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%\hautpagedroite={\hfill\sevenrm\the\titrecourant\hfill\sevenrm\folio}
%\baspagegauche={\hfil} \baspagedroite={\hfil}
%\auteurcourant={P.~D\`ebes and M.~Emsalem}
%\titrecourant={Harbater-Mumford components and towers of
%Hurwitz spaces}
%\baspagetitre={\hfill\sevenrm\folio\hfill}


%% MACROS SPECIFIQUES
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% MACROS SPECIFIQUES
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\def\unramified{unramified}

\centerline {\twelvebf Hurwitz spaces}
\bigskip

\ind {\bf 1.} In a 1891 paper, Hurwitz explains how the set of
degree $d$ simple covers (\ie, such that all fibers consist
of at least $d-1$ points) of $\P^1$ can be endowed with a
structure of complex manifold. Nowadays Hurwitz spaces 
refer more generally to moduli spaces of covers with specified
automorphism group and with certain constraints on the 
ramification.

\ind Moduli spaces of covers constitute an appropriate tool 
for studying arithmetic questions on covers of the line; for example, the 
Regular Inverse
Galois Problem amounts to finding rational points on these spaces. The 
general idea is to look
at the constraints a given question imposes on the intrinsic data of
the covers in question, as the automorphism group and the
ramification, and then to investigate whether there exist possible
solutions on the associated moduli space, first over
${\C}$, and then over the ground field. The diophantine nature of
the problem remains, but this approach somewhat classifies the
equations by abstracting their structural properties.

\ind Recent developments consider modular towers.
The motivating example is the tower of modular curves of
levels $p^n$. A modular tower is a tower of Hurwitz spaces, whose 
levels $p^n$ come from a universal Frattini construction starting 
with a given finite group and a prime $p$. The general question 
is to extend the rich arithmetic contents of modular curve
towers to general modular towers.


\bigskip

\ind {\bf 2.} Hurwitz spaces are moduli spaces of covers 
of $\P^1$ with fixed monodromy 
group\footnote{$^1$}{\baselineskip =10pt \sevenrm 
equivalently, the monodromy group is the Galois group of 
the Galois closure of the cover. \par} $G$ (given as a subgroup of the 
symmetric group $S_d$ with $d$ the degree of the cover) 
and with a fixed number 
$r \geq 3$ of branch points. The basic notation for it is ${\cal H}_{r,G}$
and a point representing a cover $f$, or more exactly its
equivalence class, is denoted by $[f]$. There are several variants 
of Hurwitz spaces, depending

\ind first, on
whether one is interested in

- {\it the mere cover situation:} the covers are not
necessarily Galois, or


- {\it the G-cover situation:} the covers are Galois covers
given with an isomorphism between their automorphism group
and the group $G$,
\smallskip
\ind and, second, on which cover equivalence is used:

-  {\it the original equivalence}: two covers $f:X\rightarrow \P^1$ and 
$g:Y\rightarrow \P^1$ are equivalent if there exists an isomorphism 
$\chi: X\rightarrow Y$  such that $g\circ \chi= f$, or

-  {\it the $\hbox{\rm PGL}_2$-reduced equivalence}: 
 $f:X\rightarrow \P^1$ and  $g:Y\rightarrow \P^1$ are equivalent if there exist two isomorphisms $\chi: X\rightarrow Y$ and  $\alpha: \P^1\rightarrow \P^1$ such that  
$g \circ \chi = \alpha \circ f$,

with for both equivalences the extra condition that $\chi: X\rightarrow Y$ be compatible 
with the actions of ${G}$ in the G-cover situation. 
For simplicity, we will not distinguish here between
these different variants.
\bigskip

