Rationality of moduli of elliptic fibrations with fixed monodromy
Fedor Bogomolov
We prove rationality results for moduli spaces of elliptic K3 surfaces and elliptic rational surfaces with fixed monodromy groups.
rat26.tex

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\begin{document}

%\input epsf

\title[Rationality of moduli]{\bf Rationality of 
moduli of elliptic fibrations with fixed monodromy}

\author{Fedor Bogomolov}
\address{Courant Institute of Mathematical
  Sciences, 251 Mercer str., New York, NY 10012-1185, USA} 
\email{bogomolo@cims.nyu.edu}
\thanks{The first author was partially 
supported by NSF grant DMS-9801591}


\author{Tihomir Petrov} 
\address{Courant Institute of Mathematical
  Sciences, 251 Mercer str., New York, NY 10012-1185, USA} 
\email{petrovt@cims.nyu.edu}
\thanks{The second author was partially supported by NSF grant
  DMS-9802154}

\author{Yuri Tschinkel}
\address{Dept. of Mathematics, Princeton University, Princeton NJ 08544-1000, USA}
\email{ytschink@math.princeton.edu}
\thanks{The third author was partially supported by NSA and by NSF
grant DMS-0100277}

%    General info
\subjclass{Primary 14J27, 14J10; Secondary 14D05}

\keywords{Elliptic fibrations, Monodromy groups, Moduli spaces}
%\date{January, 2001}
\date{\today} 

\begin{abstract}
We prove rationality results
for moduli spaces of elliptic K3 surfaces and
elliptic rational surfaces with fixed monodromy groups. 
\end{abstract}

\maketitle

%\tableofcontents

\setcounter{section}{-1}

\section{Introduction}
 
\noindent
Let $X$ be an algebraic variety of dimension $n$ over $\C$. 
One says that $X$ is rational if its function field
$\C(X)$ is isomorphic to $\C(x_1,\dots,x_n)$.
The study of rationality 
properties of fields of invariants $\C(X)^{\bG}=\C(X/\bG)$
is a classical theme in algebraic geometry.
For a finite group $\bG\subset \PGL_n$ acting on $X=\P^{n-1}$ the problem is 
referred to as Noether's problem (1916). 
It is still unsolved for $n=4$.  
Another class of examples is provided by {\em moduli spaces}. 
Birationally, they are often representable as quotients of 
simple varieties, like projective spaces or Grassmannians, 
by actions of linear algebraic groups, 
like $\PGL_2$. Rationality is known for each of the following
moduli spaces:


\begin{itemize}
\item curves of genus $\le 6$ \cite{Ka}, \cite{sh-2}, \cite{Ka-2}, \cite{K-3}, 
\cite{sh-1};
\item hyperelliptic curves \cite{Ka}, \cite{B};
\item plane curves of degrees $4n+1$ and $3n$ \cite{sh-3}, \cite{Ka-1};
\item Enriques surfaces \cite{Ko};
\item polarized K3 surfaces of degree 18 \cite{sh-2};
\item stable vector bundles (with various numerical 
characteristics) on curves, Del Pezzo surfaces, $\P^3$ 
\cite{ks}, \cite{by}, \cite{cm},\cite{N},\cite{M}; 
\end{itemize}
and in many other cases. For excellent surveys we refer to 
\cite{dol} and \cite{sh-3}. 
We will study rationality properties of moduli spaces of 
smooth non-isotrivial Jacobian elliptic fibrations
over curves 
$$
\pi\colon \E\ra C
$$  
with fixed global monodromy group 
$\tilde{\Ga}=\tilde{\Ga}(\E)\subset \SL_2(\Z)$.
In \cite{Bog} we developed techniques aimed at
the classification of possible global 
monodromies $\tilde{\Ga}$.  
The present paper gives a natural 
application of these techniques.



Let  $\cB$ be an irreducible algebraic family of 
Jacobian elliptic surfaces. 
Then the set of subgroups $\tilde{\Ga}\subset\SL_2(\Z)$ 
such that $\tilde{\Ga}$ is the (global) 
monodromy group of some $\E$ in this family is finite. 
Moreover, for every such group $\tilde{\Ga}$ the subset
of fibrations with this monodromy
$$
\cB_{\tilde{\Ga}}:=\{ b\in \cB\, \, |\, \, 
\tilde\Ga(\E_b)=\tilde\Ga\}
$$
is an algebraic (not necessarily closed) 
subvariety of $\cB$. 

Generalizing this observation, we introduce
(maximal) {\em parameter spaces} 
$\cF_{\tilde{\Ga}}$ of elliptic fibrations with
fixed global monodromy $\tilde{\Ga}$ (considered
up to fiberwise birational transformations 
acting trivially on the base of the
elliptic fibration). 
These parameter spaces can be represented
as quotients of quasi-projective varieties by
algebraic groups.
In particular, 
we can consider {\em irreducible connected components} 
of the parameter space 
$\cF_{\tilde{\Ga}}$, which we call
{\em moduli spaces}.  Even though these moduli spaces 
need not be algebraic varieties, we can still make sense 
of their birational type. 

\

\noindent
{\bf Theorem.} {\em
Let $\tilde{\Ga}\subset \SL_2(\Z)$ be a proper subgroup of finite
index. Then all moduli spaces of (Jacobian) elliptic rational or
elliptic K3 surfaces with global monodromy $\tilde{\Ga}$ are 
rational. 
}

\

\noindent
{\bf Corollary.}
{\em
For all $\tilde\Ga$ with moduli $\cF_{\tilde\Ga}$ of dimension $>0$ there
exists a number field $K$ such that
there are infinitely many nonisomorphic elliptic K3 surfaces over $K$
with global monodromy $\tilde\Ga$. 
}

\

\begin{rem}
Our method shows that many other classes of
moduli of elliptic surfaces over $\P^1$ 
with fixed monodromy are rational or unirational. 
However, we cannot expect a similar result for all moduli spaces
of elliptic surfaces over higher genus curves, since the moduli space
of higher genus curves itself is not uniruled (by a result of Harris and 
Mumford \cite{HM}). 
\end{rem}


\


We proceed to give a more 
detailed description of our approach.  
First of all, we can work not with 
the monodromy group $\tilde{\Ga}$ itself but rather 
with its image 
$$
\Ga\subset\PSL_2(\Z)
$$ 
under the
natural projection
$\SL_2(\Z)\twoheadrightarrow\PSL_2(\Z)$.
Let
$$
\cH=\{z\in \C\mid\im(z)>0\}
$$ 
be the upper half-plane and
$$
\overline{\cH}=\cH\cup\Q\cup\{\infty\}.
$$ 
The natural $j$-map
$$
  j\,:\, C\to{\P}^1={\overline {\cH}}/{\PSL_2(\Z)}
$$
decomposes as 
$$
j=j_{\Ga}\circ j_{\E},
$$ 
where
$$
\begin{array}{cccccc}
  j_{\E}\colon &  C    &  \ra  & M_{\Ga} & = & {\overline {\cH}}/{\Ga}\\
  j_{\Ga}\colon & M_{\Ga}&  \ra  & {\Pr}^1& = & 
{\overline {\cH}}/{\PSL_2(\Z)}.
\end{array}
$$
Here $M_{\Ga}$ is the 
$j$-{\em modular curve} 
corresponding
to $\Ga$; it is equipped with a
special triangulation, obtained as 
the pullback of the standard triangulation of 
${\mathbb S}^2={\Pr}^1(\C)$ (by
two triangles with vertices at $0,1$ and $\infty$) 
under the map $j_{\Ga}$
(which ramifies only over $0,1$ and $\infty$). 
We call the obtained triangulation of
$M_{\Ga}$ a $j_{\Ga}$-triangulation. 
Let $T_{\Ga}$ be the preimage in $M_{\Ga}$ 
of the closed interval $[0,1]\subset {\Pr}^1$. 
The graph $T_{\Ga}$ is our main tool in the combinatorial
analysis of $\Ga$. 

\

Denote by $\chi(\E)$ the Euler characteristic of $\E$. 
It splits equivalence classes of  Jacobian elliptic surfaces 
(modulo fiberwise birational transformations) 
into {\em algebraic families}. 
In particular, if $C={\Pr}^1$ then the {\em algebraic variety}
$\cF_r$ parametrizing (equivalence classes of)
Jacobian elliptic surfaces with given  
$\chi(\E)$ is irreducible; here we put 
$r=\chi(\E)/12.$
Our goal is to analyze the 
birational type of (irreducible components) 
$$
{\cF}_{r,\tilde\Ga}\subset {\cF}_r
$$ 
parametrizing fibrations with fixed
monodromy group $\tilde\Ga$. 
It suffices to study parameter spaces ${\cF}_{r,\Ga}$ 
corresponding to $\Ga\subset\PSL_2(\Z)$, since every 
irreducible component of ${\cF}_{r,\tilde\Ga}$
coincides with a component of ${\cF}_{r,\Ga}$. 

\

From now on we assume that $C=M_{\Ga}=\Pr^1$.  
Denote by $\cR_{d,\Ga}$ the 
space of rational maps $\Pr^1\to \Pr^1$ 
(of degree $d$) with prescribed 
ramification (encoded in $T_{\Ga}$).   
The spaces $\cF_{r,\Ga}$ are quotients
by the action of $\PGL_2\times H_{\Ga}$
of fibrations over $\cR_{d,\Ga}$ with fibers
(Zariski open subsets of) ${\Sym}^{\ell}({\Pr}^1)$ 
(for appropriate $d$ and $\ell$). 
Here $\PGL_2$ acts (on the left) by changing the 
parameter on the base $C={\Pr}^1$ and 
$H_{\Ga}$ is the group of 
automorphisms of $M_{\Ga}=\Pr^1$ 
stabilizing the embedded graph $T_{\Ga}$ (acting on the right). 
The nontriviality of $H_{\Ga}$ 
means that there is a $\Ga'\subset\PSL_2(\Z)$ containing 
$\Ga$ as a normal subgroup with $H_{\Ga} = \Ga'/\Ga$.
So in most cases in order to
prove rationality of $\cF_{r,\Ga}$ it is sufficient to 
establish it for ${\PGL_2}\backslash {\cR}_{d,\Ga}$, 
which can be deduced from general
rationality results for $\PGL_2$-quotients
(see \cite{Bogom-Katsylo}, \cite{Ka}).
To cover {\em all} cases we need to set up a rather
extensive combinatorial analysis.  

