text:
\section{Introduction}

The goal of this paper is to understand representation varieties
of Artin and Shephard groups and thereby obtain information on Serre's
problem of determining which finitely-presented groups are fundamental
groups of smooth complex (not necessarily compact)
algebraic varieties. The first
examples of finitely-presented  groups which are not fundamental
groups of smooth complex algebraic varieties were given by
J.~Morgan \cite{Morgan1}, \cite{Morgan2}.
We find a new class of such examples which consists of certain Artin and
Shephard groups. Since all Artin and Shephard groups have quadratically
presented Malcev algebras, Morgan's test does not suffice
to distinguish Artin groups from fundamental groups of smooth complex
algebraic  varieties or even from fundamental groups of
compact K\"ahler manifolds, see \mathbb S \ref{completion}.

Our main results are the following theorems (Artin and Shephard
groups are defined in \mathbb S \ref{groups} below):

\begin{thm}
\label{moremorgan}
There are infinitely many distinct Artin groups that  are not isomorphic
to fundamental groups of smooth complex algebraic varieties.
\end{thm}

\begin{thm}
\label{morereps}
For any affine variety $S$ defined over $\mathbb Q$
there are Shephard and Artin groups $G$ such that a Zariski open
subset $U$ of $S$ is biregular
isomorphic to  Zariski open subsets of the character
varieties $X(G, PO(3))= \mbox{\rm Hom}(G,  PO(3))/\!/ PO(3)$.
The subset $U$ contains all real points of $S$.
\end{thm}

The surprising thing about Theorem \ref{moremorgan} is that Artin groups
look very similar to the fundamental groups of smooth complex
quasi-projective varieties. For example the free group on $n$ letters is the fundamental group of $\mathbb C$ with $n$ points removed and it is the Artin group
associated with the graph with $n$ vertices and no edges. On the other
extreme, take a finite
complete  graph where each edge has the label $2$. The
corresponding Artin group is free Abelian, hence it is the
fundamental group of the quasi-projective variety $(\mathbb C^{\times})^n$.
Yet another example is the braid group which is the Artin group
associated with the permutation group $S_n$. Theorem \ref{moremorgan}
is a consequence of Theorems \ref{moremnev}, \ref{t3}, \ref{t4},
\ref{t6}, \ref{t7} and Corollary \ref{c5}
below. The main body of this paper is concerned with a study of the following diagram\footnote{Here and in what
follows we do not assume our varieties are reduced or irreducible, i.e. they are schemes locally of finite type.}:

\[
\begin{array}{ccc}
\framebox{\parbox{2in}
{Moduli spaces of arrangements in the projective plane}}
& \begin{array}{c}
\stackrel{\tau}{\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightarrow}\\
\stackrel{\displaystyle \leftarrow\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel}{\scriptstyle geo}
\end{array}
& \framebox{\parbox{.6in}{Varieties over $\mathbb Q$}} \\
  {\ }_{alg\Bigg\downarrow} & ~  & ~ \\
~ & ~ & ~ \\
\framebox{\parbox{2in}{Character varieties of representations
of  Shephard groups
into $PO(3)$}} & \stackrel{\mu}{\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightarrow}
&  \framebox{\parbox{2in}
{Character varieties of representations of Artin groups into $PO(3)$}}
\end{array}
\]

The arrow $\tau$ is tautological, the arrow $\mu$ is pull-back.
The arrows $geo$ and $alg$ are defined below. In
\mathbb S\S \ref{Abstract arrangements}, \ref{Projective  arrangements}
we define {\em abstract arrangements} (essentially bipartite graphs)
and their projective realizations. The space $R(A)$ of projective
realizations of a given abstract arrangement has a canonical
structure of a projective variety (i.e. projective scheme, see
the preceeding footnote) over $\mathbb Q$. In fact we refine the notion
of projective arrangement to obtain the affine variety $BR_0(A)$
of {\em finite based realizations}. The variety $BR_0(A)$ injects
as an open subvariety into
the {\em moduli space} ${\cal M}(A)=BR(A)/\!/ PGL(3)$ of the arrangement $A$.
Our version  of Mnev's theorem \cite{Mnev} is then

\begin{thm}
\label{moremnev}
For any affine algebraic variety $S$ defined over $\mathbb Q$ there is a
marked based abstract arrangement $A$ such that the varieties
$BR_0(A)$, $S$ are isomorphic.
\end{thm}

\begin{rem}
It appears that Mnev's theorem \cite{Mnev} implies only that there is a
stable homeomorphism between the sets of real points of $BR_0(A)$ and $S$.
In addition Mnev gives only an outline of the proof. For our application
to Serre's problem it is critical to prove an isomorphism on the scheme
level.
\end{rem}

The key idea  in proving Theorem \ref{moremnev} is to construct a
cross-section
$geo$ to $\tau$ (over the category of affine varieties)
by showing that one can do ``algebra via geometry'',
that is one can describe elementary algebraic operations over any
commutative ring using projective arrangements
(see \mathbb S \ref{Algebraic operations}). This idea actually goes back to the
work of von Staudt \cite{St} (the ``fundamental theorem of the
projective geometry''). The abstract arrangement $A$ corresponding
to $S$ under $geo$ depends upon a choice of
affine embedding $\tilde{S}$ (i.e. defining equations) for $S$
and upon a choice of particular formulae
describing these equations.  Moreover we obtain isomorphism $\tilde{S} \to BR_0(A)$ of affine schemes over $\mathbb Q$. We will abuse notation and use $geo$ to denote this isomorphism as well. Thus the symbol $geo$ has two meanings: $A= geo(S)$
  is the arrangement determined by the above data, but if $x\in \tilde{S}$ then $\psi= geo(x)$ is a point in $BR_0(A)$. The meaning to be assigned to $geo(\cdot)$ will be clear from its argument. We make analogous abuses of notation with mappings $\tau$ and $alg$ below.

The arrow {\em alg} encodes the points and lines of an {\em anisotropic
projective realization} (see  \mathbb S \ref{S14} for the definition)
$\psi$ of the abstract arrangement $A$ into a representation
$$
\rho= \rho_{\psi}: G^s_A \longrightarrow PO(3, \mathbb C)
$$
of the Shephard group $G^s_A$ associated to the abstract arrangement
$A$. A choice of a nondegenerate bilinear form on $\mathbb C^3$
determines anisotropic points and lines in $\mathbb P^2$ (we choose the
bilinear form  so that all real points of $\mathbb P^2$ are anisotropic).
Each anisotropic point in $\mathbb P^2$  determines the Cartan involution
in $PO(3, \mathbb C)$ around this point or the rotation of order $3$ having
this point as neutral fixed  point (i.e. a fixed  point where the
differential of the rotation has determinant 1). There are
two such rotations of order $3$, we choose one of them. Similarly
every anisotropic  line $L$
uniquely determines the reflection in $PO(3, \mathbb C)$ which keeps
$L$ pointwise fixed. Finally one can encode the incidence relation
between points and lines in $\mathbb P^2$ using algebra:
two involutions generate the subgroup $\mathbb Z/2\times \mathbb Z/2$ in $PO(3,\mathbb C)$
iff the neutral fixed point of one belongs to the fixed line of the
other, rotations of orders $2$ and $3$  anticommute iff the
neutral fixed point of the rotation of order $3$ belongs to the
fixed line of the involution, etc. We get a morphism
$$
alg: \hbox{~~based anisotropic arrangements~~} \longrightarrow
\hbox{~~representations}
$$
In the following theorem we shall identify $alg(\psi)$ with its
projection to the character variety
$$
X(G^s_A, PO(3)):= \mbox{\rm Hom}(G_A^s, PO(3))/\!/ PO(3)\ \quad . $$

\begin{thm}
\label{t3}
The mapping $alg: BR(A, \mathbb P^2_0)\to X(G_A^s, PO(3))$
is a biregular  isomorphism onto a Zariski open (and closed)
subvariety $\mbox{\rm Hom}_f^+ (G^s_A, PO(3))/\!/ PO(3)$.
\end{thm}

\smallskip
The mapping $alg$ has the following important property: Let $S$ be an
affine variety defined over $\mathbb Q$ and $O\in S$ be a rational point.
Then we can choose an arrangement $A$ so that $O$ corresponds to a
realization $\psi_0$ under the mapping $geo: S\to BR_0(A)$
such that the image of the representation
$alg (\psi_0)$ is a finite subgroup of $PO(3,\mathbb C)$ with trivial
centralizer.

There is an Artin group $G^a_A$ and a canonical epimorphism
$G^a_A\to G^s_A$ associated with the Shephard group $G^s_A$.
It remains to examine the  morphism
$\mu:  \mbox{\rm Hom}^+_f(G^s_A, PO(3)) \to \mbox{\rm Hom} (G^a_{A}, PO(3))$
given by pull-back of homomorphisms.

\begin{thm}
\label{t4}
Suppose that $A$ is an admissible based arrangement. Then the morphism
$\mu$ is an isomorphism onto a union of Zariski connected components.
\end{thm}

\begin{cor}
\label{c5}
The variety $\mbox{\rm Hom} (G^a_A, PO (3))$ inherits all the
singularities of the representation variety $\mbox{\rm Hom} (G^s_A, PO (3))$
corresponding to points of $BR(A, \mathbb P^2_0)$, whence (since all
real points of $BR(A)$ are anisotropic) to all singularities
of $BR(A)$ at real points.
\end{cor}

Combining Corollary \ref{c5} with Theorem \ref{moremnev} we obtain

\begin{thm}
\label{t6}
Let $S$ be an affine algebraic variety defined over $\mathbb Q$ and
$O\in S$ be a rational point. Then there exists an admissible
based arrangement $A$ and a
representation $\rho_0: G^a_A \to PO(3,\mathbb R)$ with finite image such that
the (analytical)
germ $(X(G^a_A, PO(3, \mathbb C)), [\rho_0])$  is isomorphic to $(S, O)$.
\end{thm}

Thus the singularities of representation varieties of Artin groups
at representations with finite image are at least as bad as germs
of affine varieties defined over $\mathbb Q$ at rational points.

On the other hand, if $M$ is a (connected) smooth complex algebraic
variety and $G$ is an algebraic Lie group, the singularities of
$\mbox{\rm Hom}(\pi_1(M), G)$ at representations with finite image are
severely limited by Theorem \ref{t7} below. We will need the following

\begin{defn}
Let $X$ be a real or complex analytic space, $x\in X$ and $G$ a Lie group acting on $X$. We say that there is a local {\em cross-section} through $x$ to the $G$-orbits if there is a $G$-invariant open neighborhood~ $U$ of $x$ and a closed analytic subspace $S\subset U$ such that the natural map $G \times S \to U$ is an isomorphicm of analytic spaces.
\end{defn}

\begin{thm}
\label{t7}
 Suppose $M$ is a smooth connected complex algebraic variety, $G$ is an
algebraic  Lie group  and $\rho: \pi_1(M)\to G$ is a representation
with finite image. Then the germ
$$(\mbox{\rm Hom}(\pi_1(M), G), \rho)$$
 is analytically isomorphic to a
quasi-homogeneous cone with generators of weights $1$ and $2$ and
relations of weights $2, 3$ and $4$.
In the case there is a local cross-section through $\rho$ to $Ad(G)$-orbits, then
the same conclusion is valid for the quotient germ
$$(\mbox{\rm Hom}(\pi_1(M), G)/\!/ G, [\rho])\quad . $$
\end{thm}

We present two proofs of this result: in  \mathbb S \ref{Hain's theorem}
we deduce it  from a theorem of  R.~Hain \cite{Hain}
 and, since Hain's paper is still in preparation, in \mathbb S \ref{direct}
we also deduce Theorem \ref{t7} from results of J.~Morgan \cite{Morgan2}
on Sullivan's minimal models of smooth complex algebraic varieties.

Note that in the case when $M$ is a {\em compact} smooth
complex algebraic variety (or more generally a compact K\"ahler manifold)
then a more general theorem holds:

\begin{thm}
\label{gms}
(W.~Goldman, J.~Millson \cite{GM}, C.~Simpson \cite{Si}). Suppose that $M$ is a compact K\"ahler manifold $G$ is an algebraic  Lie group  and
$\rho: \pi_1(M)\to G$ is a representation such that the Zariski closure of $\rho(\pi_1(M))$ is a reductive subgroup of $G$. Then the germ
$$(\mbox{\rm Hom}(\pi_1(M), G), \rho)$$
 is analytically isomorphic to a (quasi)-homogeneous cone with
generators of weight $1$ and relations of weight $2$ (i.e. a
homogeneous quadratic cone). In the case there is a local cross-section
through $\rho$ to $Ad(G)$-orbits, then
the same conclusion is valid for the quotient germ
$$(\mbox{\rm Hom}(\pi_1(M), G)/\!/ G, [\rho])\quad . $$
\end{thm}

Our proof of Theorem \ref{t7} is in a sense analogous to the proof of Theorem
\ref{gms} in \cite{GM}, \cite{Si}: we construct a differential graded Lie
algebra ${\cal Q}^{\bullet}$ which is weakly equivalent to the algebra of bundle-valued differential forms ${\cal A}^{\bullet}(M, adP)$ on $M$ so that
${\cal Q}^{\bullet}$ {\em controls} a germ which is manifestly a
quasi-homogeneous cone with the required weights.

In Figure \ref{Fig9} we describe the graph of an Artin group $G^a_A$ which
admits a representation with finite image $alg(\psi_0)= \rho_0: G^a_A\to PO(3,\mathbb C)$ such that the germ $(X(G^a_A, PO(3,\mathbb C)), [\rho_0])$
is isomorphic to the germ at $0$ defined by $x^5=0$. Thus Theorem
\ref{t7} implies that the group $G^a_A$ is not the fundamental
group of a smooth complex algebraic variety.

\begin{rem}
Our convention for Coxeter graphs is different from the standard convention
for Dynkin diagrams. Namely, if two vertices are not connected by an edge
{\bf it does not mean that corresponding generators commute}. If on our
diagram an edge has no label, we assume that the edge has the label $2$.
On the diagram for a {\bf Shephard group} if a vertex has no label this
means that the corresponding generator has {\bf infinite order}.
\end{rem}

There is a local cross-section to the $PO(3)$-orbit through the
representation $\rho_0$ (that appears is Theorem \ref{t6}),
hence we apply Theorem \ref{t7} and conclude that $G^a_A$ is not the
fundamental group of a smooth complex algebraic variety. To see that
there are infinitely many distinct examples we may proceed as follows.

Take  the varieties $V_p:= \{x^p=0\}$, $p\ge 2$ are prime numbers.
Clearly $V_p$ is not analytically isomorphic to $V_q$ for $q\ne p$.
Thus for all
$p\ge 5$ the varieties $V_p$ are not analytically  isomorphic to
quasi-homogeneous varieties described in the Theorem \ref{t7}.
Hence the Artin groups  $G^a_{A_p}$
corresponding to $V_p$ are not fundamental groups of smooth
complex quasi-projective varieties.  Note
that among the groups $G^a_{A_p}$ we have infinitely many ones
which are not  mutually isomorphic. The reason is that for any
finitely-generated group $\Gamma$ the character
 variety $X(\Gamma, PO(3, \mathbb C))$ has only finite number of isolated
singular points   whose germs are isomorphic to one of $V_p$.
This proves Theorem \ref{moremorgan}.

In \mathbb S \ref{triv} we use the results of \mathbb S \ref{completion} to show that
for every Artin group $\Gamma$ and Lie group $G$ the germ
$$
(\mbox{\rm Hom}(\Gamma, G), \rho)$$
is quadratic where $\rho$ is the trivial representation.

\medskip
{\bf Acknowledgements.} The first author is grateful to A.\ Vershik
for a lecture on Mn\"ev's result
in 1989.  The authors are grateful to E.~Bierstone, J.~Carlson, R.~Hain,
J.~Kollar, P.~Millman, C.~Simpson  and D.~Toledo for helpful conversations.

\tableofcontents

\section{Singularities}
\label{singu}

Let $\hbox{\bf k}$ be a field of zero characteristic.  Let $V$ be a variety
defined over the field {\bf k}, $o \in V$ be a point and
$\widehat{{\cal O}_{V, o}}$ the complete local ring. We denote by
$$
T^m _o (V)= \mbox{\rm Hom}_{ {\bf k}- alg}(\widehat{{\cal O}_{V, o}}~,
{\bf k}[t]/t^{m+1})$$
the  $m$-th order Zariski tangent space at $o \in V$ and by
$\pi_{mn} : T^m _o \to T_o^n (V)$ the natural projection ($m\ge n\ge 1$).
We will think of elements in $T^m_o$ as formal curves of
degree $m$ which are tangent to $V$ at $o$ up to the order $m$.
If $(V, o)$, $(W, p)$ are two germs as above and $f: (V, o)\to (W, p)$
is an analytic isomorphism of germs then it induces an
isomorphism between the corresponding
towers of Zariski tangent spaces of finite order

\[
\begin{array}{ccc}
T^m _o (V) & \longrightarrow &  T^m _p (W) \\
\pi_{mn}\Big\downarrow & ~ &  \pi_{mn}\Big\downarrow\\
T^n_o (V)  & \longrightarrow &  T^n_p (W) \\
\end{array}
\]

\begin{lem}
\label{LL3.1}
Suppose that $f: (V, o)\to (W,p)$ is a morphism of analytic germs such that
the derivatives $D_o ^{(m)}f: T^m _o (V) \to T^m _p (W)$ are isomorphisms
for all $m$. Then $f$ is an isomorphism of germs.
\end{lem}
\par\medskip\noindent{\it Proof: } Let $R:= \widehat{O_{V,o}}$ and $S:= \widehat{O_{W,p}}$ be the
complete local rings and let ${\mathfrak m}_R$ and ${\mathfrak m}_S$ be their maximal ideals.
By Artin's theorem (see \cite[Theorem 3.1]{GM}) it suffices to prove that
the induced map $f^*: S\to R$ is an isomorphism. We observe that
$D_o ^{(n+1)}f$ induces an isomorphism from $\ker(\pi_{n+1,n}:
T^{n+1}_o(V)\to T^n_o(V))$ to
 $\ker(\pi_{n+1,n}: T^{n+1}_p(W)\to T^n_p(W))$. It is clear that the above
kernels are canonically isomorphic to the dual vector spaces
$({\mathfrak m}^n_R/{\mathfrak m}_R^{n+1})^*$ and $({\mathfrak m}^n_S/{\mathfrak m}_S^{n+1})^*$. Hence $f^*$ induces an
isomorphism
$$
Gr_n(f^*): {\mathfrak m}^n_R/{\mathfrak m}_R^{n+1}\longrightarrow {\mathfrak m}^n_S/{\mathfrak m}_S^{n+1}
$$
and consequently the induced map $Gr(f^*)$ of the associated graded rings is
an isomorphism. The Lemma follows from \cite[Lemma 10.23]{AM}. $\qed$

\begin{rem}
Suppose that $(Z, 0), (W, 0)$ are {\bf minimal} germs of
varieties in ${\mathbb A}^n$ (i.e. $\hbox{\bf k}^n$ equals to the both Zariski tangent spaces
$T_0(Z)$ and $T_0(W)$), and these germs are analytically
isomorphic. Then there is an analytic diffeomorphism $f: {\mathbb A}^n\to
{\mathbb A}^n$ whose restriction to $Z$ induces an isomorphism of germs
$(Z, 0)\to (W, 0)$. See for instance \cite[Proposition 3.16]{Dimca}.
\end{rem}

\medskip
Suppose that we have a
collection of polynomials $F= (f_1, ..., f_m)$ in $\hbox{\bf k}^n$, we assume that
all these polynomials have trivial linear parts. The
polynomial $f_j$ is said to be {\em weighted homogeneous} if there
is a collection of positive integers (weights) $w_1> 0,..., w_n >0$ and
a number $u_j\ge 0$ so that
$$
f_j((x_1 t^{w_1}),..., (x_n t^{w_n}))= t^{u_j} f_j(x_1,..., x_n)
$$
for all $t\in\hbox{\bf k}$. We will call the numbers $w_i$ the {\em weights of
generators} and the numbers $u_j$  the {\em weigths of relations}.
Let $Y$ denote the variety given by the system of equations
$$
\{ f_1 =0, ...., f_m =0\}
$$
(Note that the germ $(Y,0)$  is necessarily minimal.)
We say that $(Y,0)$ is a {\em quasi-homogeneous} if we can choose
generators $f_1,..., f_m$ for its defining ideal such that all the
polynomials $f_j$ are weighted homogeneous with the same weights
$w_1,..., w_n$ (we do not require $u_j$ to be equal for distinct
$j=1,..., m$). In particular, if $(Y,0)$ is a {\em quasi-homogeneous}
then $(Y,0)$ is invariant under the $\hbox{\bf k}^{\times}$-action on $\hbox{\bf k}^n$
given by the weights $w_1,..., w_n$.

\begin{rem}
The variety $Y$ given by a system of quais-homogeneous equations
$\{ f_j=0\}$ is also called a {\bf quasi-homogeneous}
(or {\bf weighted homogeneous}) cone.
\end{rem}

We now give an intrinsic characterization of quasi-homogeneous\  germs. Suppose that
$(Y,0)$ is quasi-homogeneous\ . Let $S_m\subset \hbox{\bf k}[x_1,..., x_n]$ be the subspace of
polynomials which are homogeneous of degree $m$ (in the usual sense).
We may decompose the subspace  $S_m$ into one dimensional eigenspaces
under $\hbox{\bf k}^{\times}$ (since the multiplicative group of a field is a
reductive algebraic group). We obtain a bigrading
\[
\hbox{\bf k}[x_1,..., x_n]= \oplus_{m,n} S_{m,n}
\]
where $m$ is the degree and $n$ is the weight of polynomial under
$\hbox{\bf k}^{\times}$ ($f$ transforms to $t^n f$). We obtain  a new grading of
$\hbox{\bf k}[x_1,..., x_n]$ by weight
\[
\hbox{\bf k}[x_1,..., x_n]= \oplus_{n=1}^{\infty} S_n'
\]
where $S_n'$ is the subspace of polynomials of weight $n$. We let $I$ be
the ideal of $Y$. Then $I$ is invariant under the action of
$\hbox{\bf k}^{\times}$ (since its generators are). We claim
$$
I= \oplus_{n=1}^{\infty} I\cap S_n'
$$
This follows by decomposing the action of $\hbox{\bf k}^{\times}$ in the
finite dimensional subspaces $I \cap \oplus_{m=1}^N S_m$. Thus if
$f\in I$ we may write
$$
f= \sum_{n=1}^{\infty} f_n\ , \quad \hbox{with~~} f_n\in I\cap S_n'
$$
(the sum is of course finite). Let $R= \hbox{\bf k}[Y]= \hbox{\bf k}[x_1,.... x_n]/I$.
Then $R$ is a graded ring, $R= \oplus_{n=1}^{\infty} R_n$ with $R_0=\hbox{\bf k}$,
$R_n= S_n'/(I\cap S_n')$.

We let $\hat{R}$ be the  completion of $R$ at ${\mathfrak m}$ where ${\mathfrak m}$ is the
ideal of zero, i.e. the ideal generated by $\{x_1,..., x_n\}$. Hence
$$
\hat{R}\cong \hbox{\bf k}[[x_1,..., x_n]]/\hat{I}
$$
where $\hat{I}$ is the ideal generated by $I$ in $\hbox{\bf k}[[x_1,..., x_n]]$.
Hence $\hat{R}= \widehat{O_{Y,o}}$. The ring $\hat{R}$ is not graded but
it has a decreasing filtration $W^{\bullet}$ such that $W^N \hat{R}$ is the
closure of  $\oplus_{n=N}^{\infty} S_n'$. Define $Gr^W_{n}( \hat{R}) = W^{n}(\hat{R})/W^{n+1}(\hat{R})$.
The filtration $W^{\bullet}$ satisfies:

(i) $Gr^W_0 = \hbox{\bf k}$.

(ii) $\cap_{n=0}^{\infty}W^n = 0$.

(iii) $\dim_{\hbox{\bf k}} Gr^W_n (\hat{R})  < \infty$ for all $n$.

\noindent The inclusion $R\hookrightarrow \widehat{R}$ induces an
isomorphism $R\cong Gr^W(\hat{R})$ and we obtain:

\begin{lem}
\label{crit-qh}
(a) $\widehat{O_{Y,o}}$ admits a decreasing filtration
$W^{\bullet}$ satisfying the  properties (i)---(iii) above.

(b) There is a monomorphism of filtered rings
$Gr^W(\widehat{O_{Y,o}} )\to
\widehat{O_{Y,o}}$ with dense image, so
$\widehat{O_{Y,o}}$ is the
completion of $Gr^W(\widehat{O_{Y,o}} )$.

(c) Conversely if  $\widehat{O_{Y,o}}$ satisfies (a) and
(b) then $(Y,o)$ is quasi-homogeneous\ .
\end{lem}
\par\medskip\noindent{\it Proof: } It remains to prove (c). Define a $\hbox{\bf k}^{\times}$-action on
$Gr^W\widehat{O_{Y,o}}$ so that the elements in the $n$-graded
summand have weight $n$ for the action of $\hbox{\bf k}^{\times}$. Let ${\mathfrak m}$
be the ideal of $o$. Choose a basis of eigenvectors under $\hbox{\bf k}^{\times}$
action on ${\mathfrak m}/{\mathfrak m}^2$. Lift these vectors to eigenvectors $f_1,..., f_n$ of
${\mathfrak m}$. Then by a standard argument if we set $\pi(x_i)=f_i$  we obtain a
surjection
$$
\pi: \hbox{\bf k}[[x_1,..., x_n]] \longrightarrow \widehat{O_{Y,o}}
$$
which is $\hbox{\bf k}^{\times}$-equivariant. Hence the induced map of graded rings
$$
\pi': \hbox{\bf k}[x_1,..., x_n] \longrightarrow Gr^W \widehat{O_{Y,o}}
$$
is also surjective. Let $I$ be the kernel of $\pi'$ and
let $Y$ be the affine variety corresponding to $I$. $\qed$

\begin{defn}
We will say that a complete local ring $R$ is {\em quasi-homogeneous\  } if it
satisfies (a) and (b) as in Lemma \ref{crit-qh}. We will say a germ
$(Y,o)$ is quasi-homogeneous\  if the complete local ring $\widehat{O_{Y,o}}$ is quasi-homogeneous\ .
\end{defn}

\medskip
Here are several  examples. The polynomial $f(x, y, z)=
x^2 + y^5 + z^3$
is quasi-homogeneous with the weights of generators $15$,
$6$ and $10$ respectively. The weight of the relation is $30$.
 Let $g(x)= x^n$, then
$g$ is quasi-homogeneous for any weight $w$ of the  generator
and the weight $nw$ of the relation.

\medskip
Another example is the  germ $(V_p, 0)= (\{x^p=0\}, 0)$, $p\ge 2$ is
prime. Let's prove that this germ is not quasi-homogeneous for any
weigths of relations $< p$. Indeed, suppose $Y$ is a quasi-homogeneous
cone whose germ at zero is isomorphic to $(V_p, 0)$. Since we assume
that $Y$ is minimal, hence $Y\subset \hbox{\bf k}$. Polynomials defining
$Y$ must be monomials (since $Y$ is quasi-homogeneous). Then $Y$
clearly  can be defined by a single monomial equation $x^m=0$.
Isomorphism of germs $(Y,0)\to (V_p, 0)$ induces isomorphisms of
finite order tangent spaces, hence $m=p$.

\medskip
In a certain sense {\em generic} germs are not
quasi-homogeneous. This is discussed in details in
\cite{Arnold1}, \cite{Arnold2}. Here is one explanation,
in the case of germs in the affine plane ${\mathbb A}^2$, one which doesn't require
knowledge of singularity theory but is based on 3-dimensional
topology.
Suppose that $Y\subset {\mathbb A}^2$ is a minimal affine curve (defined over $\mathbb C$),
which is invariant under weighted action of $\mathbb C^{\times}$ on ${\mathbb A}^2$
(with the weights $w_1$, $w_2$). Then the set of complex points
$$
Y(\mathbb C) \subset \mathbb C^2
$$
is invariant under the $\mathbb C^{\times}$-action with the weights
$w_1$, $w_2$. The corresponding weighted  action of $S^1$
preserves a small sphere $S^3$ around
zero and the link $Y_{\mathbb C} \cap S^3= L$. Thus $S^3- L$ admits a
free $S^1$-action, therefore $S^3- L$ is a Seifert manifold.
{\em Generic} singularities do not have such property, see
\cite{Neu}.  For convenience of the reader we describe a way
to produce examples of singularities which do not admit
$\mathbb C^{\times}$-action. Our discussion follows \cite{Neu}.
Start with a finite ``Piuseaux series''
$$
y= x^{q_1/p_1} (a_1 + x^{q_2/p_1p_2}( a_2 + x^{q_3/p_1p_2 p_3}
(... (a_{s-1} + a_s x^{q_s/p_1... p_s})\ldots )
$$
where  $(p_i, q_i)$ are pairs of positive coprime integers.
The numbers $a_j$ are nonzero integers. Then $y, x$ satisfy
some polynomial equation $f(x, y)=0$ with integer
coefficients, the link $L$ of the singularity at zero
is an {\em iterated torus knot}, the number $s$ is
the depth of iteration, numbers $p_i, q_i$ describe
cabling invariants. The complement $S^3- L$ is not a
Seifert manifold provided that $s \ge 2$. The simplest
example is when $s=2$,
$$
y= x^{q_1/p_1} (a_1 + a_2 x^{q_2/p_1p_2})
$$
For instance take $a_1= a_2= 1$, $p_1= p_2= 2$, $q_1= q_2=3$
(the iterated trefoil knot),  then
$$
y^{2}= x^3 + x^9 + 2 x^6 \quad.
$$
Another example of a singularity which is not quasi-homogeneous\  is
$$
x^2 y^2 + x^5 + y^5 = 0
$$
see \cite[Page 122]{Dimca}.

\section{Analytic isomorphisms of algebraic varieties}

In this section we discuss the following question:

Let $\hbox{\bf k}$ be a field of zero characteristic.
Suppose that $X'\subset X, Y'\subset Y$ are subvarieties
in smooth quasi-projective varieties $X, Y$ over $\hbox{\bf k}$,
$\eta: X\to Y$ is a biregular isomorphism which carries
$X'$ bijectively to $Y'$. Does $\eta$ induce a biregular
isomorphism $X'\to Y'$ ?

Clearly the answer is ``yes'' if both subvarieties $X', Y'$
are reduced. The simple example:
$$
X= Y= \mathbb C, \quad X'= \{z^2=0\}, \quad Y'= \{z^3= 0\} ,
\quad \eta= id
$$
shows that in the nonreduced case we need some extra
assumptions to get the positive answer. Our goal is to
prove that the answer is again positive if we assume
that $\eta$ induces an {\em analytic} isomorphism
between $X', Y'$ (Theorem \ref{cbasic}).