\ind {\bf 3.} At the beginning stage, covers are considered over the
complex field $\C$. The corresponding moduli space is
then a complex smooth quasi-projective variety, denoted by
${\cal H}_{r,G}^{\infty}$. 
Given an (unordered) $r$-tuple ${\bf
C}=(C_1,\ldots,C_r)$ of conjugacy classes of $G$, we let
${\cal H}_{r,G}^{\infty}({\bf C})$ be the union of all
components\footnote{$^2$}{\baselineskip =10pt \sevenrm 
By component we mean irreducible or connected components; 
due to smoothness of $\scriptstyle {\cal H}_{r,G}^{\infty}$, these coincide. \par} of
${\cal H}^{\infty}_{r,G}$ whose points correspond to covers with
ramification type equal to ${\bf C}$ \footnote{$^3$}{\baselineskip =10pt \sevenrm 
The ramification type is the $\scriptstyle r$-tuple of 
conjugacy classes (in the monodromy group $\scriptstyle G$) of the 
monodromy branch cycles 
associated to sample loops revolving about the $\scriptstyle r$ 
branch points. It is locally constant on $\scriptstyle {\cal H}_G(\C)$, thus
is constant on each connected component 
of $\scriptstyle {\cal H}_G(\C)$. More arithmetically, the ramification type
corresponds to the inertia canonical invariant. This invariant
is the collection $\scriptstyle  (C_t)_t$ of conjugacy classes 
$\scriptstyle  C_t$ (in the Galois group of 
the Galois closure) of
distinguished generators of inertia
groups above $\scriptstyle  t$ as $\scriptstyle  t$ ranges over the
branch points of the cover. The distinguished generator 
of some inertia group
$\scriptstyle  I$, say of order
$\scriptstyle e$, is the generator that corresponds to
$\scriptstyle e^{2i\pi/e}$ in the natural isomorphism between
$\scriptstyle I$ and the group $\scriptstyle \mu_e$ of
$\scriptstyle e$-th roots of
$\scriptstyle 1$. \par}. Fixing the ramification type can be regarded analogous
to fixing the genus in the theory of curves and their moduli spaces.


\ind For each $\tau \in {\rm Aut}(\C)$, the conjugate
space ${\cal H}_{r,G}^\infty({\bf C})^\tau$ is still a Hurwitz space, which
only depends on the restriction $\tau|_{\Q^{\hbox{\fiverm ab}}}\in
\Gal(\Q^{\ab}/\Q)$; namely it is ${\cal H}_{r,G}^\infty({\bf
C}^{\chi(\tau)})$ (where $\chi$ is the cyclotomic
character and ${\bf C}^{\chi(\tau)}=(C_1^{\chi(\tau)},\ldots,
C_r^{\chi(\tau)})$). Thus the (generally reducible) varieties ${\cal
H}^{\infty}_{r,G}$ and ${\cal H}_{r,G}^{\infty}({\bf C})$ can be defined over
$\Q$ and $\Q^{\ab}$ respectively, in the sense that their (geometric)
components are permuted transitively by $\Gal(\overline \Q/\Q)$ and
$\Gal(\overline
\Q/\Q^{\ab})$ respectively. Furthermore, the Hurwitz space ${\cal
H}_{r,G}^{\infty}({\bf C})$ is itself defined over $\Q$ if ${\bf C}$ is a {\it
rational union of conjugacy classes} of $G$, {\it i.e.}, if for every integer $m$
prime to $|G|$, there exists $\sigma \in S_r$ such that $C_i^m =
C_{\sigma(i)}$. More generally, given a field $k \subset \Q^\ab$,  
the tuple ${\bf C}$ is said to be a $k$-rational union of conjugacy classes of $G$ if the
same property holds for all integers $m \equiv \chi (\tau)$ modulo
$|G|$ with $\tau \in \Gal(\Q^\ab/k)$. Under this condition, the Hurwitz
space ${\cal H}_{r,G}^\infty({\bf C})$ is defined over $k$. For example, the
field generated by all roots of unity of order $|G|$ is a 
rationality field for ${\bf C}$.