\

Here is a roadmap of the paper.
In Section~\ref{sect:covers} we discuss
finite covers $M_{\Ga}\to \Pr^1$ 
in the spirit of
Grothendieck's ``Dessins d'Enfants'' program
(see \cite{Lochak},\cite{Schneps} 
and the references therein) and 
introduce the invariants  $\GD(\Ga), \RD(\Ga)$ and $\ET(\Ga)$.
In ``ideal'' cases $\ET(\Ga)$ coincides with the
number of triangles in the 
$j_{\Ga}$-triangulation of $M_{\Ga}$
(the notation $\ET(\Ga)$ stands for
``{\em Effective Triangles}'').
In Section~\ref{sect:ellf} we recall basic facts about
elliptic fibrations and introduce the invariant $\ET(\E)$.  
For an ``ideal'' elliptic fibration 
one has $\ET(\Ga)=\ET(\E)$. 
In Section~\ref{sect:modsp} we discuss {\em moduli}
of elliptic fibrations with fixed monodromy. 
In Sections~\ref{sect:rat} and \ref{sect:rat-sp}
we formulate and prove several
rationality results for $\PGL_2$ and related
quotients. 
In Section~\ref{sect:ellmono}  
we classify families of rational elliptic
surfaces and elliptic 
K3 surfaces with different monodromy groups.
In Section~\ref{sect:comb}, we study relations between the 
combinatoric of the graph $\Ga$ and the topology of $\E$. 
And finally, in Section~\ref{sect:pictures} 
we list (certain) relevant subgroups $\Ga\subset\PSL_2(\Z)$ 
(represented by trivalent graphs $T_{\Ga}$). 
There are too many monodromy groups of elliptic K3
surfaces to be drawn on paper, but we show how to obtain 
them from our list by simple operations.






\section{Finite covers}
\label{sect:covers}


Let $\Ga$ be a subgroup of finite index in $\PSL_2(\Z)$.
The latter is isomorphic to a free product $\Z/3*\Z/2$. 
Consider the map
$$
{\overline{\cH}}/{\Ga}=M_{\Ga}\xrightarrow{j_{\Ga}}{\Pr}^1=
{\overline{\cH}}/{\PSL_2(\Z)},
$$
ramified over the points $0,1,\infty\in {\Pr}^1$.
Denote their preimages in 
$M_{\Ga}$ by $A,B$ and $I$, respectively. 
The possible ramification 
orders are $3$ or $1$ for
$A$-points, $2$ or $1$ for $B$-points and arbitrary for $I$-points.
The points $0,1$ and $\infty$ subdivide the circle
${\mathbb P}^1(\R)={\mathbb S}^1$ into three segments and, together with the
upper and lower hemisphere, define a decomposition of 
${\mathbb P}^1(\C)={\mathbb S}^2$ into three triangles.
This induces a special triangulation of $M_{\Ga}$ with
vertices in $A,B$ and $I$-points 
which we call the $j_\Ga$-{\em triangulation}. 
The preimage of the segment $[0,1]\subset{\Pr}^1$
defines a graph $T_{\Ga}$ which determines the 
$j_\Ga$-triangulation uniquely. 
Interior vertices of $T_{\Ga}$ are marked by $A_6$ and ends
are marked by either $A_2$ or $B_2$.


\begin{nota}
\label{nota:gd}
The {\em graph datum} $\GD(\Ga)$  of $T_\Ga$ is the 
formal sum 
$$
\GD(\Ga):=[a_6A_6+a_2A_2+b_2B_2],
$$
where $a_i$  ($i=6,2$) is the number of $A_i$ vertices and $b_2$ 
is the number of $B_2$-vertices. 
Denote by $\tau^0=\tau^0(\Ga)$ the number of vertices of $T_{\Ga}$
(including the ends), 
by $\tau^1=\tau^1(\Ga)$ the number of edges and by 
$\tau^2=\tau^2(\Ga)=\pi_0(M_{\Ga}\setminus T_{\Ga})$. 
\end{nota}

\

\begin{rem} 
\label{rem:uniqgr}
For given $a_2,b_2$ there
is a unique group with 
$$
\GD(\Ga)=[A_6+a_2A_2+b_2B_2].
$$  
\end{rem}

\

Forgetting the markings of $T_{\Ga}$ we obtain a connected unmarked 
{\em topological} graph $T^{u}_{\Ga}$ with (possibly some) 
ends and all interior vertices of valency $3$ --- 
a {\em trivalent graph}.





\begin{lemm} 
Let $X$ be a compact orientable Riemann surface of genus 
$g(X)$ and $T^{u}\subset X$ an 
embedding of a connected trivalent
graph such that
\begin{itemize}
\item 
the set $X\setminus T^u$ is a disjoint union of topological cells;
\item 
all interior vertices of $T^u$ are trivalent;
\item 
the ends of $T^u$ are arbitrarily marked by two colors $A_2$ and
$B_2$.
\end{itemize}
Then there exist a subgroup $\Ga\subset\PSL_2(\Z)$ 
and a unique complex structure on $X$ such that
$X=M_{\Ga}$ and $T^u=T_{\Ga}^u$.
\end{lemm}

\begin{proof}
Assume that we have an embedded graph $T^u\subset X$ satisfying the
conditions above. Mark by $A$ all trivalent vertices and
enlarge the graph $T^u$ by putting a $B$-vertex in the middle of any
edge bounded by two $A$-vertices. Put one $I$-vertex into
every connected component of $X\setminus{T}^u$ and connect all
$I$-vertices with $A$ and $B$-vertices at the boundary of the
corresponding domain. By assumption, every connected component of
$X\setminus{T}^u$ is contractible. Consider the boundary of the
individual cell. Every $A$-vertex of the boundary is
connected by edges to $B$-vertices only.  
Similarly, the  $B$-vertices are connected by 
edges only to $A$-vertices. 
Hence every triangle of the induced triangulation 
has vertices colored by three colors:
$A,B$ and $I$. This gives a $j_\Ga$-triangulation of $X$.
Following Alexander \cite{Alex}, 
we observe that a $j_\Ga$-triangulation
defines a map 
$$
{h}\,:\, X\to {\Pr}^1
$$ 
which is cyclically ramified over $A,B$ and $I$ (see \cite{Bog}). 
The trivalence of $T^u$ implies that $h$ has only 
$3$ or $1$-ramifications
over $0\in{\Pr}^1$ and only $2$ or $1$-ramifications over
$1\in{\Pr}^1$. 
Since $\PSL_2(\Z)={{\Z}/3}*{{\Z}/2}$ there is exactly
one subgroup $\Ga\subset \PSL_2(\Z)$ (of finite index) 
which corresponds to the covering $X\to{\Pr}^1$.
Any graph $T_{\Ga}^u$ constructed via a
subgroup $\Ga\subset{\PSL_2(\Z)}$ satisfies the conditions above.
Indeed, we have already described the 
$j_\Ga$-triangulation on $M_{\Ga}$. 
Triangles adjacent to a
given $I$-vertex constitute a contractible cell and the 
division of $M_{\Ga}$ into neighborhoods of $I$-vertices is a cellular
decomposition of $M_{\Ga}$. 
Hence after removing $I$-vertices with open
edges from them we obtain the preimage of $[0,1]$. If we forget the
$B$-vertices which lie between two $A$-vertices we
obtain the graph $T_{\Ga}^u$. Thus 
$T^u_{\Ga}\subset X = M_{\Ga}$ is the boundary of this
cellular decomposition and $T_{\Ga}$ is simply $T^u_{\Ga}$ with
an $A,B$-marking of the ends. 
\end{proof}

\begin{rem} 
Graphs which are isotopic in $X$ 
(modulo diffeomorphisms of $X$ of degree $1$) define 
conjugated subgroups of $\PSL_2(\Z)$.
\end{rem}


\begin{rem}
Even if we omit the condition of compactness of $X$
we still get a bijection 
between conjugacy classes of 
subgroups of finite index of $\PSL_2(\Z)$ and
embedded trivalent graphs with marked ends.
\end{rem} 


\begin{rem}
The topology of $X$ restricts the topology of $T^u_{\Ga}$. The graph 
$T^u_{\Ga}$ must contain some $1$-skeleton of $X$. 
In particular, the map 
$\pi_1(T^u_{\Ga})\to \pi_1(X)$ is surjective.
Hence $T^u_{\Ga}$ can be a tree only if $X={\mathbb S}^2$. 


For $X={\Pr}^1$ the connectedness of $T^u$ guarantees that
all the components of $X\setminus T^u$ are contractible. 
Hence we can classify graphs in $X={\Pr}^1$
by drawing them on the plane.
In general, connectedness of $T^u$ is necessary but not sufficient.  
\end{rem}



\begin{defn}
\label{defn:etg}
Define  
$$
\begin{array}{ccl}
\ET(\Ga)& := & 6{\tau}^0=6(a_6+a_2+b_2)\\
\D(\Ga) & := & 6a_6+2a_2
\end{array}
$$
\end{defn}

Thus both $\ET(\Ga)$ and $\D(\Ga)$  
depend only on the marking of the
ends but not on the embedding of the graph.
Observe that 
$\D(\Ga)$ is the {\em number of triangles} in the
corresponding $j_{\Ga}$-triangulation of $M_{\Ga}$ and that
$$
[\PSL_2(\Z):\Ga]=\D(\Ga)/2.
$$ 


\begin{rem}
If $M_{\Ga}$ arises from an elliptic fibration as
in the Introduction then $\D(\Ga)/2$ 
equals the number of {\em Dehn twists} in
$\Ga$ around the multiplicative singular fibers.
\end{rem}

\begin{nota}
\label{nota:rami-datum}
Let $f\,:\, C\to \Pr^1$ be a cover of degree $d$ and 
$p\in \Pr^1$ a ramification point of $f$. 
The {\em local ramification datum} is an $\N$-valued vector 
$v=(v_k)$, ($\sum v_k =d$), 
where $v_k$ is the order of ramification of
$f$ at a point $c_k\in f^{-1}(p)$. 
A {\em reduced local ramification datum} is a vector
$\overline{v}$ obtained from $v$ by 
omitting all entries $v_k=1$. 
The vector $v$ is defined up to 
permutation of the entries.     

For $f=j_{\E}\,:\,  C\to M_{\Ga}=\Pr^1$ 
we have distinguished ramification points,
namely those over $A$- and 
$B$-vertices of the graph $T_{\Ga}\subset M_{\Ga}$. 
The (global) $j_{\E}$-ramification datum is
the vector 
$$
\RD(j_{\E}):=[v_{1,A},\dots,v_{n,A}, v_{n+1,B},\dots,v_{n',B},
\overline{v}_{n'+1},\dots,\overline{v}_{n''}],
$$ 
where the $v_{i,A}$ are local ramification data over 
$A$-vertices for $i=1,\dots,n$,  
(resp. $v_{i,B}$ for $B$-vertices, $i=n+1,\dots,n'$) and
$\overline{v}_i$ are 
{\em reduced} local ramification data for unspecified 
other points in $M_{\Ga}$ for $i>n'$ (distinct from 
$A$- and $B$-vertices of $M_{\Ga}$). 