\bigskip
Let $R$ be a ring, $m$ is a maximal ideal in $R$, $R_m$
is the localization of $R$ at $m$ and $\hat{R_m}$ is
the completion of $R$ at $m$.

\begin{lem}
\label{basic}
Suppose $R$ is a Noetherian ring and $f\in R$ has the
property that its image in $\widehat{R_m}$ is zero for all
maximal ideals $m$. Then $f= 0$.
\end{lem}
\par\medskip\noindent{\it Proof: } By Krull's theorem \cite[Corollary 10.19]{AM},
the induced map $R_m \to
\widehat{R_m}$ is an injection. Thus the image of $f$ in
$R_m$ is zero for all maximal ideals $m$. Hence for
every such $m$ there exists $s\notin m$ with $sf=0$.
Therefore $Ann(f)$ is contained in no maximal ideal.
This implies that $Ann(f)= R$ and $f=0$. $\qed$

\begin{lem}
\label{lbas}
Let $\phi: R\to S$ be a ring homomorphism and
$I\subset R, J\subset S$ be ideals. Suppose that
for every maximal ideal $m$ in $S$ with $m\supset J$
we have
$$
\phi(I)\otimes \widehat{S_m} \subset J\otimes \widehat{S_m}
$$
Then $\phi(I)\subset J$.
\end{lem}
\par\medskip\noindent{\it Proof: } It suffices to prove this when $\phi(I)$ is
replaced by an element $f$ of $S$. Thus we assume
that the image of $f$ in $S_m$ is contained in
$J\otimes S_m$ for all maximal ideals $m\subset S$.
We want to conclude that $f\in J$. We further simplify
the situation by dividing by $J$. We have an exact sequence
$$
0\to J \to S\to S/J \to 0
$$
We use \cite[Proposition 10.15]{AM}  to conclude that
$$
\widehat{S/J_{m/J}} \cong S/J \otimes \widehat{S_m}
$$
and \cite[Proposition 10.14]{AM} to conclude
$$
S/J \otimes \widehat{S_m}\cong \widehat{S_m}/(J\otimes \widehat{S_m})
$$
Replace $f$ by its image in $S/J$. We find that the image
of $f$ in all completions of $S/J$ at maximal ideals of
$S/J$ is zero. Then $f$ is zero by the Lemma \ref{basic}.
$\qed$

\begin{thm}
\label{cbasic}
Suppose that $X, Y$ are nonsingular (quasi-)
projective varieties over $\hbox{\bf k}$ and $\eta: X\to Y$ is an isomorphism.
Let $X'\subset X, Y'\subset Y$ are subvarieties so that:
$\eta':= \eta|_{X'}: X'\to Y'$ is a bijection which is an
analytic isomorphism.  Then $\eta': X'\to Y'$ is a
biregular isomorphism.
\end{thm}
\par\medskip\noindent{\it Proof: } It is enough to check the assertion on  open subsets,
so we may as well assume that $X, Y$ are affine with coordinate
rings $S, R$, affine subvarieties $Y', X'$ are given by ideals
$I\subset R, J\subset S$. Coordinate rings of $Y', X'$ are
$R/I, S/J$. Let $m$ be a maximal ideal in $R/I$, then there
is a maximal ideal $M\subset R$ such that $m= M/I$. Thus
$$
\widehat{R/I_m} \cong \frac{\widehat{R_{M}}}{I\otimes
\widehat{R_{M}} }
$$
Let $\phi: R\to S$ be the  isomorphism induced by
$\eta: X\to Y$.
Since $\eta'$ is an analytic isomorphism it induces
isomorphisms of all completions
$$
\frac{\widehat{R_{M}}}{I\otimes  \widehat{R_{M}} } \longrightarrow
\frac{\widehat{R_{\phi(M)}}}{J\otimes \widehat{R_{M}} }
$$
Thus the assertion of Theorem follows from Lemma \ref{lbas}.  $\qed$

\section{Coxeter, Shephard and Artin groups}
\label{groups}

Let $\Lambda$ be a finite graph where two vertices are connected by at most one
edge, there are no loops (i.e. no vertex is connected by an edge to itself)
and each edge $e$ is assigned an integer
$\epsilon(e)\ge 2$. We call $\Lambda$ a {\em labelled} graph, let ${\cal V}(\Lambda)$
and ${\cal E}(\Lambda)$ denote the sets of vertices and edges of $\Lambda$.
When drawing $\Lambda$ we will omit labels $2$
from the edges (since in our examples most of the labels are $2$).
Given $\Lambda$ we construct two finitely-presented
 groups corresponding to it. The first group $G_{\Lambda}^c$ is called the
 {\em Coxeter group} with the {\em Coxeter graph} $\Lambda$, the second is the
{\em Artin group} $G_{\Lambda}^a$.
The sets of generators for the both groups are $\{ g_v, v\in {\cal V}(\Lambda)\}$.
  Relations in $G_{\Lambda}^c$ are:
$$
g_v^2= 1, v\in {\cal V}(\Lambda), \ (g_{v} g_{w})^{\epsilon(e)}= \hbox{\bf 1},
\hbox{~~over all edges~~} e= [v, w]\in {\cal E}(\Lambda)
$$
 Relations in $G_{\Lambda}^a$ are:
$$
\underbrace{g_{v} g_{w} g_v g_w ... }_{\epsilon\hbox{~multiples~}}=
\underbrace{g_{w} g_{v} g_{w} g_v ... }_{\epsilon\hbox{~multiples~}}\ , \quad
\epsilon= \epsilon(e) \hbox{,~~over all edges~~} e= [v, w]\in {\cal E}(\Lambda)
$$
We let $\epsilon(v, w)= \epsilon([v,w])$ if $v, w$ are connected by the edge
$[v, w]$ and $\epsilon(v, w)= \infty$ if $v, w$ are not connected by any edge.
For instance, if we have an edge $[v, w]$ with the label $4$, then the Artin
relation is
$$
g_{v} g_{w} g_v g_w = g_{w} g_{v} g_{w} g_v
$$
Note that there is an obvious epimorphism
$G_{\Lambda}^a \longrightarrow G_{\Lambda}^c$.
We call the groups $G_{\Lambda}^c$ and $G_{\Lambda}^a$ {\em associated}
with each other.
Artin groups above appear as generalizations of the Artin braid group.
 Each Coxeter group $G_{\Lambda}^c$ admits a canonical discrete faithful linear
 representation
$$h: G_{\Lambda}^c\longrightarrow GL(n, \mathbb R)\subset GL(n, \mathbb C)$$
where $n$ is the number of vertices in $\Lambda$. Suppose that the Coxeter
 group $G_{\Lambda}^c$ is finite, then remove from $\mathbb C^n$ the collection
 of fixed points of elements of $h( G_{\Lambda}^c- \{\hbox{\bf 1}\})$ and denote the
resulting
complement $X_{\Lambda}$. The group $G_{\Lambda}^c$ acts freely on $X_{\Lambda}$ and
the quotient $X_{\Lambda}/G_{\Lambda}^c$ is a smooth complex quasi-projective variety
with the fundamental group $G_{\Lambda}^a$, see \cite{Bri} for details. Thus the
Artin group associated to  a finite Coxeter group is  the fundamental
group of a smooth complex quasi-projective variety.

\medskip
The construction of Coxeter and Artin groups can be generalized as follows.
Suppose that not only edges of $\Lambda$, but also  its vertices $v_j$
have labels $\delta_j= \delta(v_j)\in \{0, 2, 3,...\}$.
Then take the presentation of the
Artin group $G_{\Lambda}^a$ and add the relations:
$$
g_{v}^{\delta(v)}= \hbox{\bf 1} , \quad v\in {\cal V}(\Lambda)
$$
If $\delta(v)=2$ for all vertices $v$ then we get the Coxeter group, in
general the resulting group is called the {\em Shephard group}, they were
introduced by Shephard in \cite{Sh}. Again there is a canonical epimorphism
$G^a_{\Lambda}\to G^s_{\Lambda}$.

Given a Coxeter, Artin or Shephard group $G$ associated with
the graph $\Lambda$ we define {\em vertex} subgroups
$G_v, v\in {\cal V}(\Lambda)$ and {\em edge} subgroups
$G_e= G_{vw}, e= [v,w]\in {\cal E}(\Lambda)$ as subgroups generated
by the elements: $g_v$ (in the case of the vertex subgroup)
and $g_v, g_w$ (in the case of the edge subgroup).

\medskip
We will use the fact that several of Shephard groups are
finite. Here is the list of the finite groups that we will
use (see \cite{Coxeter}):

\begin{figure}[h]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure28.eps}}
\caption{\sl Graphs for certain finite Shephard groups.}
\label{Fig28}
\end{figure}

Some of the Coxeter groups above admit extensions to {\em extended Artin}
groups, which we describe below. Suppose that $\epsilon(e)\in \{2, 4,  \infty \}$.
Enumerate vertices of $\Lambda$.
Then define a generalized Cartan matrix $N$ as follows. Diagonal
entries of $N$ are equal to   $2$. Now consider off-diagonal
entries $n_{ij}$ of $N$ assuming $i< j$.
If $\epsilon(v_i, v_j)= 2$ we let $n_{ij}= n_{ji}= 0$.
If $\epsilon(v_i, v_j)= 4$ we let $n_{ij}= -1, n_{ji}= -2$. Finally, if
$\epsilon(v_i, v_j)= \infty$ we let $n_{ij}= n_{ji}= -2$.

\begin{rem}
More generally one is also allowed to consider the case $\epsilon(e)= 3, 6$.
\end{rem}

Under these conditions there is a {\em generalized root system} associated
to $N$ and one constructs the {\em extended Artin group} $G_{\Lambda}^e$
as follows (\cite{Lo}, \cite{Le}):

Let $m$ denote the number of vertices in $\Lambda$.
Let $\tau_j, 1\le j\le m,$ denote generators of $\mathbb Z^m$ and
$g_i, 1\le i\le m,$  denote generators of $G^a$. In the
free product $G^a * \mathbb Z^m$ introduce the following relations:

(a) Suppose that $-n_{ji}= 2r$, then introduce the relations:
$$
g_i \tau_j = \tau_j \tau_i^r g_i \tau_i^{-r}
$$

(b) Suppose that $-n_{ji}= 2r +1$, then introduce the relations:
$$
g_i \tau_j = \tau_j \tau_i^{r+1} g_i^{-1} \tau_i^{-r}
$$
Note that for edges $[v_i, v_j]$ labelled $2$ we get $n_{ij}= n_{ji}=0$ and
$$
[g_i, \tau_j] =\hbox{\bf 1}, \quad [g_j, \tau_i] =\hbox{\bf 1}
$$
The resulting group is $G^e_{\Lambda}$.

The extended Artin groups arise naturally as fundamental groups of
certain Stein manifolds $M(G^e)$,  which are  complements of discriminants of
moduli  spaces of versal deformations of certain
singularities ({\em simply elliptic} and {\em cusp} singularities), see
 \cite{Lo}, \cite{Le}.
Below is an outline of the construction. Let $G^c$ be the
Coxeter group associated with $G^a$, $h: G^c\to GL(n, \mathbb R)$
be its canonical (discrete and faithful) representation.
The group $h(G^c)$ acts properly discontinuously on a
convex cone $C\subset \mathbb R^n$, which is the ``maximal'' domain
of discontinuity of $h(G^c)$. Complexify this action. The group $G^c$ acts
on $\mathbb Z^n\subset \mathbb R^n$. Identify the $G^c$-module $\mathbb Z^n \otimes \mathbb R$ with the
``imaginary part'' of $\mathbb C^n$. Thus we get a properly
discontinuous action of $G:= \mathbb Z^n \ltimes G^c$ on
$W:= V \times (\mathbb Z^n \otimes \mathbb R) \subset \mathbb C^n$. Take the
quotient $W/G$ and remove from it projections of fixed-points
of elements of $G- \{\hbox{\bf 1}\}$. The result is the manifold
$M(G^e)$.

\section{Local deformation theory of representations}
\label{defof}
Let $\hbox{\bf k}$ be a field of zero characteristic, define the Artin local
$\hbox{\bf k}$-algebra  $\Pi_m$ as $\hbox{\bf k}[t]/t^{m+1}$. Take a finitely-generated
group $\Gamma$ and
an algebraic  Lie group $G$ over $\hbox{\bf k}$, let $\hbox{\bf G}:= G(\hbox{\bf k})$ be the
set of $\hbox{\bf k}$-points
of $G$. Then the Lie group $G(\Pi_m)$ splits as the semidirect product
$G_0(\Pi_m)\rtimes \hbox{\bf G}$, where $G_0(\Pi_m)$ is the kernel of the natural
projection $p_m: G(\Pi_m)\to \hbox{\bf G}$ induced by $\Pi_m \to \hbox{\bf k}$, which is given
by $t\mapsto 0$. The group $G_0(\Pi_m)$ is $\hbox{\bf k}$-unipotent. We consider the
germ
$$
(Hom(\Gamma, \hbox{\bf G}), \rho)
$$
for a certain representation $\rho: \Gamma \to \hbox{\bf G}$. Then $m$-th order
Zariski tangent space \newline
$T^m_{\rho}( \mbox{\rm Hom}(\Gamma, \hbox{\bf G}))$ is naturally isomorphic
to the space of homomorphisms
$$
Z^1_{(m)}(\Gamma, \hbox{\bf G}; \rho):=
\{ \xi: \Gamma \to G(\Pi_m) | p_m \circ\xi = \rho\}
$$
(see \cite{GM}). If $m=1$ then $Z^1_{(m)}(\Gamma, \hbox{\bf G}; \rho)\cong
Z^1(\Gamma, ad\circ\rho)$, where $ad$ is the adjoint action of $\hbox{\bf G}$ on
its Lie algebra.

We call $\xi\in T^m_{\rho}( \mbox{\rm Hom}(\Gamma, \hbox{\bf G}))$ a {\em trivial
deformation} of $\rho$ if there is an element
$h= h(\xi)\in G(\Pi_m)$ such that
$\xi(g)= h \rho(g) h^{-1}$ for all $g\in \Gamma$. If $m=1$ then
the infinitesimal deformation $\xi$ is trivial iff the
corresponding cocycle is a coboundary.

Suppose that $G^s$ is a Shephard group, $G^a$ is the corresponding
Artin group, $q : G^a \to G^s$ is the canonical projection.
Let $V_L$ denote the set of vertices with nonzero
labels  in the graph of $G^s$. Let $\hbox{\bf G}$ be a group of $\hbox{\bf k}$-points
of an algebraic Lie group.  Consider
a homomorphism $\rho: G^s\to \hbox{\bf G}$, let $\tilde\rho= \rho \circ q$. The
projection  $q$ induces an injective
 morphism of the representation varieties
$$
q^* : \mbox{\rm Hom}(G^s, \hbox{\bf G})\longrightarrow \mbox{\rm Hom}(G^a, \hbox{\bf G})
$$
and injective morphisms of the corresponding finite order Zariski
tangent spaces
$$
D^{(n)} q^*: T^n_{\rho} \mbox{\rm Hom}(G^s, \hbox{\bf G})\longrightarrow T^n_{\tilde\rho} \mbox{\rm Hom}(G^a, \hbox{\bf G})
$$

\begin{lem}
\label{L7.1}
Suppose that $\xi\in T^n_{\tilde\rho} \mbox{\rm Hom}(G^a, \hbox{\bf G})$ is an element whose
restriction to each cyclic vertex subgroup $G_v^a, v\in V_L$
is a {\bf trivial deformation} of $\tilde\rho|_{G_v^a}$. Then $\xi$
belongs to the image of  $D^{(n)} q^*$.
\end{lem}
\par\medskip\noindent{\it Proof: } Recall that we identify $T^n_{\tilde\rho} \mbox{\rm Hom}(G^a, \hbox{\bf G})$ and
$T^n_{\rho} \mbox{\rm Hom}(G^s, \hbox{\bf G})$ with spaces of certain representations of
$G^a, G^s$ into $G(\Pi_n)$. We have the exact
sequence
$$
\hbox{\bf 1}\longrightarrow \<\< \{ g_v^{\delta(v)}, v\in V_L\}\>\> \longrightarrow G^a \longrightarrow G^s \longrightarrow \hbox{\bf 1}
$$
Then $\xi$ belongs
to the image of $D^{(n)} q^*$ if and only if $\xi(g_v^{\delta(v)})=\hbox{\bf 1}$ for
all $v\in V_L$. We assume that the restriction of $\xi$ to $G_v^a$ is
a trivial deformation, thus there is an element $h:= h_{v}\in \hbox{\bf G}(\Pi_n)$
such that $\xi(g_v)= h \tilde\rho(g_v) h^{-1}$. Since $\tilde\rho(g_v^{\delta(v)})=\hbox{\bf 1}$
we conclude that  $\xi(g_v^{\delta(v)})=\hbox{\bf 1}$ as well. $\qed$

\begin{cor}
\label{CC}
Suppose that  $H^1(G^s, ad \circ\rho)=0$ (i.e. the representation $\rho$ is
infinitesimally rigid) and for each $v\in V_L$ the restriction
homomorphism
$$
Res_v: \mbox{\rm H}^ 1(G^a, ad\circ\rho)\to \mbox{\rm H}^ 1(G_v^a, ad\circ \rho)
$$
is zero. Then
$$
q^* : (\mbox{\rm Hom}(G^s, \hbox{\bf G}), \rho) \longrightarrow (\mbox{\rm Hom}(G^a, \hbox{\bf G}), \tilde\rho)
$$
is an analytic isomorphism of germs.
\end{cor}
\par\medskip\noindent{\it Proof: } Let $Z(\rho)$ denote the centralizer of $\rho(G^s)$ if $\hbox{\bf G}$. Then
the representation variety $\mbox{\rm Hom}(G^s, \hbox{\bf G})$ is smooth near $\rho$ and is
naturally isomorphic to the quotient $\hbox{\bf G}/Z(\rho)$, see \cite[Theorem 2.6]{LM}. Now the assertion follows from Lemmas \ref{LL3.1}, \ref{L7.1}.  $\qed$

One example when the first condition of the corollary are satisfied is
the case when the Shephard group $G^s$ is finite.

\section{Projective reflections}
\label{S3}

Fix the bilinear form $\flat= x_1 y_1 + x_2 y_2 + x_3 y_3=
(x_1, x_2, x_3) \cdot (y_1, y_2, y_3)$ on the vector
space $V= \mathbb C^3$, we shall also use the notation
$\< \cdot , \cdot \>$ for $\flat$.
Let $\varphi$ denote  the quadratic form corresponding to $\flat$,
let $O(3,\mathbb C)$ be the group of automorphisms
of $\flat$. Let $\pi: V \to \mathbb P(V)$ denote the
quotient map and let
$CO(3, \mathbb C):= \mathbb C^{\times}\cdot O(3,\mathbb C) \subset End(V)$. The inclusion
$CO(3,\mathbb C) \to End(V)$ induces an embedding $PO(3,\mathbb C) \hookrightarrow \mathbb P(End~V)$. Let
$p: End(V)  \to \mathbb P(End~V)$ be the quotient map. Note that
$PSL(2, \mathbb C)\cong PO(3, \mathbb C)$ and $SO(3,\mathbb R)\cong PO(3, \mathbb R)$.

\subsection{The correspondence between  projective reflections and their
fixed points}

In this section we study projective properties of elements of order $2$ in
the group $PO(3, \mathbb C)$. Consider an element $A\in O(3, \mathbb C)$ such that the
projectivization $p(A)$ is an involution acting on $\mathbb P^2(\mathbb C)$. The
fixed-point set of $p(A)$ consists of two components: an isolated point
$a$ and a projective line  $l$ dual to $a$ (with respect to $\flat$).
Our goal is to describe the
correspondence $p(A)\leftrightarrow a$ in algebraic terms.

Let $R\subset SO(3,\mathbb C)$ be the affine subvariety of involutions. Note that
$-\hbox{\bf 1}\in O(3,\mathbb C)$ doesn't belong to $R$.
We leave the proof of the following lemma to the reader.

\begin{lem}
$PO(3, \mathbb C)$ acts transitively by conjugations
on $PR(\mathbb C)$ (the image of $R(\mathbb C)$ in $PO(3, \mathbb C)$).
\end{lem}

We now determine $\overline{p(R)}$, the Zariski closure of $p(R)$ in
$\mathbb P(End~V)$. We define a morphism $\eta: V \to End(V)$ by $\eta(v)(x)=
\varphi(v)x - 2\<v, x\> v$.
If $v$ is an anisotropic vector then $\eta(v)$ is a multiple
of the reflection through the hyperplane in $V$ orthogonal to $v$.
The reader will verify that $\eta$ is an $O(3)$-equivariant morphism, i.e.
$\eta(g v)= g \eta(v) g^{-1}$, $g\in O(3), v\in V$, and that $\eta$ induces
an equivariant embedding
$$
\eta: \mathbb P(V) \longrightarrow \mathbb P(End~V)
$$
which is an immersion of one smooth complex manifold into another.

\begin{lem}
The image of $\eta$ is contained in $\overline{p(R)}$.
\end{lem}
\par\medskip\noindent{\it Proof: } Let $V_0$ be the complement in $V$ of $X= \{ v \in V: \varphi(v)=0\}$.
Then $\mathbb P_0(V):= \pi(V_0)$ is Zariski dense in $\mathbb P(V)$. But also
$\eta(V_0)\subset \mathbb C^{\times} R$. Hence
$\overline{\mathbb P_0(V)} \subset \overline{p(R)}$. $\qed$

We now consider $\mathbb P(V)$ and $\mathbb P(End(V)$ as varieties over $\mathbb Q$.

\begin{lem}
\label{fixed point}
The morphism $\eta$ induces an isomorphism of varieties $\mathbb P(V) \cong \overline{p(R)}$.
\end{lem}
\par\medskip\noindent{\it Proof: } Since $\eta: \pi(V_0)\to p(R)$ is equivariant it is easy to verify
that it is onto (in fact a bijection). Hence $\eta(\pi(V_0))= p(R)$ and
accordingly
$\overline{\eta\pi(V_0)} = \overline{p(R)}= \eta(\mathbb P(V))$. But we have seen
that the morphism $\eta: \mathbb P(V)\to \eta(\mathbb P(V))$ is an analytic isomorphism of
smooth compact complex manifolds. Hence (by the GAGA-principle)
it is an isomorphism of projective varieties. $\qed$

\medskip
Let $N:= p^{-1} (\overline{p(R)})- p^{-1} (p(R))$. Then $N$ may be
described as follows. The bilinear form $\flat$ induces an
isomorphism $\tilde\flat: V\otimes V \to V^* \otimes V = End(V)$. Then
$N= \tilde\flat(X \otimes X)$. We note that $n_v := \tilde\flat(v \otimes v)$
is then given by $n_v(x)= \<v, x\> v$. Hence the set of real points
$N(\mathbb R)$ is empty.

Let $Q\subset End(V)$ be the affine cone defined by
$Q:= p^{-1} (\overline{p(R)})$. Hence $N \subset Q$. We define
$Q_0:= Q- N$. Then $\eta$ induces a commutative diagram
$$
\begin{array}{ccc}
V & \longrightarrow & Q  \\
\Big\uparrow & ~ & \Big\uparrow\\
V_0 & \longrightarrow & Q_0\\
\end{array}
$$

\begin{rem}
It can be shown that the cone $Q\subset End(V)$ is defined by the equations:
\begin{enumerate}
\item $X X^{\top} = X^{\top} X$,
\item $X X^{\top} E_{ij} = E_{ij}  X X^{\top}$, $1\le i, j\le 3$,
\item $X^2 E_{ij} = E_{ij} X^2$, $1\le i, j\le 3$.
\end{enumerate}
Here $E_{ij}$ is the matrix with $1$ in the $(ij)$-th position and $0$
elsewhere. The equations (1) and (2) define the closure
$\overline{PO(3)} \subset \mathbb P(End~V)$. We will not need the explicit
equations for $\overline{p(R)}$ in what follows.
\end{rem}

We let $\zeta: PQ \to \mathbb P^2$ be the inverse of $\eta$ and abbreviate
$\mathbb P(V_0)$ to $\mathbb P^2_0$. Note that \newline
$\zeta: PR = PQ_0 \to \mathbb P^2_0$\  assigns
to each projective reflection its neutral (isolated)
fixed-point. Thus we have described the correspondence
$p(A) \leftrightarrow a$ algebraically. Note that we have
$\mathbb P^2(\mathbb R)= \mathbb P^2_0(\mathbb R)$ and $PQ(\mathbb R)= PQ_0(\mathbb R)$. Let
$\hat\flat: V \to V^*$ be the isomorphism induced by $\flat$.
Define $(\mathbb P^2_0)^{\vee}$ by
$(\mathbb P^2_0)^{\vee}= \hat\flat(\mathbb P^2_0)$.
Hence the space of {\em anisotropic lines}
$(\mathbb P^2_0)^{\vee}$ is the space of lines dual to the set of
{\em anisotropic points}  $\mathbb P^2_0$.

\subsection{Fixed points}

Suppose that $g\in PO(3, \mathbb C)$ is a nontrivial element.
In this section we discuss the fixed-point set for the
action of $g$ on $\mathbb P^2(\mathbb C)$.

\begin{defn}
A fixed point $x$ for the action of $g$ on $\mathbb P^2(\mathbb C)$ is called {\bf neutral}
if the determinant of the differential of $g$ at $x$ is equal to 1.
\end{defn}

There are two classes of nontrivial elements $g\in PO(3, \mathbb C)$:
(a) $g$ is unipotent, (b) $g$ is semi-simple.

\medskip
{\bf Case (a).} In this case $g$ has a single fixed
point $a\in \mathbb P^2$, the point $a$ belongs to an
invariant projective line $L\subset \mathbb P^2$. If we
choose coordinates on $L$ such that $a= \infty$, then
$g$ acts on $L- \{a\}$ as a translation.  The flag
$(L, a)$ is determined by the element $g$ uniquely.
On the other hand, the flag $(L, a)$ uniquely
determines the 1-parameter maximal unipotent
subgroup in $PO(3)$ which contains $g$. Finally,
the fixed point $a$ of $g$  uniquely determines
the line $L$. It is easy to see that $a$ is the {\em neutral}
fixed point of $g$.

The collection $P\subset \mathbb P^2$ of fixed points of all
unipotent elements in $PO(3,\mathbb C)$ is the projectivization of the
cone $\{ \overrightarrow{x}\in \mathbb C^3 : \varphi(\overrightarrow{x})= 0\}$.

\medskip
{\bf Case (b).} Each semisimple element of $O(3)$
is conjugate (in $GL(3, \mathbb C)$) to
$$
A= \left( \matrix  \pm \lambda & 0 & 0\\
0 & \lambda^{-1} & 0\\
0 & 0 & 1\endmatrix \right).
$$
(up to $\pm$). We have two possible cases depending on whether
or not $A^2=\hbox{\bf 1}$. If $A^2\ne \hbox{\bf 1}$ then $A$ has three distinct
complex eigenvectors, one of them $\overrightarrow{f}$
 is fixed by $A$. Thus, in the case the
transformation $p(A)$ has three fixed points on $\mathbb P^2(\mathbb C)$,
one of them $f= p(\overrightarrow{f})$ is the {\em neutral} fixed point.
The maximal torus in $PO(3, \mathbb C)$
containing $p(A)$ is uniquely determined by the (neutral)
fixed point $f$ in the both real and complex cases.

Finally, consider the case $A^2= \hbox{\bf 1}$. Then  $p(A)$ has one
isolated fixed point $f$ on $\mathbb P^2(\mathbb C)$ (which is
{\em neutral}) and the fixed projective line disjoint from $f$.

\medskip
We will realize the Lie algebra $so(3, \mathbb C)$ of $PO(3, \mathbb C)$
as the subalgebra in the space of vector-fields on $\mathbb P^2$.
Again, each nonzero element $\zeta\in so(3,\mathbb C)$ has either one
or $3$ zeroes on $\mathbb P^2(\mathbb C)$. One of them is the {\em neutral zero},
it corresponds to the neutral fixed point of the  subgroup
$\exp(\mathbb C \cdot \zeta)\subset PO(3,\mathbb C)$. The maximal abelian subalgebra
$\mathbb C \cdot \zeta$ of $so(3,\mathbb C)$ containing $\zeta$  is determined
by the neutral zero of $\zeta$.

\subsection{Commuting and anticommuting elements}
\label{Commuting}

Suppose that $\alpha, \alpha'$ are  involutions in $PO(3,\mathbb C)$, thus
they have isolated fixed points $\lambda, \lambda'$ and fixed
projective lines $\Lambda, \Lambda'\subset \mathbb P^2(\mathbb C)$.

\begin{lem}
The elements $\alpha, \alpha'$ commute if and only if either:

 (1) $\alpha= \alpha'$ and $\lambda= \lambda', \Lambda= \Lambda'$, or (2) $\lambda\in \Lambda'$,
$\lambda' \in \Lambda$ and $\Lambda$ intersects $\Lambda'$ orthogonally
(with respect to the quadratic form $\varphi$).
\end{lem}
\par\medskip\noindent{\it Proof: } Proof is obvious and is left to the reader. $\qed$

\medskip
Suppose that $\alpha$ is an involution in $PO(3,\mathbb C)$ and $\beta\in
PO(3,\mathbb C)- \{\hbox{\bf 1}\}$.
We say that the elements $\alpha, \beta$ {\em anticommute} if
$\alpha \beta \alpha \beta = \hbox{\bf 1}$ (i.e. $\alpha \beta \alpha^{-1}=  \beta ^{- 1}$) and
$\alpha\ne \beta$.

\begin{lem}
\label{anticom}
The elements $\alpha, \beta$ anticommute iff the neutral fixed point
$\mu$ of $\beta$  belongs to the fixed projective line $\Lambda$ of
$\alpha$.
\end{lem}
\par\medskip\noindent{\it Proof: } If $\alpha, \beta$ anticommute then $\mu$ must belong to the
fixed-point set of $\alpha$. If $\mu$ is the neutral point of $\alpha$
then $\alpha$ and $\beta$ commute (and $\beta^2\ne \hbox{\bf 1}$)
or $\alpha= \beta$ which contradicts our assumptions.
$\qed$

\begin{rem}
Suppose that $\alpha, \beta$ are anticommuting elements as above.
Then they satisfy the relation
$$
\alpha \beta \alpha \beta = \beta \alpha \beta \alpha
$$
(since both right- and left-hand side are equal to $\hbox{\bf 1}$).
\end{rem}

\section{Cohomology computations}
\label{elem}

In this section we consider cohomologies of certain {\em elementary}
Artin and Shephard groups (their graphs have only two vertices). The section
is rather technical, its material will be needed in Section \ref{S14}.

Let $G$ be a finitely-generated group, $\rho: G\to PO(3, \mathbb C)$ be
a representation. We will denote the action of elements $g$ of $G$
on vectors $\xi$ in the Lie algebra $so(3,\mathbb C)$ by
$$
g\xi:= ad\circ\rho(g) \xi$$

\subsection{2-generated Abelian group}

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure21.eps}}
\caption{\sl Graph $\Gamma$ for 2-generated free Abelian group.}
\label{Fig21}
\end{figure}

Let $\Gamma$ be the graph with two vertices $v, w$ connected by the edge
$e$ with the label $2$ (Figure \ref{Fig21}).
The corresponding Artin group $G^a$ is free
Abelian group on two generators $a= g_v, b= g_w$. Take two distinct
commuting involutions $\alpha, \beta \in PO(3)$ and the homomorphism
$\rho_0: G^a\to PO(3,\mathbb C)$ given
by $\rho_0: a\mapsto \alpha$, $\rho_0: b\mapsto \beta$.

\begin{lem}
\label{L5.3}
The representation $\rho_0$ is infinitesimally rigid, i.e.
$\mbox{\rm H}^ 1(G^a, ad\circ \rho_0)= 0$. The point
$\rho_0\in \mbox{\rm Hom}(G^a, PO(3,\mathbb C))$ is nonsingular.
\end{lem}
\par\medskip\noindent{\it Proof: } $G^a= \pi_1(T^2)$ (where $T^2$ is the 2-dimensional torus)
and the group $PO(3)$ is reductive, hence Poincare duality gives
an isomorphism
 $\mbox{\rm H}^ 0(G^a, ad \circ \rho_0)\cong \mbox{\rm H}^ 2( G^a, ad \circ \rho_0 )$.
The centralizer of $\rho_0( G^a)$ in $PO(3,\mathbb C)$ is trivial.
Therefore,
$$
0= \mbox{\rm H}^ 0(G^a, ad \circ \rho_0)\cong \mbox{\rm H}^ 2( G^a, ad \circ \rho_0 )
$$
Since the Euler characteristic of $T^2$ (and hence of $G^a$)
equals zero, we conclude that $0= \mbox{\rm H}^ 1(G^a, ad \circ \rho_0 )$.
The second assertion of Lemma follows from \cite[Theorem 2.6]{LM}.
$\qed$

\medskip
Note that the associated Coxeter group $G^c$ is the
finite group $\mathbb Z/2 \times \mathbb Z/2$. Let $\rho^c_0$
denote the homomorphism $G^c\to PO(3,\mathbb C)$ corresponding to $\rho_0$.

\begin{cor}
\label{C1}
We have natural isomorphism between germs of representation
varieties:
$$
(\mbox{\rm Hom}(G^c, PO(3,\mathbb C)), \rho_0^c)\cong (\mbox{\rm Hom}(G^a, PO(3,\mathbb C)), \rho_0)
$$
given by composing homomorphisms $G^c\to PO(3,\mathbb C)$ with
the projection $G^a \to G^c$.
\end{cor}
\par\medskip\noindent{\it Proof: } Follows from the Lemma \ref{L5.3} and Corollary
\ref{CC}. $\qed$

\medskip
Now we consider the global structure of $\mbox{\rm Hom}(G^a, PO(3))$.

\begin{lem}
\label{L5.8}
The variety $\mbox{\rm Hom}(G^a, PO(3, \mathbb C))$ is the disjoint union
of two Zariski closed subsets:

(1) Reducible representations
$S_1:= \{\rho : \dim \mbox{\rm H}^ 0(G^a, ad\circ \rho)\ge 1\}$;

(2) The orbit  $S_2:= Ad(PO(3,\mathbb C)) \rho_0$, where $\rho_0$
is the  representation  whose image is the group
$\mathbb Z/2 \times \mathbb Z/2$.
\end{lem}
\par\medskip\noindent{\it Proof: } First we verify that $\mbox{\rm Hom}(G^a, PO(3, \mathbb C))$ is the
union $S_1\cup S_2$. Let $A\subset PO(3,\mathbb C)$ be an abelian
subgroup, $\bar{A}$ is the Zariski
closure of $A$. If $A$ is infinite then the abelian group
$\bar{A}$ is the finite extension of a 1-dimensional
connected abelian Lie
subgroup $\bar{A}^0$ of $PO(3,\mathbb C)$. Hence $\bar{A}^0$
is either a maximal torus
or a maximal unipotent subgroup of $PO(3,\mathbb C)$: both are
maximal abelian subgroups in $PO(3,\mathbb C)$, thus
$A\subset \bar{A}^0$. We apply this to the
group $A= \rho(G^a)$ and conclude that in this case
  $\dim \mbox{\rm H}^ 0(G^a, ad\circ \rho)=1$.

Now we consider the case when $A$ is finite, hence, after
conjugation, we get: $A\subset SO(3,\mathbb R)$. It follows from
classification of finite subgroups in $SO(3,\mathbb R)$ that $A$
either has an invariant vector in $\mathbb R^3$ (which again means
that $\dim \mbox{\rm H}^ 0(G^a, ad\circ \rho)\ge 1$ ) or
$A\cong \mathbb Z/2 \times \mathbb Z/2$ is generated by two involutions
with orthogonal axes. Hence $S_1\cup S_2=\mbox{\rm Hom}(G^a, PO(3, \mathbb C))$.