\bigskip

\ind {\bf 4.} Denote the configuration space for
finite subsets of $\P^1$ of cardinality $r$ by ${\cal U}_r$ and then by 
$\Psi_{r}: {\cal H}_{r,G}^{\infty} \rightarrow
{\cal U}_r \otimes_{\Z}\C$ the map sending each point
$[f]\in {\cal H}_{r,G}^{\infty}(\C)$ to the branch point set ${\bf
t}=\{t_1,\ldots,t_r\}\in {\cal U}_r(\C)$ 
of the cover $f$. A key to the theory, based on Riemann's existence theorem,
is that this map is an etale cover (of algebraic varieties); furthermore 
there is a one-one correspondence
between the fiber $\Psi_{r}^{-1}({\bf t})$ and the set

$$\hbox{\rm ni}({\bf C})^{\bullet}=\left\{ (g_1,\ldots,g_r)
\in G^r
\left|
 \matrix{\ \uplex g1r =1 \hfill \cr\ <\uple g1r>\hskip 2pt
= G
\hfill \cr\ g_i\in C_{\sigma(i)}, i=1,\ldots,r\
\hbox{for some}\ \sigma \in S_r \hfill \cr}\right.
\right\}\ /\ \sim$$

Here by ``$/\ \sim$'', we mean that the tuples $(g_1,\ldots,g_r)$
are regarded up to componentwise conjugation by elements of 
a certain subgroup of $S_d$ (depending on the situation: for example it
is $G$ for G-covers with the original equivalence, it is the normalizer $\hbox{\rm
Nor}_{S_d}(G)$ for mere covers, etc.). The identification 
$\Psi_{r}^{-1}({\bf t})\simeq \hbox{\rm ni}({\bf C})^{\bullet}$ is
given by the monodromy representation:
for any choice 
of a topological bouquet\footnote{$^4$}{\baselineskip =10pt \sevenrm 
i.e., a $\scriptstyle r$-tuple $\scriptstyle \underline
\Gamma= (\Gamma_1,\ldots,\Gamma_r)$ of homotopy classes of sample loops
based at some point $\scriptstyle t_o\notin {\bf
t}$ generating the topological fundamental
group $\scriptstyle \pi_1^{\hbox{\sevenrm top}} (\P^1(\C) \setminus {\bf t},
t_0)$ with the unique relation
$\scriptstyle \Gamma_1 \cdots \Gamma_r = 1$ (plus some other technical 
conditions). \par}
$\underline \Gamma=(\Gamma_1,\ldots,\Gamma_r)$ for $\P^1\setminus\({\bf t}\}$ 
based at some point $t_0\notin {\bf
t}$, the
map ${\rm BCD}_{\underline
\Gamma}$ \footnote{$^5$}{\baselineskip =10pt \sevenrm 
where BCD stands for ``branch cycle
description''. \par}, sending each complex cover $f:X\rightarrow
\P^1$ to the $r$-tuple
with entries the monodromy permutations of
$f^{-1}(t_0)$ associated with  $\Gamma_1,
\ldots,\Gamma_r$, provides the correspondence $\Psi_{r}^{-1}({\bf t})\rightarrow \hbox{\rm ni}({\bf C})^{\bullet}$.
There is a classical outer action of the Hurwitz braid 
group\footnote{$^6$}{\baselineskip =10pt \sevenrm 
The Hurwitz braid group $\scriptstyle H_r$ has a classical presentation: 
it is the group on 
$\scriptstyle r-1$ generators $\scriptstyle \uple Q1{r-1}$ with 
relations $\scriptstyle Q_iQ_j=Q_jQ_i \hbox{ for } |i-j|>1$, 
$\scriptstyle Q_{i+1}Q_iQ_{i+1}=Q_iQ_{i+1}Q_i \hbox{ for } 1\le i\le r-2$
and $\scriptstyle Q_1\cdots Q_{r-1}Q_{r-1}\cdots Q_1 =1$.  \par}
$H_r=\pi _1^{\hbox{\sevenrm top}}({\cal U} _r,\bf t )$ on $\pi
_1^{\hbox{\sevenrm top}}(\P ^1 (\C ) \setminus {\bf t}, t_0)$, which
induces an action on the fiber $\Psi_r ^{-1} (\bf t)$, and on ${\rm
ni}(\bf C) ^\bullet $ {\it via} maps ${\rm BCD} _{\underline \Gamma}$.
This induced action on $\Psi_r ^{-1} (\bf t)$ is the monodromy action
corresponding to the topological cover
$\Psi_{r}: {\cal H}_{r,G}^{\infty}(\C) \rightarrow {\cal U}_r(\C)$.
It can be explicitly determined: $\pi_1({\cal U}_r,{\bf t})^{\hbox{\sevenrm top}}$ has
generators $Q_1,\ldots,Q_{r-1}$ whose action on $\Psi_r^{-1}({\bf
t})$, when computed relative to some suitable topological bouquet
$\underline \Gamma$, corresponds to the following action on
$\hbox{\rm ni}({\bf C})^{\bullet}$:
\smallskip
\centerline{$\displaystyle (g_1,\ldots, g_r) \rhfl{Q_i}{}
(g_1,\ldots,g_{i-1},g_i g_{i+1}g_i^{-1}, g_i, g_{i+2},\ldots,g_r),
\hskip 5mm i=1,\ldots,r-1$.}
\smallskip