For $f=j_{\Ga}\,:\, M_{\Ga}\ra \Pr^1$ 
the distinguished (and the only) ramification points are $0,1,\infty$. 
We write
$$
\RD(j_{\Ga}):=[v_{0},v_{1},{v}_{\infty}]
$$ 
for the global $j_{\Ga}$-ramification datum.
\end{nota}





\begin{exam}
Assume that $\GD(\Ga)=[nA_6+A_2+B_2]$ is the graph datum 
of $T_{\Ga}\subset M_{\Ga}$ and 
let $j_\E\,:\, {\Pr}^1\to M_{\Ga}={\Pr}^1$ be a finite cover. 
Then the $j_{\E}$-ramification datum 
\begin{align*}
\RD(j_\E)=[(2,3)_A,(2,2,1)_B,(2),(2)]
\end{align*}
means that $\deg(j_\E)=5$, that $j_\E$  
has ramification points of order $2$ and $3$ over one point 
$A_2\in T_{\Ga}$ and $(2,2,1)$ over one $B_2$-point
and ramifications of order $2$ over 
two other unspecified points in $M_{\Ga}$. 
\end{exam}














\section{Elliptic fibrations}
\label{sect:ellf}


\no
In this section 
we briefly recall some basic facts of Kodaira's theory
\cite{Kod} of elliptic fibrations.
For more details we refer to 
\cite{Bart}, \cite{Fried} and
\cite{Shafar}. Let 
$$
\pi\colon \E\ra C
$$ 
be a smooth non-isotrivial relatively minimal
Jacobian elliptic fibration over a smooth 
projective curve $C$. This means that:
\begin{itemize}
\item 
$\E$ is a smooth compact complex 
projective surface and $\pi$ is a proper holomorphic map;
\item 
the generic fiber of $\pi$ is a smooth curve of genus $1$;
\item 
the fibers of $\E$ do not contain exceptional curves of the
first kind, i.e., rational curves $F$ such that $(F^2)=-1$ ({\em
relative minimality});
\item 
there exists a (global) zero section 
$s\,:\, C\to \E$ ({\em Jacobian elliptic fibration});
\item 
the $j$-function which assigns to each smooth fiber
${\pi}^{-1}(p)=\E_p\subset \E$ its 
$j$-invariant is a non-constant
rational function on $C$ ({\em non-isotriviality}).
\end{itemize}
\

\no
It is well known that $s^2<0$. We define  
$$
\ET(\E):= -24s^2.
$$

\begin{lemm}
\label{lemm:ett}
We have  
$$
\ET(\E)/2=-12s^2 =\chi(\E)=c_2(\E).
$$
\end{lemm}

\begin{proof}
Well known, but we decided to include an argument.
Since $\E$ is smooth and relatively minimal the
canonical bundle $K_\E$ of $\E$ is 
induced from a one-dimensional bundle $K$ on the base $C$. 
The sheaf $\pi^*K(C)$ is a subsheaf of $K_\E$. 
Since there are singular fibers we have the following
equality
\begin{align*}
  h^0(\E,\Omega^1)= h^1(\E,\mathcal O)=g
\end{align*}
where $g$ is the genus of $C$. By Riemann-Roch we obtain
\begin{align*}
  \chi({\mathcal O})=1-g+h^0(\E,K_\E)=\chi(\E)/12.
\end{align*}
We also know that $s^2+sK_\E-2g+2=0$ (genus formula). 
Therefore, 
\begin{align*}
  1-g+h^0(\E,K_\E)=\deg(K)-2g+2=\chi(\E)/12
\end{align*}
since $\deg(K)>2g-2$ and hence $h^1(C,K)=0$. 
Further, 
$$
sK_\E=\deg(K).
$$ 
Thus $s^2 +sK_\E-2g+2=0$ 
transforms to $s^2+\chi(\E)/12=0$.
\end{proof} 


Let $C^{\rm sing}=\{p_1,\dots,p_k\}\subset C$ be 
the set of points on the base 
corresponding to singular fibers. 
The topological Euler characteristic 
$\chi(\E)=c_2(\E)$ is equal to the sum of Euler
characteristics of the singular fibers 
$\E_{p_i}={\pi}^{-1}(p_i)$ 
(since every generic fiber has Euler characteristic equal
to $0$). Therefore, 
$$
\ET(\E)=\sum_{p_i\in C^{\rm sing}}\ET(\E_{p_i}),
$$
where the summation runs over all singular fibers of $\E$
and $\ET(\E_{p_i})$ is the contribution from the
corresponding singular fiber. 
Since the fibration is 
Jacobian every singular fiber
has a unique representative from 
Kodaira's list and it is defined by
the local monodromy.
The possible types of singular fibers 
and their $\ET$-contributions are:
$$
\begin{array}{c|c|c|c}
          & \ET  &            & \ET \\
 \hline
{\rm I}_0 &     & {\rm I}_0^* &  12    \\ 
{\rm I}_n & 2n  & {\rm I}_n^* & 2n+12 \\
{\rm II}  &  4  & {\rm IV}^*  &  16  \\
{\rm III} &  6  & {\rm III}^* &  18  \\
{\rm IV}  &  8  & {\rm  II}^* &  20
\end{array}
$$
Here ${\rm I}_0$ is a smooth fiber, ${\rm I}_n$ is a multiplicative
fiber with $n$-irreducible components. The types ${\rm II},{\rm III}$
and ${\rm IV}$ correspond to the case of potentially good reduction.
More precisely, 
the neighborhood of such a fiber is a
(desingularization of a) quotient of a
local fibration with smooth 
fibers by an automorphism of finite
order. 
The corresponding order is $4$ for the case ${\rm III}$ and $3$
in the cases ${\rm II},{\rm IV}$. 
The fibers of type ${\rm I}_0^*,$ 
(resp. ${\rm I}_n^*,{\rm II}^*,{\rm III}^*,{\rm IV}^*)$ 
are obtained from fibers ${\rm I}_0$ 
(resp. ${\rm I}_n,{\rm IV},{\rm III},{\rm II})$ 
(after changing the local automorphism by the involution
$x\mapsto -x$ in the local group 
structure of the fibration). We shall
call them $*$-{\em fibers} in the sequel.


\begin{rem}
The local invariant $\ET(\E_{p})$ has a monodromy interpretation.
Namely, every element of a local monodromy at $p\in C^{\rm sing}$ has a
minimal representation as a product of elements conjugated to 
$\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$
in $\SL_2(\Z)$. The length of this representation equals 
$\ET(\E_p)/2$. This explains the equality
$\ET(\E_p^*)= \ET(\E_p)+12$ --- the element
$\left(\begin{smallmatrix} {-1}& 0 \\0 &
      {-1}\end{smallmatrix}\right)\in\SL_2(\Z)$ 
is a product of $6$ elements conjugated to
$\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1
\end{smallmatrix}\right)$ (elementary Dehn twists).
\end{rem}














\section{Moduli spaces}
\label{sect:modsp}

\no
Every Jacobian elliptic fibration  $\E\to\Pr^1$
admits a {\em Weierstrass model}
$\bar{\E}$. 
Its geometric realization is given as follows: there
exists a pair of sections 
$$
\begin{array}{ccc}
g_2& \in &  H^0({\Pr}^1,{\mathcal O}_{{\Pr}^1}(4r)), \\
g_3& \in & H^0({\Pr}^1,{\mathcal O}_{{\Pr}^1}(6r))
\end{array}
$$ 
such that $\E$ is given by 
\begin{equation}
\label{weier}
  y^2z=4x^3-g_2xz^2-g_3z^3,
\end{equation}
inside $\Pr({\mathcal O}_{\Pr^1} \oplus {\mathcal O}_{{\Pr}^1}(2r)\oplus 
{\mathcal O}_{{\Pr}^1}(3r))$,
subject to conditions 
\begin{itemize}
\item
the discriminant $\D=g_2^3-27g_3^2$ is not identically $0$;
\item
for every point $p\in{\Pr}^1$ we have 
\begin{equation}\label{eqn:ineq}
\min(3\nu_p(g_2),2\nu_p(g_3))<12,
\end{equation}
where $\nu_p$ is the valuation corresponding to $p\in{\Pr}^1$
\end{itemize}
(see \cite{Fried} or \cite{F}, Section 7).

\

Two pairs $(g_2,g_3)$ and $(g'_2,g'_3)$ define 
isomorphic Jacobian elliptic surfaces 
$(\E,s)$ and $(\E',s')$ iff there exits an 
$h\in \GL_2(\C)$ transforming $(g_2,g_3)$ into $(g'_2,g'_3)$
under the natural action of $\GL_2$ on 
(the $\GL_2$-linearized) $\mathcal O_{\Pr^1}(r)$.
We define $\cF_{r}$ as the set of isomorphism classes
of pairs $(g_2,g_3)$ subject to the conditions above.

The parameter space 
$\cF_r$ has a natural structure of a (categorical) 
quotient of some open subvariety $U_r$ of the sum of two linear 
$\GL_2$-representations
$$
H^0({\Pr}^1,{\mathcal O}_{{\Pr}^1}(4r))
\oplus H^0({\Pr}^1,{\mathcal O}_{{\Pr}^1}(6r))
$$ 
by the action of $\GL_2$. 
Equivalently, $\cF_r$ is a (categorical) 
quotient of the open subvariety 
$U_r'=U_r/\gm$ of the weighted projective space 
$$
{\Pr}_{4r,6r}(4r+1,6r+1)
$$ 
by the action of $\PGL_2$.

\begin{lemm}
The variety $U_r'$ is a disjoint union of locally closed
subvarieties $U_{r,\tilde{\Ga}}'$, each preserved under 
the action of $\PGL_2$, such that for every $u\in U_{r,\tilde\Ga}'$
one has $\tilde\Ga(\E_u)=\tilde\Ga$. 
\end{lemm}

\begin{proof}
The set of  $(g_2,g_3)$ (of bounded degree) 
such that the $j$-map 
decomposes as $j_{\Ga}\circ j_{\E}$ 
is a closed algebraic variety (not necessarily irreducible).
Its intersection with the open subvariety defined by the conditions
\ref{eqn:ineq} is also closed. 
It contains a {\em finite} number of 
proper locally closed subvarieties corresponding 
to {\em proper} finite index subgroups of $\Ga$ 
(their number is bounded
as a function of $r$).  
The complement to these subvarieties
consists of finitely many irreducible components, 
each preserved under the action of $\PGL_2$. 
For $u$ in any of these components the lift of $\Ga$ to
the monodromy group of the corresponding fibration $\E_u$
is constant. The number of such possible lifts is finite. 
This gives a finite decomposition of $U_r'$ as claimed. 
\end{proof}



The unstable points of the $\PGL_2$-action
on the weighted projective space correspond to 
sections $g_2,g_3$ with high order of vanishing 
at some point $p$. Namely
$\nu_p(g_2)>2r,\nu_p(g_3)>3r$. 
However, the inequality~(\ref{eqn:ineq})
implies that $6r<12$. Thus, for $r\ge 2$, 
$\cF_r$ is a $\PGL_2$-quotient of some open
subvariety of the semistable locus 
$$
{\Pr}_{4r,6r}^{\rm ss}(4r+1,6r+1)\subset 
{\Pr}_{4r,6r}(4r+1,6r+1). 
$$
It follows that $\cF_r$ is a quasi-projective algebraic variety.
This variety is clearly unirational and in fact rational by
\cite{Ka}.