\medskip
It's clear that $S_1$ and $S_2$ are disjoint.
Since the representation $\rho_0$ is locally rigid it
follows that the orbit
$Ad(PO(3,\mathbb C) )\rho_0$ is open, \cite[Theorem 2.6]{LM}.
The representation $\rho_0$ is {\em stable} (see \cite[Theorem 1.1]{JM})
since $\rho_0(G^a)= \mathbb Z/2 \times \mathbb Z/2$ is
not contained in a proper parabolic subgroup of $PO(3, \mathbb C)$.
Hence the orbit  $Ad(PO(3,\mathbb C) )\rho_0$ is closed. $\qed$

\subsection{Finite elementary Shephard groups}

Take $\Gamma$ be any of the labelled graphs from the Figure
\ref{Fig28}. The corresponding Shephard group
$G:= G^s_{\Gamma}$ is finite, hence we have

\begin{prop}
\label{L5.4}
All  representations of $G$ to $PO(3)$ are
infinitesimally rigid and \newline
$\mbox{\rm Hom}(G, PO(3))$ is smooth.
\end{prop}

We will need a slight modification of the above proposition. Let
$L$ be an algebraic Lie group with the Lie algebra ${\cal L}$,
$G$ is a finitely-generated group such that all representations
$\rho: G\to L(\mathbb C)$ are infinitesimally rigid, pick elements
$a\in G$ and $\alpha\in L(\mathbb C)$ and consider the subvariety
$$
F= F_{a, \alpha}(G, L)=
\{ \rho: G\to L| \rho(a)= \alpha \} \subset \mbox{\rm Hom}(G, L)
$$

\begin{prop}
\label{PF}
The subvariety $F$ is smooth.
\end{prop}
\par\medskip\noindent{\it Proof: }
The space $\mbox{\rm Hom}(G, L(\mathbb C))$ is the union of $L(\mathbb C)$-orbits
of representations $\rho_j, 1\le j\le m$. If $\alpha\ne \rho(a)$
for all $\rho\in \mbox{\rm Hom}(G, L(\mathbb C))$
then there is nothing to prove. Otherwise we can assume that
$\alpha= \rho_j(a)$, $j\in J\subset\{1,..., m\}$.
Since the representations $\rho_j$ are locally rigid we get
$$
F(\mathbb C)\cong \bigcup_{j\in J} Z_{L(\mathbb C)}(\<\alpha\>)/ Z_{L(\mathbb C)}(\rho_j(G))
$$
where $Z_{L(\mathbb C)}(H)$ denotes the centralizer of a subgroup
$H\subset L(\mathbb C)$.
It is enough to verify smoothness of $F$ at the representations
$\rho_j, j\in J$.  Consider the Zariski tangent space
$T= T_{\rho_j}F(\mathbb C)$. It is naturally isomorphic to
$$
\{ \xi\in Z^1(G, {\cal L}_{ad\rho_j}) : \xi(a)=0\}
$$
However infinitesimal rigidity of $\rho_j$ implies
that\footnote{Recall  that ${\cal L}^H$ denotes the subspace of
$H$-invariant vectors for a group $H$ acting on ${\cal L}$.}
$T\cong {\cal L}^{\<a\>}/{\cal L}^G$,
where $G$ and $\<a\>$ act on ${\cal L}$ via the adjoint
representation $ad\rho_j$. Hence the dimension of $F(\mathbb C)$
(as a complex manifold) at $\rho_j$
is equal to the  dimension of its Zariski tangent space at
$\rho_j$, which implies that $F$ is smooth at $\rho_j$. $\qed$

As a particular case we let $a$ be one of the generators of
$G= G^s_{\Gamma}$, pick an element $\alpha\in PO(3,\mathbb C)$ and
consider the subvariety
$$
F_{a, \alpha}(G, PO(3))
= \{ \rho: G\to PO(3,\mathbb C)| \rho(a)= \alpha \} \subset \mbox{\rm Hom}(G, PO(3,\mathbb C))
$$

\begin{cor}
\label{PFC}
Suppose that $G= G^s_{\Gamma}$ is a Shephard group as above, $a$ is
one of the generators of $G$, $\alpha\in PO(3, \mathbb C)$. Then the
$F_{a, \alpha}(G, PO(3))$ is smooth.
\end{cor}

\subsection{The infinite cyclic group}
\label{cy}

Consider the infinite cyclic group $G= \<b\>$ and a representation
$\rho: \<b\>\to PO(3,\mathbb C)$ so that $\rho(b)=\beta$ is a nontrivial
semisimple element.

Note that the Lie algebra $so(3,\mathbb C)$ has $ad\circ \rho(b)$-invariant splitting $L\oplus L^{\perp}$ where $L$ consists of vectors vanishing at the neutral
fixed point point $B\in \mathbb P^2(\mathbb C)$ of the element $\beta$. The action
of $b$ on $L^{\perp}$ has no nonzero invariant subspaces. Thus
$\mbox{\rm H}^ 1(G, L^{\perp})= 0$ and
$$
\mbox{\rm H}^ 1(G, L)\oplus \mbox{\rm H}^ 1(G, L^{\perp})  = \mbox{\rm H}^ 1(G, L)\cong  \mbox{\rm H}^ 1(G, so(3,\mathbb C))
$$

This proves the following

\begin{prop}
\label{cycl}
Any cocycle $\sigma \in Z^1(G, ad\circ \rho)$ has the form $\sigma(b)= \tau +
b\xi - \xi$ where $b\tau= \tau$. The element $\tau$ depends only on the cohomology class of $\sigma$.
\end{prop}

\subsection{Elementary Shephard groups with the edge-label $4$.}

Consider the graph with two vertices $v, w$ connected by the edge with
the label $4$, we put the label $2$ on the vertex $v$, see
Figure \ref{Fig23}.
The corresponding Shephard group $G$ has the presentation
$\< a= g_v , b= g_w | a^2= \hbox{\bf 1}, (ab)^2= (ba)^2\>$.

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure23.eps}}
\caption{\sl Graph for a Shephard group}
\label{Fig23}
\end{figure}

 Consider a representation $\rho: G \to PO(3, \mathbb C), \  \rho: a\mapsto
\alpha, \  \rho: b \mapsto \beta$\ ,
where we choose $\alpha$ to be an involution, $\beta\ne \hbox{\bf 1}$ is an
element such that $\alpha,\beta$ anticommute, i.e. $\alpha \beta \alpha \beta =\hbox{\bf 1}$ and
$\alpha \ne \beta$. We define another group $\bar{G}$ (which is
 a Coxeter group) by
$$
\bar{G}= \< a, b|  a^2=\hbox{\bf 1}, (ab)^2=\hbox{\bf 1} \>\cong \mathbb Z/2 * \mathbb Z/2
$$
Let $q: G\to \bar{G}$ denote the natural projection.
It's clear that we have a homomorphism
$\bar{\rho}: \bar{G}\to PO(3, \mathbb C)$ such
that $\bar\rho\circ q=\rho$. Thus we have the induced monomorphism
$$
q^*: \mbox{\rm H}^ 1(\bar G, ad\circ \bar\rho)\longrightarrow \mbox{\rm H}^ 1(G, ad\circ \rho)
$$

\begin{prop}
\label{L5.5}
The mapping $q^*$ is an isomorphism. Any cocycle
is cohomologous to $\sigma\in Z^1(G, ad\circ\rho)$ of the form
$$
\sigma(a)= 0, \quad \sigma(b)= \tau, \hbox{~~so that~~} b\tau= \tau
$$
If $\sigma|_{\< b\>}$ is a coboundary then $\sigma=0$.
\end{prop}
\par\medskip\noindent{\it Proof: } Let $\sigma\in Z^1(G, ad\circ\rho)$,
 by normalizing $\sigma$ via coboundary we can assume that $\sigma(a)= 0$,
 hence there is an element
$\xi\in so(3, \mathbb C)$ such that $\sigma(b)= b\xi - \xi + \tau$, where
$b\tau= \tau$ (see Proposition \ref{cycl}).  Now we need

\begin{lem}
The cocycle $\sigma$ can be chosen within its cohomology class in
$\mbox{\rm H}^ 1(G, ad\circ\rho)$ so that $\sigma(a)=0, a\xi= - \xi$.
\end{lem}
\par\medskip\noindent{\it Proof: } Let $\theta:= (\xi + a \xi)/2$, then $a\theta= \theta$. Let
$\xi':= \xi - \theta$, let $\sigma'(a):= 0, \sigma'(b):= b\xi' - \xi' + \tau$.
Then its clear that $\sigma'$ is cohomologous to $\sigma$ and a direct
computation shows that  $a\xi'= -\xi'$. $\qed$

Now we suppose that $\sigma$ is chosen as in the above lemma.
Since $\alpha$ anticommutes with $\beta$ it also anticommutes with all elements of
the 1-parameter subgroup $B\subset PO(3, \mathbb C)$ containing $\beta$. Suppose for
a moment that $\tau \ne 0$, since $b\tau = \tau$ we conclude that
$B= \exp(\mathbb C \cdot \tau)$; hence $a\tau = -\tau$. The same of course is true if
$\tau=0$. Thus $a\tau= -\tau$ in any case. The Artin relation in $G$
is $ab ab  = ba ba $. Therefore we must have:
$$
\sigma(ab ab )= \sigma( ba ba)
$$
Direct calculation shows that
$$
\sigma(ab ab)= - 2 b^{-1} \xi +2\xi ~~, \quad \sigma(ba ba)= 2b\xi - 2\xi
$$
Thus we must have: $b\xi + b^{-1}\xi - 2\xi= 0$, equivalently
$b\xi - \xi= \xi- b^{-1}\xi$. Let $\eta:= b\xi - \xi$, then the above
equation implies:
$$
b^{-1} \eta= \eta , \quad b  \eta= \eta
$$
On the other hand $a \eta= \eta$ because
$$
a \eta = ab \xi - a \xi = b^{-1} a \xi + \xi =
\xi- b^{-1}\xi = b\xi - \xi=
\eta $$
Since $\alpha, \beta$ anticommute we conclude that $\eta=0$, which implies that
$b \xi= \xi$.
 Thus $\sigma(b)= \tau$. Let's check that the cocycle $\sigma$ lies in
the image of the mapping $q^*$, for this it's enough to verify that
$\sigma(ab ab)=0$. Indeed, $\sigma(ab ab)= -\tau + \tau=0$.
It is left to prove that the assmption that $\sigma$ is a coboundary
on $\<b\>$ implies that $\tau=0$.
Let $L\subset so(3, \mathbb C)$ be the subspace fixed by the adjoint action of
$\beta$ (as in Section \ref{cy}). By Proposition \ref{cycl},
if cocycle $\sigma$ is a coboundary on $\<b\>$ then there exists
$\eta\in L$ such that $\sigma(b)= \tau= b\eta - \eta= \eta -\eta=0$. $\qed$

\begin{rem}
\label{smo}
Note that the cocycles $\sigma$ described in the above Lemma are
integrable, they
correspond to deformations $\rho_t$  of the representation $\rho$
which fix $\rho(a)= \alpha$ and
change the element $\rho(b)= \beta$ within the corresponding
1-parameter subgroup
in $PO(3, \mathbb C)$. For such  representations the elements
$\rho(a), \rho_t(b)$ anticommute.
\end{rem}

\subsection{Elementary  Artin group with the edge-label $6$.}

Now consider 2-generated Artin group $G^a$ given by  the relation
$(ab)^3= (ba)^3 $, see Figure \ref{Fig25}.
Take a representation $\rho: G^a\to PO(3,\mathbb C)$ which maps
$a$ and $b$ to elements $\alpha, \beta$ so that $\alpha^2=1$,
$[\alpha ,\beta^2] \ne \hbox{\bf 1}$ and the product $\alpha\beta$ has the order $3$.
(In particular $\beta \ne \hbox{\bf 1}$.)

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure25.eps}}
\caption{\sl Graph for the Artin group $G^a$}
\label{Fig25}
\end{figure}

\begin{lem}
\label{L5.6}
Let $\sigma\in Z^1(G, ad\circ\rho)$ be a cocycle  such that
$\sigma(a)=0, \sigma(b)= \tau$, where $b\tau= \tau$. Then $\tau= 0$.
\end{lem}
\par\medskip\noindent{\it Proof: } Direct calculation shows that
$$
\sigma(ab ab ab)= a \tau + \tau + aba \tau= a\tau + \tau + b^{-1} a\tau$$
$$
 \sigma(ba ba ba)= \tau + ba \tau + a\tau
$$
Therefore $ad(\rho(b^{-1} a))\tau = ad(\rho( ba)) \tau$.
We suppose that $\tau\ne 0$, then it has unique zero in $\mathbb P^2(\mathbb C)$
which is the same as the neutral fixed point of $\beta$. This implies that
$$
\beta^{-1}\alpha \ \gamma\ (\beta^{-1}\alpha )^{-1} = \beta \alpha \ \gamma\ (\beta \alpha)^{-1}
$$
for all elements $\gamma$ of the 1-parameter subgroup
$\exp(\mathbb C \cdot \tau)$, in
particular it is true for $\beta= \gamma$. Since $\alpha^2=1$,
$(\alpha\beta)^3= (\beta \alpha)^3 =1$ we conclude that
$\beta^{-2} \alpha = \alpha \beta^{-2}$,
which contradicts our restrictions on $\alpha, \beta$. $\qed$

\subsection{A non-elementary Shephard group.}

Now suppose that we have a group $G^s$ with the presentation:
$$
\< a_1, a_2, a_3,   b | a_j^2= \hbox{\bf 1}, \quad j=1, 2, 3\ ; \quad
(a_i b)^2=  (b a_i)^2 , i=1,2 ; ~~~ (a_3 b)^3= (b a_3)^3\>
$$
See the graph on the Figure \ref{Fig26}.

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure26.eps}}
\caption{\sl Graph for the nonelementary Shephard group $G^s$}
\label{Fig26}
\end{figure}

Consider a representation $\rho_0: G^s \to PO(3, \mathbb C)$ so that

\begin{enumerate}
\item $\rho_0(a_j)\ne \hbox{\bf 1}$, $1\le j \le 3$;
\item $\rho_0(b)^3= \hbox{\bf 1}$, $\rho_0(b_3)\ne \hbox{\bf 1}$;
\item the neutral fixed points of the elements
$\rho(a_1), \rho(a_2), \rho(b)$ do not belong to a common projective
line in $\mathbb P^2$;
\item $\rho_0(a_3 b)^3 =\hbox{\bf 1}$.
\end{enumerate}

Take the  subvariety $V\subset \mbox{\rm Hom}(G, PO(3,\mathbb C))$ which contains
$\rho_0$ and consists of homomorphisms that are constant on the
generators $a_j, 1\le j\le 3$.

\begin{lem}
\label{L5.10}
The point $\rho_0$ is an isolated reduced point in $V$.
\end{lem}
\par\medskip\noindent{\it Proof: } Note that $\rho_0(b^2)= \rho_0(b^{-1})$. Thus, if $\rho_0(b^2)$ commutes
with $\rho_0(a_3)$ then $\rho_0(a_3)^3=1$ which is impossible.
Take a cocycle
$\sigma\in Z^1(G^s, ad\circ\rho)$ which is tangent to the variety $V$. Hence
$\sigma(a_j)=0, 1\le j\le 3$. According to Proposition \ref{cycl} the value of $\sigma$ on the generator $b$ equals $\tau + b\xi -\xi$. On the other hand, by
Proposition \ref{L5.5} we can find coboundaries $\delta_{\theta_j} \in
B^1(\< a_j, b\>, so(3,\mathbb C))$, $j=1,2$ so that:
$$
\sigma_j:= \sigma - \delta_{\theta_j}, \sigma_j(a_j)=0, \sigma_j(b)= \tau
$$
The coboundary $\delta_{\theta_j}$ is given by
$$
\delta_{\theta_j}(x)=  x\theta_j -\theta_j, x\in \< a_j, b\>,
\theta_j\in so(3,\mathbb C)
$$
Thus
$$
a_j \theta_j = \theta_j, b(\theta_1- \theta_2)= \theta_1- \theta_2
$$
Note however that the condition (3) on the representation $\rho_0$ implies that
the (1-dimen\-si\-onal) fixed-point sets for the adjoint actions of
$\rho_0(a_1), \rho_0(a_2), \rho_0(b)$ on $so(3,\mathbb C)$ are linearly independent. Therefore we conclude that $\theta_j=0$, $j=1,2$, thus
$$
\sigma(b)= \tau
$$
and by Lemma \ref{L5.6} we have $\sigma(b)=0$. Hence $\sigma=0$ and the Zariski tangent space to $\rho_0$ in $V$ is zero. $\qed$

\subsection{Nondegenerate representations}
\label{appli}

Let $\Gamma$ be a labelled graph where all vertices and edges have nonzero even
labels. Let $G^s$ denote the Shephard group corresponding to the graph $\Gamma$.
The following technical definition will be used in
Section \ref{Representations of Shephard groups}.

\begin{defn}
A representation $\rho: G^s \to PO(3,\mathbb C)$ will be called
{\bf nondegenerate on the edge} $e= [v, w]\subset \Gamma$ if the element
$\rho(g_v g_w)$ has the order
$$
\left\{ \begin{array}{cc}
\epsilon(e) \ \ , & \hbox{ if $\delta(v)= \delta(w)=2$} \\
\epsilon(e)/2 \ \  , & \hbox{otherwise} \end{array} \right.
$$
A representation
$\rho$ will be called
{\bf nondegenerate  on the vertex} $v\in V(\Gamma)$ if $\rho(g_v)\ne \hbox{\bf 1}$. A
representation $\rho$ will be called {\bf nondegenerate} if it is
nondegenerate on each edge and each vertex of $\Gamma$. Let
$\mbox{\rm Hom}_f(G^s, PO(3,\mathbb C))$ denote the space of all nondegenerate
 representations.
\end{defn}

\begin{prop}
\label{P6}
Suppose that for each edge $e\subset \Gamma$ the corresponding edge subgroup
$G_e\subset G^s$ is finite. Then
$\mbox{\rm Hom}_f(G^s, PO(3,\mathbb C))$  is Zariski open and closed in  $\mbox{\rm Hom}(G^s, PO(3,\mathbb C))$.
\end{prop}
\par\medskip\noindent{\it Proof: } Since each $G_v, G_e \subset G^s$ is finite,
$\mbox{\rm Hom}(G_v, PO(3,\mathbb C))$, $\mbox{\rm Hom}(G_e, PO(3,\mathbb C))$ are disjoint unions
of finite numbers of $PO(3, \mathbb C)$-orbits of rigid representations.
Since each orbit is Zariski open the proposition follows in the
case $G^s= G_v, G_e$.
Let
$$Res_e: \mbox{\rm Hom}(G^s, PO(3,\mathbb C)) \to \mbox{\rm Hom}(G_e, PO(3,\mathbb C)),$$
$$Res_v: \mbox{\rm Hom}(G^s, PO(3,\mathbb C)) \to \mbox{\rm Hom}(G_v, PO(3,\mathbb C))
$$
 be the restriction morphisms. Then $\mbox{\rm Hom}_f(G^s, PO(3,\mathbb C)) = $
$$
\bigcap_{v\in {\cal V}(\Gamma)} Res_v^{-1}
\mbox{\rm Hom}_f(G_v, PO(3,\mathbb C)) \cap \bigcap_{e\in {\cal E}(\Gamma)} Res_e^{-1}
\mbox{\rm Hom}_f(G_e, PO(3,\mathbb C))
$$
and the proposition follows.  $\qed$

\section{Arrangements}
\subsection{Abstract arrangements}
\label{Abstract arrangements}

An {\it abstract arrangement} $A$ is a disjoint
union  of two finite sets $A = {\cal P} \sqcup {\cal L}$, with the  set of
``points'' ${\cal P} = \{v_1\ , \, v_2\ , \cdots\}$
and the set of ``lines'' ${\cal L} = (l_1\ , l_2\ , \cdots)$
together with the {\em incidence  relation}
$\iota = \iota_A \subset {\cal P} \times {\cal L};
\iota (v, l)$ is interpreted to mean ``the point $v$ lies on the
line $l$~". We may represent the arrangement $A$ by a
bipartite graph, $\Gamma =\Gamma_A$:
vertices of $\Gamma$ are elements of $A$, two vertices are connected
by an edge if and only if the corresponding elements of $A$ are
incident ($\Gamma$ is also called the {\em Hasse diagram} of
the arrangement $A$).

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure1.eps}}
\caption{\sl Standard triangle.}
\label{Fig1}
\end{figure}

\begin{conv}
When drawing an arrangement we shall draw {\em points} as {\em solid points} and {\em lines} as {\em lines}. If $\iota(v, l)$ then we shall draw the point
$v$ on the line $l$.
\end{conv}

An example of an abstract arrangement is the {\em standard triangle}
$T$ described on the Figure \ref{Fig1} (this is a triangle with the complete set of bisectors):
$$
T = \{v_{00}, v_x, v_y , v_{1,0}, v_{01}; l_x, l_y, l_{\infty},
l_d, l_{y1}, l_{x1}: \iota(v_{00}, l_x), \iota(v_{01}, l_y),
\iota(v_x, l_x), \iota(v_x, l_{\infty}),$$
$$
  \iota(v_y, l_y), \iota(v_y, l_{\infty}),
 \iota(v_{11}, l_d), \iota(v_{00}, l_d), \iota(v_{00}, l_y), \iota (v_{10},
l_{y1}), \iota(v_{01}, l_{x1}), $$
$$
\iota(v_{10}, l_x),
\iota(v_y, l_{y1}), \iota (v_x, l_{x1}), \iota(v_{11}, l_{y1}), \iota(v_{11}, l_{x1}) \}
$$

\medskip
Here is another example of arrangement (Figure \ref{Fig2}), we take
$A= \{ v_1, v_2; l_1, l_2\}$ with the incidence relation:
$$
\iota(v_1, l_1), \iota(v_1, l_2), \iota(v_2, l_1), \iota(v_2, l_2) \quad .
$$

Suppose that $(A, \iota_A), (B, \iota_B)$ are abstract arrangements,
$\phi: B\to A$ is a map which sends points to points and lines to lines.
We say that $\phi$ is a {\em morphism} of arrangements if $\iota_B(x, y)$
implies $\iota_A(\phi(x), \phi(y))$. A {\em monomorphism} of arrangements
is an injective morphism.
An {\em isomorphism} of arrangements is an invertible morphism. Suppose
that $\phi: B \to A$ is a monomorphism of arrangements, we call the
image $\phi(B)$ a {\em subarrangement} in $A$. Note that if
 we work with the corresponding bipartite graphs $\Gamma_A, \Gamma_B$,
then morphisms $A\to B$ are morphisms of these bipartite graphs
which send points to points, lines to lines.

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure2.eps}}
\caption{\sl Bigon.}
\label{Fig2}
\end{figure}

\begin{defn}
 An abstract {\bf based arrangement} $A$ is an arrangement together with\newline  a
monomorphism of the standard triangle
$T\hookrightarrow A$ that we call the {\bf canonical} {\bf embedding}.
\end{defn}

\medskip
\noindent An arrangement $A$ is called {\em admissible} if it satisfies the axiom:

\medskip
{\bf (A1)} {\em Every element of $A$ is incident to at least two
distinct elements.}  (I.e. every \newline point belongs to at least two
lines and every line contains at least two
points.)\footnote{This axiom will be needed only in the Section
\ref{repart}.}

\medskip
Suppose that $A$, $B$, $C$ is a triple of arrangements and
$\phi: C\to A, \psi: C\to B$ are monomorphisms. We define the
 {\em fiber sum} $A\times_C B$ as follows. First we take
the disjoint union of the arrangements $A$ and $B$. Then
identify in $A\cup B$ the elements $\phi(c), \psi(c)$ for
all $c\in C$. If $A, B$ are based arrangements, $C$ is as above,
then their join $A*_C B$ is defined as $A\times_{T\sqcup C} B$, where
$T$ is the standard triangle with canonical embedding into $A, B$.
If $C$ is an  arrangement which consists of a single {\em point} $c$ and
$\phi(c)= a\in A, \psi(c)=b\in B$, then we denote
$$
A*_{a\equiv b} B := A *_C B
$$

\subsection{Fiber products}
\label{3.2}

We remind the reader of the  definition of the  {\em fiber product}
of varieties (recall that our varieties are neither reduced nor
ireducible). Let $f: X\to Z, g: Y\to Z$ be morphisms. Then the
{\em fiber product} $X\times_Z Y$ of $X$ and
$Y$ with respect to $Z$ is a variety $X\times_Z Y$ together
with canonical morphisms  $\Pi_X: X\times_Z Y \to X$ and
$\Pi_Y: X\times_Z Y \to Y$
 such that the fol\-lo\-wing diagram is commutative
\[
\begin{array}{ccc}
X\times_Z Y & \longrightarrow & X\\
 \downarrow & ~ & \downarrow \\
Y & \longrightarrow & Z\\
\end{array}
\]
These data satisfy the
property that given a
variety $W$ and a commutative diagram
\[
\begin{array}{ccc}
W & \longrightarrow & X\\
 \downarrow & ~ & \downarrow \\
Y & \longrightarrow & Z\\
\end{array}
\]
we obtain a commutative diagram
\[
\begin{array}{ccccc}
W &  \joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightarrow &  X\\
~& \parbox[l]{.2in}{{\large $\searrow$}}
\quad \quad\quad\quad \quad \parbox[r]{.3in}
{{\large $\nearrow$}} & ~ \\
\Big\downarrow &  X\times_Z Y &  \Big\downarrow\\
~ & \hbox{{\large $\swarrow$ \quad\quad \quad\quad  \quad }} &  ~ \\
Y &  \joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightarrow &  Z\\
\end{array}
\]
The fiber product of quasi-projective (resp. projective) varieties
is again a quasi-projective (resp. projective).

We recall  how to describe graphs of morphisms via fiber
products. Suppose that \newline $f: X= {\mathbb A}^n \to {\mathbb A}^m= Z$ is a morphism.
Let $g: Y= {\mathbb A}^m \to {\mathbb A}^m$ be the identity morphism. Define
$\Gamma_f$, graph of $f$, to be the subvariety
$$
\Gamma_f= X\times_Y Y = \{ (x, y)\in X\times Y: f(x)= g(y)= y\}
$$
Clearly $\pi_X : \Gamma_f \to X$ is an isomorphism of affine varieties.

Let $f: X= {\mathbb A}^n \to Y= {\mathbb A}^m$ be a morphism, $\Gamma_f$ its graph and
$\pi_X: \Gamma_f\cong {\mathbb A}^n$ be the canonical projection. We split
${\mathbb A}^n$ as ${\mathbb A}^{n-k}\times {\mathbb A}^k$, so $x\in {\mathbb A}^n$ is written as \newline
$x= (x' , x'')$, $x'\in {\mathbb A}^{n-k}, x''\in {\mathbb A}^k$. We obtain projections
$$
\Pi'_X: \Gamma_f \to {\mathbb A}^{n-k} \hbox{~~and~~} \Pi''_X: \Gamma_f \to {\mathbb A}^{k}$$
 defined by $\Pi'_X(x)= x'$ and $\Pi''_X(x)= x''$. Now let
$g: {\mathbb A}^s \to {\mathbb A}^{n-k}$ be a morphism with graph
$\Gamma_g$, let $y\in {\mathbb A}^s$ denote the variable in this space.
Using the morphism\newline
$\Gamma_g\to {\mathbb A}^{n-k}$ (the second projection) and
$\Pi'_X: \Gamma_f \to {\mathbb A}^{n-k}$
we form the fiber product
$$\Gamma_g \times_{{\mathbb A}^{n-k}} \Gamma_f\quad .$$
Now let $h: {\mathbb A}^s \times {\mathbb A}^k \to {\mathbb A}^m$ be the morphism given by
$$
h(y, x'')= f(g(y), x'')
$$

\begin{lem}
\label{fib}
The projection map
$$
p: ({\mathbb A}^s \times {\mathbb A}^{n-k})\times ({\mathbb A}^{n-k}\times {\mathbb A}^k) \times {\mathbb A}^m \to
{\mathbb A}^s \times {\mathbb A}^k \times {\mathbb A}^m = T
$$
given by
$$
p((y, x'), (u', u''), z)= (y, u'', z)
$$
induces an isomorphism
$$
\Gamma_g \times_{{\mathbb A}^{n-k}} \Gamma_f\longrightarrow \Gamma_h \quad .$$
\end{lem}
\par\medskip\noindent{\it Proof: } Obvious. $\qed$

\begin{cor}
The morphism
$$
q: \Gamma_g \times_{{\mathbb A}^{n-k}} \Gamma_f\longrightarrow {\mathbb A}^s \times {\mathbb A}^k$$
 (given
by the restriction of $p$ to $\Gamma_g \times_{{\mathbb A}^{n-k}} \Gamma_f$ and the
the projection on the first and second factor of $T$) is an isomorphism.
\end{cor}

\begin{cor}
\label{C3.2}
Suppose that  $k=0$, then the composition of $q$  with the
projection on the first factor
$$
r: \Gamma_g \times_{{\mathbb A}^{n-k}} \Gamma_f\longrightarrow {\mathbb A}^s
$$
is an isomorphism.
\end{cor}

\subsection{Intersection operations in the projective plane}
\label{Intersection}

Let ${\cal R}$ be a commutative ring. We recall that the projective space $\mathbb P(M)$
for a projective ${\cal R}$-module $M$  of rank $n$ is defined by
$$
\mathbb P(M):= \{ V\subset M: V ~~~ \hbox{is a projective submodule of rank
1 such that~} M/V \hbox{~is projective}\}
$$
We then define $\mathbb P^n({\cal R})$ and $\mathbb P^n({\cal R})^{\vee}$ by
$$
\mathbb P^n({\cal R}):= \mathbb P({\cal R}^{n+1}), \quad \mathbb P^n({\cal R})^{\vee}:= \mathbb P(\mbox{\rm Hom}_{{\cal R}}
({\cal R}^{n+1}, {\cal R}))
$$
We refer to \cite[\mathbb S 1.3.4, \mathbb S 1.3.9]{DG}, to see that this is consistent
with the usual definition of $\mathbb P^n$.

Note that an element
$\alpha\in \mathbb P^2({\cal R})^{\vee}$ gives rise to a projective
${\cal R}$-submodule $L\subset {\cal R}^3$ of rank $2$, $L:= \ker(\alpha)$, such that
${\cal R}^3/L$ is projective. We will call both $\alpha$ and $L$ {\em lines} in $\mathbb P^2({\cal R})$. We say that a point $V\in \mathbb P(M)$ belongs to a line $L$ (corresponding to $\alpha\in \mathbb P^2({\cal R})^{\vee}$ if and only if $V\subset L$; equivalently $\alpha(V)=0$. Suppose $V\in \mathbb P^2({\cal R})$ correspond to rank one
{\em free} submodule of ${\cal R}^3$ with the basis $u= (x, y, z)$, then
 we will write $V:= [x: y: z]$ (these are the homogeneous
coordinates of $V$).

We now show how to do projective geometry over ${\cal R}$. We define two
elements $Span(u_1)$, $Span(u_2)\in \mathbb P^2({\cal R})$
to be {\em independent} if the submodule
$$
L= Span\{u_1, u_2\} := {\cal R} u_1 + {\cal R} u_2$$
 is a projective summand of ${\cal R}^3$ of rank $2$. In this case we will also
say that $u_1, u_2$ are independent.

\begin{lem}
If $Span(u_1)$ and $Span(u_2)$ are independent over ${\cal R}$ then
$Span\{u_1,u_2\}$ is the unique projective summand of ${\cal R}^3$ containing
$u_1$ and $u_2$.
\end{lem}
\par\medskip\noindent{\it Proof: } Let $M$ denote $Span\{u_1,u_2\}$. Suppose that $N$ is a projective
summand containing $u_1$ and $u_2$. Then $N$ contains $M$. We want to
prove that $M=N$. We may assume that ${\cal R}$ is local whence $M$ and $N$ are
free. Let $\{v_1, v_2\}$ be a basis for $N$. Note that $\{u_1,u_2\}$
is a basis for $M$. Write
$$
u_1 = av_1 + cv_2 , \quad u_2 = bv_1 + dv_2
$$
Let $\hbox{\bf k}$ be the residue field of ${\cal R}$. The images of $u_1$ and $u_2$ in
$\hbox{\bf k}^3$  are independent so the image of $ad-bc$ in $k$ is nonzero.
Hence $ad-bc$ is a unit in ${\cal R}$. $\qed$

\medskip
Thus two independent points $U_1= Span(u_1), U_2= Span(u_2) \in \mathbb P^2({\cal R})$
belong to the unique line $L= Span\{u_1,u_2\}$ in $\mathbb P^2({\cal R})$. We shall use
the notation
$$
L:= U_1 \bullet U_2
$$
for the line $L$ through the points $U_1, U_2$.
If $u\in {\cal R}^3$ we let $u^{\vee}$ denote the
element of $({\cal R}^3)^{\vee}$ given by $u^{\vee}(v):= u\cdot v$.
We have the following sufficient condition  for  independence:

\begin{lem}
 Suppose that there exists
$u_3= (x_3, y_3, z_3)\in{\cal R}^3$ such that
$(u_1 \times u_2)\cdot u_3= 1$. Then $u_1, u_2$ are independent, moreover
$$
Span\{u_1, u_2\} = \ker (u_1 \times u_2)^{\vee}
$$
\end{lem}
\par\medskip\noindent{\it Proof: } The determinant of the matrix with the columns $u_1, u_2, u_3$ equals $1$, whence \newline
$\{u_1, u_2, u_3\}$ is a basis for ${\cal R}^3$. Furthermore, suppose
$v= a u_1 + bu_2 + cu_3$. Then $(u_1 \times u_2)\cdot v= c$, so $c=0$ if
and only if $v\in \ker (u_1 \times u_2)^{\vee}$. $\qed$

\begin{rem}
We observe that $u_3$ as above always exists (and hence $u_1, u_2$
are independent) if one of the coordinates of $u_1\times u_2$ is a
unit in ${\cal R}$. In this case we will say that $u_1$ and $u_2$ satisfy
 the {\bf cross-product test} for independence.
\end{rem}

Dual to the correspondence $\bullet: \mathbb P^2 \times \mathbb P^2 \to (\mathbb P^2)^{\vee}$
there is  an operation of intersection of lines in $\mathbb P^2$. Namely, if
$\lambda, \mu$ are  lines in
$\mathbb P^2$ such that $\lambda^{\vee}, \mu^{\vee}$ are independent points in $\mathbb P^2$,  then we let
$(\lambda\bullet \mu)^{\vee}= \lambda^{\vee}\bullet \mu^{\vee}$.
Clearly $\lambda\bullet \mu= \ker(\lambda) \cap \ker(\mu)$. We have

\begin{lem}
If $u_1, u_2\in \mathbb R^3$ above satisfy  the {\bf cross-product test},
then $V_1= Span(u_1)$, $V_2= Span(u_2)$ can be joined by the
unique projective line $V_1\bullet V_2$ in $\mathbb P^2({\cal R})$
corresponding to $(u_1\times u_2)^{\vee}$.
\end{lem}

Dually let $L_j \in (\mathbb P^2)^{\vee}$ are dual to rank one free modules
with bases $\sigma_j= (\alpha_j, \beta_j, \gamma_j)$, ($j=1,2$). We will write
$L_j= [ \alpha_j: \beta_j: \gamma_j]$. We have

\begin{lem}
If $\sigma_1, \sigma_2$ above are independent then $L_1, L_2$ intersect in the
unique point $L_1\bullet L_2$ with the homogeneous coordinates
$[\sigma_1\times \sigma_2]$.
\end{lem}

The {\em incidence variety}
${\cal I}\subset \mathbb P^2 \times (\mathbb P^2)^{\vee}$ is given by the equations:
$$
\{ (p, l)\in \mathbb P^2 \times (\mathbb P^2)^{\vee} | l(p)=0\}
$$
Let $x, y, z$ be the coordinate functions on
$\mathbb C^3$ relative to the standard basis and $\alpha,\beta, \gamma$ be the
coordinate functions on $(\mathbb C^3)^*$ relative to the basis dual
to $\overrightarrow{e_1}, \overrightarrow{e_2}, \overrightarrow{e_3}$.
Then the homogeneous coordinate ring of ${\cal I}$ is isomorphic to
$$
\frac{\mathbb C[x, y, z; \alpha, \beta, \gamma]}{(x \alpha + y\beta + z\gamma)}
$$
For a general commutative ring ${\cal R}$ the set of ${\cal R}$-points
$I({\cal R})\subset \mathbb P^2({\cal R})\times \mathbb P^2({\cal R})^{\vee}$
consists of pairs $(V, \alpha)$ such that $V\subset L= \ker(\alpha)$.

