Algebraic
Structure on a Hurwitz space
Our applications are to interpreting points on Hurwitz spaces as a
surrogate for finding covers of the projective line with given
arithmetic/geometric monodromy groups. For example, interpretation of
the
R(egular) I(nverse) G(alois) P(roblem) that requires knowing properties
of the spaces – their fields of definition, components and cusps – that
don't make sense unless there exist
– versus you need them at hand – algebraic equations for the spaces.
The spaces have abstract defined definitions, through Nielsen classes. That brings up how we
can get
our hands on desirable properties. Also, there are two
famous theoretical issues.
- One is considered an unsolved problem: production of branch cycles for a cover without
using topology.
- The other is psychological: A disturbing feeling that
working with such abstract spaces without equations is an airy
pursuit, unlike combinatorial pursuits yielding pretty formulas or coefficient
anomolies that focus attention on a visually clear starting point.
Fortunately, both objections in #2 have answers. The Hurwitz monodromy
(a combinatorial quotient of the Artin Braid) group is the source
of tremendous combinatorics. Acquainting yourself with its use
and
the permutation representations that produce various versions of
Hurwitz spaces is akin to picking up a version of Galois
Theory.
As for learning comfort in dealing without equations, we say this. Once
mastered, the combinatorial approach to cusps gives a
direct line to properties of the spaces by imitating the classical
example of modular curves. Still, there are occasional subtle questions
for which no one has yet found a work-around to having equations.
Usually, however, that means the question has been answered only in
special cases, and there is no possibility of writing equations for all
relevant Hurwitz spaces.
It takes some doing to acquaint yourself with the use of the Hurwitz
monodromy group (a combinatorial quotient of the Artin Braid group) and
the permutation representations that produce various versions of
Hurwitz spaces, depending on the equivalence one uses on covers. Still,
as one's skill grows on that, it becomes a version of using Galois
Theory.
Here are our topics.
- Galois closure of a cover:
- Branch cycles and Nielsen classes:
- Definition fields of Hurwitz spaces from the B(ranch) C(ycle)
L(emma):
- Hurwitz space components and their definition fields:
- Cusps and the boundary of Hurwitz spaces:
I. Galois closure of a cover: We need the n-fold fiber
product of a degree n cover with itself.
II. Branch cycles and Nielsen classes: Given a group G and a collection of conjugacy
classes C, the Nielsen class
Ni(G, C) has a simple seeming theorem
attached to it: G is the
monodromy group of a cover, f: X → P1z, in the Nielsen class if
and only if the Nielsen class is nonempty.
II.a. A Nielsen class mystery: One‟if part” requires
explanation: Its
statement requires no topology. That is, suppose you have such an f
(defined by algebraic equations, even) and you also know the conjugacy C classes defined by its branch
points z={z1,…, zr}. Then, how can you
assert there are g=(g1,…, gr) with generation (<g1,…, gr>=G) and product
-one (g1… gr=1) without using classical generators?
II.b. Nielsen class as a generalization of genus: Fixing a
Nielsen
class is an analog of fixing the genus in the theory of curves and
their moduli
spaces. That is, given a cover f: X → P1z in a Nielsen class for
inner or absolute equivalence, you automatically know the genus of X from the R(iemann)-H(urwitz)
formula for its genus (see Nielsen-ClassesCont.html).
II.c. Covers of P1z are algebraic: That
is equivalent to the following simple phrase, if f has degree nf,
then
III. Definition fields of Hurwitz spaces from the B(ranch) C(ycle)
L(emma): For each τ ∈ Aut(C), the conjugate space
H∞(C)τ is still a Hurwitz space, which
r,G
only depends on the restriction τ|Qab ∈ Gal(Qab/Q);
namely it is H∞(Cχ(τ)) (where χ is
r,G
the cyclotomic character and Cχ(τ) =(C1
χ(τ),...,Crχ(τ))). Thus the (generally
reducible) varieties H∞and H∞(C) can be
de�ned over Q and Qab respectively, in the sense that
r,G r,G
their (geometric) components are permuted transitively by Gal(Q/Q) and
Gal(Q/Qab) respectively. Furthermore, the Hurwitz space H∞(C)
is
itself de�ned over Q if C is a
r,G
rational union of conjugacy classes of G, i.e., if for every integer m
prime to |G|, there exists σ ∈ Sr such that Cm =
Cσ(i). More
generally, given a �eld k ⊂ Qab, the tuple C is
i
said to be a k-rational union of conjugacy classes of G if the same
property holds for all integers m ≡ χ(τ)
modulo |G| with τ ∈
Gal(Qab/k). Under this condition, the Hurwitz space H∞(C) is
de�ned
over k. For example, the �eld generated by all roots of
unity of
r,G
order |G| is a rationality �eld for C.
IV. Hurwitz space components and their definition fields:
V. Cusps and the boundary of Hurwitz spaces: S.
Wewers has given
a general construction of Hurwitz spaces, which leads to a
de�nition of Hr,G and of some compacti�cation
Hr,G as schemes over Spec(Z[1/|Gp|]). For each prime p not dividing
|G|, the corresponding �bers above p are denoted by Hand
r,G
Hp This includes the case p = ∞ for which one recovers the
space H∞
r,G. r,G. There is good reduction of Hr,G at those primes p
� ||G|: the �ber Hp is a (reducible)
r,G
smooth variety de�ned over Fp and its components correspond
to those of H∞through
r,G
the reduction process. Furthermore, each Hp is a moduli space, for
covers of P1 with r
r,G
branch points and monodromy group G, over algebraically closed
�elds of characteristic p.
The compacti�cation Hr,G is locally the quotient of a smooth
variety by a �nite group Components in Hr,G are closures of
components in Hr,G. The natural ´etale morphism
Ψr : Hr,G →Ur extends to a rami�ed cover
Hr,G →Ur. Points on the boundary Ur \Ur represent degenerations
of tuples t =(t1,...,tr) when two or more of the ti
“coalesce� (i.e. become equal). More formally they
correspond to stable marked curves of genus 0 with a root, i.e. trees
of curves of genus 0 with a distinguished component T0 — the
root <>— equipped with an isomorphism P1
� T0 and at least three marked points (including
the double points) on any component but the root. Points on the
boundary Hr,G \Hr,G represent admissible covers (in a certain sense) of
stable marked curves of genus 0.