Moreover, for $r\ge 2$ we can define
a set of {\em subvarieties} $\cF_{r,{\tilde{\Ga}}}\subset \cF_r$ 
such that for every 
$b\in \cF_{r,\tilde{\Ga}}$ the corresponding Jacobian elliptic 
surface $(\E_b,s)$ has global monodromy group $\tilde{\Ga}$.

\

\begin{rem}
\label{rem:degen}
Notice that the maps $j_{\E}$ for 
elliptic  fibrations corresponding 
to different points of the same irreducible component
of $\cF_{r,\tilde{\Ga}}$ can have different $\RD(j_\E)$, even 
over the $A_2$ or $B_2$-ends of $T_{\Ga}\subset M_{\Ga}$. 
Thus, for a given irreducible component, we have the notion of 
a {\em generic} ramification datum $\RD(j_{\E})$ and 
its {\em degenerations}. 
\end{rem}

\


The case $r=1$,  corresponding to rational elliptic surfaces, is more subtle -
the subvariety $U_1'$ contains unstable points. 
The quasi-projective locus of semistable points 
${U_r^{\rm ss}}'$ is a disjoint union of locally closed
$\PGL_2$-semistable subsets
${U_{r,\tilde{\Ga}}^{\rm ss}}$; taking quotients we 
obtain varieties $\cF_{1,\tilde{\Ga}}$ parametrizing rational 
elliptic fibrations with global monodromy $\tilde{\Ga}$. 

Let $W_1'=U_1'-{U_1^{\rm ss}}'$ be
the complement.
It consists of pairs $(g_2,g_3)$ with 
$$
g_2 = l^3 f_2,\,\,\, g_3 =l^4f_3,
$$ 
where $l$ is a linear form (vanishing at a point $p$ and) 
coprime to $f_2,f_3$ and 
$\deg(f_2)=1,\deg(f_3)=2$. For $w\in W_1'$ we have
$\deg(j)\le 4$. The case of $\tilde\Ga\neq \SL_2(\Z)$ 
corresponds to $\deg(j_{\Ga})\ge 2$. Thus we have to consider
two cases: 
\begin{itemize}
\item
$\deg(j_{\Ga}) = \deg(j_{\E})=2$;
\item 
$\deg(j_{\Ga}) \le 4, \deg(j_{\E})=1$. 
\end{itemize}
The first case does not occur since $j^{-1}(0)$ has
ramification of type $(3,1)$ 
(by the assumption that $f_2$ is coprime to $l$
and that $3\nu_p(g_2)<12$).
Thus the $j$-map cannot be decomposed 
even locally into a product of two
maps. 
The second case leads to  

\begin{lemm}
If $w\in W_1'$ and $\tilde\Ga(\E_w)\neq \SL_2(\Z)$ 
then $\deg(j_{\E})=1$ and one has one of the 
following graph and ramification data:
$$
\begin{array}{l|l}
\GD(\G)        &  \RD(j_{\Ga}) \\
   \hline 
\lbrack A_6+A_2\rbrack   & \lbrack (3,1)_0,(2,2)_1, (3,1)_{\infty}\rbrack \\ 
\lbrack A_6+A_2+2B_2\rbrack &\lbrack (3,1)_0,(2,1,1)_1,(4)_{\infty}\rbrack \\ 
\lbrack A_6+3B_2 \rbrack   & \lbrack (3)_0,(1,1,1)_1,(3)_{\infty} \rbrack   \\
\lbrack A_6+B_2 \rbrack   & \lbrack (3)_0,(2,1)_1,(2,1)_{\infty} \rbrack   
\end{array}
$$ 
\end{lemm}

\begin{proof}
The formula $j = lf_2^3/(lf_2^3 - f_3^2) $ shows that
$j_{\Ga}$ has a point with local ramification datum $(3,1)$ or $(3)$,  
corresponding to 
$$
[\PSL_2(\Z):\Ga(\E)]=4\,\,{\rm or }\,\, 3.
$$
Since only two more branch points are allowed and one of them is $1$
(with local ramifications 1 or 2), 
the Euler characteristic computation gives 
the ramification data listed in the statement plus one more:
$$
[(3,1)_0,(2,2)_1,(2,2)_{\infty}].
$$
However, this datum is impossible for topological reasons 
(the only possible graph datum is $[A_6+A_2]$ and there
is a unique embedded graph $T_{\Ga}$ with this datum). 

If $\deg (j)=3$ then one has a
cyclic point of order $3$, leading to the data above.  
\end{proof}


\begin{coro}
Every irreducible component $W_{1,\tilde\Ga}'\subset W_1'$ such that
$\tilde\Ga(\E_w)\neq \SL_2(\Z)$ for $w\in W_{1,\tilde\Ga}'$ 
is rational. 
\end{coro}
Consider an irreducible component $\cF_{r,\tilde\Ga}$ and the
corresponding decomposition $j=j_\Ga\circ j_{\E}$. Here 
$$
j_{\E}=(j_{\E,2},j_{\E,3})\,:\,\Pr^1\ra M_{\Ga}=\Pr^1
$$
is a pair of homogeneous polynomials in 2 variables. 
Let 
$$
\cG=\{ (g_2,g_3)\}\subset H^0(\Pr^1,\cO_{\Pr^1}(4r))
\oplus H^0(\Pr^1,\cO_{\Pr^1}(6r))
$$
be the subset corresponding to smooth elliptic fibrations. 
Put
$$
\cJ_{\Ga}:=\{ j \, |\, \exists j_{\E}\,:\, \Pr^1\ra M_{\Ga}\,\, 
s.t.\,\, j=j_{\Ga}\circ j_{\E}\}.
$$

\begin{lemm}
\label{lemm:jj}
If $j\in \cJ_{\Ga}\cap \cJ_{\Ga'}$ with 
$\Ga\neq \Ga'$ then there exist
an $h\in \PSL_2(\Z)$, a group  
$\Ga''\subset \Ga\cap h\Ga'h^{-1}$ and a map 
$j_{\E}''\,:\, \Pr^1\to M_{\Ga''}$ such that
$j=j_{\E''}\circ j_{\Ga''}$. 
\end{lemm}

\begin{proof}
The monodromy group and its image in $\PSL_2(\Z)$ are 
uniquely determined by the smooth part of 
the elliptic fibration. 
Therefore,  in any smooth family of elliptic surfaces
$$
\Ga({\rm generic}\,\, {\rm fiber})\supseteq
\Ga({\rm special}\,\, {\rm fiber}).
$$
Since $\Ga$ is defined modulo conjugation by elements in $\SL_2(\Z)$
the claim follows.
\end{proof}

\begin{coro}
We have a decomposition $\cG=\bigsqcup \cG_{\Ga}$ into a finite (disjoint) union
of algebraic $\GL_2$-stable subvarieties such that for all 
$g=(g_2,g_3)\in \cG_{\Ga}$  the monodromy group 
$\tilde{\Ga}(\E_g)\subset \SL_2(\Z)$ is a subgroup of a central 
$\Z/2$-extension of $\Ga$.  
\end{coro}

\begin{rem}
For a given $g\in \cG_{\Ga}$ the map $j_{\E}$ is not unique. 
Let $j_{\E}$ and $j_{\E}'$ be two such maps. 
Then $j_\E=h_{\Ga}\circ j_{\E}'$, where $h_{\Ga}\in \Aut(T_{\Ga})$
is an automorphism of $M_{\Ga}$, preserving $T_{\Ga}$.
\end{rem}





\begin{lemm}
\label{lemm:decom}
We have a decomposition 
$$
\cG_{\Ga}=\bigsqcup_k \cG_{\tilde{\Ga},k}
$$
into a  finite union of algebraic {\em irreducible} 
$\GL_2$-stable subvarieties
such that $\tilde{\Ga}(\E_g)=\tilde{\Ga}$ for all $g\in \cG_{\tilde{\Ga},k}$. 
\end{lemm}

\begin{proof}
Assume that some $g\in \cG_{\Ga}$ belongs to 
$\cG_{\tilde{\Ga},1}\cap \cG_{\tilde{\Ga},2}$, where
$\cG_{\tilde{\Ga},1},\cG_{\tilde{\Ga},2}$ are different (non-conjugated)
lifts of $\Ga$ into $\SL_2(\Z)$. 
Lemma~\ref{lemm:jj} implies that there exists a {\em proper}
subgroup $\Ga''\subset\Ga$ such that 
$g$ belongs to $\cG_{\Ga''}$, contradiction.
\end{proof}



Let $\cG_{\tilde{\Ga}}=\cG_{\tilde{\Ga},k}$ 
be an irreducible component of $\cG_{\Ga}$
as in Lemma~\ref{lemm:decom} and $g\in \cG_{\tilde{\Ga}}$ its generic point. 
It determines a set of $*$-fibers on the base $\Pr^1$. We denote their
number by $\ell$. 
Choose (one of) the 
$j_{\E_g}$, with ramification datum $\RD=\RD(j_{\E_g})$.  
We get a map 
$$
\phi_{\cU}\,\,:\,\, \cU_g\to \cU_{j_g}\times (\Pr^1)^{\ell},
$$
where $\cU_g\subset \cG_{\tilde{\Ga}}$ is a neighborhood of $g$ and 
$\cU_{j_g}\subset \cR(\RD)$
is a neighborhood of the map $j_g=j_{\E_g}$ in the space 
$$
\cR(\RD):=\{ j\,:\, \Pr^1\to \Pr^1 \, |\,\, \RD(j)=\RD \}
$$
of rational maps with ramification datum $\RD$. 