Pick a point $t\in \mathbb P^2 (\mathbb C)$. The {\em relative} incidence variety
${\cal I}(t)\subset  (\mathbb P^2)^{\vee}(\mathbb C)$ is given by the equation:
$$
\{ l\in  (\mathbb P^2)^{\vee}(\mathbb C) | l(t) =0\}
$$
By dualizing we define the {\em relative} incidence variety
${\cal I}(l)$ for any element $l \in (\mathbb P^2)^{\vee}(\mathbb C)$.
We define {\em anisotropic} incidence varieties
${\cal I}_0$, ${\cal I}_0(t)$ and ${\cal I}_0(l)$
by intersecting with $\mathbb P^2_0 \times (\mathbb P^2_0)^{\vee}$.
The proof of the following lemma is a straightforward calculation
and we leave it to the reader.

\begin{lem}
\label{is smooth}
For any $t$ and $l$ the varieties ${\cal I}$, ${\cal I}(t)$, ${\cal I}(l)$
are smooth.
\end{lem}

\begin{Not}
We will make the following convention about inhomogeneous
coordinates of points in $\mathbb P^2$: if $q= [x: y: 1]$ then we let $q:= (x, y)$,
if $q= [0: 1: 0]$ we let $q:= (0, \infty)$, if  $q= [1: 0: 0]$ we
let $q:= (\infty, 0)$ and if $q= [1:1:0]$ we let $q:= (\infty, \infty)$.
\end{Not}

\subsection{Projective  arrangements}
\label{Projective  arrangements}
A {\em geometric realization} of the abstract arrangement
$A= {\cal P} \sqcup {\cal L}$ is a map
$$
\phi : {\cal P} \sqcup {\cal L} \to \mathbb P^2(\mathbb C) \sqcup (\mathbb P^2(\mathbb C))^{\vee} $$
which sends {\em points} to points and {\em lines} to lines. This map must
satisfy the following condition:
\begin{equation}
\iota(v, l) \Rightarrow \quad [\phi(v) \in \phi(l) \iff \quad
\phi(l)^{\vee}\cdot  \phi(v) = 0\ ]
\end{equation}
The image of $\phi$ is a {\em projective  arrangement} in $\mathbb P^2$. Usually
we shall denote {\em lines} of arrangements $A$ by uncapitalized letters
($l, m, $ etc.), and their images under geometric realizations by
the corresponding capital letters ($L, M, $ etc.).

\begin{ex}
Consider the  standard triangle $T$. Define a geometric realization
$\phi_T$ of $T$ in $\mathbb P^2$ so that:
$$
\phi_T(v_{00})= (0,0), \phi_T(v_{x})= (\infty,0), \phi_T(v_{y})= (0,\infty),
\phi_T(v_{11})= (1,1)
$$
This realization uniquely extends to the rest of $T$.
See Figure \ref{Fig11}. We call
$\phi_T$ the {\em standard} realization of $T$.
\end{ex}

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure11.eps}}
\caption{\sl Standard realization of the standard triangle.}
\label{Fig11}
\end{figure}

\medskip
The {\em configuration space} of an abstract arrangement $A$ is the
space $R(A,\mathbb P^2(\mathbb C))$ of all geometric realizations of $C$.
The space $R(A,\mathbb P^2(\mathbb C))$
is the set of $\mathbb C$-points of  a projective variety $R(A)$
defined over $\mathbb Z$  with equations determined by the condition (1).

\begin{rem}
We will consider $R(A)$ as a variety over $\mathbb Q$.
\end{rem}

We now give a concrete description of the homogeneous coordinate ring
of $R(A)$ and the functor of points $R(A)(\bullet)$. Recall that $x, y, z$
be the coordinate functions on
$\mathbb C^3$ relative to the standard basis and $\alpha,\beta, \gamma$ are the
coordinate functions on $(\mathbb C^3)^*$ relative to the basis dual
to $\overrightarrow{e_1}, \overrightarrow{e_2}, \overrightarrow{e_3}$.
Now let $\Gamma$ be the (bipartite) graph corresonding to the abstract
arrangement $A$. We let $\{v_1,..., v_m\}$ denote
the vertices of $\Gamma$ corresponding to the {\em points} of $A$ and let
$\ell_1,..., \ell_n$ denote the vertices of $\Gamma$ corresponding to
the {\em lines} of $A$. Let $P$ be the polynomial ring defined by
$$
P:= \mathbb C[x_1, y_1, z_1, ..., x_m, y_m, z_m; \alpha_1, \beta_1, \gamma_1,..., \alpha_n,
\beta_n,\gamma_n]
$$
If $v_i$ and $\ell_j$ are incident we impose the relation $x_i \alpha_j +
y_i \beta_j + z_i \gamma_j=0$. The resulting quotient of $P$ is the
homogeneous coordinate ring of $R(A)$ which we denote $\mathbb C\, [R(A)]$.
We next describe the functor of points ${\cal R}\to R(A)({\cal R})$ where ${\cal R}$ is a
commutative ring. It follows from the description of $\mathbb P^2({\cal R})$ in
\mathbb S \ref{Intersection} that the set $R(A)({\cal R})$ is described by

\begin{lem}
\label{descr of points}
Let ${\cal R}$ be a commutative ring. Then an ${\cal R}$-point $\psi\in R(A)({\cal R})$
consists of an assignment of a {\em point} $V_i\in \mathbb P^2({\cal R})$ for each
{\em point}-vertex $v_i\in \Gamma$ and a {\em line} $L_j \in \mathbb P^2({\cal R})^{\vee}$
for each {\em line}-vertex $\ell_j\in \Gamma$ such that $\iota(v_i, \ell_j)$
implies $V_i\subset L_j$.
\end{lem}

\medskip
A {\em based realization} is a realization $\phi$ of a based
abstract arrangement such that the restriction of $\phi$ to
the canonically embedded triangle  $T$ is the {\em standard}
realization $\phi_T$.

The space of based realizations of an arrangement $C$ will be denoted
by  $BR(A, \mathbb P^2(\mathbb C))$. It also is the set of  $\mathbb C$-points of a projective
variety $BR(A)$ defined over $\mathbb Q$. Suppose that $A$ is a based arrangement
which has distinguished set of {\em points} $\nu=\{v_1,..., v_n\}$ which
lie on the {\em line} $l_x\in T$. We call $A_{\nu}$ a {\em marked} arrangement.
A {\em morphism of marked arrangements} is a mapping
$h: A_{\mu}\to B_{\nu}$
such that $h(\mu)\subset \nu$. Note that fiber sum of
two marked arrangements has the natural structure of a marked
arrangement.  If  $\nu=\{v_1,..., v_n\}$  is a marking
we define the space of {\em finite realizations} as
$$
BR_0(A_{\nu})= \{\psi\in BR(A): \quad \psi(v_j)\notin L_{\infty}\ ,
\quad v_j\in \nu \}
$$
Clearly $BR_0(A)$ is a quasi-projective subvariety in $BR(A)$.

\begin{lem}
The maps $R$, $BR$ and $BR_0$ define  contravariant functors from the
category of abstract arrangements, based abstract arrangements and
marked arrangements to the category of projective
and quasi-projective varieties defined over $\mathbb Q$.
\end{lem}
\par\medskip\noindent{\it Proof: } Obvious. $\qed$

We leave the description of the coordinate rings and functors of
points of $BR(C)$ and $BR_0(C)$ to the reader.

\begin{thm}
\label{ftof}
Under the functors $R$, $BR$ and $BR_0$ the operation of fiber
sum of arrangements corresponds to the operation of fiber
product of projective varieties and quasi-projective varieties.
\end{thm}
\par\medskip\noindent{\it Proof: } Note that fiber sum of arrangements fits into the
following commutative diagram of monomorphisms  of arrangements:
\[
\begin{array}{ccc}
~ & A & ~ \\
~ & \nearrow \quad  \quad \searrow & ~  \\
~& C \longrightarrow A\times_C B & ~\\
 ~ & \searrow \quad  \quad \nearrow & ~  \\
~ & B & ~ \\
\end{array}
\]
We shall consider the case of the functor $BR$, the other two
cases are similar. It  follows from the previous lemma that we
have commutative diagram of morphisms of varieties:
\[
\begin{array}{ccc}
~ & BR(A) &  ~\\
 ~ & \nearrow \quad \quad \quad \quad \searrow & ~ \\
~ & BR(A\times_C B) \joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightarrow  BR(C) & ~ \\
 ~ & \searrow \quad \quad \quad \quad \nearrow & ~  \\
~ & BR(B) & ~ \\
\end{array}
\]
Thus, by the universal property we get a  morphism
$$
\lambda: BR(A\times_C B) \to BR(A)\times_{BR(C)} BR(B)
$$
To see that $\lambda$ is an isomorphism we have only to check that it
induces bijections of ${\cal R}$-points, for each commutative ring ${\cal R}$
(see \cite[Proposition IV-2]{EH}).
This is clear by Lemma \ref{descr of points}. $\qed$

\subsection{The moduli space of a projective arrangement}

In this section we will construct a distinguished Mumford quotient
$R(A, \mathbb P^2(\mathbb C))/\!/ PGL(3,\mathbb C)$ which we will refer to as the
{\em moduli space} ${\cal M}(A, \mathbb P^2(\mathbb C))$ for a based arrangement
$A$. Since the equations defining
$R(A, \mathbb P^2(\mathbb C))$ are invariant under $PGL(3,\mathbb C)$ it suffices to construct a
Mumford quotient of
$$
(\mathbb P^2)^m \times ((\mathbb P^2)^{\vee})^n
$$
where $m$ is the number of {\em points} in $A$ and $n$ is the number of
{\em lines}.
The {\em Mumford quotient} $R(A, \mathbb P^2(\mathbb C))/\!/ PGL(3,\mathbb C)$ of
$R(A, \mathbb P^2(\mathbb C))$ will be then the subvariety of
the quotient variaty cut out by the incidence equations. In the
next section we will identify ${\cal M}(A, \mathbb P^2(\mathbb C))$ with $BR(A, \mathbb P^2(\mathbb C))$
which will be seen
to be a cross-section to the action of $PGL(3,\mathbb C)$ on an open subvariety
of $R(A, \mathbb P^2(\mathbb C))$.

To construct a (weighted) Mumford quotient of  $(\mathbb P^2(\mathbb C))^m \times
((\mathbb P^2(\mathbb C))^{\vee})^n$ we
need a projective embedding. Such an embedding corresponds to a choice
of polarizing line bundle over each factor of the product.
Since the group of isomorphism classes of line bundles on $\mathbb P^2(\mathbb C)$  is
infinite cyclic this amounts to assigning a positive integer weigth to
each factor (i.e. to each vertex of the graph $\Gamma$ of $A$). It will be
more convenient to assign positive rational weigths to each vertex,
then the integer weights are obtained by clearing the denominators.
We choose a small positive rational number $\epsilon$ and assign the
weight $\frac{1}{4}- \epsilon$  to each of the four {\em point vertices}
$v_{00}, v_{x}, v_{y}, v_{11}$ of $T\subset A$ and the weight $\epsilon$
to all other vertices. Let $W:= \{v_{00}, v_{x}, v_{y}, v_{11}\}$.

Note that all semistable configurations for
the four-point case are stable, see below.
Thus it is clear that if $\epsilon$ is small enough
the calculation of stable and semistable points will reduce to the
corresponding calculation for $PGL(3,\mathbb C)$ acting on the product of
four copies of $\mathbb P^2(\mathbb C)$ corresponding to the four {\em point}-vertices
described above. This calculation is well-known (see \cite{Newstead}).
A configuration is stable (resp. semistable) iff less than (resp.
no more than) $1/3$ of the total weight is concentrated on any point
and less than $2/3$ (resp. no more than $2/3$) of the total weight
is concentrated on any line. We obtain

\begin{lem}
\label{L8.6}
All semistable configurations in $R(A, \mathbb P^2(\mathbb C))$ are stable.
A realization
$\psi: A\to \mathbb P^2(\mathbb C) \sqcup (\mathbb P^2(\mathbb C))^{\vee}$ is stable if and only if
no three points of $\psi(W)$ lie on the same projective line in $\mathbb P^2(\mathbb C)$.
\end{lem}

\subsection{The moduli space of the standard triangle}

As an important example we consider the configuration space
and moduli space of the standard triangle $T$. First of all
we note that $R(T, \mathbb P^2(\mathbb C) )$ is not irreducible. Here is the reason.
Let $\phi_T\in R(T, \mathbb P^2(\mathbb C))$ be the standard realization. Then all
$\phi\in R(T, \mathbb P^2(\mathbb C))$ nearby are equivalent to $\phi_T$ under some
projective transformation. However there are other (degenerate)
realizations $\psi_d\in R(T, \mathbb P^2(\mathbb C))$.  Namely, send all the {\em points}
of $T$ to the origin $(0,0)\in {\mathbb A}^2\subset \mathbb P^2$. The triangle
$T$ has 6 {\em lines}, which can be sent to any 6 lines in $\mathbb P^2$
passing through $(0,0)$. This gives us a 6-parameter family
$F$ of degenerate realizations (which is the product of six
copies of $\mathbb P^1$). After we mod out by the stabilizer of
$(0,0)$ in $PGL(3, \mathbb C)$ we get  3-dimensional quotient. There
are several other components which interpolate between $\phi_T$
and $\psi_d$, namely when three of the {\em points}
$v_{00}, v_x, v_y, v_{11}$ belong to a common projective line. To
remedy the problem we consider the {\em Mumford quotient}
$R(T, \mathbb P^2(\mathbb C))/\!/ PGL(3, \mathbb C)$ where we assign  weights as in the
previous section.  Let $R_{ss}(T, \mathbb P^2(\mathbb C))$ be
the set of semi-stable points with respect to these weights.
Our definition of weights  prevents images  of any three of the
{\em points}  $v_{00}, v_x, v_y, v_{11}$ from belonging to a
common projective line in $\mathbb P^2(\mathbb C)$. It is clear that
$R_{ss}(T, \mathbb P^2(\mathbb C))= \ PGL(3, \mathbb C) \{\phi_T\}$. Thus we get

\begin{lem}
The weighted quotient ${\cal M}(T)(\mathbb C) = R(T, \mathbb P^2(\mathbb C))/\!/ PGL(3, \mathbb C)$
consists of a
single point which we can identify with the cross-section $\{\phi_T\}$
for the action of  $PGL(3, \mathbb C)$ on $R_{ss}(T, \mathbb P^2(\mathbb C))$.
\end{lem}

\noindent Suppose now that $A$ is a general based arrangement, we assign
weights as above. Then $BR(A, \mathbb P^2(\mathbb C))\subset
R_s(A, \mathbb P^2(\mathbb C))= R_{ss}(A, \mathbb P^2(\mathbb C))$. By Lemma \ref{L8.6},
$BR(A, \mathbb P^2(\mathbb C))$ is a cross-section to the action of $PGL(3, \mathbb C)$ on
$R_s(A, \mathbb P^2(\mathbb C))$. We obtain

\begin{thm}
\label{T8.7}
The inclusion $BR(A, \mathbb P^2)\hookrightarrow R(A)$ induces an isomorphism
$BR(A, \mathbb P^2(\mathbb C))$ $\cong {\cal M}(A, \mathbb P^2(\mathbb C))$ between projective varieties.
\end{thm}

\subsection{Functional arrangements}

Suppose that $A$ is a based marked arrangement
with the {\em marking} $\nu=\{v_1,..., v_n\}$ (see \mathbb S
\ref{Projective  arrangements}).
 We call the points in $\nu$ the {\em input} points and we shall assume
that $\nu \cap T= \emptyset$. We also suppose that $A$ has the second marking
$\mu= \{w_1,..., w_s\}$, which consists of distinct
{\em output} points  $\{w_1,..., w_s\}$. (The sets $\mu$ and
$\nu$ can intersect and we allow $\mu\cap T\ne \emptyset$.) Recall
that $\iota(v_i, l_x), \iota(w_j, l_x)$ for all $i, j$.

Define the {\em projection maps} from the spaces of finite
realizations of  $\Pi: BR_0(A_{\nu}) \to {\mathbb A}^n = \mathbb C^n$,
$\Delta: BR_0(A_{\nu}) \to \mathbb P^s$ by
$$
\Pi: \phi \mapsto (\phi(v_1),..., \phi(v_n))= (z_1,..., z_n)\in {\mathbb A}^n
$$
$$
\Delta: \phi \mapsto (\phi(w_1),..., \phi(w_s))= (y_1,..., y_s)\in \mathbb P^s
$$
(here we identify $L_x - \{(\infty, 0)\}$ with the affine line ${\mathbb A}$).

\begin{defn}
Suppose that arrangement $A$ above satisfies  the following axioms:

\begin{itemize}
\item {\bf (A2)} $BR_0(A_{\nu})\subset BR_0(A_{\mu})$, i.e.
$$
\psi(v_j)\notin L_{\infty} , 1\le j\le n\quad \Rightarrow \quad
\psi(w_i)\notin L_{\infty} , 1\le i\le s~.
$$
\item {\bf (A3)}  The projection $\Pi$ is a biregular isomorphism of
the variety $BR_0(A_{\nu})$ onto ${\mathbb A}^n$.
\end{itemize}
Such arrangement $A$ is called a {\bf functional} arrangement on
$n$ variables.
\end{defn}

Each functional arrangement  {\em defines}
 a vector-function $f: \mathbb C^n \to \mathbb C^s$ by
$$
f(z_1,..., z_n) := \Delta(\phi) = (\phi(w_1),..., \phi(w_s))$$
where $\phi\in BR_0(A_{\nu})$ corresponds to $(z_1,..., z_n)$ under
the map $\Pi$. We shall record this by writing $A= A^f$.
It's easy to see that the  vector-function $f$ must be polynomial.
Later on we will give some examples of functional arrangements
and we will prove that {\bf any} $m$-tuple of polynomials in
$\mathbb Z[x_1,..., x_n]$ can be defined by a functional arrangement.

\begin{lem}
The space $BR_0(A^f)$ is biregular isomorphic to the graph
$\Gamma_f$ of the function $f: {\mathbb A}^n \to {\mathbb A}^s$.
\end{lem}
\par\medskip\noindent{\it Proof: } Indeed, the natural projection $\pi: \Gamma_f \to {\mathbb A}^n$ is
an isomorphism. Compose it with the isomorphism $\Pi^{-1}$. The
result is the required isomorphism $\Gamma_f \to BR_0(A^f)$. $\qed$

\medskip
Now suppose that we are given two functional arrangements
$A^f , A^g$
which define the functions $f: \mathbb C^n \to \mathbb C^s$ and $g: \mathbb C^t \to \mathbb C$.
We denote the variables for $f$ by  $(z_1,..., z_n)$ and the variables for
$g$ by $(x_1,..., x_t)$. We assume that they correspond to the
{\em input points} $p_1,..., p_n$ and $q_1,..., q_t$ respectively.
Denote by $w_0\in A^g$ the {\em output
point}. We would like to find an arrangement which {\em defines} the
function $h= f(g(x_1,..., x_t), z_2,..., z_n)$. To do this we
let $A^h$ be the join $A^f *_{p_1\equiv w_0} A^g$.

\begin{lem}
\label{comp}
The arrangement $A^h$ is functional and it {\em defines} the polynomial
$h$ which is the above composition of the functions $f, g$.
\end{lem}
\par\medskip\noindent{\it Proof: } The Axiom (A3) for $A_h$ follows from the Axiom (A2)
for the arrangements $A_{f}, A_g$. The Axiom (A4) follows from
the fact that $BR_0(A^h)$ is the fiber product of the varieties
$BR_0(A^f)$, $BR_0(A^g)$, see Lemma \ref{fib}.  $\qed$

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure3.eps}}
\caption{\sl The abstract arrangement $C_M$ for multiplication.}
\label{Fig3}
\end{figure}

\section{Algebraic operations via   arrangements}
\label{Algebraic operations}

The following theorem goes back to the work of von Staudt \cite{St} in the middle of the last century:

\begin{thm}
\label{add}
There are admissible functional arrangements $C_A, C_M$ which {\em  define}
the functions
$$
A(z_1, z_2) =  z_1 + z_2~, \quad M(z_1, z_2) =  z_1 \cdot z_2$$
\end{thm}
\par\medskip\noindent{\it Proof: } Consider the functional arrangement
$C_M$ described on Figure \ref{Fig3}. We omitted from the
figure several (inessential) {\em lines} and {\em points} of the
standard triangle $T$, however we still assume that $C_M$ is a
{\em based arrangement}. A generic projective realization $\psi$ of
this arrangement is described on Figure \ref{Fig4}.
Then the point of intersection of the line $M_1:= \psi(m_1)$ and the
$x$-axis $L_x$ is equal to $ab$. (See \cite[page 45]{Hart}.)
The addition is {\em defined} via the abstract arrangement on the Figure
\ref{Fig5}. Generic  projective realization of $C_A$ is described
on the Figure \ref{Fig6}. (See \cite[page 44]{Hart}.)
We will prove that $C_M$ is a functional arrangement, leaving
similar case of $C_A$ to the reader. The Axioms {\bf (A1)}, {\bf (A2)}
are clearly satisfied by the arrangement $C_M$\ , it is also
easy to see that the morphism $\Pi: BR_0(C_M)\to {\mathbb A}^2$ is
a bijection of complex points.

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure4.eps}}
\caption{\sl Projective arrangement for multiplication.}
\label{Fig4}
\end{figure}

The problem is to prove that $\Pi$ is invertible
as a morphism. The example that the reader should keep in
mind is the following. Consider the identity map
$$
id: \{ x= 0: x\in \mathbb C\} \longrightarrow \{x^2= 0: x\in \mathbb C\}
$$
Then $id$ is a  morphism which is bijective on complex points but
not invertible as a morphism.

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure5.eps}}
\caption{\sl The abstract arrangement $C_A$ for  addition.}
\label{Fig5}
\end{figure}

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure6.eps}}
\caption{\sl Projective arrangement for addition.}
\label{Fig6}
\end{figure}

We will prove that $\Pi: BR_0(C_M)\to {\mathbb A}^2$ induces a bijection
$\Pi_R$ of ${\cal R}$-points for any commutative ring ${\cal R}$. This will
imply that $\Pi$ is an isomorphism by \cite[Proposition IV-2]{EH}.

Let $A_{\nu}$ be a based marked arrangement.
We first interpret the finiteness condition
$\psi(v_j)\notin L_{\infty}, v_j\in\nu$,
scheme-theortically for any ${\cal R}$-valued point $\psi\in BR_0(A)({\cal R})$.
We will construct a subfunctor $U\subset \mathbb P^2$ corresponding to the
affine plane $\mathbb P^2- L_{\infty}$. Let ${\cal R}$ be a commutative ring,
$\{e_1, e_2, e_3\}$ be the standard basis of ${\cal R}^3$ and let
$M\subset {\cal R}^3$ be a submodule defined by $M= {\cal R} e_1 + {\cal R} e_2$. We define
$U({\cal R})\subset \mathbb P^2({\cal R})$ by
$$
U({\cal R})= \{V \in \mathbb P^2({\cal R}) : V \ \ \hbox{is a complement to}\ \   M\}
$$
It is then immediate, see \cite[ \mathbb S 1.3.9]{DG}, that $V\in U({\cal R})$
implies that $V$ is free and contains a unique vector
(necessarily a basis) $u$ of the form
$u= x e_1 + y e_2 + e_3$. The map $\omega: U({\cal R})\to {\cal R}^2 + e_3$
given by $\omega(V)= (x, y,1)$ is a natural (with respect to
${\cal R}$) bijection. Consequently $U$ is represented by the polynomial
algebra $\mathbb Z[X,Y]$. By \cite{DG}, loc. cit., $U$ corresponds
to an open subvariety, again denoted $U$, of $\mathbb P^2$
isomorphic to ${\mathbb A}^2$ (by the previous sentence). We observe
that $U(\mathbb C)= P^2(\mathbb C) - L_{\infty}$. Thus $U$ gives us the
scheme-theoretic definition of $\mathbb P^2- L_{\infty}$.

We now give the scheme-theoretic definition of the space of
{\em finite} realizations $BR_0(A_{\nu})$ by requiring that
$\psi(v_j)\in U$, $v_j\in \nu$. In the case
$\psi\in BR_0(A_{\nu})({\cal R})$ we have
$$
\omega(\psi(v_j))= (x_j, y_j, 1), v_j \in \nu
$$
Moreover, since $\iota(v_j, l_x)$ we have: $y_j=0, v_j \in \nu$. Thus we
have associated a point $(x_1,..., x_n)\in {\cal R}^n$ to each
$\psi\in BR_0(A_{\nu})({\cal R})$. We will see in what follows that
the coordinates $(x_1,..., x_n)$ will play the same role as
complex or real variables in the classical arguments.

We are now ready to prove

\begin{prop}
$\Pi_{{\cal R}}: BR_0(C_M)({\cal R}) \longrightarrow {\cal R}^2$ is a bijection.
\end{prop}
\par\medskip\noindent{\it Proof: } Let $\psi\in BR_0(C_M)({\cal R})$. We will prove that $\psi$ is determined
by $\psi(v_1), \psi(v_2)$ and the incidence relations in $C_M$. If
$u= (x, y,z)$ we will use $[x: y : z]$ below to denote both the
element in $\mathbb P^2({\cal R})$ with the basis $u$ and the element in
$\mathbb P^2({\cal R})^{\vee}$ with the basis $u^{\vee}$; the meaning will be
clear from the context.

We now  prove the proposition. Let $\omega(\psi(v_1))= (a, 0, 1)$
and $\omega(\psi(v_2))= (b, 0, 1)$. Now
$\omega(\psi(v_1))\times (0, 1, 0)= (1, 0, -a)$.
Hence by the cross-product test $[1: 0: -a]$ is the unique
line joining $\psi(v_1)$ and $[0, 1, 0]$. But
$\psi(l_1)= \psi(v_1)\bullet L_y$, whence
$\psi(l_1)= [1: 0: -a]$. We continue in this way, at each stage the
cross-product test applies and we find in order
$$
\psi(v)= \psi(l_1)\bullet \psi(l_d)= [a: a: 1]
$$
$$
\psi(l_2)= \psi(v_{11})\bullet \psi(v_2)= [1: b-1: -b]
$$
$$
\psi(u)= \psi(l_2)\bullet \psi(l_{\infty})= [b-1: -1: 0]
$$
$$
\psi(m_1)= \psi(u)\bullet \psi(v)= [-1: 1-b: ab]
$$
$$
\psi(w_1)= \psi(m_1)\bullet \psi(l_x)= [ab: 0: 1]
$$
This concludes the proof of Theorem \ref{add} in the case of the
arrangement $C_M$. The argument in the case of $C_A$ is
similar and is left to the reader.   $\qed$

\begin{lem}
\label{min}
For the function $D(x, y)= x-y$ there is a functional
arrangement $C_D$ which defines the function $D(x, y)$.
\end{lem}
\par\medskip\noindent{\it Proof: } Take the arrangement $C_A$ corresponding to the function
$A(x, y)= x+y$ and reverse the roles of input-output {\em points}
$v_2, w_1$. $\qed$

\begin{rem}
\label{use}
Suppose that $A$ is one of the arrangements $C_A, C_M, X_D$ and \newline
$\psi\in BR(A,\mathbb P_2(\mathbb C))$  is a realization such that
$\Pi(\psi)= (0, 0)$, hence $\psi(w_1)= 0$, $\psi(m_1)= L_y$.
Then $\psi(A)\subset \phi_T(T)\cup \{(\infty, \infty)\}$.
(Recall that $\phi_T$ is the
standard realization of the standard triangle.) Let's
verify this for $C_M$. If $\psi(v_1)= 0, \psi(v_2)= 0$
then $\psi(l_1)= \psi(l_y), \psi(l_2)= \psi(l_d)$.
Hence $\psi(u)= (\infty,\infty)$ and $\psi(v)= (0,0)$.
\end{rem}

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure7.eps}}
\caption{\sl Projective arrangement for the constant function
$f_-(z)= -1$.}
\label{Fig7}
\end{figure}

\begin{lem}
\label{const}
There exist admissible functional arrangements $C^+$, $C^-$ which define
the constant functions
$$
f_+(z) = 1, f_- (z) = -1$$
\end{lem}
We describe configuration only for the function $f_-$, the
other case is similar. Consider the arrangement on the Figure
\ref{Fig7}, as usual we omit inessential lines and points
form the standard triangle. Let $v_1$ be the input and  $w_1$
be the output points.
Under each realization $\phi$ of $C^-$ the image $\phi(w_1)$
is the point $(-1, 0)$. The image of $l_1$ is any vertical line in ${\mathbb A}^2$. We
leave the proof of the fact that the arrangement $C^-$ is
functional to the reader. $\qed$

\begin{cor}
\label{polyn}
For any polynomial $f$ of $n$ variables with integer coefficients
there exists a functional arrangement $A^f$ which {\em defines} $f$.
\end{cor}
\par\medskip\noindent{\it Proof: } Any such polynomial is a composition of the constant functions $f_{\pm}$, addition and multiplication.
Thus the asssertion is a straightforward  application of Lemmas \ref{comp},  \ref{add}, \ref{min}, \ref{const} and Corollary \ref{ftof}.   $\qed$

\medskip
Now we  construct arrangements corresponding to polynomial
vector-functions defined over $\mathbb Z$.

\begin{lem}
\label{eq}
Suppose that $f_1,..., f_m \in \mathbb Z [x_1,..., x_n]$. Then there
exists an admissible abstract functional arrangement $A^F$ which {\em defines} the vector-function $F= (f_1,..., f_m)$.
\end{lem}
\par\medskip\noindent{\it Proof: } By Corollary \ref{polyn} there exist functional
arrangements $A^{f_1},..., A^{f_m}$ which {\em define} the polynomials
$f_j$. Let $v_{ij}$ be input point of $A^{f_i}$ corresponding to
the variable $x_j$, $1\le i\le m$, $1\le j \le n$. Let
$C= \{v_1,..., v_n\}$ be the arrangement which consists only of
{\em points} $v_j$ and has no {\em  lines}, $B= C \cup T$. For each
{\em point}  $v_j$ define $\psi_i : v_j \mapsto v_i$, this gives
us embeddings $\psi_i: C\hookrightarrow A^{f_i}$. Using these embeddings
define  $A= (A^{f_1}* ... * A^{f_m}) \times_{C} B$. In  the
arrangement $A$ we have $v_{ij}\equiv v_{kj}\equiv v_j$,
$1\le j, k\le n$, these are the {\em input points} of $A$.
The output points $w_1,..., w_m$ correspond to the {\em  output
points} of the functional arrangements  $A^{f_1},..., A^{f_m}$.
Then the fact that $A= A^F$ is the functional arrangement
{\em defining} the vector-function $F$ follows by iterated application
of Corollary \ref{C3.2} and Theorem \ref{ftof} similarly to the
proof of Lemma \ref{comp}. $\qed$

\section{Systems of polynomial equations}

Suppose that we have a system of polynomial equations defined
over $\mathbb Q$:
$$
\Sigma= \left\{ \matrix f_1(x)= 0\cr f_2(x)= 0\\ \vdots\\ f_m(x)= 0 \endmatrix
\right.
$$
where $x= (x_1,..., x_n)$, $x_j \in \mathbb C$. These equations determine an affine
variety $S\subset {\mathbb A}^n$ defined over $\mathbb Q$. In the previous section we had
constructed a functional arrangement $A= A^F$ which {\em  defines}
the vector-function $F= (f_1,..., f_m)$.  Recall that we have
two projection morphisms
$$
\Pi: BR_0(A, \mathbb P^2)\to {\mathbb A}^n\ , \quad\Delta: BR_0(A, \mathbb P^2)\to {\mathbb A}^m
$$
so that the diagram of morphisms
\[
\begin{array}{ccc}
BR_0(A, \mathbb P^2) & \stackrel{\Pi}{\longleftrightarrow} & \mathbb C^n\\
\Delta {\big\downarrow} & ~ & F{\big\downarrow} \\
\mathbb C^m & \= & \mathbb C^m\\
\end{array}
\]
is commutative. By the Axiom (A3) the projection $\Pi$ is an
isomorphism. Let $T\subset A$ be the standard triangle,
$w= v_{00}$ be its vertex. Define the new abstract arrangement
$A^{\Sigma}$ as the join
$$
A^{\Sigma}=(\ldots ((A*_{w_1\equiv w} T)*_{w_2\equiv w}T)
\ldots *_{w_m\equiv w}T)
$$
Then $BR_0(A^{\Sigma})$ is the fiber product
$\{\psi \in BR_0(A, \mathbb P^2): \Delta(\psi)= 0\}$
and $BR_0(A^{\Sigma}) \cong \{x \in {\mathbb A}^n : F(x)=0\}$, where the
isomorphism is given by the restriction $\tau$ of $\Pi$
to $BR_0(A)$. Thus we get the following

\begin{thm}
\label{systems}
For any system of polynomial equations with integer coefficients
$$
\Sigma=\left\{ \matrix f_1(x)= 0\cr f_2(x)= 0\cr \vdots\cr f_m(x)= 0
\endmatrix \right.
$$
there is an admissible based  arrangement $A= A^{\Sigma}$ such that
$\tau: BR_0(A, \mathbb P^2)\cong S$
is an isomorphism of quasi-projective varieties over $\mathbb Q$.
\end{thm}

\begin{defn}
We call the morphism $geo= \tau^{-1}$ {\bf geometrization}: it allows us to do
algebra  (i.e. solve the system $\Sigma$ of algebraic equations) via geometry
(i.e. by studying the space of projective arrangements).
\end{defn}

Note that the arrangement $A= A^{\Sigma}$ is not uniquely determined by the affine
variety $S$ but also by its affine embedding
(the system of polynomial equations) and particular formulae used to describe
these equations. For instance, the equation $x^5=0$ can be described as
$(x^2 \cdot x^2)\cdot x$ as well as $x\cdot (x^2 \cdot x^2)$ and
$(x\cdot (x^2))\cdot x^2$, etc.