Components of ${\cal H}_{r,G}^{\infty}({\bf C})$ correspond to
orbits of the Hurwitz braid group action.

\bigskip

\ind {\bf 5.} S.~Wewers has given a general construction of Hurwitz
spaces, which leads to a definition of ${\cal H}_{r,G}$ and of some
compactification
$\overline {{\cal H}_{r,G}}$ as schemes over $\hbox{\rm
Spec}(\Z[1/|G|])$. For each prime $p$ not dividing $|G|$, 
the corresponding fibers above $p$ are denoted by ${\cal H}^p_{r,G}$ and
$\overline {{\cal H}^p_{r,G}}$. This includes the case  $p=\infty$ for which one recovers the space ${\cal
H}^\infty_{r,G}$.

\ind There is good reduction of ${\cal H}_{r,G}$ at those primes
$p\not| \hskip 1pt|G|$: the fiber ${\cal H}^p_{r,G}$ is a
(reducible) smooth variety defined over $\overline \F_p$ and its
components correspond to those of ${\cal
H}^\infty_{r,G}$ through the reduction process. Furthermore, each
${\cal H}^p_{r,G}$ is a moduli space, for covers of $\P^1$ with $r$
branch points and monodromy group $G$, over algebraically
closed fields of characteristic $p$.

\ind The compactification
$\overline {{\cal H}_{r,G}}$ is locally the quotient of a smooth variety by a finite group
Components in $\overline{{\cal H}_{r,G}}$ are closures
of components in ${\cal H}_{r,G}$. The natural \'etale
morphism $\Psi_{r}: {\cal H}_{r,G} \rightarrow {\cal U}_r$
extends to a ramified cover $\overline {{\cal H}_{r,G}}\rightarrow
\overline {\cal U}_r$. Points on the boundary $\overline {\cal
U}_r\setminus{\cal U}_r$ represent degenerations of 
tuples ${\bf t} = (t_1,\ldots,t_r)$ when two or more of the $t_i$
``coalesce'' ({\it i.e.} become equal). More formally they correspond  
to stable marked curves of genus $0$ with a root, {\it i.e.} trees of curves of genus
$0$ with a distinguished component $T_0$ --- the {\it root}
--- equipped with an isomorphism $\P ^1 \simeq T_0$ and at
least three marked points (including the double points) on
any component but the root. 
Points on the boundary $\overline {{\cal H}_{r,G}}\setminus {{\cal
H}_{r,G}}$ represent {\it admissible covers} (in a certain sense) 
of stable marked curves of genus $0$.
\end