\begin{lemm}
\label{lemm:locall}
The map $\phi_{\cU}$ is a local (complex analytic) surjection.  
\end{lemm}

\begin{proof}
First observe that on $M_{\Ga}$ there is a projective
local system $(\Z\oplus\Z)/(\Z/2)$ which induces
a projective local system on an open part of $\Pr^1$. 
The obstruction to the extension of this system to
a linear $\Z\oplus \Z$-system is an integer modulo 2
which depends only on the topological type of the projective
system. Therefore, it doesn't change under a small variation
of maps $j$ with fixed $\RD$. Now it suffices to apply Kodaira's
main theorem which guarantees the existence and uniqueness
of an elliptic fibration with a given linear $\Z\oplus\Z$-system.  
\end{proof}


\begin{coro}
Let $\cF_{r,\tilde\Ga}'\subset \cF_{r,\tilde\Ga}$ 
be an (irreducible) component with 
generic ramification datum $\RD$. Then 
$\cF_{r,\tilde\Ga}'$ surjects (rationally) onto
the quotient of the variety of
rational maps $\cR(\RD)$ by $H_{\Ga}$.
\end{coro}

\begin{proof}
Since both  $\cF_{r,\tilde\Ga}'$ and $\cR(\RD)$ 
are algebraic varieties the local complex analytic surjection
from Lemma~\ref{lemm:locall} extends to an algebraic correspondence.
Moreover, two decompositions of the map $j$ as
$j=j_{\Ga}\circ j_{\E}$ differ by an element in $H_{\Ga}$. 
This gives a map to the quotient space, which is a (global) rational
surjection.  
\end{proof}





\begin{prop}
\label{lemm:struct}
Every irreducible component $\cF_{r,\tilde\Ga}$ contains an open
part $\cF_{r,\tilde\Ga}'$ with the following properties:
\begin{itemize}
\item  
$\cF_{r,\tilde\Ga}'$ is a quotient of an algebraic variety 
$U_{r,\tilde\Ga,\ell}'$
by the (left) action of $\PGL_2$ and (right) action of a subgroup $H_{\Ga}$ 
of $\Aut(T_{\Ga}) ;$
\item  
$U_{r,\tilde\Ga,\ell}' $ admits a fibration with fiber (an open 
subset of) ${\Sym}^\ell({\Pr}^1)$ 
and base the variety  $\cR_{r,\Ga}$ of maps $f\,:\, \Pr^1 \to M_{\Ga}$
with fixed local ramification data over 
$A_2$ and $B_2$-points of $T_{\Ga}\subset M_\Ga ;$
\item  
the action of $\PGL_2$ on $U_{r,\tilde\Ga,\ell}'$ is induced from the
standard $\PGL_2$-action on ${\Pr}^1 ;$
\item  
the group $\Aut(T_{\Ga})$ is a subgroup of 
$\PGL_2$ (acting on $M_{\Ga}$).
\end{itemize}
\end{prop}

\begin{proof}
Elliptic surfaces parametrized by a 
smooth irreducible variety have
the same $\ET(\E)$, which 
depends on the number $\ell$ of 
$*$-fibers in $\E$, on the degree of $j_{\E}$
and on the ramification properties over 
the ends of $T_{\Ga}$.
Once $\ell$ is fixed, for any given $j_{\E}$, 
the $*$-mark can be placed over
arbitrary $\ell$-points of ${\Pr}^1$. 
Their position defines a unique surface $\E$.
This implies that $U_{r,\tilde\Ga,\ell}'$ is fibered with 
fibers (birationally) 
isomorphic to ${\Sym}^{\ell}({\Pr}^1)={\Pr}^{\ell}$. 
The ramification properties of $j_{\E}$ 
remain the same on the open
part of $U_{r,\tilde\Ga,\ell}'$ (since the number of $*$-fibers
remains the same). 
Thus the base of the 
above fibration is the space of rational
maps $f\,:\, {\Pr}^{1}\to {\Pr}^{1}=M_{\Ga}$ 
with fixed ramification locus. Any such map 
defines an elliptic surface  $\E$
with given $\Ga$ (see \cite{Bog}). The $\PGL_2$-action
on $U_{r, \tilde\Ga,\ell}'$ 
identifies points corresponding to isomorphic
surfaces $\E$. Additional nontrivial 
isomorphisms correspond to exterior automorphisms of $\Ga$, 
coming from the action on $M_{\Ga}$, i.e., 
automorphisms of the graph $T_{\Ga}$. 
\end{proof}



\begin{rem}
\label{rem:free}
If the $\PGL_2\times\Aut({T_{\Ga}})$-action on $U_{r,\tilde\Ga,\ell}'$ is
almost free 
then the rationality of $\PGL_2\backslash U_{r,\tilde\Ga,0}'/\Aut(T_{\Ga})$ 
implies the rationality the corresponding quotients
for all $\ell$. In the other cases the degree of 
$j_{\E}$ is small and they are handled separately (see Section~\ref{sect:ratio}). 
\end{rem}



Most of the graphs 
$T_{\Ga}$ have trivial automorphisms. In particular,
any nontrivial automorphism acts on the ends of the graph.
In general, automorphisms of the pair
$(M_{\Ga},T_{\Ga})$ correspond to elements of $\Ga'/\Ga$ where
$\Ga'\subset \PSL_2(\Z)$ is a 
maximal subgroup with the property that
$\Ga$ is a normal subgroup of $\Ga'$.

\begin{lemm} 
\label{lemm:autt}
The group 
$\Aut({T_{\Ga}})$ acts freely on the set of ends and
end-loops.
\end{lemm}

\begin{proof}
Consider $j_{\Ga}\,:\, M_{\Ga}\ra \Pr^1$. Then 
$h\in\Aut({T_{\Ga}})$ is any element in $\PGL_2(\C)$ such that
$j_{\Ga}(hz)=j_{\Ga}(z)$ for all $z\in T_{\Ga}\subset M_{\Ga}$.
If $h$ stabilizes an end or an end-loop of $T_{\Ga}$ then 
it stabilizes the unique adjacent vertex and its other end.
Any element of $\PGL_2(\C)$ preserving a closed interval is 
the identity.
\end{proof}

\begin{coro}
\label{coro:hgamma}
For $r\le 2$, the only possible groups $\Aut(T_{\Ga})$ are 
cyclic, dihedral or subgroups of $\sS_4$. More precisely, 
for graphs with one end $\Aut(T_{\Ga})=1$ and graphs with two ends
$\Aut(T_{\Ga})$ is a subgroup of $\Z/2$.
\end{coro}


\begin{lemm}
\label{lemm:str}
Let 
$$
\cR:=\{f\,:\,{\Pr}^1\to {\Pr}^1\}
$$
be the space of rational maps 
with ramifications over exactly $0$ and $\infty$. 
Then $\cR$ is a ${\mathbb G}_m$-fibration over the
product of symmetric spaces $\Sym^{m_i}({\Pr}^1)$.
\end{lemm}

\begin{proof}
Indeed any two cycles $c_1$ and $c_2$ of fixed degree are
equivalent on ${\Pr}^1$. Therefore, 
there is a rational function $f$
on ${\Pr}^1$ with $c_1=f^{-1}(0)$ and $c_2=f^{-1}(\infty)$. If
$c_1,c_2$ do not intersect then $\deg(f)=\deg (c_1)=\deg(c_2)$. The
function $f$ is defined modulo multiplication by a constant. The
space of cycles 
$c_1=\sum_i n_i p_i$ 
is a product of symmetric powers 
$\Sym^{m}({\Pr}^1)$ where $m$ is the number of
equal $n_i$. 
\end{proof} 











\section{Combinatorics}
\label{sect:comb}


In this section we investigate relations between 
$\ET(\E)$ and $\ET(\Ga)$. 
We keep the notations of the previous sections. 


\begin{lemm}
\label{lem3}
Let $j\,:\, \E\to C$ be an elliptic fibration. Then  
\begin{align}\label{for2}
  \ET(\E)=\deg (j_{\E})\D(\Ga)+8\alpha_2+4\alpha_1+6\beta_1+12 \ell.
\end{align}
Here $\alpha_1$ and $\alpha_2$ equal 
the number of points over $A_2$-ends of
$T_{\Ga}$ with ramification multiplicity  
$1\ (\mod 3)$ and $2\ (\mod 3)$, respectively, $\beta_1$ is the 
number of odd ramification points over
the $B_2$-ends and $\ell$ is the number of $*$-fibers of $\E$.
\end{lemm}

\begin{proof}
The summand $\deg (j_{\E})\D(\Ga)$ corresponds to 
multiplicative fibers of $\E$.
The next summands are the contributions of those singular fibers
of $\E$ which are in the preimage of $A_2$ or $B_2$-ends of $T_{\Ga}$. 
If the ramification order at a point $p$ over a 
$B_2$-end is even then the
corresponding fiber with minimal $\ET$ is smooth and hence 
does not contribute to $\ET(\E)$. If it 
is odd then the fiber with minimal $\ET$ is of type $\rm III$
and we have to add $6\beta_1$. Similarly,
for the preimages of $A_2$-ends and $*$-twists. 
\end{proof}
 

\begin{coro}
\label{coro:ete}
In particular, 
$$
\ET(\E)\le \deg(j_{\E})\ET(\Ga)+12\ell,
$$
with equality if 
$$
\begin{array}{rcl}
2\alpha_2 +\alpha_1& = & a_2\cdot \deg(j_{\E})\\
\beta_1            & = & b_2 \cdot \deg(j_{\E}).
\end{array} 
$$
\end{coro}



\begin{defn}
We call $T_{\Ga}$ {\em saturated} if all vertices of
$T_{\Ga}^u$ are trivalent and a {\em tree} if it is
contractible.
\end{defn}

\begin{rem}
\label{rem:pi}
For saturated graphs $\D(\Ga)=12\rk \pi_1(T_{\Ga})$, 
where  
$$
\rk \pi_1(T_\Ga)=\rk H_1(T_\Ga)
$$ 
is the number of independent closed loops of $T_{\Ga}\subset M_{\Ga}$. 
\end{rem}

The following simple procedures produce new
graphs:

\begin{itemize}
\item If $T_1$ and $T_2$ are (unmarked) trivalent graphs
we can join $T_1$ and $T_2$ along two edges.  
For the resulting graph $T'$ we have
$$
\ET(T')=\ET(T_1)+ \ET(T_2)+12.
$$ 
If $T_i$ are marked and 
the marking of the ends of $T'$ is induced from 
the marking of the corresponding ends of $T_1$ and $T_2$ 
then
$$
\D(T')=\D(T_1)+\D(T_2)+12.
$$
\item We can glue an end $p$ of $T_1$ to an edge of $T_2$.
In this case 
$$
\ET(T')=\ET(T_1)+\ET(T_2).
$$ 
The change of $\D$ depends on the 
marking of the end:
$$
\D(T')=
\begin{cases} 
\D(T_1)+\D(T_2)+6 &\text{if $p=B_2$}\\            
\D(T_1)+\D(T_2)+4 &\text{if $p=A_2$}.
\end{cases}
$$
\end{itemize}

\begin{rem}
Any connected graph $T$ can be 
uniquely decomposed into a union
of a saturated graph and a union of trees.
\end{rem}


\begin{lemm} 
\label{lemm:12}
$\ET(\Ga)$ is divisible by $12$.
\end{lemm}



\begin{proof}
Every vertex of $T_{\Ga}$ 
has either one or three incoming edges.
Therefore, the number of edges
$$
{\tau}^1=\frac{1}{2}({\tau}^0_1+{\tau}^0_3),
$$ 
(${\tau}^0_i$ is
the number of vertices with $i$-edges). Thus
${\tau}^0={\tau}^0_1+{\tau}^0_3$ 
is even and since
$\ET(\Ga)=6{\tau}^0$ we are done.
\end{proof}