\medskip
Suppose that the system of equations $\Sigma$ is defined over $\mathbb Z$ and
has no constant terms. Then
the only subarrangements involved in construction of the arrangement
$A^{\Sigma}$ are arrangements for multiplication, addition and subtraction
(described in Lemmas  \ref{add}, \ref{min}) and we do not need
arrangements for the constant functions $\pm 1$. Let
$\psi_0\in BR(A, \mathbb P^2)$ be the realization corresponding to the point
$0\in S$ under the isomorphism $\tau$. Take any {\em  line} $l\in A- T$ and a
{\em  point} $v\in A-T$. Then by using Remark \ref{use} and the fact
that arrangement for the composition of functions is a join of their
arrangements (so it has no new points or lines) we conclude
that the following holds:

\begin{lem}
\label{noco}
$\psi_0(l)$ is one of the {\em lines} $L_x, L_y, L_d$
and $\psi_0(v)$ is one of the {\em points}
$$(0,0)\ , (0,\infty)\ , (\infty, 0)\ , (\infty, \infty)$$
for each {\em line} $l\in A- T$  and each {\em point} $v\in A- T$.
\end{lem}

Suppose now that $Q$ is an affine variety defined over $\mathbb Q$ and $q\in Q$ is
a rational point. Then we can realize $Q$ as an affine subvariety
$S \subset {\mathbb A}^N$ defined over $\mathbb Z$ so that $q$ goes to zero. Hence we get the following

\begin{cor}
\label{csystems}
For any affine variety $Q$ defined over $\mathbb Q$ and
a rational point $q\in Q$ there exists an abstract admissible arrangement
$A$ and an isomorphism\footnote{of varieties defined over $\mathbb Q$}
$geo: Q \to BR_0(A)$ so that the point $q$ corresponds to a realization
$\psi$ so that  $\psi(l)$ is one of the {\em lines} $L_x, L_y, L_d$
and $\psi(v)$ is one of the {\em points}
$$(0,0)\ , (0,\infty)\ , (\infty, 0)\ , (\infty, \infty)$$
for each {\em line} $l\in A- T$  and each {\em point} $v\in A- T$.
\end{cor}

\section{Groups corresponding to abstract arrangements}
\label{gaa}

We will define several classes of groups corresponding to
abstract arrangements. Let $\Gamma$ be the bipartite graph corresponding to
an abstract arrangement $A$.
We first construct  the Coxeter group $G^c_A$ without assuming
that $A$ is a based arrangement: we assign the label $2$ to all edges of
 $\Gamma$ and let $G^c_A:= G^c_{\Gamma}$.

From now on we suppose that $A$ is a based arrangement.
We start by identifying the {\em point} $v_{00}$ with the {\em line}
$l_{\infty}$, the {\em point} $v_x$ with the {\em line} $l_y$ and the {\em point}
$v_y$ with the {\em line} $l_x$ in the standard triangle $T$. We also
introduce  the new edges
$$
[v_{10}, v_{00}], \quad [v_{01}, v_{00}], \quad [v_{11}, v_{00}]
$$
(Where $v_{10}, v_{00}, v_{11}, v_{01}$ are again {\em points} in the standard triangle $T$.) We will use  the notation $\Lambda$ for the resulting graph.
Put the following labels on the edges of $\Lambda$:

1) We assign the label $4$ to the edges $[v_{10}, v_{00}],
[v_{01}, v_{00}]$ and all the edges which contain $v_{11}$
as a vertex (with the exception of $[v_{11}, v_{00}]$). We put
the label $6$ on the edge
$[v_{11}, v_{00}]$.

2) We assign the label $2$ to  the rest of the edges.

\noindent Let $\Delta:= \Lambda- [v_{11}, v_{00}]$.
Now we have  labelled graphs and we use the procedure
from the Section \ref{groups} to construct:

\medskip
(a) The {\em Artin group} $G_A^a:= G_{\Lambda}^a$\ .

\medskip
(b) We assign the label $3$ to the vertex $v_{11}$ and labels
$2$ to the rest of the vertices. Then we get the {\em Shephard}
group $G_A^s:= G_{\Lambda}^s$\ .

\medskip
\noindent We will denote generators of the above groups $g_v, g_l$,
where $v, l$ are elements of $A$ (corresponding to vertices of
$\Lambda$).

\begin{rem}
Suppose that $A$ is a based arrangement. Then all the group
$G_A^s$ admit an epimorphism onto free products
of at least 3 copies of $\mathbb Z/2$ and the group $G^a_A$ has an epimorphism
onto a free group of rank $r\ge 3$, where $r+3$ is the number of {\em lines}
in $A$. Let's check this for $G= G_A^a$. Construct a new arrangement
$B$ by removing all the {\em points} in $A$ (and the {\em lines}
$l_x, l_y, l_{\infty}$). Then
$G_B^s\cong \mathbb Z * ...* \mathbb Z$ is the $r$-fold free product,
where $r\ge 3$ is the number of lines in $A- \{l_x, l_y, l_{\infty}\}$.
It's clear that we have an epimorphism $G_A^a \to G_B^a$. Hence
all the groups   $G_A^a, G_A^s$  are not virtually solvable.
Suppose that $G^a_A$ is the fundamental group of a smooth
complex quasi-projective surface $X$. Then it is known
that $X$
admits a nonconstant holomorphic mapping onto a Riemann surface
$Y$ of negative Euler characteristic.
\end{rem}

As an illustration we describe an example of a labelled graph
corresponding to the based functional arrangement {\em defining} the
function $x\mapsto x^2$, see the Figure \ref{Fig10}.

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=3.5in \epsfbox{figure10.eps}}
\caption{\sl Labelled graph of the functional arrangement for the
function $x^2$. Identify vertices with the same labels. The {\em point}
$v$ is the ``input'', the {\em point} $w$ is the ``output''. }
\label{Fig10}
\end{figure}

\section{Representations associated with projective arrangements}
\label{S14}

This section is in a sense the heart of the paper.
We start with an
outline of the main idea behind it\footnote{Certain versions of this idea
were used previously in our papers \cite{KM1}, \cite{KM2}, \cite{KM3}.}.
A projective arrangement $\psi$  is {\it anisotropic}
of $\psi (v) \in \mathbb P^2_0, \psi (l) \in (\mathbb P^2_0)^{\vee}$,
all points and lines $v, l\in A$. The anisotropic condition
defines Zariski open subsets
of the previous arrangement varieties to be denoted $R (A, \, \mathbb P^2_0),
BR(A, \mathbb P^2_0)$ and $BR_0(A, \mathbb P^2_0)$ respectively.
 As we already saw in the Section
\ref{S3}, the correspondence between involutions in $\mathbb P^2$ and their
isolated fixed points is a biregular isomorphism between $PQ_0$ and
$\mathbb P^2_0$: any point $x$ in $\mathbb P^2_0$ uniquely determines the
``Cartan involution'' around this point, i.e. the involution such
that $x$ is the isolated fixed point. The point $x= (1,1)$ in
${\mathbb A}^2\subset \mathbb P^2$
also determines (uniquely up to inversion) the
rotation of the order 3 around $x$, so that $x$ is the neutral fixed
point. (We will choose one of these rotations once and for all.)
 Similarly any line $L\in (\mathbb P^2_0)^{\vee}$  uniquely
determines the reflection which keeps  $L$ pointwise fixed. Finally we
encode the incidence relation between points and lines in $\mathbb P^2$
using algebra: two involutions generate the subgroup
$\mathbb Z/2\times \mathbb Z/2$ in $PO(3)$ iff the isolated fixed point
of one belongs to the fixed line of another, rotations of the
orders $2$ and $3$  anticommute iff the neutral fixed point
of the rotation of the order $3$ belongs to the fixed line of the
involution, etc. Thus, given a geometric object (projective
arrangement) we can constructs an algebraic object (a representation
of the associated Shephard group). We call the
 mapping
$$
alg: \hbox{~~arrangements~~} \longrightarrow \hbox{~~representations}
$$
{\em algebraization}. This mapping is the key in passing from
realization spaces of projective arrangements to representation
varieties. The fact that this correspondence is a homeomorphism
between sets of $\mathbb C$-points of
appropriate subvarieties will be more or less obvious,
however we will prove more: the {\em algebraization} is a
biregular isomorphism of certain (quasi-) projective varieties,
the latter requires more work.

\subsection{Representations of Shephard groups}
\label{Representations of Shephard groups}

Let $A$ be an abstract based arrangement with the graph $\Gamma_A$
and $G^s_A$ be the corresponding Shephard group with the graph $\Lambda_A$.
For all edges $e$ in the graph $\Lambda_A$
the edge subgroups $G_e\subset G^s_A$ are finite. Recall that in
Section \ref{appli}
we had defined the space $\mbox{\rm Hom}_f(G, PO(3, \mathbb C))$ of nondegenerate
representations of Shephard groups $G$. Thus as a direct
corollary of Proposition \ref{P6} we get

\begin{cor}
\label{zaropen}
$\mbox{\rm Hom}_f(G^s, PO(3,\mathbb C))$ is Zariski open and closed in $\mbox{\rm Hom}(G^s, PO(3))$.
\end{cor}

The significance of nondegenerate representations of Shephard groups
is that they correspond to projective arrangements under the ``algebraization''
morphism $alg$.

\begin{defn}
Suppose that $\rho\in \mbox{\rm Hom}_f(G^s, PO(3,\mathbb C))$ is a representation,
then we associate a projective arrangement
$\psi= \alpha(\rho)\in R(A, \mathbb P^2_0(\mathbb C))$ as follows:

(a) If $v\in A$ is a point then we let $\psi(v)$ be the neutral fixed point
of $\rho(g_v)$.

(b) If $l\in A$ is a line then we let $\psi(l)$ be the fixed line
of the involution $\rho(g_v)$.
\end{defn}

Clearly $\psi(v)\in \mathbb P^2_0$ and $\psi(l)\in (\mathbb P^2_0)^{\vee}$.
Now we verify that $\psi$ respects the incidence relation.
Suppose that $e= [v, w]$ is an edge in $\Lambda_A$ where $v, w, e$
have the label $2$. Then,
$\rho\in  \mbox{\rm Hom}_f(G_e, PO(3,\mathbb C))$ implies that $\rho(g_v), \rho(g_w)$
{\em anticommute} and hence $\psi(v)\cdot \psi(l)= 0$ (see
\mathbb S \ref{Commuting}). All other edges $e= [v, l]$ in $\Lambda_A$ are labelled
as: $\delta(v)=3, \delta(l)=2, \epsilon(e)= 4$, thus $\rho\in
\mbox{\rm Hom}_f(G_e, PO(3,\mathbb C))$ again implies that
$\rho(g_v), \rho(g_l)$ {\em anticommute}:
$$
(\rho(g_v)\rho(g_l))^2 = \hbox{\bf 1}
$$
and hence $\psi(v)\cdot \psi(l)= 0$ (see \mathbb S  \ref{Commuting}).
Note that the mapping
$$
\alpha: \mbox{\rm Hom}_f(G^s, PO(3,\mathbb C)) \to R(A, \mathbb P^2_0(\mathbb C)), \quad
\rho\mapsto \psi= \alpha(\rho)
$$
is  2-1. Namely, we can modify any representation $\rho \in
\mbox{\rm Hom}_f(G^s, PO(3,\mathbb C))$ by taking
$\rho_-(g_{v_{11}}):= \rho(g_{v_{11}})^{-1}$ and
$\rho_-(g_{w}):= \rho(g_{w})$ for all $w\in V(\Gamma)- \{v_{11}\}$, then
$\alpha(\rho)= \alpha(\rho^-)$. We denote the involution $\rho \mapsto \rho_-$ by
$\omega$. The mapping  $\alpha$ is far from being onto $R(A, \mathbb P^2_0(\mathbb C))$
because of extra identifications and edges in the graph $\Lambda$
(comparing to the $\Gamma_A$). The mapping $alg$ will be the
right-inverse to $\alpha$ if we restrict the targent of $\alpha$
to {\em based realizations}:

\begin{defn}
Suppose that $A$ is an abstract based arrangement,
$\psi\in BR(A, \mathbb P^2_0)$ is a realization.\footnote{In particular
$\psi(v_{00})= \psi(l_{\infty})^{\vee}, \psi(v_x)=
\psi(l_y)^{\vee}, \psi(v_y)= \psi(l_x)^{\vee}$.}
We construct a homomorphism
$$alg(\psi)= \rho_{\psi}: G^s_A\to PO(3,\mathbb C)$$
as follows.
If $v$ is a {\em point} in $A- \{v_{11}\}$, let $\rho(g_v)$ be the
rotation of order $2$ in $\mathbb P^2$ with the neutral fixed point
$\psi(v)$. (Such rotation exists since $\psi(v)$ is anisotropic.)
If $v=l$ is a {\bf line} in $A$ we let $\rho(g_v)$ be the
reflection in the line $\psi(l)$ (equivalently this is
the rotation of order $2$ with the isolated fixed  point
$\psi(l)^{\vee}\in \mathbb P^2_0$. For the vertex
$v= v_{11}$   we take the rotation of
the order $3$ around the point $(1,1)$.
There are two such rotations, so we shall
choose $\rho(g_v): \psi(v_{00})\mapsto \psi(v_x)$. We call the
mapping $alg: \psi \mapsto \rho_{\psi}$,  {\bf algebraization}.
\end{defn}

Below we verify that $\rho$ respects relations in $G^s$ and determines
 a nondegenerate representation.  For all $v\in {\cal P}- \{v_{11}\}\subset A$
and each {\em line} $l \in {\cal L} \subset A$ we have
$$
\psi(v)\in \psi(l) \Rightarrow  [\rho(g_l),\rho(g_v)]=\hbox{\bf 1}, \quad
\rho(g_l)\ne \rho(g_v)
$$
Hence all the relations in $G^s$ corresponding to the
edges labelled by $2$ are preserved by $\rho$ and $\rho$ is
{\em nondegenerate} on such edges. Since the spherical distances
between the points $(0,0)$ and $(1, 0)$, and points
$(0,0)$ and $(0, 1)$ in $\mathbb R\mathbb P^2$ equal $\pi/4$ we conclude that
$$
(g_{v_{00}} g_{v_{10}})^4= \hbox{\bf 1}~~, \quad (g_{v_{00}}
g_{v_{01}})^4= \hbox{\bf 1} , (g_{v_{00}} g_{v_{10}})^2\ne \hbox{\bf 1}, (g_{v_{00}}
\ne g_{v_{01}})^2\ne \hbox{\bf 1}
$$
This implies that the Artin relations for the edges $[v_{00}, v_{10}],
[v_{00}, v_{01}]$ are preserved by $\rho$ and $\rho$ is nondegenerate
on these edges.

For each {\em line} $l$ incident to the {\em point}
$v_{11}$ the rotations $\rho(g_l)$ and $\rho(g_{v_{11}})$
anticommute by Lemma \ref{anticom} and  we get:
$$
\rho(g_{l}) \rho(g_{v_{11}}) \rho(g_{l})
\rho(g_{v_{11}})= \hbox{\bf 1}~,\quad
\rho(g_{v_{11}}) \rho(g_{l}) \rho(g_{v_{11}})
\rho(g_{l}) = \hbox{\bf 1}
$$
Hence the  Artin relations for the edges $[l, v_{11}]\subset
\Lambda$ are preserved by $\rho$.
It's  easy to check that the order of the element
$\rho(g_{v_{00}} g_{v_{11}})$ equals $3$. Recall that the
edge $e= [v_{00}, v_{11}]$ in $\Lambda$ has the label $6$.
Thus  the relation
$$
(g_{v_{00}} g_{v_{11}})^3 = (g_{v_{11}} g_{v_{00}})^3
$$
associated with the edge $e$ is preserved by $\rho$. Thus we have proved

\begin{prop}
The  mapping $alg: BR(A, \mathbb P^2_0(\mathbb C)) \to \mbox{\rm Hom}_f(G^s_A, PO(3,\mathbb C))$ is
such that $\alpha \circ alg = id$. The space $BR(A, \mathbb P^2_0(\mathbb C))$ lies in
the image of the mapping $\alpha$.
\end{prop}

Let $Res_T: \mbox{\rm Hom}_f(G^s_A, PO(3,\mathbb C)) \to \mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))$
be the restriction homomorphism. Define the varieties
$$
\mbox{\rm BHom}_f(G^s_A, PO(3,\mathbb C)):= Res_T^{-1} \{\rho_{\phi_T} , \omega( \rho_{\phi_T})\}
$$
$$
\mbox{\rm BHom}_f^+(G^s_A, PO(3,\mathbb C)):= Res_T^{-1} \{\rho_{\phi_T}\}
$$
of {\em based representations}. Clearly $\mbox{\rm BHom}_f^+(G^s_A, PO(3,\mathbb C))$ is
the image $alg(BR(A, \mathbb P^2_0(\mathbb C)))$ (as a set) and the mapping
$$alg: BR(A, \mathbb P^2_0(\mathbb C)) \to \mbox{\rm BHom}_f^+(G^s_A, PO(3,\mathbb C))$$
 is a bijection.

\begin{lem}
\label{fina}
The representation $\rho= \rho_{\phi_T}: G_T^s \to PO(3,\mathbb R)\subset
PO(3,\mathbb C)$ corresponding
to the canonical realization $\phi= \phi_T$ of the standard triangle
$T$ has finite image.  The centralizer of the group $\rho(G^s_T)$ in
$PO(3,\mathbb C)$ is trivial.
\end{lem}

\par\medskip\noindent{\it Proof: } It is clear that the group $\rho(G^s_T)$ has invariant
finite set
$$\Sigma = \{(1,1), (-1, 1), (-1, -1), (1, -1)\}\subset
\mathbb P^2(\mathbb R)\subset \mathbb P^2(\mathbb C)\quad .$$
 Any three distinct points of $\Sigma$ do not lie on a
projective line in
$\mathbb P^2(\mathbb C)$, thus if $g\in PO(3, \mathbb C)$ fixes $\Sigma$ pointwise
then $g= id$. This implies finiteness of $\rho(G^s_T)$. It is
easy to check
that $\rho(G^s_T)$ equals $A_4$, the alternating group of
the 4-element set $\Sigma$. Any element of $g\in PO(3, \mathbb C)$
centralizing $\rho(G^s_T)$ must
fix $\Sigma$ pointwise, hence $g=\hbox{\bf 1} $. $\qed$

\begin{prop}
\label{lorigid}
The group $PO(3,\mathbb C)$ acts
simply-transitively by conjugations on the set $\mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))$.
\end{prop}
\par\medskip\noindent{\it Proof: }
Suppose that $\rho\in \mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))$. We note that $G^s= G^s_T$ contains the abelian subgroup $\mathbb Z/2 \times \mathbb Z/2 \times \mathbb Z/2$ generated by
the involutions $g_{v_{00}}, g_{v_{x}}, g_{v_{y}}$.
Since $\rho$ is nondegenerate
we conclude that  the restriction of $\rho$ to this subgroup is
injective. Thus, by the classification of finite subgroups of
$PO(3, \mathbb C)$, we conclude that $\rho$ can be conjugate to a
representation (which we again denote by $\rho$) so that
the projective arrangement
$\psi= \alpha(\rho)$ has the property:
$$
\psi(v_{00})= (0,0), \psi(v_{x})= (0, \infty), \psi(v_{y})= (\infty , 0)
$$
Thus necessarily: $\psi(l_x)= L_x, \psi(l_y)= L_y, \psi(l_{\infty})= L_{\infty}$.
Let $G_{10}$ denote the subgroup of $G^s$ generated by
$g_{v_{00}}, g_{v_{y}}= g_{l_{x}}, g_{v_{10}}$. The restriction of
$\rho$ to $G_{10}$ factors through the finite Coxeter group
$G_{10}^c= G_{10}/\<\< (g_{v_{00}} g_{v_{10}})^2\>\>$.
There are only two homomorphisms \newline
$\rho^c: G_{10}^c \to PO(3, \mathbb C)$
such that
$$
\rho^c(g_{v_{00}})= \rho(g_{v_{00}}), \rho^c(g_{v_{y}})= \rho(g_{v_{y}}),
\rho^c(g_{v_{10}})\ne  \rho^c(g_{v_{00}}) , \rho^c(g_{v_{10}})\ne
\rho^c(g_{v_{y}})
$$
Namely, for one of them the isolated fixed point of $\rho^c(g_{v_{10}})$
has the affine coordinates $(1,0)$, for the second it has the affine
coordinates $(-1,0)$. The reflection in the line $L_y$ conjugates
one representation to the other and commutes with the elements
$\rho(g_{v_{00}}), \rho(g_{v_{x}}), \rho(g_{v_{y}})$. Thus, after
adjusting $\rho$ by this conjugation (if necessary), we conclude that
\newline
$\alpha(\rho)= \psi:  v_{10} \mapsto (1, 0)$. Similar argument works for the vertex $v_{01}$.
It remains to determine $\psi(l_{x1}), \psi(l_{y1}), \psi(v_{11})$ and $\psi(l_d)$. Since $\rho(g_{l_{x1}})$ commutes with the elements $\rho(g_{v_{01}})$ and $\rho(g_{v_{\infty 0}})$ and does not coinside with either, the line $\psi(l_{x1})$ contains the points $(0,1)$ and $(\infty, 0)$. Similarly the line $\psi(l_{y1})$ contains $(0,1)$ and $(0, \infty)$.

We next determine $\psi(v_{11})$. Since the edge $[v_{11}, l_{y1}]$ has label 4, the elements $\rho(g_{v_{11}})$ and $\rho(g_{l_{y1}})$ anticommute, so $\psi(v_{11}) \in L_{y1}$ (the line in $\mathbb P^2$ joining $(1,0)$ and $(0,\infty)$. Similarly $\psi(v_{11})\in L_{x1}$. Finally we determine $\psi(l_d)$. Since $\rho(g_{l_d})$ commutes with $\rho(g_{v_{00}})$ we have $(0,0)\in \psi(l_d)$. Also $\rho(g_{l_d})$ anticommutes with $\rho(g_{v_{11}})$, so $(1,1) \in \psi(l_d)$. We conclude that
$$
\psi= \alpha(\rho)= \phi_T
$$
and either $\rho= \rho_{\phi_T}$ or  $\rho= \omega (\rho_{\phi_T})$.
It now follows from Lemma \ref{fina} that the action of the group
$PO(3,\mathbb C)$ on $Hom_f(G^s_T, PO(3,\mathbb C))$ is free. $\qed$

\begin{cor}
$\mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))$ equals the $Ad PO(3,\mathbb C)$-orbit of the
set\newline
  $\{\rho_{\phi_T}, \omega (\rho_{\phi_T})\}$.
In particular, both $\rho_{\phi_T}$ and $\omega (\rho_{\phi_T})$ are
locally rigid.
\end{cor}

\noindent Note however that this corollary doesn't apriori imply
that the variety $\mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))$ is smooth since it
could be nonreduced. To prove smoothness we need the following:

\begin{prop}
\label{C11.3}
The representations $\rho_{\phi_T}, \omega (\rho_{\phi_T}):
G_T^s \to PO(3,\mathbb C)$ are
infinitesimally rigid.
\end{prop}

We will prove the proposition only for $\rho_{\phi_T}$, the
second case easily follows. Proposition \ref{C11.3} will
immediately follow from the more general

\begin{prop}
\label{rigid triangle}
Let $G^a= G_T^a$, define $\tilde\rho: G^a \to PO(3,\mathbb C)$ by composing
$\rho_{\phi_T}$ with the canonical projection  $G_T^a\to G^s_T$. Then
the representation $\tilde\rho$ is infinitesimally rigid.
\end{prop}
\par\medskip\noindent{\it Proof: } Our proof is based on the results of Section \ref{elem}. The reader
will notice that the proof follows the lines of the proof of
Proposition \ref{lorigid}. We first consider the subgroup
$F$ in $G^a$ generated by $g_{v_{00}}, g_{l_x}, g_{l_y}$, these
generators mutually commute, hence the subgroup is abelian.
Let $\sigma\in Z^1(G^a, ad\circ\tilde\rho)$. By Lemma \ref{L5.3} the
restriction of $\sigma$
to each cyclic subgroup $\<g_{v_{00}}\>, \<g_{l_x}\>, \<g_{l_y}\>$
is cohomologically trivial, thus $\sigma|_F$ comes from a cocycle on
the finite Coxeter group
$$
F/\<\<\ g_{v_{00}}^2\ , \ g_{l_x}^2 \ ,\  g_{l_y}^2\ \>\>
$$
which implies that $\sigma|_F$ is a coboundary. By adjusting the
cocycle $\sigma$ by a coboundary we may assume that $\sigma|_F =0$.
Now we consider the subgroup $F_{10}$ generated by
$g_{l_x}, g_{v_{10}}$. Because of the Artin relations in
$G^a_T$ this is again an abelian subgroup whose image under
$\tilde\rho$ is dihedral. Hence $\sigma|_{F_{10}}$ is also a
coboundary. Let $H_{10}$ denote the subgroup  generated by elements of
$F_{10}$ and $g_{v_{00}}$, we recall that
$(g_{v_{00}} g_{v_{10}})^2= (g_{v_{10}}g_{v_{00}} )^2$.

Since the restriction of $\sigma$ to each generator of
$H_{10}$ is exact, the cocycle $\sigma|_{H_{10}}$ comes
from the finite Coxeter group
$H_{10}/\<\< g_{v_{00}}^2, g_{l_x}^2, g_{v_{10}}^2\>\>$.
This implies that $\sigma|_{H_{10}}$ is a
coboundary. Since $\sigma(g_{l_x})=0, \sigma( g_{v_{00}})=0$
and the fixed points of $\tilde\rho(g_{l_x} ), \tilde\rho(g_{v_{00}} )$
are distinct we conclude  that $\sigma|_{H_{10}}=0$.
The same argument implies that $\sigma(g_{v_{01}})=0$. We
repeat our argument for the two abelian subgroups
generated by $g_{l_x}, g_{v_{10}}, g_{l_{y1}}$ and
by $g_{l_x}, g_{v_{01}}, g_{l_{x1}}$, it follows
that $\sigma( g_{l_{y1}})= 0, \sigma( g_{l_{x1}})= 0$.

Then we use Lemma \ref{L5.10}, where $a_1= g_{l_{x1}},
a_2= g_{l_{y1}}, a_3=
g_{v_{00}}, b= g_{v_{11}}$, to conclude that
$\sigma(g_{v_{11}})=0$. Finally $\sigma(g_{l_{d}})=0$
since the Shephard subgroup of $G^s$ generated by
$g_{v_{11}},  g_{l_{d}}, g_{v_{00}}$ is again finite. $\qed$

\begin{cor}
\label{cross-sec}
The variety $\mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))$ is smooth. The variety
\newline
$\mbox{\rm BHom}_f(G^s_T, PO(3,\mathbb C))$ is a (scheme-theoretic) cross-section for
the action of $PO(3,\mathbb C)$ by conjugation  on the variety
$\mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))$.
\end{cor}
\par\medskip\noindent{\it Proof: } Smoothness of $\mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))$ follows from
infinitesimal rigidity of the representations
$\rho_{\phi_T}, \omega \rho_{\phi_T}$, see \cite[Theorem 2.6]{LM}.
In Proposition
\ref{lorigid} we proved that the morphism
$$
\lambda: PO(3,\mathbb C)\times  \mbox{\rm BHom}_f(G^s_T, PO(3,\mathbb C)) \hookrightarrow
\mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))
$$
given by the action of $PO(3,\mathbb C)$ by conjugation, is a bijection.
Thus $\mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))$ is also smooth, which implies
that the morphism $\lambda$ is actually an isomorphism of varieties.
$\qed$

\begin{cor}
\label{cro-section}
The variety $\mbox{\rm BHom}_f(G^s_A, PO(3,\mathbb C))$ is a (scheme-theoretic)
cross-section for the action
of $PO(3,\mathbb C)$ by conjugation  on the variety $\mbox{\rm Hom}_f(G^s_A, PO(3,\mathbb C))$.
\end{cor}
\par\medskip\noindent{\it Proof: } Consider the $Ad (PO(3,\mathbb C))$-equivariant restriction morphism
$$
Res_T: \mbox{\rm Hom}_f(G^s_A, PO(3,\mathbb C)) \longrightarrow \mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))
$$
It was proven in Corollary \ref{cross-sec} that the subvariety
 $\mbox{\rm BHom}_f(G^s_T, PO(3,\mathbb C))$ is a cross-section for the action of $PO(3,\mathbb C)$
by conjugation on $\mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C))$. Thus the pull-back variety
$$
Res^{-1}_T \mbox{\rm Hom}_f(G^s_T, PO(3,\mathbb C)) \equiv  \mbox{\rm BHom}_f(G^s_A, PO(3,\mathbb C))
$$
is a cross-section as well. $\qed$

\medskip
Our goal is to show that the mapping
$alg: BR(A, \mathbb P^2_0(\mathbb C)) \to \mbox{\rm BHom}_f^+(G^s_A, PO(3,\mathbb C))$ is
an isomorphism of varieties over $\mathbb Q$, this will be proven in the next section.

\subsection{$alg$ is an isomorphism of varieties}

We first establish that $alg: BR(A, \mathbb P^2_0(\mathbb C)) \to
\mbox{\rm BHom}_f^+(G^s_A, PO(3,\mathbb C))$ is an isomorphism of varieties in
two elementary cases. Let $C$ be an arrangement whose graph $\Gamma_C$
has only one edge $e= [v, l]$, $m$ is the number of isolated vertices $v_j$
in $\Gamma_C$. Let $A$ be a based arrangement which is the disjoint
union of the standard triangle and $C$.

\begin{lem}
\label{above}
$alg: BR(A, \mathbb P^2_0(\mathbb C)) \to \mbox{\rm BHom}_f^+(G^s_A, PO(3,\mathbb C))$ is
an isomorphism of varieties over $\mathbb Q$.
\end{lem}
\par\medskip\noindent{\it Proof: } We already know that $alg$ is a bijection.
It is clear that the restriction morphisms
$$
Res: \mbox{\rm BHom}_f^+(G^s_A, PO(3,\mathbb C)) \to \mbox{\rm Hom}_f(G^s_C, PO(3,\mathbb C))$$
$$
res: BR(A, \mathbb P^2(\mathbb C)) \to BR(C, \mathbb P^2(\mathbb C)) $$
are isomorphisms of varieties. Let
$alg: BR(C, \mathbb P^2(\mathbb C)) \to \mbox{\rm Hom}_f(G^s_C, PO(3,\mathbb C))$
denote the mapping induced by the restriction. The group
$G^s_C$ is the free product

$$
G^s_e * \underbrace{\mathbb Z/2 * ... * \mathbb Z/2}_{m\hbox{~times}}
$$
Hence
$$
\mbox{\rm Hom}_f(G^s_C, PO(3))\cong \mbox{\rm Hom}_f(G^s_e, PO(3)) \times
( \mbox{\rm Hom}_f(\mathbb Z /2, PO(3)) )^m
$$
\noindent Since the edge group $G^s_e$ and the vertex groups $\mathbb Z/2$ are
finite, the variety $\mbox{\rm Hom}_f(G^s_C, PO(3))$ is smooth
(Proposition \ref{L5.4}). The quasi-projective variety
$R(C, \mathbb P^2_0)$ again splits as the direct product
$$
{\cal I}_0 \times  \mathbb P^2_0 \times ... \times  \mathbb P^2_0 \ ,
$$
\noindent where ${\cal I}_0$ is the anisotropic incidence variety, see
 Section \ref{Projective arrangements}. The anisotropic incidence variety
is smooth by  Proposition \ref{is smooth}, hence the product smooth as well.

