

1

(b). Let .C(G)

be the collection of conjugacy classes

=f.

{1} of

G,

and let £(r) be all r-tuples

(0"1 , ...

,O"r)

E

Er

where each

C

E

.C(G) is represented exactly r/I.C(G)I times

(2.13) When commutators generate the Schur multiplier of G. For

the rest of section 2 assume that

r

is a multiple of I.C( G) I and suitably large

[FV1, Appendix], and the Schur multiplier of

G

is generated by commutators.

We explain the latter condition. Let

R

be a group of maximal order with the

property that

R

has a subgroup

M :::; R'

n

Z(R)

satisfying

R/M

~

G.

Then

M

n

{g- 1h- 1ghl g, hER}

generates

M,

the Schur multiplier of

G.

Fix b

E

Ur and ')'1

, ... ,

/'r as above. By [FV1, §1.3] there is a (unique)

connected component

H

ofHin containing {[b,O,f]l

(!(1'1), ... ,f(!'r))

E

£(rl}.

Let

H

= A(H) be its image in Hab. We call

H

and

H

Hurwitz spaces. By

[FV1, Thm. 1] they are absolutely irreducible algebraic varieties defined over

Q.

Moreover, since IJI:

H---* Ur

and

~:

H

---*

Ur

are finite normal covers of an affine

variety, Hand Hare affine [H, Exc.

III.4.1].

(2.14) Automorphisms of 1-(in---* Hab. For A

E

Aut( G) (acting from the

left on G), let 8

A:

H ---* H be the map sending the point

[x,

h] to

[x,

A o h ]. Then

8A

is an automorphism of the covering A:

H---*

H.

It depends only on the class

of

A

modulo Inn( G). In fact, A is a Galois covering, and the map

A

f--+

8

A

induces an isomorphism

8:

Out( G)= Aut(G)/Inn(G)---*

Aut(H/H)

[FV1, §6.1].

Furthermore,

8A

is a morphism defined over Q [FV1, §6.2]. In the description

of Hin in (2.4),

8A

sends the point [a, a0

,

f] to [a, ao,

A

of]. As A: Hin ---* Hab is

an unramified covering (2.8),

8A

has no fixed points.

For the rest of this section assume that

G

has trivial center. Accordingly,

identify

G

with the subgroup Inn( G) of Aut( G) (acting from the left on G). Let

p

E

Hand let

K

~

L

be subfields of C such that A(p)

E

H(K)

and

L

= K(p).

(2.15) Fields of definition of covers. Write p as p =

[x,

h]. Then, the

cover

x:

X ---*

JP1

can be defined over L (in a unique way) such that all automor-