\begin{exam}
If $T_{\Ga}$ is a tree with $k+2$ vertices then 
$$
\begin{array}{cl}  
\ET({\Ga}) & = 12k+12  \\
\D({\Ga})  &  
\left\{\begin{array}{ccl} = 6k & {\rm if}\,\, 
      {\rm  all}\,\, {\rm ends}\,\,
{\rm are}\,\, B_2, \\
                          > 6k & {\rm otherwise}.
\end{array}
\right.
\end{array}
$$
\end{exam}



\begin{lemm}
\label{lemm:ineq}
For all $\Ga$ one has
$$
\D(\Ga)\geq \ET(\Ga)/2+6(\rk H_1(T_\Ga) -1).
$$
\end{lemm}

\begin{proof} 
A direct computation shows that
for saturated graphs one has an equality.
Suppose that $T_{\Ga}$ is a concatenation 
of a saturated graph
$T_{\rm sat}$ and a tree $T_{\rm tree}$. 
The number of ends drops by one and the number of 
$A_6$ vertices increases by $1$.
Thus the tree will add $12k+12$ to $\ET({\Ga})$ but 
$\D(\Ga)$ will change by $6k+6$. 
Finally, the ratio $\D(\Ga)/{\ET(\Ga)}$ only increases if 
we change $B_2$- to $A_2$-markings 
for some ends. Indeed, $\D(\Ga)$ increases
without changing $\ET(\Ga)$.
\end{proof}

\begin{coro}
\label{coro:useful}
If $\D(\Ga)={\ET(\Ga)}/2$ then 
$T_{\Ga}$ is a concatenation
of a loop $L$ and some trees. Moreover, 
all the ends of $T_{\Ga}$ are of type $B_2$.
\end{coro} 



\begin{prop}
\label{prop:<}
Let $\E\to C$ be an elliptic fibration with 
$\ET(\E)<\ET(\Ga)$. Then: 
\begin{itemize}
\item  
$M_{\Ga}=\Pr^1$ and $T_{\Ga}$ is a tree without $A_2$-ends
and with $\ET(\Ga)>24 ;$
\item 
$\deg(j_{\E})=2$ and it is 
ramified in all ($B_2$) ends of $T_{\Ga}$
(and, possibly, some other points);
\item
$\E$ has $1$ or $2$ singular fibers of type ${\rm I}_n$.
\end{itemize}
\end{prop}

\begin{proof}
From \ref{lem3} and \ref{lemm:ineq} we conclude that
$\rk H_1(T_{\Ga})=0$ which implies that $T_{\Ga}$ is a tree
and $M_{\Ga}=\Pr^1$. By Lemma~\ref{lemm:ett} 
and our assumption, $\ET(\Ga)>24$, which implies that $\deg(j_{\E})\le 2$. 
If $\deg(j_{\E})=1$, we apply Corollary~\ref{coro:ete} and 
get a contradiction to the assumption. 
For $\deg(j_\E)=2$ combine Definition~\ref{defn:etg} and \eqref{lem3}:
$$
\ET(\E)=\ET(\Ga)+4a_2+4\alpha_1+8\alpha_2+6\beta_1 -12.
$$
Since $\alpha_1$, resp. $\beta_1$ is twice the number of unramified 
$A_2$, resp. $B_2$-ends, and $\alpha_2$ is the number of ramified $A_2$-ends
we see that if at least one of them is not zero, then 
$\ET(\E)\ge \ET(\Ga)$.  
The claim follows. 
\end{proof}


\begin{coro}
\label{coro:use}
For every elliptic fibration $\E\to \Pr^1$
one has 
$$
\ET(\E)\ge \ET(\Ga).
$$
Further, if  $\deg (j_{\E})=2$ and $j_\E$ is ramified
over only one $B_2$-point then
\begin{align*}
  \ET(\E)\geq 2\ET(\Ga)-12.
\end{align*}
\end{coro}

\begin{proof}
If $\deg(j_\E)=2$ and $\C=\Pr^1$ then $j_\E$ ramifies in two 
points. If neither of these points is $B_2$ then, by Lemma~\ref{lem3},
$\ET(\E)\ge 2\ET(\Ga)$. If both of these points are $B_2$-points
then the covering $j_\E$ corresponds to a subgroup $\Ga'$ 
of index 2 in $\Ga$ and $C=M_{\Ga'}$, contradiction. 
Otherwise, the claimed inequality follows from Lemma~\ref{lem3}.  
\end{proof}

\section{Elliptic K$3$ surfaces with $\deg(j_{\E})>1$}
\label{sect:ellmono}

In this section we assume that $C=\Pr^1$, that 
$j_{\E}>1$ and that $\Ga$ is a proper subgroup of $\PSL_2(\Z)$. 
We consider 
$$
\begin{array}{cccc}
{\rm general \,\, families:} &  \ET(\E)-12\ell  & = & \deg (j_{\E})\ET(\Ga),\\
{\rm special\,\, families:}  &  \ET(\E)-12 \ell & < & \deg (j_{\E}) \ET(\Ga).
\end{array}
$$

In Section~\ref{sect:modsp}
we showed that the main building block in the construction
of moduli space of elliptic surfaces with fixed $\Ga$ 
is the space of rational maps 
$j_{\E}\,:\,  C\to M_{\Ga}$ 
of fixed degree and ramification restrictions over certain points. 
For a {\em general} family 
there are no such restrictions and the corresponding moduli spaces 
are rational by classical results of invariant
theory for actions of $\PGL_2$ and its 
algebraic subgroups (see Section~\ref{sect:rat}).
For {\em special} families the
corresponding space of rational maps is more complicated.


\begin{lemm}\
\label{lemm:48}
There are no special families of elliptic 
{\rm K}$3$ surfaces with 
$$
\ET(\Ga)=48, 36.
$$
\end{lemm}

\begin{proof}

\

\begin{itemize}
\item If $\ET(\Ga)=48$ then 
$\D(\Ga)\geq 18$ and $\deg(j_{\E})\le 2$.
However, $\deg (j_{\E})=2$ contradicts Corollary~\ref{coro:use}
($\ET(\E)\geq 96-24>48$).
\item
If $\ET(\Ga)=36$ and $\D(\Ga)>16$ then $\deg (j_{\E})=2$,
contradicting  to \ref{coro:use}.  
We are left with $\D(\Ga)=16,14,12$ for 
$\deg (j_{\E})=3$ and $\D(\Ga)=12$ for $\deg(j_\E)=4$.
\item 
If $\deg(j_\E)=4$ then $T_\Ga$ is a tree with $\GD(\Ga)=[2A_6+4B_2]$. 
By Lemma~\ref{lem3}, all ramifications over the $B_2$-ends
are even, which contradicts $C=\Pr^1$ (compute $\chi(C)$).
\item   
If $\deg (j_{\E})=3$ then $T_\Ga$ is a tree (by \ref{lemm:ineq})
and  
$$
\GD(\G)=[2A_6+ a_2A_2+(4-a_2)B_2]
$$
with $a_2\leq 2$. We have 
\begin{align*}
48\geq  \ET(\E) \geq 3(12+2a_2)+ 4\alpha_1+8\alpha_2 + 6\beta_1,
\end{align*}
where $\beta_1\ge 2$ (since $\deg(j_\E)$ is odd
there is odd ramification over some $B_2$-end).  
Therefore, $a_2=0$ and consequently, $\beta_1\ge 4$, contradiction. 
\end{itemize}
\end{proof}


\begin{lemm}
\label{lemm:24-sp}
If $T_{\Ga}$ is not a tree and $j_{\E}$ is special 
(and generic for the corresponding irreducible component of
$\cF_{2,\tilde{\Ga}}$) then 
$$
\begin{array}{c|c|l|l}
\ET(\Ga)& \deg(j_{\E}) & \GD(\Ga)& \RD(j_{\E}) \\
\hline    
24 & 4   &   [2A_6+2B_2]     &  [(2,2)_B,(2,2)_B,(2),(2)] \\
24 & 3   &   [2A_6+2B_2]     &  [(2,1)_B,(2,1)_B,(2),(2)]\\  
24 & 3   &   [2A_6+A_2+B_2]  &  [(3)_A,(2,1)_B,(2)]\\
12 & 6   &   [A_6+A_2]       &  [(3,3)_A,(3,3)_A,(2),(2)]\\
12 & 5   &   [A_6+A_2]       &  [(3,1,1)_A]\\
12 & 5\le d\le 8&  [A_6+B_2] &  [\beta=(\beta_i)_B, (2)_B^{d'}],
\end{array}
$$
where
$$
\beta_i\in \N, \,\, \sum \beta_i=d,\,\, \# {\rm odd} \,\,\beta_i \le 8-d
$$
and 
$$
d'=2d - \# {\rm nonzero} \,\,\beta_i.
$$  
\end{lemm}

\begin{proof}
Follows from Lemma~\ref{lem3}. 
First observe that $\Delta(\Ga)\le 16$, which implies that 
$a_6=2$ and $a_2\le 2$.
If $a_2=2$ then $\Delta(\Ga)=16$ and
$$
\alpha_1=\alpha_2=\beta_1=0.
$$
Hence both $A_2$-ends have a 3-cyclic ramification
and the cover corresponds to a subgroup $\Ga'\subset \Ga$ of index 3. 
This excludes $\GD(\Ga)=[2A_6+2A_2]$. 
If $\deg(j_\E)=4$ then $\Delta(\Ga)=12$ which implies that 
all preimages of $B_2$-ends have even ramification.
The description of all other ramification 
data follows similarly from  Lemma~\ref{lem3}.
Notice that the (omitted) possibilities 
$$
\begin{array}{c|c|l|l}
\ET(\Ga)& \deg(j_{\E}) & \GD(\Ga)& \RD(j_{\E}) \\
\hline     
12 & 6   &   [A_6+A_2]       &  [(6)_A,(3,3)_A]\\
12 & 5   &   [A_6+A_2]       &  [(3,2)_A]
\end{array}
$$
are degenerations of the listed cases
(see Remark~\ref{rem:degen}).
\end{proof}