 Let $B$ denote the arrangement
obtained by removing the incidence relation between $v$ and $l$ in $C$
and $G_B^s$ be the corresponding Coxeter group. Then
$$
alg : R(B, \mathbb P^2_0) \longrightarrow \mbox{\rm Hom}_f(G_B, PO(3))
$$
is an isomorphism of smooth varieties (the left-hand side is
$(\mathbb P^2_0)^{m+2}$ and the right-hand side is $p(R)^{m+2}$,
see Lemma \ref{fixed point}). Thus
$alg:  BR(A, \mathbb P^2_0)\longrightarrow \mbox{\rm BHom}_f(G, PO(3))$ is an
isomorphism of smooth varieties. $\qed$

\medskip
We now consider a relative version of the above lemma. Suppose that
a based abstract arrangement $D$ is the fiber sum $T\times_{w\equiv v} C$
where $C$ is the arrangement above, $w, v$ are elements of
$T$ and $C$ respectively.

\begin{lem}
\label{below}
$alg: BR(D, \mathbb P^2_0(\mathbb C)) \to \mbox{\rm BHom}_f^+(G^s_D, PO(3,\mathbb C))$ is
an isomorphism of varieties.
\end{lem}
\par\medskip\noindent{\it Proof: }
Let $\alpha = \rho_{\phi_T}(g_v)$,  $G_e\subset G^s_D$ is the  edge subgroup
corresponding to $e=[v, l]$.
It is clear that the restriction morphism
$$
\mbox{\rm BHom}^+_f(G^s_A, PO(3,\mathbb C))\stackrel{Res}{\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightarrow}
\{\rho\in \mbox{\rm Hom}_f(G_C^s, PO(3,\mathbb C)): \rho(g_v)= \alpha \}$$
$$
\quad \quad \quad =: F_{g_v, \alpha}(G_C^s, PO(3, \mathbb C))$$
is an isomorphism of varieties. The group $G_e$ is finite, hence by
Proposition \ref{L5.4} the variety $F_{g_v, \alpha}(G_C^s, PO(3, \mathbb C))$ is smooth.
Similarly let $q:= \phi_T(v)$, the restriction morphism
$$
res: BR(D, \mathbb P^2_0(\mathbb C)) \to \{\psi \in R(C, \mathbb P^2_0(\mathbb C)): \psi(v)= q\} =:
F_{v, q}(C, \mathbb P^2_0(\mathbb C))
$$
is an isomorphism of varieties. Then $F_{v, q}(C, \mathbb P^2_0(\mathbb C))$ is
isomorphic the product of the {\em relative anisotropic incidence variety}
$$
{\cal I}_0(q)= \{ l \in (\mathbb P^2_0)^{\vee} : q \cdot l =0 \}
$$
(see \mathbb S \ref{Projective  arrangements}) with $m$ copies of
$\mathbb P^2_0$. It is clear that the mapping
$alg$ induces a bijection of the sets of complex points
$$
alg: {\cal I}_0(q) (\mathbb C) \longrightarrow F_{g_v, \alpha}(G_e, PO(3, \mathbb C))
$$
According to Lemma \ref{is smooth} the variety ${\cal I}(q)$ is
smooth and we  repeat the arguments from the proof of
Lemma \ref{above}. $\qed$

\bigskip
Now we consider the case when $A$ is a general based arrangement.
Let $X:= BR(A, \mathbb P^2_0)$, $Y:= \mbox{\rm BHom}_f^+(G_A^s, PO(3))$.

\begin{thm}
\label{T7.2}
The mapping $alg: X\to Y$ is a
biregular  isomorphism of quasi-projective varieties.
\end{thm}
\par\medskip\noindent{\it Proof: } Let $F$ (resp. $R$) be the functor of points of $X$ (resp. $Y$).
Then $alg$ is an isomorphism of the varieties $X$ and $Y$ if and only if
$F$ and $R$ are naturally isomorphic (see \cite[Proposition IV-2]{EH}). Let
$B$ be the based arrangement obtained by removing all edges from the
graph of $A$ that are not in the graph of $T$ (and retaining all vertices).
Let $G^s_B$ be the corresponding Shephard group, clearly
$$
G^s_B\cong G^s_T * \underbrace{\mathbb Z/2 * ... * \mathbb Z/2}_{m \hbox{~~times}}
$$
where $m$ is the cardinality of ${\cal V}(A- T)$. Thus
$$BR(B, \mathbb P^2_0(\mathbb C))\cong \mathbb P^2_0(\mathbb C)^m, \mbox{\rm BHom}^+_f(G^s_B, PO(3))\cong
\mbox{\rm Hom}_f(\mathbb Z/2, PO(3, \mathbb C))^m$$
 are smooth varieties and
$$
\widetilde{alg} : \tilde{X}:= BR(B)\longrightarrow \tilde{Y}:= \mbox{\rm BHom}^+_f(G^s_B, PO(3))
$$
is an isomorphism of smooth varieties (see the proof of  Lemma \ref{above}).
We let $\tilde{F}$ and
$\tilde{R}$ be the  functors of points of $\tilde{X}$ and $\tilde{Y}$. The isomorphism
$\widetilde{alg}: \tilde{X}\to \tilde{Y}$ induces a natural isomorphism of functors
$\eta: \tilde{F}\to \tilde{R}$. The functors $F$ and $R$ are subfunctors of
$\tilde{F}$ and $\tilde{R}$. We now make explicit the
inclusions $F\subset \tilde{F}, R\subset \tilde{R}$.

Let ${\cal E}_0$ be the collection of edges of $\Gamma_A$
that are not edges of $\Gamma_T$.
Suppose $e_{ij}= [v_i, \ell_j]\in {\cal E}_0$ is such an edge.
Let $X_{ij}$ be the subvariety of $BR_0(B)$ defined by
$$
x_i \alpha_j + y_i \beta_j + z_i \gamma_j=0
$$
where $[x_i: y_i: z_i]$ and $[\alpha_j: \beta_j :\gamma_j]$ are
homogeneous coordinates on $BR_0(B)$ corresponding to
$v_i, l_j$ respectively.
We let $F_{ij}$ be the subfunctor of $\tilde{F}$ which as the functor of points
associated to the subvariety $X_{ij}$. Then we have
\begin{equation}
F = \bigcap_{[v_i l_j]\in {\cal E}_0} F_{ij}  \subset\tilde{F}
\end{equation}
Similarly if the edge $e_{ij}$ has the label $\epsilon_{ij}$ then we let
$R_{ij}$ be the subfunctor of $\tilde{R}$ corresponding to the subvariety
$Y_{ij}$ defined by the Artin relation
$$
(g_{v_i} g_{\ell_j})^{\epsilon_{ij}}= (g_{\ell_j}g_{v_i} )^{\epsilon_{ij}}\ .$$
We have
\begin{equation}
R= \bigcap_{(ij)\in {\cal E}} R_{ij} \subset \tilde{R}
\end{equation}

\begin{lem}
$\eta: \tilde{F}\to \tilde{R}$ induces an isomorphism from $F_{ij}$ to $R_{ij}$.
\end{lem}
\par\medskip\noindent{\it Proof: } Let $X_{ij}$ and $Y_{ij}$ be the subvarieties  of
$BR(B, \mathbb P^2_0), \mbox{\rm BHom}_f^+(G^s_B, PO(3))$ corresponding to the
subfunctors $F_{ij},  R_{ij}$. Then Lemmas \ref{above}
and \ref{below} imply that $alg$ induces
an isomorphism of smooth varieties $X_{ij}\to Y_{ij}$.
Hence $\eta$ induces an isomorphism  of the corresponding
subfunctors. $\qed$

The above lemma and equations (2), (3) immediately imply that
$\eta$ induces a natural isomorphism from $F$ to $R$. Theorem
\ref{T7.2} follows.  $\qed$

\medskip
Now we can prove one of the two main results of this paper. Let
$S \subset \mathbb C^n$ be an affine variety defined over
$\mathbb Q$. We will consider $S$ as a quasi-projective variety
in $\mathbb P^n(\mathbb C)$.

\begin{thm}
\label{main}
For any variety $S$ as above there exists a Zariski open
subset $U\subset S(\mathbb C)$ containing all real points and a
based arrangement
$A$ so that the corresponding  Shephard  group $G^s_A$ has the property:

There is a Zariski open subset $W$ in $\mbox{\rm Hom}(G^s_A, PO(3, \mathbb C))/\!/ PO(3, \mathbb C)$
which is biregular isomorphic to $U$.
\end{thm}

\begin{rem}
$W$ is never Zariski dense in the character variety $X(G^s_A, PO(3,\mathbb C))$.
\end{rem}

\par\medskip\noindent{\it Proof: } Given the variety $S$ we construct an abstract based
arrangement $A$ such that $BR_0(A, \mathbb P^2)$ is biregular isomorphic
to $S$ via the isomorphism $geo$ (Theorem \ref{systems} and Corollary \ref{csystems}). Let
$U:= \tau (BR_0(A, \mathbb P^2_0(\mathbb C)))$, it is  Zariski open in
$S(\mathbb C)$ and contains all real points
since the subvariety $BR_0(A, \mathbb P^2_0)$ is Zariski open in
$BR_0(A, \mathbb P^2)$ and contains all real points.  Theorem \ref{T7.2} implies that we have
an isomorphic embedding with
Zariski open image $\mbox{\rm BHom}_f^+(G^s_A, PO(3, \mathbb C))_0$
$$
alg: BR_0(A, \mathbb P^2_0) \hookrightarrow  \mbox{\rm BHom}_f^+(G^s_A, PO(3, \mathbb C))
$$
Corollary \ref{cro-section} implies that $\mbox{\rm BHom}_f^+(G^s_A, PO(3))$
is a cross-section for the action of $PO(3,\mathbb C)$ by conjugation on
the Zariski component $\mbox{\rm Hom}_f^+(G^s_A, PO(3))$ of
$\mbox{\rm Hom}(G^s_A, PO(3))$. All representations
in the variety $\mbox{\rm Hom}_f(G^s_A, PO(3, \mathbb C))$ are {\em stable} in the
sense of \cite[ Theorem 1.1]{JM}, thus we get an open
monomorphism of varieties
$$
\mbox{\rm BHom}_f^+(G^s_A, PO(3,\mathbb C))_0 \hookrightarrow X(G^s_A, PO(3,\mathbb C))=
\mbox{\rm Hom}(G^s_A, PO(3,\mathbb C))/\!/ PO(3,\mathbb C)
$$
so that the image is a Zariski open subvariety $W$ in the
character variety. Thus we get the composition
$$
\theta: U \stackrel{geo}{\longrightarrow} BR_0(A, \mathbb P^2(\mathbb C))
\stackrel{alg}{\longrightarrow}
\mbox{\rm BHom}_f^+(G^s_A, PO(3,\mathbb C))_0 \longrightarrow W \subset  X(G^s_A, PO(3,\mathbb C))
$$
is the required isomorphism onto a Zariski open subvariety $W$ of the
character variety. $\qed$

\begin{prop}
\label{sfina}
Suppose that  $S$ is an affine variety over $\mathbb Q$ and $q\in \mathbb Q$ is a rational point. Then there is an abstract arrangement $A$ (as in Theorem \ref{main})
so that the representation
$\rho= alg\circ geo(q)$ has finite image. The centralizer of the subgroup
$\rho(G^s_A)\subset PO(3,\mathbb C)$ is trivial.
\end{prop}
\par\medskip\noindent{\it Proof: } Follows from Corollary \ref{csystems} and Lemma \ref{fina}. $\qed$

\medskip
As an example we consider the configuration space of the
arrangement\footnote{
This is not a based arrangement.}
$$A= \{v; l_1, l_2: \iota(v, l_1), \iota(v, l_2)\} $$
Take the corresponding Coxeter group $G= G_A^c$. Then
$R(A, \mathbb P^2_0)\cong \mathbb P^2_0 \times \mathbb P^1_0 \times \mathbb P^1_0$ corresponds to
the representations of $G$ which are {\em nondegenerate},
i.e. the elements $\rho(g_v), \rho(g_{l_1}), \rho(g_{l_2})$,
$\rho(g_v g_{l_1})$,  $\rho(g_v g_{l_2})$ have order 2. However
there are some other components of \newline
$\mbox{\rm Hom}(G, PO(3,\mathbb C))$ which are
described by assigning which of the elements\newline
$\rho(g_v), \rho(g_{l_1}), \rho(g_{l_2}), \rho(g_v g_{l_1})$,
$\rho(g_v g_{l_2})$ are equal to $\hbox{\bf 1}$. If $\rho(g_v)= \hbox{\bf 1}$ then
any representation factors through the free product
$\mathbb Z/2 * \mathbb Z/2$ and the corresponding component of
$\mbox{\rm Hom}(G, PO(3,\mathbb C))$ is isomorphic to $\mathbb P^2_0 \times \mathbb P^2_0$.
The reader will verify that besides
$$
\mbox{\rm Hom}_f(G, PO(3,\mathbb C))\cong \mathbb P^2_0 \times \mathbb P^1_0 \times \mathbb P^1_0$$
and the above component isomorphic to
$\mathbb P^2_0 \times \mathbb P^2_0$ there are 4 components isomorphic to
$\mathbb P^2_0\times \mathbb P^1_0$, 4 components isomorphic to $\mathbb P^2_0$ and
one component which consists of a single reduced point
(the last corresponds to the trivial representation).

\subsection{Representations of Artin groups}
\label{repart}

Our next goal is to prove a theorem analogous to Theorem \ref{main}
for Artin groups. Take a based admissible arrangement $A$, consider
the germ $(\mbox{\rm Hom}(G^a_A, PO(3, \mathbb C)), \tilde\rho)$, where $\tilde\rho= \mu(\rho)$,
$\rho\in \mbox{\rm Hom}_f(G^s_A, PO(3,\mathbb C))$.

\begin{thm}
\label{iso}
The morphism $\mu: (\mbox{\rm Hom}(G^s_A, PO(3,\mathbb C)), \rho)
\to (\mbox{\rm Hom}(G^a_A, PO(3)), \tilde\rho)$ is an analytical
isomorphism of germs.
\end{thm}
\par\medskip\noindent{\it Proof: } Let $m\ge 1$ be an integer, consider the $m$-th
order Zariski tangent spaces $T^m_{\rho}$ of the variety
$\mbox{\rm Hom}(G^s_A, PO(3,\mathbb C))$ at $\rho$ and $T^m_{\tilde\rho}$
of the variety $\mbox{\rm Hom}(G^a_A, PO(3,\mathbb C))$ at $\tilde\rho$, there
is a well-defined mapping $\mu_m: T^m_{\rho} \to T^m_{\tilde\rho}$
induced by $\mu$. We will think of elements of $m$-th order
Zariski tangent spaces as {\em formal curves} or homomorphisms
from $G^s, G^a$ to
$PO(3)({\cal A})$ which is the set of  ${\cal A}$-points of $PO(3)$
and ${\cal A}$ is a certain Artin local $\mathbb C$-algebra, see \mathbb S \ref{defof}.

Our goal is to prove that $\mu_m$ is an isomorphism for all $m$.

\begin{lem}
$\mu_m$ is injective.
\end{lem}
Suppose that $\alpha\ne \beta\in T^m_{\tilde\rho}$ are formal
curves of the order $m$. We think of them
as representations of $G^s$ into $PO(3)({\cal A})$.
Then $\mu_m(\alpha), \mu_m(\beta)$ are homomorphisms of
$G^a$ into $PO(3)({\cal A})$ defined by composing
with the projection $G^a\to G^s$. Clearly $\mu_m(\alpha)\ne \mu_m(\beta)$.
$\qed$

\begin{lem}
$\mu_m$ is surjective.
\end{lem}
\par\medskip\noindent{\it Proof: } Let $\xi\in T^m_{\tilde\rho}(G^a, PO(3,\mathbb C))$ and $v$ is a
{\em point} in $A- \{v_{11}\}$.
To prove that $\xi$ belongs to the image of $\mu_m$ it is enough to check that
the restriction of $\mu_m$ to each vertex-subgroup of $G^a$ is a {\em trivial deformation} (see Lemma \ref{L7.1}).  We first consider the {\em points}
$v\in A$ distinct from $v_{11}$. Then $v$
is incident to at least one {\em line} $l\in A$ and hence  the image of the
edge-subgroup $\rho( G_{vl}^s)= \tilde\rho(G^a_{vl})$ is generated by two
distinct  commuting involutions (since $\rho$ is ``nondegenerate'').
Thus by Lemma \ref{L5.3} the restriction $\rho|G_{vl}^a$ is
infinitesimally rigid, in particular $\xi|G^a_v$ is a
{\em trivial deformation}. In the case $v=v_{11}$ we use
Proposition \ref{rigid triangle} to conclude that the restriction of
$\xi$ to the vertex-subgroup
$G_v^a$ is a trivial deformation as well. Suppose that
$l\in A$ is a {\em line}. Then admissibility of the arrangement
$A$ implies that $l$ is incident to at least one {\em point}
$w$ in $A- \{v_{11}\}$. Hence
we repeat the same argument as in the case of {\it points} in $A$. $\qed$

Thus, we established that the morphism $\mu$ induces an
isomorphism of Zariski tangent spaces of all orders. Hence
Theorem \ref{iso} follows from Lemma \ref{LL3.1}. $\qed$

\begin{cor}
\label{inhe}
For any admissible based arrangement $A$ the morphism
$$\mu: \mbox{\rm Hom}_f(G_A^s, PO(3))\to \mbox{\rm Hom}(G_A^a, PO(3))$$
is open (in the classical topology)
and is an analytic isomorphism onto its image.
\end{cor}

\begin{cor}
For any admissible based arrangement $A$ the character variety of
$G_A^a$ inherits all
singularities of the representation variety of the
group $G_A^s$ corresponding to points of $BR(A, \mathbb P^2_0(\mathbb C))$.
\end{cor}

The image $\mbox{\rm Hom}_f(G^a_A, PO(3,\mathbb C)):= \mu(\mbox{\rm Hom}_f(G_A^s, PO(3,\mathbb C)))$
is a constructible set.

\begin{prop}
$\mbox{\rm Hom}_f(G^a_A, PO(3,\mathbb C))$ is closed in $\mbox{\rm Hom}(G^a_A, PO(3,\mathbb C))$
\end{prop}
\par\medskip\noindent{\it Proof: } The image $Z$ of the monomorphism
$$
\mbox{\rm Hom}(G_A^s, PO(3,\mathbb C)) \longrightarrow \mbox{\rm Hom}(G_A^a, PO(3,\mathbb C))
$$
is described by the equations
$$
\rho(g_v)^{\delta(v)}= \hbox{\bf 1} , \quad v\in {\cal V}(\Lambda_A)
$$
where $\delta(v)$ is the label of the vertex $v$. Thus $Z$ is
closed. On the other hand, by Corollary \ref{zaropen},
the space $\mbox{\rm Hom}_f(G^s_A, PO(3,\mathbb C))$ is closed in
$\mbox{\rm Hom}(G_A^s, PO(3,\mathbb C))$, the proposition follows. $\qed$

\begin{cor}
\label{open-artin}
For any admissible based arrangement the space
$\mbox{\rm Hom}_f(G^a_A, PO(3,\mathbb C))$ is a union of Zariski connected components of
$\mbox{\rm Hom}(G^a_A, PO(3, \mathbb C))$.
\end{cor}
\par\medskip\noindent{\it Proof: } The set $\mbox{\rm Hom}_f(G^a_A, PO(3,\mathbb C))$ is Zariski open
since it is constructible and open in the classical
topology. $\qed$

\begin{thm}
The morphism $\mu: \mbox{\rm Hom}_f(G_A^s, PO(3))\to  \mbox{\rm Hom}_f(G_A^a, PO(3)))$
is a biregular isomorphism of varieties.
\end{thm}
\par\medskip\noindent{\it Proof: } We will first prove that the  reduced  varieties corresponding to
$$
X:= \mbox{\rm Hom}_f(G_A^s, PO(3)) \hbox{~~and~~} Y:= \mbox{\rm Hom}_f(G_A^a, PO(3))$$
 are isomorphic. We construct nonsingular varieties $U, W$ containing $X, Y$
and an extension of $\mu$ to an isomorphism $U\to W$ which
carries $X$ to $Y$ bijectively as sets. Let $C$ be the
arrangement obtained from $A$ by removing the incidence
relations everywhere outside of the standard triangle.
Thus we get two nonsingular varieties $Q= \mbox{\rm Hom}_f(G^a_C, PO(3))$
and $U:= \mbox{\rm Hom}_f(G^s_C, PO(3))$ with $X\subset U$ and $Y\subset Q$.

Clearly $\mu: U \to V$ is a biregular isomorphism and it
bijectively carries $X$ to $Y$. Thus the reduced
varieties $X^r, Y^r$ are isomorphic via $\mu$. Hence the assertion
of Theorem follows from combination of Corollary
\ref{inhe} and Theorem \ref{cbasic}. $\qed$

\medskip
As a corollary we get the following generalization of Theorem \ref{main}:

\begin{thm}
\label{Artin}
Let $S$ be an affine variety defined over $\mathbb Q$. Then
there exists an admissible based arrangement
$A$ so that the corresponding  Artin group $G^a_A$ has the property:

There is a Zariski open subset in $S$ (containing
all real points) which is biregular isomorphic to a Zariski open and
closed subvariety in $\mbox{\rm Hom}(G^a_A, PO(3,\mathbb C))/\!/ PO(3,\mathbb C)$. If
$q\in \mathbb Q$ is a rational point. Then the abstract arrangement
$A$ can be chosen so that the representation
$\rho= \mu\circ alg\circ geo(q)$ has finite image. The
centralizer of the subgroup $\rho(G^a_A)\subset PO(3,\mathbb C)$ is trivial.
\end{thm}

\section{Differential graded algebras and Lie algebras}
\label{dgla}

In this section we discuss differential graded Lie algebras, differential algebras  and their Sullivan's minimal models. These definitions will be used in the following two sections.
Let $\hbox{\bf k}$ be the ground field. A {\em graded Lie algebra} over $\hbox{\bf k}$ is a $\hbox{\bf k}$-vector space
$$
L^{\bullet}= \oplus_{i\ge 0}\ L^i
$$
graded by (nonnegative) integers and a family of bilinear mappings
$$
[\cdot , \cdot]: L^i \otimes L^j \longrightarrow L^{i+j}
$$
satisfying graded skew-commutativity:
$$
[\alpha, \beta] + (-1)^{ij}[\beta, \alpha] =0
$$
and the graded Jacobi identity:
$$
(-1)^{ki}[\alpha, [\beta, \gamma]] + (-1)^{ij}[\beta, [\gamma, \alpha]]
+ (-1)^{jk}[\gamma, [\alpha, \beta]]=0
$$
where $\alpha\in L^i , \beta\in L^j , \gamma\in L^k$. We say that $L$ is
{\em bigraded}  if $L$ is graded as above and
$$
L= \oplus_{q\ge 0} L_q= \oplus_{i,q\ge 0} \ L^i_q
$$
where $ L^i_q= L^i\cap L_q$. We also require:
$$
[\cdot , \cdot]: L_q \otimes L_p \longrightarrow L_{p+q}
$$

A {\em differential graded Lie algebra} is a pair $(L, d)$ where
$d: L \to L$ is a {\em derivation} of the degree $\ell$, i.e.
$$
d: L^i \longrightarrow L^{i+\ell}, \quad d\circ d=0, \quad \hbox{and}
$$
$$
d[\alpha, \beta]= [d\alpha, \beta] + (-1)^{i\ell}[\alpha, d\beta]
$$
\begin{rem}
In this paper we will use only the degree $\ell=1$.
\end{rem}

Suppose that $L= L^{\bullet}_{\bullet}$ is a bigraded Lie algebra, then the differential graded Lie algebra~
$(L, d)$ is called a  {\em differential bigraded Lie algebra} if
there is a number $s$ such that $d$ has the bidegree $(\ell, s)$:
$$
d: L_q \longrightarrow L_{q+s}
$$
\begin{rem}
In this paper we will use only the bidegree $(\ell,s)=(1,0)$. Similarly one defines {\em trigraded} differential Lie algebras $L^{\bullet}_{\bullet,\bullet}$
\end{rem}

Let $L^{\bullet}$ be a differential graded Lie algebra, and suppose that
${\cal J}\subset L^{\bullet}$ is a vector subspace. Then ${\cal J}$
is an {\em ideal} in $L^{\bullet}$ if:
\begin{itemize}
\item ${\cal J}$ is {\em graded}, i.e. ${\cal J}= \oplus_{n=0}^{\infty}
({\cal J} \cap L^n)$;
\item $d({\cal J}) \subset {\cal J}$;
\item For each $\gamma\in {\cal J}, \alpha\in L$ we have: $[\gamma, \alpha]\in {\cal J}$.
\end{itemize}

\begin{rem}
If $L$ is bigraded then we require ideals in $L$ to be bigraded
as well, i.e.
$$
{\cal J}= \oplus_{n,q\ge 0}\ ({\cal J} \cap L^n_q)
$$
\end{rem}

\begin{lem}
\label{big}
If ${\cal J} \subset L$ is an ideal in a differential graded (bigraded)
Lie algebra then the quotient $L/{\cal J}$ has the natural structure
of a differential graded (bigraded) Lie algebra so that the
natural projection $L\to L/{\cal J}$ is a morphism.
\end{lem}
\par\medskip\noindent{\it Proof: } Proof is straightforward and is left to the reader. $\qed$

\medskip
Let $(L^{\bullet},d)$ be a differential graded Lie algebra~ and ${\mathfrak g}$ be a Lie algebra.
An {\em augmentation} is a homomorphism $\epsilon: L \to {\mathfrak g}$
such that $\oplus_{i\ge 1} L^i \subset \ker(\epsilon)$ and $\epsilon \ne 0$.
An {\em augmentation ideal} of $\epsilon$ is the kernel of $\epsilon$.

We recall the definition of the complete local $\hbox{\bf k}$-algebra
$R_{L^{\bullet}}$ (see \cite{M}) associated to a differential graded Lie algebra~
$L^{\bullet}$ over $\hbox{\bf k}$. We will give a definition which
conveys the intuitive meaning of $R_{L^{\bullet}}$ but has
the defect of being non-functorial. We have the sequence
$$
0\longrightarrow L^0 \stackrel{d}{\longrightarrow} L^1 \stackrel{d}{\longrightarrow}
L^2 \stackrel{d}{\longrightarrow} ...
$$
Choose a complement $C^1\subset L^1$ to the 1-coboundaries
$B^1\subset L^1$.
Let ${\cal F}: C^1 \to L^2$ be the polynomial mapping given by
${\cal F}(\eta)= d\eta + \frac{1}{2} [\eta, \eta]$. Then $R_{L^{\bullet}}$
is the completion at $0$ of the coordinate ring of the affine
subvariety of $C^1$ defined by the equation ${\cal F}(\eta)= 0$.

Since $L^1$ and $C^1$ are infinite dimensional in many applications,
further definitions of affine variety and completion are needed in
infinite dimensional vector spaces. These are given in \cite[\mathbb S 1]{BuM}.
Suppose next that $(X, x)$ is an analytic germ. We will say that
$L^{\bullet}$ {\em controls} $(X,x)$ if $R_{L^{\bullet}}$ is
isomorphic to the complete local ring $\widehat{O_{X,x}}$.

\begin{defn}
Let $A^{\bullet}, B^{\bullet}$ be cochain complexes and $\rho: A^{\bullet}\to
B^{\bullet}$ is a morphism. Then $\rho$ is called a {\bf quasi-isomorphism}
if it induces an isomorphism of all cohomology groups. The morphism is
called a {\bf weak equivalence} if it induces an isomorphism of
$\mbox{\rm H}^0, \mbox{\rm H}^1$ and a monomorphism of $\mbox{\rm H}^2$.
\end{defn}

Weak equivalence induces an equivalence relation on the category of differential graded Lie algebras: algebras $A^{\bullet}_1, A^{\bullet}_m$
are (weakly) equivalent if there is a chain of weak equivalences:
$$
A^{\bullet}_1 \rightarrow A^{\bullet}_2 \leftarrow A^{\bullet}_2 \rightarrow ...
\leftarrow A^{\bullet}_m
$$

In the following two sections we will use the following theorem about controlling differential graded Lie algebras~ proven in \cite{GM}:

\begin{thm}
\label{GolMil}
Suppose that $L^{\bullet}$ and $N^{\bullet}$ are weakly equivalent
differential graded Lie algebras~. Then the germs controlled by the algebras
$L^{\bullet}$ and $N^{\bullet}$ are analytically isomorphic.
\end{thm}

A {\em differential graded algebra} $A^{\bullet}$
is defined similarly to a differential graded Lie algebra~: just instead of the Lie bracket
$[\cdot , \cdot]$ satisfying graded Jacobi identity we have an
associative multiplication:
$$
\wedge : A^i \otimes A^j \longrightarrow A^{i+j}
$$
satisfying the properties:

\begin{itemize}
\item $d: A^i \to A^{i+1}$, $d\circ d=0$;
\item $d(\alpha \wedge \beta)= d\alpha \wedge \beta + (-1)^i \alpha \wedge d\beta$, for all
$\alpha\in A^i$;
\item $\alpha\wedge\beta= (-1)^{ij}\beta \wedge \alpha$, for all
$\alpha\in A^i$, $\beta \in A^j$;
\item $A$ has the unit $1\in A^0$.
\end{itemize}

To get a differential graded Lie algebra~ from a differential graded algebra $A^{\bullet}$
take a Lie algebra ${\mathfrak g}$ and let
$L^{\bullet}= A^{\bullet} \otimes {\mathfrak g}$ (see \cite{GM} for details).

Suppose that $V$ is a vector space over $\hbox{\bf k}$, $(A^{\bullet}, d)$ is a
differential graded algebra and $f: V \to Z^{2}$ is a linear mapping, where $Z^2$ is the space of 2-cocycles of $A^{\bullet}$.
Then the {\em Hirsch extension} $A^{\bullet}\otimes_f V$ is a
differential graded algebra which (as an algebra) equals to
$A^{\bullet}\otimes \Lambda(V)$ and the restriction of the
differential on $A^{\bullet}\otimes_f V$ to $A^{\bullet}$
equals $d$ and the restriction to $V$ equals $f$.

A differential graded algebra ${\cal M}^{\bullet}$ is called {\em 1-minimal} if:

(a) ${\cal M}^0 = \hbox{\bf k}$;

(b) ${\cal M}^{\bullet}$ is the increasing union of differential subalgebras:
$$
\hbox{\bf k} = {\cal M}_{[0]} \subset {\cal M}_{[1]} \subset {\cal M}_{[2]}\subset ...
$$
with each ${\cal M}_{[i]} \subset {\cal M}_{[i+1]}$ a Hirsch extension;

(c) The differential $d$ on ${\cal M}$ is decomposable, i.e. for each $\alpha\in {\cal M}$ we have:
$$d(\alpha)= \sum_{j,i} \beta_j \wedge\gamma_i\ , \ \
\beta_j\ , \gamma_i \in \oplus_{s\ge 1}{\cal M}^s$$

\begin{defn}
Suppose that $A^{\bullet}$ is a differential graded algebra. Then a
{\bf 1-minimal model} for $A^{\bullet}$ is a morphism
$\rho: {\cal M}^{\bullet} \to A^{\bullet}$ such that:
\begin{itemize}
\item The differential graded algebra ${\cal M}^{\bullet}$ is 1-minimal;
\item The morphism $\rho$ is a weak equivalence.
\end{itemize}
\end{defn}

We refer the reader to \cite{Sul}, \cite{GrM}, \cite{Morgan2}
for further discussion of the definition, properties and
construction of 1-minimal models.

\section{Hain's theorem and its applications}
\label{Hain's theorem}

In this section we give an exposition of a work of
R.\ Hain \cite{Hain} which shows that the singularities
in representation varieties of fundamental  groups of smooth
complex algebraic varieties
are quasi-homogeneous. In fact some assumption on the
representation $\rho$ which is being deformed is  also
required. In \cite{Hain} the analogue of our Theorem
\ref{hain} is proven under the assumption that $\rho$
was the monodromy representation of an admissible
variation of a mixed Hodge structure. One doesn't obtain a
restriction on weights working in this generality.

Let $M$ be a smooth  connected
manifold with the fundamental group $\Gamma$. Let
$\hbox{\bf G}$ be the Lie group of real points of an algebraic group $G$ with the
Lie algebra ${\mathfrak g}$ defined over $\mathbb R$ and $\rho: \Gamma \to \hbox{\bf G}$ is a
homomorphism. Let $P$ be the flat bundle over $M$ associated to $\rho$
and $adP$ the associated ${\mathfrak g}$-bundle. Then ${\cal A}^{\bullet}(M, adP)$, the
complex of smooth $adP$-valued differential forms on $M$ is a differential graded Lie algebra~. We
define an augmentation $\epsilon:
{\cal A}^{\bullet}(M, adP)\to {\mathfrak g}$ by evaluating degree zero forms
at a base-point $x\in M$ and sending the rest of forms to zero.
Let ${\cal A}^{\bullet}(M, adP)_0$ be the kernel of $\epsilon$. The following theorem
follows immediately from \cite[Theorem 6.8]{GM}.

\begin{thm}
\label{con germ}
${\cal A}^{\bullet}(M, adP)_0$ controls the germ $(\mbox{\rm Hom}(\Gamma, \hbox{\bf G}),\rho)$.
\end{thm}

The point of this section is that if $M$ is smooth  connected  complex
algebraic variety and $\rho$ has finite image then
${\cal A}^{\bullet}(M, adP)_0$ is quasi-isomorphic to a differential graded Lie algebra~ which
has a structure  of a mixed Hodge complex.
By a theorem of Hain this implies that
$$
R_{{\cal A}^{\bullet}(M, adP)_0\otimes \mathbb C}
$$
is a quasi-homogeneous\  ring.  We now give details.

A {\em real mixed Hodge complex} (abbreviated MHC) is a
pair of complexes $K_{\mathbb R}^{\bullet}$ and $K_{\mathbb C}^{\bullet}$ (real
and complex respectively),
together with a quasi-isomorphism $\alpha: K_{\mathbb R}^{\bullet}
\otimes \mathbb C \to K_{\mathbb C}^{\bullet}$ such that  $K_{\mathbb R}^{\bullet}$
is a complex of real vector spaces equipped with an
increasing filtration $W_{\bullet}$ (called the {\em weight}
filtration) and $K_{\mathbb C}^{\bullet}$ is equipped with an increasing
(weight) filtration $W_{\bullet}$ and a decreasing filtration $F^{\bullet}$
( called the {\em Hodge} filtration). The data
$$
K_{\mathbb R}^{\bullet}, K_{\mathbb C}^{\bullet}, \alpha, W_{\bullet}, F^{\bullet}
$$
satisfies the axioms described in \cite[Scholie 8.1.5]{D2} (take $A= \mathbb R)$.

