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-
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