\begin{lemm}
\label{lemm:table-tree}
If $T_\Ga$ is a tree and $j_{\E}$ is special
(and generic for the corresponding irreducible component of
$\cF_{2,\tilde{\Ga}}$) then
$$
\begin{array}{c|c|l|l}
       & \deg(j_{\E}) & \GD(\Ga)& \RD(j_{\E}) \\
\hline    
j_1    & 4    &   [A_6+A_2+2B_2]  & [(1,1,1,1)_A,(2,2)_B,(2,2)_B,(2),(2)]\\
j_2    & 4    &   [A_6+A_2+2B_2]  & [(3,1)_A,(2,2)_B,(2,2)_B] + *\\
j_3    & 4    &   [A_6+A_2+2B_2]  & [(3,1)_A,(2,2)_B,(2,1,1)_B,(2)]\\
j_4    & 3    &   [A_6+2A_2+B_2]  & [(3)_A,(1,1,1)_A,(2,1)_B]\\ 
j_5    & 3    &   [A_6+A_2+2B_2]  & [(1,1,1)_A,(2,1)_B,(2,1)_B,(2),(2)]\\ 
j_6    & 3    &   [A_6+A_2+2B_2]  & [(3)_A,(1,1,1)_B,(2,1)_B,(2)] 
\end{array}
$$
or $\GD(\Ga)=[A_6+3B_2]$ and 
$$
\begin{array}{c|c|l}
       & \deg(j_{\E}) & \RD(j_{\E}) \\
\hline 
j_7    & 8    & [(2,2,2,2)_B,(2,2,2,2)_B,(2,2,2,2)_B,(2),(2)]\\
j_8    & 6    & [(2,2,2)_B,(2,2,2)_B,(2,2,1,1)_B,(2),(2)] \\
j_{9} & 6    & [(2,2,2)_B,(2,2,2)_B,(2,2,2)_B,(2)] + *\\
j_{10} & 5    & [(2,2,1)_B,(2,2,1)_B,(2,2,1)_B,(2),(2)]\\
j_{11} & 4    & [(2,1,1)_B,(2,1,1)_B,(2,2)_B,(2),(2)]\\
j_{12} & 4    & [(2,1,1)_B,(2,2)_B,(2,2)_B,(2)]+ *\\
j_{13} & 3    & [(1,1,1)_B,(2,1)_B,(2,1)_B]\\
j_{14} & 3    & [(2,1)_B,(2,1)_B,(2,1)_B,(2)] + * 
\end{array}
$$
or $\ET(\Ga)=12$ and $\GD(\Ga)=[2A_2]$ with $\deg(j_{\E})=4-10, 12$.

(In the above tables,  $+*$ means that there exists a moduli space of elliptic
surfaces with the same $\RD(j_{\E})$ and  with an additional $*$-fiber over
an unspecified point.) 

\end{lemm}

\begin{proof}
Assume that $\ET(\Ga)=24$ and $T_\Ga$ is a tree with
$$
\GD(\Ga)\neq [A_6+3B_2].
$$ 
First observe that $\deg(j_\E)\le 6$, since $\D(\Ga)\ge 8$.
If $\deg(j_\E)\ge 5$ then, by \ref{lem3},  
$\GD(\Ga)=[A_6+A_2+2B_2]$. If $\deg(j_\E)=6$ then 
$j_\E$ has to be completely ramified over all ends and 
no other ramifications are allowed by Euler characteristic
computation. Therefore, it is a group-covering and can't be $j_\E$.  
If $\deg(j_\E)=5$ then there are two odd ramifications over $B_2$-ends, 
and by \ref{lem3}, $\ET(\Ga)>48$. 

We are left with 
$$
\begin{array}{ccl}
\GD(\Ga) & = &  [A_6+3A_2],\\
        & = &  [A_6+2A_2+B_2],\\
        & = &  [A_6+A_2+2B_2]
\end{array}
$$
and $3\le \deg(j_\E)\le 4$.  
If there are at least two $A_2$-ends without
$3$-cyclic ramification points over them
then $\ET(\E)> 48$ (see \ref{lem3}). 
The first case is impossible: $\deg(j_\E)=4$ does not occur
(the degree is not divisible by 3), if $\deg(j_\E)=3$ 
and there is at most one $3$-cyclic ramification over an $A_2$-end then, 
by \ref{lem3}, $\ET(\E)>48$, contradiction. 
Consider the second case and $\deg(j_\E)=4$. Then $\D(\Ga)=10$ and
$4\alpha_1+8\alpha_2+6\beta_1\le 8$.  Since $\alpha_1\ge 2$ 
we have $\alpha_2=\beta_1=0$ and $\alpha_1=2$.  
The only possible 
$$
\RD(j_\E)=[(3,1)_A,(3,1)_A,(2,2)_B],
$$ 
which corresponds to a group covering, contradiction. 

Similarly, if $\GD(\Ga)=[A_6+A_2+2B_2]$ and $\deg(j_{\E})=4$
then $\D(\Ga)=8$ and $4\alpha_1+8\alpha_2+6\beta_1\le 16$. 
We have $\alpha_1\ge 1$ and $8\alpha_2+4\alpha_1=16$ or $4$. 
In the first case, both $B_2$-ends are completely ramified, 
and we get $j_1$. The second case splits into subcases:
$\beta_1=0$ or $2$, leading to $j_2$, resp. $j_3$. 
If $\deg(j_{\E})=3$, then if $\GD(\Ga)=[A_6+2A_2+B_2]$ 
then exactly one of the $A_2$-ends has cyclic ramification.  
It follows that $\beta_1=1$, which leads to $j_4$. 
If $\GD(\Ga)=[A_6+A_2+2B_2]$ there are two subcases:
there is cyclic ramification over the $A_2$-end or not. 
In the first subcase, possible $\RD(j_\E)$
include $[(2,1)_B,(2,1)_B]$, which is excluded as
it gives a group covering. The other case leads to $j_6$. 
In the second subcase, we get $j_5$.  


\

Consider the case $T_{\Ga}=A_6+3B_2$. Here $\D(\Ga)=6$ and 
$$
\ET(\E)\geq 6\deg(j_{\E})+6n,
$$ 
where $n$ is a number of points with odd ramification
over $B_2$-vertices.
It follows that 
$$
48\ge 6\deg(j_{\E})+6\beta_1
$$
and $\beta_1\ge 3$ if $\deg(j_{\E})$ is odd 
and the number
of odd ramifications over {\em each} $B_2$-end is 
congruent to $\deg(j_{\E})$ modulo 2. 


If $\deg (j_{\E})=8$ then all preimages of $B_2$-vertices are
$2n$-ramified. 
If $\deg(j_\Ga)$ is odd then $\ET(\E)\geq 6\deg(j_\Ga)+18$, 
which excludes $\deg(j_\Ga)=7$.
Now assume $\deg(j_\Ga)= 6$. The number of possible odd
ramifications over any $B_2$-end is even by~\ref{lem3} and 
it cannot exceed $2$. There are two possibilities listed above.
Assume that $\deg(j_{\E})=5$. The minimal possible
ramifications are $(2,2,1)$ over all 
$B_2$-ends. Since $10-6=4$ we can add two more points.


In $\deg(j_{\E})=4$ we could have further 
$\RD$: 
$$
\begin{array}{ccl}
\RD(j_\E)& = & [(2,2)_B,(2,1,1)_B,(2,2)_B,(2)],\\
         & = & [(2,2)_B,(2,1,1)_B,(2,1,1)_B,(2),(2)],\\
         & = & [(2,2)_B,(2,1,1)_B,(2,1,1)_B,(3)]
\end{array}
$$ 
but they are obtained as degenerations of $j_{12}$ and $j_{13}$. 


The only $\GD(\Ga)$ which allow $\deg(j_{\E})\ge 12$ are
$[A_2+B_2]$ and $[2A_2]$.
The first case corresponds to $\PSL_2(\Z)$ (which we don't consider). 
The second case corresponds to subgroups $\Ga\subset \PSL_2(\Z)$ of 
index $2$. 
For a generic $\E$ in each moduli space
the ramification datum $\RD(j_{\E})$
is one of the following:
$$
\RD(j_{\E})=[(3,\dots,3_{n_1},1,\dots,1)_A,(3,\dots,3_{n_1},1,\dots,1)_A, (2)^d] + *,
$$
where $n_1,n_2,d$ are non-negative integers such that
$$
\deg(j_{\E})-(n_1+n_2)\le 4,
$$ 
$$
3n_1, 3 n_2\le \deg(j_{\E})\,\, {\rm and},\,\,
d\le 2(\deg(j_{\E}-(n_1+n_2+1))). 
$$
(In particular, $d\le 4$).
\end{proof}


\section{Rational elliptic surfaces with $\deg(j_{\E})>1$}
\label{sect:rat-surf}


\begin{lemm}
There are no special families of rational elliptic surfaces with $\ET(\Ga)=24$.
\end{lemm}

\begin{proof}
If $\deg(j_{\E})=2$ then $j_{\E}$ cannot be ramified over more than one $B_2$-end
(otherwise it is a group covering). Therefore, we can apply 
Corollary~\ref{coro:use} and get $\ET(\E)>2\cdot 24 -12 >24$, 
contradiction (to \ref{lemm:ett}).  
Thus  $\deg(j_{\E})=3$ or 4 and $a_6 =1$. Moreover, $\D(\Ga)\le 8$. 
This leaves
the cases: 
$$
\begin{array}{ccl}
\GD(\Ga) & = & [A_6+3B_2],\\
        & = & [A_6+A_2+2B_2].
\end{array}
$$  
In the first case $\deg(j_\E)=3$ is impossible, and $\deg(j_\E)=4$ leads
to 
$$
\RD(j_{\E})=[(2,2)_B,(2,2)_B,(2,2)_B,(2,2)_B]
$$
which corresponds to a group covering. 
In the second case $\deg(j_\E)\neq 4$ (since $\D(\Ga)=8$) 
and $\deg(j_\E)=3$ implies that $\beta_1\ge 2$ and $\ET(\E)\ge 36$, 
contradiction. 
\end{proof}




\begin{lemm}
\label{lemm:table-rat}
If $j_{\E}$ is special
(and generic for the corresponding irreducible component of
$\cF_{1,\tilde{\Ga}}$) then
$$
\begin{array}{c|c|l|l}
       & \deg(j_{\E}) & \GD(\Ga)& \RD(j_{\E}) \\
\hline    
j_{15}    & 6    &   [2A_2]    & [(3,3)_A,(3,3)_A,(2),(2)]\\
j_{16}    & 4    &   [2A_2]    & [(3,1)_A,(3,1)_A,(2),(2)]\\
j_{17}    & 3    &   [2A_2]     & [(3)_A,(1,1,1)_A,(2),(2)]\\
j_{18}    & 4    &   [A_6+B_2]  & [(2,2)_B,(2),(2),(2),(2)]\\ 
j_{19}    & 3    &   [A_6+B_2]  & [(2,1)_B,(2),(2)]\\ 
j_{20}    & 3    &   [A_6+A_2]  & [(3)_A,(2),(2)] 
\end{array}
$$
\end{lemm}

\begin{proof}
If $a_6\ge 1$ then $\deg(j_{\E})=4$ or $3$. 
In the first case $a_2=0$ and $\GD(\Ga)=[A_6+B_2]$ and 
we have complete ramification over the $B_2$-end. This gives $j_{18}$.
In the second case the ramification over $B_2$ is $(2,1)_B$ and 
we get $j_{19}$. If $\GD(\Ga)=[A_6+A_2]$ then $\deg(j_{\E})=3$ and 
$\alpha_1=\alpha_2=0$, leading $j_{20}$. 