By a theorem of Deligne (\cite[ Scholie 8.1.9]{D2})
the cohomology of a MHC has a
mixed Hodge structure, \cite[Scholie 2.3.1]{D}. It
is important in what follows that the filtrations on
$\mbox{\rm H}^ {\bullet}(K_{\mathbb C})$ induced by $W_{\bullet}$ and $F^{\bullet}$
can be canonically split, \cite[Section 1.2.8]{D}.
Thus the filtration  $W_{\bullet}$ induces a canonical grading
 on $\mbox{\rm H}^ {\ell}(K_{\mathbb C})$ and consequently on $\mbox{\rm H}^ {\ell}(K_{\mathbb C})^{*}$,
$\ell= 0, 1, 2$.

If $V$ is a finite-dimensional vector space over
$\mathbb C$ we will let $\mathbb C\, [[V]]$ denote the completion
of the symmetric algebra $\mathbb C\, [V]$ at the maximal ideal
corresponding to $0$. Thus $\mathbb C\, [[\mbox{\rm H}^ {\ell}(K_{\mathbb C})^*]]$ has
a canonical decreasing filtration (as an algebra) induced by the grading
 of $\mbox{\rm H}^ {\ell}(K_{\mathbb C})^{*}$, $\ell= 0, 1, 2$.

We will say that a MHC is a {\em mixed
Hodge differential graded Lie algebra} if the
complexes  $K_{\mathbb R}^{\bullet}$ and $K_{\mathbb C}^{\bullet}$
are differential graded Lie algebras~ (\cite[\mathbb S 1.1]{GM}), such that $\alpha$ is
bracket preserving and the filtrations satisfy

(i) $[W_p (K_{\mathbb R}^{\bullet}), W_q (K_{\mathbb R}^{\bullet})]\subset W_{p+q}
(K_{\mathbb R}^{\bullet})$;

(ii) $[F^p (K_{\mathbb C}^{\bullet}), F^q (K_{\mathbb C}^{\bullet})]\subset F^{p+q}
(K_{\mathbb C}^{\bullet})$.

\smallskip
In this case we will use $L^{\bullet}_{\mathbb R}$ and
$L^{\bullet}_{\mathbb C}$ in place of $K_{\mathbb R}^{\bullet}$
and $K_{\mathbb C}^{\bullet}$. Let ${\mathfrak m}$ denote the maximal ideal of
$\mathbb C\, [[\mbox{\rm H}^ 1( L^{\bullet}_{\mathbb C})^*]]$. We now have

\begin{thm}
\label{ha}
(Hain's Theorem.) Suppose $L^{\bullet}= (L^{\bullet}_{\mathbb R},
L^{\bullet}_{\mathbb C}, \alpha, W_{\bullet}, F^{\bullet})$ is a mixed
Hodge differential graded Lie algebra~ with $\mbox{\rm H}^ 0(L^{\bullet}_{\mathbb R} )= 0$. Then $R_{L^{\bullet}_{\mathbb C}}$
 is quasi-homogeneous\  (see Section \ref{singu}).
Moreover there exists a morphism of graded vector spaces
$$
\delta: \mbox{\rm H}^ 2(L^{\bullet}_{\mathbb C})^* \to Gr^W  \mathbb C\, [[ \mbox{\rm H}^ 1(L^{\bullet}_{\mathbb C})^*]]
$$
with image of $\delta$ contained in ${\mathfrak m}^2$
such that $R_{L^{\bullet}_{\mathbb C}}$  is the quotient
of $\mathbb C\, [[ \mbox{\rm H}^ 1(L^{\bullet}_{\mathbb C}) ^* ]]$ by the graded ideal
generated by the image of $\delta$.
\end{thm}

\begin{rem}
Hain further proves that $\delta\equiv Q^* \ (mod ~~{\mathfrak m}^3)$,
$Q^*$ is dual to $Q$ where
$$
Q: S^2 \mbox{\rm H}^ 1(L^{\bullet})\to \mbox{\rm H}^ 2(L^{\bullet})
$$
is given by the cup-product $Q(\eta)= [\eta, \eta]$.
\end{rem}

We will say an element $\eta\in \mbox{\rm H}^ i(L^{\bullet}_{\mathbb C} )$ has
{\em weight} $n$ if
$\eta\in W_n \mbox{\rm H}^ i( L^{\bullet}_{\mathbb C})$ but
$\eta\notin W_{n-1} \mbox{\rm H}^ i( L^{\bullet}_{\mathbb C})$. We will combine Hain's
theorem with the following theorem to obtain our desired result
about the singularities in representation varieties of fundamental
groups of smooth complex algebraic varieties.  Suppose now that
$M$ is a smooth  connected  complex algebraic variety, a representation
$\rho: \pi_1(M)\to \hbox{\bf G}$ with finite image, the bundle $adP$, etc., are
as above.

\begin{thm}
\label{we}
Under the conditions above there is a filtration $W_{\bullet}$ on
$A^{\bullet}(M, ad P)$ and a filtration $F^{\bullet}$ on
$A^{\bullet}(M, ad P_{\mathbb C})$
such that for the canonical map \newline
$\alpha: A^{\bullet}(M, ad P)\otimes \mathbb C \to
A^{\bullet}(M, ad P_{\mathbb C})$ the algebra
$$
L^{\bullet} = ( A^{\bullet}(M, ad P), A^{\bullet}(M, ad P_{\mathbb C}),
\alpha, W_{\bullet}, F^{\bullet})$$
is a mixed Hodge differential graded Lie algebra~. Moreover the weights of
$\mbox{\rm H}^ 1( A^{\bullet}(M, ad P_{\mathbb C}) )$ are $1$ and $2$
and the weights of $\mbox{\rm H}^ 2( A^{\bullet}(M, ad P_{\mathbb C}) )$
are $2, 3$ and $4$.
\end{thm}
\par\medskip\noindent{\it Proof: } Let $\tilde{M}$ be the finite cover of $M$ corresponding
to $\ker(\rho)$. Let $\Phi\cong \rho(\Gamma)$ be the group of
covering transformations.
By \cite{Su} there exists an equivariant completion $\tilde{N}$
of  $\tilde{M}$. But according to \cite{BM} there is also a
canonical resolution of singularities $N$ of $\tilde{N}$ so that the
complement $N- \tilde{M}$ is a divisor $D= D_1 \cup ... \cup D_m$ with normal crossings.

Hence the action of $\Phi$ extends to $N$, which is a
smooth $\Phi$-equivariant completion  of $\tilde{M}$.
Hence $\Phi$ acts on the log-complex of $\tilde{M}$
defined using the compactification   $N$. It is a
basic result of Deligne (\cite[Theorem 1.5]{D}) that
one may use the log-complex to define:

(a) A subcomplex $A^{\bullet}(\tilde{M})\subset {\cal A}^{\bullet}(\tilde{M})$
so that the inclusion is a quasi-isomorphism.

(b) Filtrations $W_{\bullet}, F^{\bullet}$ on $A^{\bullet}(\tilde{M})$ and
$A^{\bullet}(\tilde{M})\otimes \mathbb C$ which satisfy the axioms
of MHC.

By construction these filtrations are
$\Phi$-invariant.  We tensor with ${\mathfrak g}$ (regarded
as a mixed Hodge differential graded Lie algebra~ concentrated in degree zero).
Since $\Phi$ acts on
${\mathfrak g}$ via $ad \rho$, it also acts on the tensor
product. We obtain the required mixed Hodge
differential graded Lie algebra~ $L^{\bullet}$ by taking $\Phi$-invariants.
To derive weight restrictions we use results of
 Morgan \cite{Morgan1}, \cite{Morgan2}, who proved it
for  $A^{\bullet}(\tilde{M})\otimes \mathbb C$, etc. The operations of
tensoring with ${\mathfrak g}$ and taking $\Phi$-invariants
will not change  these restrictions on weights. $\qed$

\begin{rem}
Choose a point $m\in X$ and a point $\tilde{m}\in \tilde{M}$ over
$m$. We define augmentations (see \cite[\mathbb S 3.1]{GM})
$\epsilon: A^{\bullet}(M, ad P) \to {\mathfrak g}$ and
$\tilde\epsilon: A^{\bullet}(\tilde{M})\otimes {\mathfrak g}\to {\mathfrak g}$ by
evaluation at $m$ and $\tilde{m}$ respectively. We let
$A^{\bullet}(M, ad P)_0$ and $A^{\bullet}(\tilde{M})_0\otimes {\mathfrak g}$
be the augmentation ideals. Then all statements in Theorem \ref{we}
hold when   $A^{\bullet}(M, ad P)$ and $A^{\bullet}(M, ad P_{\mathbb C})$
are replaced by $A^{\bullet}(M, ad P)_0$ and
$A^{\bullet}(M, ad P_{\mathbb C})_0$. We abbreviate the corresponding
mixed Hodge differential graded Lie algebras~ by $L^{\bullet}_0$.
\end{rem}

Let $\Gamma, \rho, G$ be as above. Let $Z$ be
the representation variety $\mbox{\rm Hom}(\Gamma, G)$.
By combining Theorems \ref{ha}, \ref{we} we obtain

\begin{thm}
\label{hain}
The germ $(Z_{\mathbb C}, \rho)$ is analytically equivalent to a
quasi-homogeneous cone with generators of weights
$1$ and $2$ and relations of weights $2, 3$ and $4$.
Suppose that there is a  local
cross-section $S$ through $\rho$ to $G$-orbits. Then the  conclusion
is valid  not just for the germ $(Z_{\mathbb C}, \rho)$ but
also for $(\mbox{\rm Hom}(\Gamma, G)/\!/ G, [\rho])$.
\end{thm}
\par\medskip\noindent{\it Proof: } We apply Theorems \ref{ha} and \ref{we} to deduce
that the complete local $\mathbb C$-algebra $(R_{L^{\bullet}_0})_{\mathbb C}$
has a presentation of the required type. But by
Theorem \ref{con germ}, $(R_{L^{\bullet}_0})_{\mathbb C}$
 is isomorphic to the
complete local ring associated to the germ $(Z_{\mathbb C}, \rho)$.
We obtain the corresponding result for
$(\mbox{\rm Hom}(\Gamma, G_{\mathbb C})/\!/ G_{\mathbb C}, [\rho])$ by
replacing $L^{\bullet}_0$ by
$L^{\bullet}$ and applying \cite[Theorem 2.4 ]{KM3}.  Note that if $S$
exists then $\mbox{\rm H}^ 0(L^{\bullet})=0$. $\qed$

\medskip
There are infinitely many germs $(Y, 0)$, where
$Y\subset \mathbb C^n$ is an affine variety defined over
$\mathbb Z$, so that   $(Y, 0)$ is not quasi-homogeneous
with the weights of relations between $2$ and $4$.
We can even assume that $0$ is an isolated singular point, see
the Section \ref{singu}.
Thus as a consequence of Theorems \ref{hain},
\ref{main} we obtain the following

\begin{thm}
\label{T10.3}
Among the Artin groups $G_A^a$ there are infinitely
many mutually nonisomorphic groups which are not
isomorphic to fundamental groups of smooth complex
algebraic varieties.
\end{thm}
\par\medskip\noindent{\it Proof: } Let $Y$ be an affine variety defined over $\mathbb Q$ and $y\in Y$
be a rational point. Assume that the analytical germ $(Y, x)$ is
not quasi-homogeneous\  (with the weights of variables $1,2$ and weights of generators $2,3,4$). Let $A$ be an affine arrangement corresponding to the pair
$(Y, y)$ as in Proposition \ref{sfina}, so that the representation
$\rho_s: G^s_A\to PO(3,\mathbb C)$ corresponding to $y$ has finite image and
the group $\rho_s(G^s_A)$ has trivial centralizer in $PO(3,\mathbb C)$. Let
$\rho= \mu(\rho_s): G^a_A\to PO(3,\mathbb C)$, where $G_A^a$ is the {\em Artin
group} of the arrangement $A$. Recall that we have an open embedding
$$
\mu \circ alg \circ geo: Y\cong BR_0(A)\hookrightarrow \mbox{\rm Hom}(G_A^a, PO(3))/\!/ PO(3) = X(G_A^a, PO(3))
$$
Suppose that $G_A^a$ is
the fundamental group of a smooth complex algebraic
variety.  Then Theorem \ref{hain} can be applied to
the germ $(X(G_A^a, PO(3)), [\rho])$
provided we can construct a local cross-section through $\rho$ to the $PO(3,\mathbb C)$-orbits. In the definition of local cross-section we take
$$
U:= \mbox{\rm Hom}_f(G_A^a, PO(3,\mathbb C)) \hbox{~~and~~}
S:= \mu \mbox{\rm BHom}_f(G_A^s, PO(3,\mathbb C))\ .
$$
 Then $U$ is open by Corollary \ref{open-artin} and $S$ is a
cross-section because $\mbox{\rm BHom}_f(G_A^s, PO(3,\mathbb C))$ is a
cross-section for the action of $PO(3,\mathbb C)$ on
$\mbox{\rm Hom}_f(G_A^s, PO(3,\mathbb C))$ and the morphism
$\mu: \mbox{\rm Hom}_f(G_A^s, PO(3,\mathbb C)) \to U$ is an isomorphism. We get a
contradiction. To see that there
are infinitely many nonisomorphic examples we refer to
the argument at the end of the introduction.  $\qed$

\medskip
As the simplest example of $(Y,0)$ we can take the germ
$(\{x^5=0\}, 0)$.   We describe the Coxeter graph of the
Artin group corresponding to this singularity on the
Figure \ref{Fig9}. We let $x^5= (x^2)^2 \cdot x$, to get the
labelled graph\footnote{See \mathbb S\S \ref{groups}, \ref{gaa}.}
$\Lambda$ of $G^a$  from this diagram identify vertices marked
by the same symbols.

\begin{figure}[tbh]
\leavevmode
\centerline{\epsfxsize=5.5in \epsfbox{figure9.eps}}
\caption{\sl Labelled graph of an Artin group.}
\label{Fig9}
\end{figure}

\section{Sullivan's minimal models and singularities of
representation varieties}
\label{direct}

The goal of this section is to give a direct proof of Theorem \ref{hai} below
(which is Theorem \ref{t7} of the Introduction).

\begin{thm}
\label{hai}
Let $M$ be a smooth connected
complex algebraic variety with the fundamental group $\Gamma$. Let $\hbox{\bf G}$ be
the Lie group of real points of an algebraic group $G$ defined over
${\mathbb R}$~; let ${\mathfrak g}$ be the  Lie algebra of $\hbox{\bf G}$.
 Suppose that $\rho: \Gamma\to \hbox{\bf G}$ is a representation
with finite image. Then the germ  $(\mbox{\rm Hom}(\Gamma, G), \rho)$
 is analytically isomorphic to a
quasi-homogeneous cone with generators of weights $1$ and $2$ and
relations of weights $2, 3$ and $4$.
In the case there is a local
cross-section through $\rho$ to $Ad(G)$-orbits, then
the same conclusion is valid for the quotient germ
$(X(\Gamma, G), [\rho])$ of the character variety.
\end{thm}
\par\medskip\noindent{\it Proof: } We begin the proof by choosing a smooth $\Phi$-equivariant compactification $\tilde{N}= \tilde{M}\cup D$ of $\tilde{M}$ as in the proof of Theorem \ref{we}.

In what follows we shall use the following simple lemma:

\begin{lem}
\label{fi}
Suppose that $\Phi$ is a finite group and
$$
\begin{array}{ccccc}
~ & ~ & H\\
~ & ~ & g {\Big\downarrow}\\
E & \stackrel{f}{\longrightarrow} & F
\end{array}
$$
is a diagram of morphisms of $\Phi$-modules over $\mathbb C$ such that
$f(E)\subset g(H)$. Then $f$ admits a $\Phi$-equivariant lifting
$\tilde{f}: E \to H$.
\end{lem}
\par\medskip\noindent{\it Proof: } Since $f(E)\subset g(H)$ there exists a linear mapping $h: E \to H$ which lifts $f$. Then we let
$$
\tilde{f}= Av(h):= |\Phi|^{-1}\sum_{\phi\in\Phi} \phi \circ h \circ \phi^{-1} \quad \qed
$$

Recall that Morgan in \cite{Morgan2} defines a {\em mixed Hodge diagram}
$$
{\cal E}(log D) \stackrel{\varphi}{\longleftarrow}  E_{C^{\infty}}(\tilde{M}) \otimes \mathbb C \stackrel{\bar\varphi}{\longrightarrow} \bar{{\cal E}}(log D)
$$
associated with the pair $(\tilde{N}, D)$. The log-complex ${\cal E}(log D)$ is a subcomplex of $\mathbb C$-valued differential forms on $\tilde{M}$ and the complex $\bar{{\cal E}}(log D)$ is the log-complex with the {\em opposite} complex structure.

The mixed Hodge diagram must satisfy
certain properties described in \cite{Morgan2}.
In particular, it has a structure
of a mixed Hodge complex, i.e.  $E_{C^{\infty}}(\tilde{M})$ has an increasing
filtration $W$ and ${\cal E}(log D)$ has a pair of filtrations: an increasing
weight filtration $W$ and decreasing Hodge filtration $F$; $\varphi, \bar\varphi$ must preserve the weight filtrations and be quasi-isomorphisms.

\begin{prop}
\label{mhd}
There exists a $\Phi$-invariant mixed Hodge diagram with
$\Phi$-invariant structure of a mixed Hodge complex. The identity embedding
$$
id: {\cal E}(log D) \hookrightarrow {\cal A}(\tilde{M})\otimes\mathbb C
$$
is a quasi-isomorphism.
\end{prop}
\par\medskip\noindent{\it Proof: } First we describe  the {\em log-complex} ${\cal E}(log D)$ on
$\tilde{M}$ associated with the
compactification $N$. Let $z\in D$ be a point of $p$-fold intersection
$$
z\in D_{i_1} \cap ...\cap D_{i_p}
$$
where each $D_{i_j}$ is locally (near $z$) is given by the equation
$z_{i_j}=0$. Then elements $\sigma$ of ${\cal E}$ are $\mathbb C$-valued differential forms
on $\tilde{M}$ which can be (locally with respect to the $z_j$-coordinates)
written as
$$
\sum_J \eta_J \frac{dz_{i_1}}{z_{i_1}} \wedge ... \wedge \frac{dz_{i_p}}{z_{i_p}}
$$
where each $\eta_j$ extends to a $C^{\infty}$-form in a neighborhood~ of $z$ in $N$.
Thus near any generic point $z\in D_j$ the form $\sigma$ has at worst a simple
pole. Since the group $\Phi$ acts holomorphically on $N$ leaving $D$
invariant we conclude that this group acts naturally on the log-complex
${\cal E}(log D)$. The complex ${\cal E}(log D)$ has the {\em weight} and {\em Hodge} filtrations which are defined in a canonical way. Recall that
$W_{\ell}({\cal E}(log D))$ consists of differential forms of the type
$$
\omega = \sum_J \omega_J \wedge \frac{dz_{j_1}}{z_{j_1}}\wedge ... \wedge \frac{dz_{j_t}}{z_{j_t}} \ ,\ t\le \ell
$$
and $F^p({\cal E}(log D))$ consists of differential forms of the type
$$
\omega = \sum_J \omega_J \wedge dz_{j_1}\wedge ... dz_{j_s} \wedge  \frac{dz_{j_{s+1}}}{z_{j_{s+1}}}\wedge ... \wedge \frac{dz_{j_t}}{z_{j_t}} \ , \  t\ge p
$$
where the forms $\omega_J$ extend smoothly over the divisor $D$. Thus both filtrations are $\Phi$-invariant.

Now we describe the second complex
$E_{C^{\infty}}(\tilde{M})$ associated to $(N, D)$. For each component $D_j$ of $D$ choose a regular neighborhood~ $N_j$ in $\Phi$-invariant way, i.e. if $\phi\in \Phi$ and
$\phi: D_i \to D_j$ then $\phi: N_i \to N_j$. Let
$[\omega_j]\in \mbox{\rm H}^2(N_j, \partial N_j; \mathbb R)$ be the Thom's class. By using Lemma \ref{fi}
we choose a 2-form $\omega_j$ representing the class $[\omega_j]$ so that for $\phi\in \Phi$ mapping $D_i$ to $D_j$ we have: $\phi^*\omega_j =\omega_i$. Then Morgan takes 1-forms $\gamma_j \in {\cal E}(log D)$ supported on $N_j$ so that
$d\gamma_j= \omega_j$. We can choose $\gamma_j$ so that $\phi^*\gamma_j =\gamma_i$ for each $\phi\in \Phi$ such that $\phi: D_i \to D_j$ by using Lemma \ref{fi} again.

\medskip
The complex $E_{C^{\infty}}(\tilde{M})$ consists of global
sections of a certain sheaf ${\cal S}$ that we will describe
below. Let $U\subset \tilde{M}$ be an open subset missing all regular
neighborhoods~ $N_j$. Then sections of  ${\cal S}$ over $U$ are real-valued
infinitely differentiable differential forms on $U$.

Let $N^p$ denote subset of $\tilde{N}$ consisting of $p$-fold intersections between the regular neighborhoods~ $N_j$. Take a connected open subset $U\subset N^p$ which is disjoint from $N^{p+1}$.  We suppose that $U$ is contained in the $p$-fold intersection $N_{i_1} \cap ... \cap N_{i_p}$. Then sections of
${\cal S}$ over $U$ are elements of the Hirsch extension:
$$
{\cal A}^{\bullet}(U) \otimes_d \Lambda(\tau_{i_1},..., \tau_{i_p})
$$
where ${\cal A}^{\bullet}(U)$ is the complex of real-valued differential
forms on $U$ and
$$
d\tau_{i_j}= \omega_{i_j}|_U
$$
It is clear that the group $\Phi$ acts on the sheaf ${\cal S}$ and on the
complex $E_{C^{\infty}}(\tilde{M})$ of its sections as well. The complex
 $E_{C^{\infty}}(\tilde{M})$ has a canonically defined {\em weight}
filtration $W_{\bullet}$ which is therefore invariant under the action of
$\Phi$. (This filtration is similar to the weight filtration on
the log-complex, just use $\tau_j$-s instead of the forms $dz_j/z_j$.)
Finally Morgan defines a morphism
$$
\alpha: E_{C^{\infty}} (\tilde{M}) \otimes \mathbb C \to {\cal E}(log D)
$$
by mapping $\tau_j$ to $\gamma_j$ and differential forms supported in $\tilde{M}$
to themselves. Morgan proves that this morphism and the identify embedding
$$
{\cal E}(log D) \hookrightarrow {\cal A}^{\bullet}(\tilde{M})\otimes \mathbb C
$$
induce isomorphisms of cohomology groups. The mixed
Hodge structure on the diagram is $\Phi$-invariant by the construction.
This finishes the proof of Proposition \ref{mhd}. $\qed$

\begin{rem}
In what follows we shall use the notation  $A^{\bullet}= A^{\bullet}(\tilde{M})$ to denote the complex $E_{C^{\infty}}(\tilde{M})\otimes \mathbb C$.
\end{rem}

Given the weight filtration $W$ on $A^{\bullet}$ Morgan defines an increasing
{\em Dec-weight  filtration} $Dec\, W_{\bullet}(A^{\bullet})$ as
$$
Dec\, W_{\ell}(A^{k})= \{ x : x\in W_{k-\ell}\, (A^k), dx \in W_{k-\ell+1}\, (A^{k-1})\}
$$
Let $\nu: {\cal N}\to A^{\bullet}$ denote a 1-minimal model:
$$
{\cal N}= (\ 0\longrightarrow {\cal N}^0= \mathbb C \stackrel{0}{\longrightarrow} {\cal N}^1 \stackrel{d}{\longrightarrow} {\cal N}^2 \stackrel{d}{\longrightarrow} ...\ )
$$
Recall the basic properties of ${\cal N}$ and $\nu$ proven in
\cite[\S6 and Lemma 7.2]{Morgan2}:

(a) The $Dec$-weight filtration $Dec\, W_{\bullet}(A^{\bullet})$ of $A^{\bullet}$
pulls back to ${\cal N}$ to a {\em weight} filtration $W_{\bullet}({\cal N})$ which
{\em splits} so that ${\cal N}$ becomes {\em bigraded} with the differential
of the bidegree $(1,0)$:

${\cal N}^j= \oplus_{i\ge 0} {\cal N}_i^j$, ${\cal N}^j_i= {\cal N}^j\cap {\cal N}_i$,
 where ${\cal N}_i$ consists of elements of the {\em weight} $i$,
$$
d: {\cal N}_i \to {\cal N}_i, \ \ \wedge :
{\cal N}_i\otimes {\cal N}_j \to {\cal N}_{i+j}\quad ,$$
${\cal N}_0= {\cal N}^0$ and each ${\cal N}^i_j$ is finite-dimensional.

(b) The weight filtration on ${\cal N}$ induces a weight filtration on
$\mbox{\rm H}^{\bullet}({\cal N})$ so that the induced weights on $\mbox{\rm H}^1({\cal N})$
are $1, 2$, the induced weights on $\mbox{\rm H}^2({\cal N})$ are $2, 3, 4$.

(c) The homomorphism $\nu: {\cal N}\to A^{\bullet}$ is a weak equivalence.

(d) ${\cal N}$ is a 1-minimal differential algebra. Together with (a) it
implies that the restriction of the differential
$d$ to ${\cal N}^1_1$ is identically zero.

\begin{prop}
\label{lifmin}
Suppose that $A^{\bullet}$ is a differential graded commutative algebra
(over the ground field $\hbox{\bf k}= \mathbb C$ or $\mathbb R$) and
$\Phi$ is a finite group acting on $A^{\bullet}$. Then
the action of $\Phi$ on $A^{\bullet}$ lifts
to its action on a certain 1-minimal model ${\cal N}$ for
$A^{\bullet}$.
\end{prop}
\par\medskip\noindent{\it Proof: } We will prove the proposition by constructing ${\cal N}$ in
$\Phi$-invariant way. Let ${\cal N}^0:= \hbox{\bf k}$. Take ${\cal N}_{[1]}^1:= \mbox{\rm H}^1(A^{\bullet})$
and ${\cal N}_{[1]}$ be the differential graded algebra freely generated by
${\cal N}^0= {\cal N}_{[1]}^0$  and ${\cal N}_{[1]}^1$. We need a homomorphism
$$
\nu= \nu_{[1]}: {\cal N}_{[1]} \to A^{\bullet}
$$
which induces an isomorphism of the 1-st cohomology groups. The group $\Phi$ naturally
acts on $\mbox{\rm H}^1(A^{\bullet})$. We have the epimorphism of
$\Phi$-modules
$$
Z^1(A^{\bullet}) \longrightarrow \mbox{\rm H}^1(A^{\bullet})
$$
By Lemma \ref{fi} this epimorphism admits a $\Phi$-invariant splitting
$\nu^1: \mbox{\rm H}^1(A^{\bullet}) \to Z^1(A^{\bullet})$. Thus we let $\nu$ be the identity
embedding of ${\cal N}^0=\hbox{\bf k}$ to $A^0$ and $\nu|{\cal N}_{[1]}^1$ be $\nu^1$.
We continue the construction of $({\cal N}, \nu)$ by induction. Suppose
that  $({\cal N}_{[i]}, \nu_{[i]})$
are constructed and the homomorphism of $\Phi$-modules
$\nu_{[i]}: {\cal N}_{[i]} \to A^{\bullet}$ induces an isomorphism
of $\mbox{\rm H}^1$ but the induced mapping of $\mbox{\rm H}^2$ has nonzero kernel.
This kernel is canonically isomorphic to the relative cohomology group $\mbox{\rm H}^2({\cal N}_{[i]}^{\bullet}, A^{\bullet})$. Choose a $\Phi$-equivariant section $\sigma_{[i]}$ to the projection from the space of relative cocycles
$Z^2({\cal N}_{[i]}^{\bullet}, A^{\bullet})$ onto $\mbox{\rm H}^2({\cal N}_{[i]}, A^{\bullet})$. Let
$$
p_1: Z^2({\cal N}_{[i]}^{\bullet}, A^{\bullet}) \to Z^2({\cal N}_{[i]}^{\bullet})$$
and
$$
p_2: Z^2({\cal N}_{[i]}^{\bullet}, A^{\bullet}) \to A^1$$
be the projections. Both are $\Phi$-equivariant. Define
$$
d: \mbox{\rm H}^2({\cal N}_{[i]}^{\bullet}, A^{\bullet}) \to Z^2({\cal N}_{[i]})
$$
by $d= p_1 \circ \sigma_{[i]}$ and let
$$
\nu: \mbox{\rm H}^2({\cal N}_{[i]}^{\bullet}, A^{\bullet}) \to A^1\ , \quad \nu= p_2 \circ \sigma_{[i]}
$$
Define
$$
{\cal N}_{[i+1]}:= {\cal N}_{[i]} \otimes_d \mbox{\rm H}^2({\cal N}_{[i]}, A^{\bullet})
$$
and extend $\nu$ to $\nu_{[i+1]}: {\cal N}_{[i+1]} \to A^{\bullet}$ multiplicatively.
The group $\Phi$ acts on ${\cal N}_{[i+1]}$ in the natural way and the homomorphism $\nu_{[i+1]}$ is $\Phi$-equivariant.
$\qed$

\medskip
We now assume that we have a mixed Hodge diagram:
$$
{\cal E} \stackrel{\varphi}{\longleftarrow} A_{\mathbb C}
\stackrel{\overline{\varphi}}{\longrightarrow} \overline{\cal E}
$$
and a finite group $\Phi$ acting on the diagram compatibly with the differential graded algebra~ structures so that $\varphi$ and $\overline{\varphi}$ are $\Phi$-equivariant.
In \cite[\mathbb S 6]{Morgan2}, Morgan constructs a trigraded 1-minimal model
\[
\begin{array}{ccccc}
 {\cal E} & \stackrel{\varphi}{\longleftarrow} & A_{\mathbb C} &
\stackrel{\overline{\varphi}}{\longrightarrow} & \overline{\cal E} \\
~ &  \nwarrow & \uparrow & \nearrow \\
~ & ~ & {\cal N}^{\bullet}_{\bullet,\bullet} & ~ & ~   \\
\end{array}
\]
for the above mixed Hodge diagram (see \cite[Page 270]{Morgan2} for definition).
We will say that ${\cal N}^{\bullet}_{\bullet,\bullet}$ is a $\Phi$-equivariant
trigraded 1-minimal model for the above mixed Hodge diagram if all three morphisms with the source ${\cal N}$ are $\Phi$-equivariant and the trigrading of ${\cal N}$ is $\Phi$-invariant.

\begin{rem}
Morgan calls ${\cal N}^{\bullet}_{\bullet,\bullet}$ a {\bf bigraded} minimal model.
\end{rem}

\begin{prop}
\label{lif}
There exists a $\Phi$-equivariant trigraded
1-minimal model for the mixed Hodge diagram
$$
{\cal E} \stackrel{\varphi}{\longleftarrow} A_{\mathbb C}
\stackrel{\overline{\varphi}}{\longrightarrow} \overline{\cal E}
$$
The filtration $Dec\, W_{\bullet}(A)$ pulls back to a filtration $Dec\, W_{\bullet}({\cal N})$ given by
$$
Dec\, W_{q}({\cal N}):= \oplus_{r+s\le q}\ {\cal N}_{r,s}
$$
Consequently the filtration $Dec\, W_{\bullet}({\cal N})$ is $\Phi$-equivariantly split.
\end{prop}
\par\medskip\noindent{\it Proof: } We will check that Morgan's construction can be made $\Phi$-equivariant. To do this we examine Morgan's induction step when he passes from a trigrading on ${\cal N}_{[i]}$ to one on ${\cal N}_{[i+1]}$. This step is carried on the page 176 of Morgan's paper and involves a study of the diagram
\[
\begin{array}{ccccc}
Z^2({\cal N}_{[i]}, {\cal E}) & \stackrel{j}{\longleftarrow} & Z^2({\cal N}_{[i]}, A_{\mathbb C}) & \stackrel{\overline{j}}{\longrightarrow} & Z^2({\cal N}_{[i]}, \overline{\cal E}) \\
~ &  \searrow \! \nwarrow s & \downarrow \!\uparrow p &  s'\nearrow\! \swarrow
\\~ & ~ & \mbox{\rm H}^2({\cal N}_{[i]}, A_{\mathbb C}) & ~ & ~   \\
\end{array}
\]

By induction $\mbox{\rm H}^2({\cal N}_{[i]}^{\bullet})$ has a $\Phi$-equivariant mixed Hodge structure. Since $\mbox{\rm H}^2({\cal N}_{[i]}^{\bullet}, A_{\mathbb C}^{\bullet})$ is the kernel of the canonical (thus $\Phi$-equivariant) morphism
$$
\mbox{\rm H}^2({\cal N}_{[i]}^{\bullet})\to \mbox{\rm H}^2(A_{\mathbb C}^{\bullet})
$$
it inherits a $\Phi$-equivariant mixed Hodge structure. Consequently by Deligne's Theorem (see \cite[Proposition 1.9]{Morgan2})
$\mbox{\rm H}^2({\cal N}_{[i]}^{\bullet}, A_{\mathbb C}^{\bullet})$ has a canonical
(hence $\Phi$-invariant) bigrading. Thus it suffices to check that the cross-sections $s, p$ and $s'$ as well as the maps $h, h'$ on page 176 of \cite{Morgan2} can be chosen $\Phi$-equivariant. The cross-sections $s, p, s'$ are required to satisfy the linear conditions (1)--(3) of
\cite[Page 176]{Morgan2}. It is immediate that our averages
$Av(s), Av(p), Av(s')$ (defined as in Lemma \ref{fi} with respect
to the action of $\Phi$) again satisfy (1)--(3). Finally the maps
$$
h: \mbox{\rm H}^2({\cal N}_{[i]}^{\bullet}, A_{\mathbb C}^{\bullet}) \to Dec W_{r+s}({\cal E})
$$
and
$$
h': \mbox{\rm H}^2({\cal N}_{[i]}^{\bullet}, A_{\mathbb C}^{\bullet}) \to Dec W_{r+s}(\overline{\cal E})
$$
must satisfy certain lifting conditions. By Lemma \ref{fi} we can take $h, h'$ to be $\Phi$-equivariant.

By definition (see Proposition \ref{lifmin}) we have
$$
{\cal N}_{[i+1]}= {\cal N}_{[i]}\otimes_d \mbox{\rm H}^2({\cal N}_{[i]}^{\bullet}, A^{\bullet}_{\mathbb C})
$$
we extend the trigrading from ${\cal N}_{[i]}$ and $\mbox{\rm H}^2({\cal N}_{[i]}^{\bullet}, A^{\bullet}_{\mathbb C})$ to ${\cal N}_{[i+1]}$ multiplicatively. We obtain a $\Phi$-equivariant  trigrading on  ${\cal N}_{[i+1]}$ and a new diagram
\[
\begin{array}{ccccc}
 {\cal E} & \longleftarrow & A_{\mathbb C} &
\longrightarrow & \overline{\cal E} \\
~ &  \nwarrow & \uparrow & \nearrow \\
~ & ~ & {\cal N}_{[i+1]} & ~ & ~   \\
\end{array}
\]
of equivariant maps satisfying Morgan's axioms. This completes the
proof of the proposition. $\qed$

\medskip
We will no longer need the trigrading on ${\cal N}$ and instead will consider the bigrading:
$$
{\cal N}^k_{q}:= \oplus_{r+s=q}\ {\cal N}^k_{r,s}
$$
which defines a splitting of the Dec-filtration $Dec\, W_{\bullet}({\cal N})$.

\medskip

 Let $P$ be the flat bundle over $M$ associated
to $\rho$ and $adP$ the associated ${\mathfrak g}$-bundle; similarly
$ad\tilde{P}= \tilde{M}\times {\mathfrak g}$. We let $adP_{\mathbb C}$, $ad\tilde{P}_{\mathbb C}$
denote the complexifications of these vector bundles.
Let ${\cal A}^{\bullet}(M, adP)$, ${\cal A}^{\bullet}(\tilde{M}, ad\tilde{P})$ denote the
complexes of $adP$, $ad\tilde{P}$--valued differential forms.
According to  \cite[Theorem 2.4]{KM3} we have:

\begin{thm}
\label{con germ1}
If there is a local cross-section through $\rho$ to the $Ad(G)$-orbits then the
differential graded Lie algebra ${\cal A}^{\bullet}(M, adP_{\mathbb C})$ {\bf controls} the
germ $(X(\Gamma, G(\mathbb C)),[\rho])$.
\end{thm}

We tensor $A^{\bullet}$ with the Lie algebra ${\mathfrak g}$
(regarded as a differential graded Lie algebra concentrated in the
degree zero). Since
$\Phi$ acts on ${\mathfrak g}$ via $ad \rho$, it also
acts on the tensor product.
Clearly ${\cal A}^{\bullet}(M, adP_{\mathbb C})\cong
{\cal A}^{\bullet}(\tilde{M}, ad\tilde{P}\otimes\mathbb C)^{\Phi}$.
Since the inclusion $A^{\bullet}(\tilde{M})\hookrightarrow {\cal A}^{\bullet}(\tilde{M}, \mathbb C)$
induces an isomorphism of cohomology groups,  the inclusion
$$
A^{\bullet}(\tilde{M})\otimes {\mathfrak g}\hookrightarrow {\cal A}^{\bullet}(\tilde{M}, adP_{\mathbb C}) $$
also induces an isomorphism of cohomology groups of the differential
graded Lie algebras. Hence the inclusion
$$
(A^{\bullet}(\tilde{M})\otimes {\mathfrak g})^{\Phi}\hookrightarrow
({\cal A}^{\bullet}(\tilde{M}, ad\tilde{P}_{\mathbb C})^{\Phi}) \cong
{\cal A}^{\bullet}(M, adP_{\mathbb C})
$$
will also induce an isomorphism of cohomology groups. We conclude that
 under the conditions of Theorem  \ref{con germ1} the differential
graded Lie algebra
$(A^{\bullet}(\tilde{M})\otimes {\mathfrak g})^{\Phi}$ controls the germ of the
character variety $(X(\Gamma, G),[\rho])$ (see Theorem \ref{GolMil}).

\medskip
We will need similar results for the representation variety
$\mbox{\rm Hom}(\Gamma, G)$ itself. Pick a point $m \in \tilde{M}$.
We define an augmentation
$\epsilon: {\cal A}^{\bullet}(M, adP_{\mathbb C})\to {\mathfrak g}\otimes\mathbb C$
by evaluating degree zero forms
at a base-point $m\in M$ and sending the rest of forms to zero.
Let $A^{\bullet}(M, adP_{\mathbb C})_0$ be the kernel of $\epsilon$.
Recall that by Theorem \ref{con germ}
${\cal A}^{\bullet}(M, adP_{\mathbb C})_0$ controls the germ $(\mbox{\rm Hom}(\Gamma, G(\mathbb C)),\rho)$.

\medskip
We lift the augmentation $\epsilon$ to
${\cal A}^{\bullet}(\tilde{M}, ad\tilde{P}_{\mathbb C})$ as follows.
Let $\tilde{m}$ be a point in $\tilde{M}$ which projects to $m$. Then for each
$\omega \in {\cal A}^{0}(\tilde{M}, ad\tilde{P}_{\mathbb C})$ let
$$
\tilde\epsilon(\omega):= |\Phi|^{-1}\sum_{\gamma\in \Phi} \omega(\gamma\tilde{m})
$$
where $|\Phi|$ is the order of the group $\Phi$. We extend $\tilde\epsilon$ to
the rest
of ${\cal A}^{\bullet}(M, ad\tilde{P}_{\mathbb C})$ by zero.
It is clear that the restriction of $\tilde\epsilon$ to
$$
{\cal A}^{\bullet}(\tilde{M}, ad\tilde{P}_{\mathbb C})^{\Phi}\cong {\cal A}^{\bullet}(M, adP_{\mathbb C})$$
is the same as $\epsilon$. We let
$A^{\bullet}_0:= A^{\bullet}\cap \ker(\tilde\epsilon)$.
Again the inclusion
$$
(A^{\bullet}_0\otimes {\mathfrak g} )^{\Phi}\hookrightarrow
{\cal A}^{\bullet}(\tilde{M}, ad\tilde{P}_{\mathbb C})^{\Phi}_0
$$
is a quasi-isomorphism (i.e. it induces an isomorphism on cohomologies).

The action of the finite group $\Phi$ lifts from $A^{\bullet}\otimes{\mathfrak g}$
to the tensor product ${\cal N} \otimes {\mathfrak g}$ (recall that the action on the
Lie algebra ${\mathfrak g}$ is induced by the adjoint representation $ad\rho$
of the group $\pi_1(M)$).

Let ${\cal M} \subset {\cal N} \otimes {\mathfrak g}$ denote the subalgebra defined as:
$$
 {\cal M}  = ({\cal N})^{\Phi} \hbox{~~is the space of~~} \Phi-\hbox{invariants.}
$$
Let $\mu$ denote the restriction of $\nu$ to ${\cal M}$, the image of
$\mu$ lies in the algebra of $\Phi$-invariants
$(A^{\bullet}\otimes{\mathfrak g})^{\Phi}$. Similarly we let  ${\cal L} \subset {\cal M}$ denote
the kernel of $\tilde\epsilon\circ \mu$.

\begin{lem}
The homomorphisms $\mu: {\cal M} \to (A^{\bullet}\otimes {\mathfrak g})^{\Phi},
\mu: {\cal L} \to (A^{\bullet}\otimes {\mathfrak g})_0^{\Phi}$ induce
isomorphisms of $\mbox{\rm H}^0, \mbox{\rm H}^1$ and monomorphisms of $\mbox{\rm H}^2$.
\end{lem}
\par\medskip\noindent{\it Proof: } Standard. $\qed$

\begin{rem}
Note that $\mbox{\rm H}^0({\cal A}^{\bullet}(M, adP)_0)\cong \mbox{\rm H}^0((A^{\bullet}\otimes
{\mathfrak g})^{\Phi}_0) \cong \mbox{\rm H}^0({\cal L})\cong 0$ and, under the assumption
of Theorem \ref{con germ1},
$\mbox{\rm H}^0({\cal A}^{\bullet}(M, adP))\cong \mbox{\rm H}^0((A^{\bullet}\otimes
{\mathfrak g})^{\Phi}) \cong \mbox{\rm H}^0({\cal M})\cong 0$.
\end{rem}

\begin{cor}
The differential graded Lie algebra ${\cal L}$ controls the germ
$(\mbox{\rm Hom}(\Gamma, G), \rho)$ and (under the conditions of Theorem
\ref{con germ1}) the differential graded Lie algebra ${\cal M}$ controls
the germ $(X(\Gamma, G), [\rho])$.
\end{cor}

The split weight filtration on ${\cal N}$ defines a split weight filtration
on ${\cal N} \otimes {\mathfrak g}$ (by taking tensor products of the
components ${\cal N}_i$ with ${\mathfrak g}$). Using Proposition
 \ref{lif} restrict the split weight  filtration from ${\cal N}$ to  ${\cal M}$
and ${\cal L}$. Clearly these split filtrations of ${\cal M}$ and ${\cal L}$ satisfy the
properties (a)-(c), where in (a) instead of $\wedge$ we take the Lie
bracket. The property (d) can apriori fail, however we
will need only its weak version:

(d') The restriction of the differential to ${\cal M}^1_1$ and ${\cal L}^1_1$
is identically zero.

\medskip
\noindent Our further arguments are the same in the cases of ${\cal M}$ and ${\cal L}$
so we will discuss only ${\cal M}$.

\medskip
Note that ${\cal J}_5:= \oplus_{i=5}^{\infty} {\cal M}_i$
is an {\em ideal} in ${\cal M}$. We let ${\cal J}$ be the ideal generated by ${\cal J}_5$ and ${\cal M}^1_4$:
$$
{\cal J} = {\cal J}_5 \oplus {\cal M}^1_4 \oplus d({\cal M}^1_4)
$$
The quotient ${\cal M}/{\cal J}$ is again a differential graded Lie algebra.
The property (b) of ${\cal M}$ (weight restrictions
on the cohomology groups) implies that the projection morphism
${\cal M} \to {\cal Q}:= {\cal M}/{\cal J}$
induces an isomorphism of the 1-st and 2-nd cohomology groups. Hence by Theorem \ref{GolMil} the differential graded Lie algebras ${\cal Q}$ and ${\cal M}$ control
germs which are analytically isomorphic. So it is enough to prove that
${\cal Q}$ controls a quasi-homogeneous germ with the correct weights and we shall consider the differential graded Lie algebra~ ${\cal Q}$ from now on.

\begin{rem}
We shall use the notation ${\cal Q}_p$ to denote
the projection of ${\cal M}_p$ ($p\le 4$), elements of ${\cal Q}_p$
will be denoted $\eta_p$. Lemma \ref{big} implies
 that  ${\cal Q}= \oplus_p {\cal Q}_p$ so that:
\begin{itemize}
\item $d: {\cal Q}_p \to {\cal Q}_p$ for each $p$;
\item $[\cdot , \cdot ]: {\cal Q}_p \otimes {\cal Q}_q \to {\cal Q}_{p+q}$\ ;
\item The induced weights on $\mbox{\rm H}^1({\cal Q}^{\bullet})$ are $1, 2$ and the induced weights on $\mbox{\rm H}^2({\cal Q}^{\bullet})$ are $2, 3, 4$;
\item $d({\cal Q}^1_1) =0$.
\item ${\cal Q}^0 = {\cal Q}_0= 0$.
\end{itemize}
\end{rem}

\noindent We split each vector space ${\cal Q}^k_p$ into the direct sum
${\cal H}^k_p \oplus B^k_p \oplus C_p^k$, where

\begin{itemize}
\item The space of coboundaries $B^k_p$ is the image of
$d_p: {\cal Q}^{k-1}_p \to {\cal Q}^k_p$;

\item  The space ${\cal H}^k_p$ of ``harmonic forms'' is
a complement to $B^k_p$ in \newline
$Z^k_p:= \ker(d_p: {\cal Q}^k_p \to {\cal Q}^{k+1}_p)$;

\item  $C_p^k$ is a complement to $Z^k_p$ in ${\cal Q}^k_p$.
\end{itemize}

We let $\beta_p: {\cal Q}^k_p \to B^k_p$ denote the projection
with the kernel ${\cal H}^k_p \oplus C_p^k$ (``coclosed $k$-forms''). We
let $I_p: B^k_p \to C^{k-1}_p$ denote the inverse to the differential $d_p$.
This allows us to define the ``co-differential''
$$
\delta_p: {\cal Q}^k_p \to C^{k-1}_p \subset {\cal Q}^{k-1}_p, \quad \delta_p= I_p\circ \beta_p
$$
(whose kernel is ${\cal H}^k_p \oplus C_p^k$). Let
$\Pi_p: {\cal Q}_p \to {\cal H}_p$ denote the projection with the kernel
$B_p \oplus C_p$. Clearly the projection
$$
{\cal H}^k_p \to \mbox{\rm H}^k({\cal Q}_p^{\bullet})
$$
is an isomorphism of vector spaces. Notice that
${\cal Q}^1 = {\cal Q}^1_1 \oplus {\cal Q}^1_2 \oplus {\cal Q}^1_3$.  Consider the variety
$V\subset {\cal Q}^1$ given by the equation
\begin{equation}
\label{e1}
d\eta + [\eta, \eta]/2 =0 , \quad \eta \in {\cal Q}^1
\end{equation}
The algebra ${\cal Q}$ controls the germ $(V, 0)$ (see \mathbb S \ref{dgla})
and our goal is to show that this
germ is quasi-homogeneous with correct weights.
Since $\eta= \eta_1 + \eta_2 + \eta_3$ the equation (\ref{e1})
is equivalent to the system:
$$
d\eta_1 = 0, \ \ d\eta_2 + [\eta_1, \eta_1]/2=0, \ \
d\eta_3 + [\eta_1, \eta_2]=0, \ \
[\eta_1, \eta_3] + [\eta_2, \eta_2]/2= 0
$$
Recall that the differential $d$ is identically zero on ${\cal Q}^1_1$ and the
restriction
$$
d:{\cal Q}^1_3 \to {\cal Q}^2_3
$$
has zero kernel (since $\mbox{\rm H}^1({\cal Q}^{\bullet})$ has no weight 3 elements). Therefore
the equation $d\eta_3 + [\eta_1, \eta_2]=0$ is equivalent to the system:
$$
[\eta_1, d\eta_2]=0, \quad \Pi_3[\eta_1, \eta_2]=0, \quad \eta_3 + \delta_3 [\eta_1, \eta_2]=0
$$
Note however that the equation $d\eta_2 + [\eta_1, \eta_1]=0$ together with
the graded Lie identity imply that $[\eta_1, d\eta_2]=0$.
Thus we eliminate the variable $\eta_3$ and the system of equations
(\ref{e1}) is equivalent to:
\begin{equation}
\label{e2}
 d\eta_2 + [\eta_1, \eta_1]/2=0,\  \
  \Pi_3[\eta_1, \eta_2]=0,  \ \
-[\eta_1, \delta_3 [\eta_1, \eta_2]] + [\eta_2, \eta_2]/2= 0
\end{equation}
The mappings $d, \delta_3, \Pi_3$ are linear and the bracket
$[\cdot , \cdot]$ is quadratic. We conclude that the system of
equations (\ref{e2}) is quasi-homogeneous where
the weights of the generators (i.e. the components of) $\eta_j$ are $j=1,2$
and the weights of the relations are $2, 3$ and $4$. The only problem is that
the first polynomial equation has nonzero linear term. To resolve this problem
we let $\eta_2 = \eta'_2 + \eta''_2$, where
$\eta'_2 \in Z^1_2, \eta''_2 \in C^1_2$.
Thus (similarly to the case of $\eta_3$) we get:

the equation $d\eta_2 + [\eta_1, \eta_1]/2 =0$ is equivalent to the system:
$$
\Pi_2[\eta_1, \eta_1]=0, \quad \eta_2'' + \delta_2 [\eta_1, \eta_1]/2=0
$$
Thus we have $\eta_2= \eta_2' - \delta_2 [\eta_1, \eta_1]/2$ and
instead of the system of equations (\ref{e2}) we get the system
$$
 \Pi_2[\eta_1, \eta_1]=0,
  \Pi_3[\eta_1, \eta_2' - \delta_2 [\eta_1, \eta_1]]=0, $$
$$
-[\eta_1, \delta_3 [\eta_1, \eta_2' - \delta_2 [\eta_1, \eta_1]/2 ]] +
[\eta_2 ' - \delta_2 [\eta_1, \eta_1]/2, \eta_2' -
\delta_2 [\eta_1, \eta_1]/2]/2= 0
$$
which is quasi-homogeneous with the required weights.
Theorem \ref{hai} follows. $\qed$

\section{Malcev Lie algebras of Artin groups}
\label{completion}

Out discussion of the  material below follows \cite{Kah}.
Let {\bf k} be
a field of zero characteristic and $\Gamma$ be a group. We define the
{\bf k}-{\em unipotent completion} (or {\em Malcev completion}) $\Gamma\otimes${\bf k} of $\Gamma$  by the following universal property:

\begin{itemize}
\item There is a homomorphism $\eta\otimes $ {\bf k} $:\Gamma \to \Gamma\otimes
${\bf k}.
\item For every {\bf k}-unipotent Lie group $U$ and any homomorphism
$\rho: \Gamma\to U$ there is a lift $\tilde\rho: \Gamma\otimes ${\bf k}$ \to U$
so that $\tilde\rho \circ (\eta\otimes ${\bf k}$)= \rho$.
\item $\Gamma\otimes ${\bf k} and $\eta\otimes ${\bf k} are unique up to
an isomorphism.
\item  $\Gamma\otimes\hbox{\bf k}$ is $\hbox{\bf k}$-pro-unipotent.
\end{itemize}

\begin{rem}
Recall that any group $U$ above is torsion-free and nilpotent.
\end{rem}

\begin{defn}
The group $\Gamma\otimes${\bf k}
has a {\em pro}-{\bf k}-{\em nilpotent} Lie algebra ${\cal L}(\Gamma, \hbox{\bf k})$.
This algebra is called the {\bf k}-Malcev Lie  algebra of $\Gamma$.
\end{defn}

We will take $\hbox{\bf k}=\mathbb R$ in what follows. Thus we shall denote
${\cal L}(\Gamma):= {\cal L}(\Gamma, \mathbb R)$, etc.

\begin{ex}
Suppose that the group $\Gamma$ has a generating set consisting of elements of finite order. Then  ${\cal L}(\Gamma)=0$ because $\Gamma\otimes\mathbb R= \{1\}$. In particular, if $\Gamma$ is a Shephard group where all vertices have nonzero
labels then  ${\cal L}(\Gamma)=0$.
\end{ex}

Let $H$ be a finite-dimensional real vector space, $L(H)$ is the free
Lie algebra spanned by $H$. It can be described as follows. Consider the
tensor algebra $T(H)$ of tensors of all possible degrees on $H$, define the
Lie bracket of $T(H)$ by $[u, v] =u\otimes v- v\otimes u$. Then $L(H)$ is
the Lie subalgebra in $T(H)$ generated by elements of $H$.

Let $F_r$ be a free group of rank $r$ and $H$ be the $r$-dimensional
real  vector-space, then
$$
L(H)\cong {\cal L}(F_r) \quad .
$$
An element $u\in L(H)$ is said to have the degree $\le d$ if
$u\in \oplus_{i\le n} H^{\otimes i}$. The {\em degree}  of $u$ equals
$d$ if $deg(u)\le d$ but $deg(u)$ is not $\le d-1$. I.e. the degree of
$u$ is the highest degree of monomial in the expansion of $u$ as a
linear combination of tensor products of elements of $H$. For instance,
{\em quadratic elements} of $L(H)$ are elements of the degree $2$, i.e.
they have the form of nonzero linear combinations
$$
\sum_j [u_j, v_j], \quad u_j~, v_j~\in H
$$

A {\em quadratically presented} Lie algebra is the quotient $L(H)/J$ where
$J$ is an ideal generated by a (possibly empty) set of quadratic elements.

\begin{thm}
(P.\ Deligne, P.\ Griffith, J.\ Morgan, D.\ Sullivan, \cite{DGMS}.)
 Suppose that $M$ is a compact K\"ahler manifold, then the Malcev Lie
algebra ${\cal L}(\pi_1(M))$ is quadratically presented.
\end{thm}

\begin{thm}
(J.\ Morgan, \cite{Morgan1}, \cite{Morgan2}. ) {\bf ``Morgan's test.''}
Suppose that $M$ is a smooth complex algebraic variety. Then the
Malcev Lie algebra
${\cal L}(\pi_1(M))$ is the quotient $L(H)/J$, where $L(H)$ is a free
Lie algebra and the ideal $J$ is generated by elements of degrees
$2\le d \le 4$.
\end{thm}

\begin{rem}
Until now Morgan's theorem was the only known restriction on fundamental
groups of smooth complex algebraic varieties, besides finite presentability.
Much more restrictions are known in the case of smooth {\bf complete}
varieties and compact K\"ahler manifolds, see \cite{Kah}.
\end{rem}

Below we compute Malcev algebras of Artin groups. Suppose that
$G^a$ is an Artin group. Let $n$ be the number of generators of
$G^a$. Define a Lie algebra over $\mathbb R$
$$
{\cal L}:= \< X_1,..., X_n | [X_i, X_j]=0 \hbox{~if~} \epsilon(i,j)\ne \infty
\hbox{~is even~},  X_i=  X_j \hbox{~if~} \epsilon(i,j)\ne \infty \hbox{~is odd~}
\>$$
where $[X, Y]$ denotes the Lie algebra commutator. Clearly this Lie
algebra is quadratically presentable.

To compute Malcev completions we will need the following two lemmas

\begin{lem}
\label{LMa}
Suppose that $\rho: G^a\to N$ is a homomorphism to a torsion-free
 $s$-step nilpotent group. Then for all $x_i, x_j$ such
that $2q_{ij}= \epsilon(i,j)\ne \infty$ we have  $[\rho(x_i), \rho(x_j)]=\hbox{\bf 1}$.
 For all $x_i, x_j$ such that $2q_{ij} +1= \epsilon(i,j)\ne \infty$ we
have  $\rho(x_i)= \rho(x_j)$.
\end{lem}
\par\medskip\noindent{\it Proof: } The assertion is obvious if $N$ is Abelian. So we assume that
the assertion is valid for all $(s-1)$-step nilpotent torsion-free
groups $\bar{N}$. Let $Z(N)$ be the center of $N$, let
$\bar{N}:= N/Z(N)$ and $p: N\to \bar {N}$ be the projection.
Then by the induction hypothesis:

\begin{itemize}
\item $[p(\rho(x_i)), p(\rho(x_j))]=\hbox{\bf 1}$, provided that
$2q_{ij}= \epsilon(i,j)\ne \infty$.
\item $p(\rho(x_i))= p(\rho(x_j))$, provided that
$2q_{ij} +1= \epsilon(i,j)\ne \infty$.
\end{itemize}

(1) Consider the case $2q_{ij}= \epsilon(i,j)$. Then
$\rho(x_i) \rho(x_j)= \rho(x_j) \rho(x_i)z $, for some  $z\in  Z(N)$.
Thus the relation
$$
(x_i x_j)^{q_{ij}} =   (x_j x_i)^{q_{ij}}
$$
implies that
$$
z^{q_{ij}} \rho[(x_j x_i)^{q_{ij}}]=  \rho[(x_j x_i)^{q_{ij}}]
$$
Since $N$ is torsion-free we conclude that $\rho( [x_i, x_j])= \hbox{\bf 1}$.

(2) Another case is when $2q_{ij} +1= \epsilon(i,j)$. Then
$\rho(x_i)=z \rho(x_j)$
, for some  $z\in  Z(N)$. The relation
$$
(x_i x_j)^{q_{ij}} x_i =   (x_j x_i)^{q_{ij}} x_j
$$
implies that $z=1$. $\qed$

\medskip
\begin{rem}
In our paper we use only Artin groups with even labels.
\end{rem}

\begin{lem}
\label{Lcom}
Suppose that $U$ is a unipotent group, $a, b\in U$ are commuting
elements. Then
$[\log(a), \log(b)]=0$ in the Lie algebra of $U$.
\end{lem}
\par\medskip\noindent{\it Proof: } Since $U$ is unipotent we can think of $U$ as the subgroup of
the group of upper-triangular matrices with $1$-s on the diagonal.
Then for any $g\in U$ we have: $\log(g)= \log(\hbox{\bf 1} - (\hbox{\bf 1}- g))$,
$h= \hbox{\bf 1}- g$ is a nilpotent matrix, thus $\log( \hbox{\bf 1} -h)$ is a polynomial
of $h$. Since matrices $a, b$ commute, then any polynomial functions
of them commute as well. Thus $[\log(a), \log(b)]=0$.
$\qed$

\begin{thm}
\label{quad}
Under the above conditions ${\cal L}\cong {\cal L}(G^a)$ is the Malcev
Lie algebra of $G^a$.
\end{thm}
\par\medskip\noindent{\it Proof: } Let $F$ denote the free group on $x_1,..., x_n$ and $\pi: F\to G^a$
be the quotient map. Let $F\otimes \mathbb R$ be the $\mathbb R$-unipotent completion of
$F$ and $\eta: F\to F\otimes \mathbb R$ be the canonical homomorphism. Let
${\cal L}(F, \mathbb R)$ be the Lie algebra of $F\otimes \mathbb R$. Put $g_i= \eta(x_i)$,
$1\le i\le n$ and $X_i:= \log(g_i)$. Let ${\cal I}$ be the ideal in
${\cal L}(F, \mathbb R)$ generated by the commutators $[X_i, X_j]$, $\epsilon(i, j)\ne \infty$
is even and the elements
$X_i-  X_j$ if $\epsilon(i,j)\ne \infty$ is odd. Let ${\cal Q}:=
{\cal L}(F, \mathbb R)/{\cal I}$ be the quotient Lie algebra and $Q$ be the
corresponding pro-unipotent Lie group over $\mathbb R$. Let
$\hat{\pi}: F\otimes \mathbb R \to Q$ be the quotient map. Put
$\bar{g}_i := \hat{\pi}(g_i)$ and $\bar{X}_i:= d\hat{\pi}(X_i)$.
Then $\bar{g}_i  = \exp(\bar{X}_i)$, $1\le i\le n$. Consequently
$[\bar{g}_i, \bar{g}_j]= \hbox{\bf 1}$ for all vertices $i,j$ connected by an
edge with even label and $\bar{g}_i= \bar{g}_j$ for all vertices
$i,j$ connected by an edge with odd label. Hence we have a
commutative diagram
\[
\begin{array}{ccccc}
F & \stackrel{\eta}{\longrightarrow} & F\otimes \mathbb R & \longrightarrow & {\cal L}(F, \mathbb R) \\
\pi {\Big\downarrow} & ~ & \hat{\pi}{\Big\downarrow} & ~ & {\Big\downarrow}\\
G^a & \stackrel\tau{\longrightarrow} & Q & \longrightarrow & {\cal Q}\\
\end{array}
\]
where $\tau(\pi(x_i))= \bar{g}_i$, $1\le i\le n$. We claim that $Q$
is the Malcev completion of $G^a$. It is clear that any homomorphism
$\rho: Q\to U$ from $Q$ to a unipotent group $U$ is determined by
its pull-back to $G^a$ (because its pull-back to $F\otimes \mathbb R$ is
determined by its further pull-back to $F$). So let $\rho: G^a\to U$
be a homomorphism with $U$ a unipotent group over $\mathbb R$. The
homomorphism $\pi^* \rho$ extends to a morphism $\hat{\rho}:
F \otimes \mathbb R \to U$.  Note that $U$ is necessarily nilpotent
and torsion-free.

\medskip
According to Lemma \ref{LMa} for each pair of
$(i,j)$ such that $\epsilon(i, j)$ is even we have:
$$
\hat\rho ([g_i, g_j])= \pi^* \rho([x_i, x_j])= \rho([x_i, x_j])= \hbox{\bf 1}
$$
Hence $\log(\hat\rho(g_i))$ and  $\log(\hat\rho(g_j))$ commute in
the Lie algebra $u$ of $U$ (see Lemma \ref{Lcom}). Therefore
$d\hat\rho(X_i)$ and
$d\hat\rho(X_j)$ commute in $u$. The case of odd labels
$\epsilon(i, j)$ is similar. This implies that $d\hat\rho$
descends to ${\cal Q}$ and consequently $\hat\rho$ descends
to $Q$. $\qed$

\begin{cor}
\label{squad}
If $G^a$ is any Artin group then ${\cal L}(G^a)$ is quadratically presented.
\end{cor}

Thus the Artin groups constructed in Theorem \ref{T10.3} satisfy
 {\em Morgan's test} of being fundamental groups of smooth
complex algebraic varieties.

\section{Representation varieties near the trivial representation}
\label{triv}

The second author would like to thank Carlos Simpson for explaining
Theorem \ref{clarify} in this section.

Let $\Gamma$ be a finitely-generated group, $\Gamma \otimes \mathbb R$ is
its Malcev completion, $\eta\otimes \mathbb R: \Gamma \to
 \Gamma \otimes \mathbb R$ is the canonical homomorphism. Let
${\cal L}(\Gamma)$ denote the Malcev Lie algebra of $\Gamma$. Let
$\hbox{\bf G}$ be the set of real points of an algebraic
group $G$ defined over $\mathbb R$, ${\mathfrak g}$ be the Lie algebra of $\hbox{\bf G}$.
The homomorphism $\eta\otimes \mathbb R$ induces the pull-back morphism
$\eta^*: \mbox{\rm Hom}(\Gamma\otimes\mathbb R, \hbox{\bf G}) \to \mbox{\rm Hom}(\Gamma, \hbox{\bf G})$.
Let $\rho_0: \Gamma \otimes\mathbb R\to \hbox{\bf G}$ be the trivial
representation .

\begin{thm}
\label{clarify}
\begin{enumerate}
\item If the Lie algebra ${\cal L}(\Gamma)$ is quadratically
presentable then the variety $\mbox{\rm Hom}({\cal L}(\Gamma), {\mathfrak g})$
is given by homogeneous quadratic equations.
\item The varieties $\mbox{\rm Hom}(\Gamma\otimes\mathbb R, \hbox{\bf G})$ and
$\mbox{\rm Hom}({\cal L}(\Gamma), {\mathfrak g})$ are naturally isomorphic.
\item The morphism $\eta^*$ induces an isomorphism of germs
$$
(\mbox{\rm Hom}(\Gamma\otimes\mathbb R, \hbox{\bf G}),\rho_0) \longrightarrow
( \mbox{\rm Hom}(\Gamma  , \hbox{\bf G}), \eta^*(\rho_0))
$$
\end{enumerate}
\end{thm}
\par\medskip\noindent{\it Proof: } The property (1) is obvious. To prove (2) note
that the group $\Gamma\otimes\mathbb R$ is $\mathbb R$-pro-unipotent, thus
we have a natural isomorphism between the representation variety
of $\Gamma\otimes\mathbb R$ and of its Lie algebra.

Now consider (3). Let ${\cal A}$ be an Artin local $\mathbb R$-algebra.
We recall  that $G({\cal A})$ is the set of
${\cal A}$-points of $G$, algebraically the group $G({\cal A})$ is
the semidirect product $N_{{\cal A}} \rtimes \hbox{\bf G}$, where
$N_{{\cal A}}$ is a certain $\mathbb R$-unipotent group (kernel of the
natural projection
$p_0: G({\cal A})\to \hbox{\bf G}$). Consider the space
$$
\mbox{\rm Hom}_0(\Gamma, G({\cal A})):= \{ \rho: \Gamma \to G({\cal A}) |
p_0(\rho)= \rho_0\}\cong \mbox{\rm Hom}(\Gamma, N_{{\cal A}})
$$
Thus, by the definition of $\Gamma\otimes\mathbb R$, for all Artin local
$\mathbb R$-algebra ${\cal A}$ the morphism $\eta^*$ induces a
 natural bijection between the $\mathbb R$-points of the varieties
$\mbox{\rm Hom}(\Gamma, N_{{\cal A}})$ and $\mbox{\rm Hom}(\Gamma\otimes\mathbb R, N_{{\cal A}})$. Then we have an
induced isomorphism between functors
$$
\mbox{\rm Hom}(\Gamma, G) : {\cal A} \mapsto set\ ,  \quad
\mbox{\rm Hom}(\Gamma\otimes\mathbb R, G) : {\cal A} \mapsto set
$$
of ${\cal A}$-points. Therefore (by \cite{GM}) $\eta^*$ induces an
isomorphism of the germs
$$
(\mbox{\rm Hom}(\Gamma\otimes\mathbb R, \hbox{\bf G}),\rho_0~) \longrightarrow
( \mbox{\rm Hom}(\Gamma  , \hbox{\bf G}), \eta^*(\rho_0))
$$
$\qed$

\begin{prop}
Let $\Gamma$ be a Coxeter group or a Shephard group (where all vertices
have labels). Then the
trivial representation $\rho_0: \Gamma\to \hbox{\bf G}$ is
infinitesimally rigid (and hence is an isolated reduced point)
in $\mbox{\rm Hom}(\Gamma, \hbox{\bf G})$.
\end{prop}
\par\medskip\noindent{\it Proof: } The group $\Gamma$ is generated by elements of finite order.
Let $\zeta$ be a cocycle in $Z^1(\Gamma, {\mathfrak g})$. Then
$\zeta|_{\<g_j\>}$ is a coboundary for each generator $g_j$
of $\Gamma$ (since $g_j$ has finite order). However $\rho_0(g_j)=\hbox{\bf 1}$,
hence $\zeta(g_j)=0$ for all $j$. We conclude that $\zeta=0$.
$\qed$

\begin{thm}
 Let $\Gamma$ be any  Artin group. Then the representation
variety
$\mbox{\rm Hom}(\Gamma, \hbox{\bf G})$ has at worst quadratic singularity at
the trivial representation.
\end{thm}
\par\medskip\noindent{\it Proof: } Combine Corollary \ref{squad} and Theorem \ref{clarify}.
$\qed$

redpajama_set_name: RedPajamaArXiv