It remains to consider $\GD(\Ga)=[2A_2]$. 
We apply the same formulas as in the proof of Lemma~\ref{lemm:table-tree}, 
with the inequality 
$$
\deg(j_{\E})-n_1-n_2\le 2.
$$
We have $\deg(j_\E)\le 6$ and
$\alpha_1=\alpha_2=0$. Notice that $\deg(j_\E)=5$ is impossible. 
\end{proof}





\section{General rationality results}
\label{sect:rat}



\begin{nota}
\label{nota:rat}
We will denote by $\sS_n$ the symmetric group on $n$ letters, 
by $\sA_n$ the alternating group, by $\sD_n$ the dihedral group
and by $\sC_n=\Z/n$ the cyclic group.
In particular, $\sS_2=\sC_2=\Z/2$ and $\sD_2=\Z/2 \times \Z/2$
(sometimes we prefer the notation $\sS_2$ over $\sC_2$ to 
stress that the action is by permutation).
We write $\Gr(k,n)$ for the Grassmannian of $k$-planes in a vector
space of dimension $n$ and $V_{d}$ for 
the space of binary forms of degree $d$. 
We will denote by $\GL_2,\PGL_2,\gm$ etc. the corresponding 
{\em complex} algebraic groups.
For a group $\bG$, we 
denote by  $\bZ_g$ the centralizer of $g\in \bG$ and by
$\bZ_{\bG}$ its center.
We denote by $\M2=V_1\oplus V_1$ the space of $2\times 2$-matrices.
We write $\cV\stackrel{V}{\longrightarrow}X$ or simply 
$\stackrel{V}{\longrightarrow}X$ for a 
locally trivial (in Zariski topology) fibration $\cV$ over $X$ with
generic fiber $V$.
We will often write $\bG$-map (etc.), 
instead of $\bG$-equivariant map. 
\end{nota}

\

\no
We say that two algebraic varieties $X$ and $X'$ are birational, and 
write $X\sim X'$, if $\C(X)=\C(X')$. 
A variety $X$ of dimension $n$ is {\em rational} if 
$X\sim \A^n$, {\em $k$-stably rational} 
if $X\times \A^k\sim \A^{n+k}$ and {\em stably rational}
if there exists such a $k\in \N$. We say that $X$ is  
{\em unirational} if $X$ is dominated by $\A^n$. 
The first basic result, a theorem of Castelnuovo from 
1894, is:

\

\begin{thm}
\label{thm:q}
A unirational surface is rational.
\end{thm}


Already in dimension three,
one has strict inclusions
$$
{\rm rational}\subsetneq {\rm stably\,\, rational}\subsetneq {\rm unirational}
$$
(see the counterexamples in \cite{im}, \cite{am},\cite{cg},\cite{bcs}). 
There is a very extensive literature on 
rationality for various classes of varieties. 
We will use the following facts:

\

\begin{lemm}
\label{lemm:cb}
Let $S\ra B$ be a ruled surface with base $B$ and 
$\pi\,:\, \cC\ra S$ a conic bundle over $S$.
Assume that the restriction of $\pi$ to a generic $\P^1\subset S$
is a conic bundle with at most three singular fibers.
Then $\cC\sim\A^2\times B$.
\end{lemm}
 
\

\begin{lemm}
\label{lemm:conic}
Let $\pi\,:\, \cC\ra S$ be a conic bundle over an irreducible variety  $S$ and
$Y\subset \cC$ a subvariety such that the restriction of
$\pi$ to $Y$ is a surjective finite map of odd degree. 
Then $\cC$ has a section and $\cC\sim S\times \A^1$. 
\end{lemm} 


\


Let $\bG$ be an algebraic group. 
A (good) {\em rational action} of $\bG$
is a homomorphism 
$$
\rho_{\rm rat}\,:\, \bG\ra {\rm Bir}(X)
$$
such that there exists a birational model $X'$ of $X$ 
with the property that $\rho_{\rm rat}$ extends to a (regular)
morphism  $\bG\times X'\ra X'$. We consider only rational 
actions. We write $X \sim_{\bG} Y$ for a $\bG$-birational 
($=\bG$-equivariant birational)
isomorphism between $X$ and $Y$.
We will denote by $\bG\ba X$ a model for the field of invariants 
$\C(X)^{\bG}$.


\

Let $E\ra X$ be a vector bundle. 
A {\em linear} action of $\bG$ on $E$ is 
a rational action which
preserves the subspace of fiberwise linear functions on $E$. 
In particular, there is a linear $\bG$-action on
regular and rational sections of $E$.




\

We are interested in rationality properties of quotient spaces 
for the actions of $\PGL_2$, its subgroups and products of $\PGL_2$ with
finite groups. The finite subgroups of $\PGL_2$ are
$$
\sC_n,\sD_n,\sA_4,\sS_4,\sA_5.
$$
We denote by $\tilde{\sC}_n,\tilde{\sD}_n$ etc. their lifts to $\GL_2$
(as central $\sC_2$-extensions). 
We denote by
$$
\bB,\bT=\C^*, \bN_{\bT}
$$
the upper-triangular group, the standard 
maximal torus and the normalizer of this
torus in $\PGL_2$ and by
$$
\tilde{\bB},\tilde{\bT},\bN_{\tilde{\bT}} 
$$
the corresponding subgroups in $\GL_2$ (or $\SL_2$).




\



Let $V$ be an $n$-dimensional vector space,
$\tilde{\bG}\subset \GL(V)$ a  subgroup
and $\bG$ its projection to $\PGL(V)$, acting naturally on $\P(V)$.   
Determining the rationality of quotients 
$\bG\ba \P(V)$ (at least for finite groups) 
is known as Noether's problem.   

\

\begin{coro}[of Theorem~\ref{thm:q}]
\label{coro:dim}
For all $n\le 3$ the space $\bG\ba\P(V)$ is rational. 
\end{coro}

\


\begin{thm}\cite{M},\cite{V}
\label{thm:qq}
A quotient of $\bP(V)$ by a (projective) action of
a connected solvable group, a torus or a finite abelian subgroup of a torus
is rational.
\end{thm}

\

A fundamental rationality result is the following theorem of Katsylo:


\


\begin{thm}\cite{K}
\label{thm:kats}
For any representation $V$ of $\GL_2$ or $\PGL_2$ the quotient 
$\PGL_2\ba \P(V)$ is rational. 
\end{thm}


\

In general, the quotients need not be rational 
(see Saltman's counterexamples in \cite{S}). 
We now describe some partial results for $n=4$, which 
we will use later on. 

\

\begin{defn}
\label{defn:prim}
A finite group $\tilde{\bG}\subset \GL_n=\GL(V)$ 
is called {\em imprimitive} if 
there exists a decomposition $V=\oplus_{\al} V^{\al}$ such that
for all $\al$ and $\tilde{g}\in \tilde{\bG}$ there is an $\al'$ with 
$\tilde{g}V^\al=V^{\al'}$. Otherwise, $\bG$ is called {\em primitive}.
\end{defn}

\begin{rem}
There are 29 types of primitive subgroups of $\GL_4$. 
For some of them, like  
$$
\sA_6,\sA_7,\PSL_2({\bF}_7),\sS_6,
$$ 
rationality of the quotient is still unknown.  
\end{rem}

\begin{thm}
\cite{km}
\label{thm:kolp}
For every primitive solvable subgroup
$\bG\subset \PGL_4$ the quotient $\bG\ba \P^3$ is rational. 
\end{thm}

\begin{rem}
\label{rem:km}
In \cite{km} it is shown that 
$$
\bG\ba \P^3\sim_{\bG} \bG'\ba X_3,  
$$
where $X_3$ is the Segre cubic threefold and 
$\bG'$ is a quotient of $\bG$.
The problem is then reduced to the 
(easy) case of imprimitive actions. 
\end{rem}


\

We will also need to consider quotients by {\em nonlinear} actions.

\

\begin{lemm}
\label{lemm:qqq}
The quotient of $\GL_2$  (or $\PGL_2$)
by the involution $i\,:\, x\mapsto x^{-1}$
is rational.
\end{lemm}

\begin{proof}
The involution decomposes 
as a product $i=i_1\circ i_2$, where 
$$
i_1\,:\, 
x:=\left(\begin{array}{cc} a & b \\ c & -a +d \end{array}\right)\mapsto
\left(\begin{array}{cc} -a+d & -b \\ -c & a\end{array}\right)
$$
and
$$
i_2\, :\, y\mapsto y\cdot \det(y)^{-1}.
$$
are two commuting involutions.
Another set of independent generators of  
$\C(a,b,c,d)$ is given by $\{ a,b,c,\det(x)\}$ (write 
$d = (\det(x) + bc+a^2)/a$).
Now the involutions take the form
$$
i_1\, :\, (a,b,c)\to (-a ,-b,-c)
$$ 
and 
$$
i_2\, :\, \det(x)\to \det(x)^{-1}
$$ 
and we can write down 
independent generators of the field of invariants. 
If 
$$
D := \frac{\det(x) +1}{\det(x)-1}
$$ 
then 
$$
\begin{array}{cccc}
i_2 :  & D         &  \mapsto & -D \\
i_1 :  & (a,b,c,D)&  \mapsto & (-a ,-b,-c, - D).
\end{array}
$$
This finishes the proof.
\end{proof}

\


A {\em (rational) slice} for the action of 
$\bG$ is a subvariety $S\subset X$
such that the general $\bG$-orbit intersects $S$ in exactly one point. 
(The slice $S$ need not be a rational variety.
To avoid confusion, we will always refer to $S$ as a slice.)
A subvariety $Y\subset X$ is called a $(\bG,\bH)$-{\em slice} 
(where $\bH\subset \bG$ is a subgroup) if $\bG\cdot Y\sim X$ and $gy\in Y$ implies 
that $g\in \bH$. Clearly, $\bG\ba X \sim \bH\ba Y$. Moreover, if
$f\,:\, X\ra X'$ is a $\bG$-equivariant morphism 
and $Y'$ is a $(\bG,\bH)$-slice in $X'$ then 
$f^{-1}(Y')$ is a $(\bG,\bH)$-slice in $X$.
    
\

\begin{nota}
\label{nota:stab}
For (a reductive group) $\bG$ acting (rationally) on $X$
we denote by 
$$
\Stab_{gen}=\Stab_{gen}(\bG,X)
$$ 
the generic stabilizer (defined up to conjugacy). 
The action is called an $af$-action ({\em almost free})
if $\Stab_{gen}$ is trivial.
\end{nota}



We use a more precise version of Theorem~\ref{thm:kats}:


\begin{thm}
\cite{K}
\label{thm:katsylo}
Let $\rho\,:\, \PGL_2\ra \PGL(V)$ be a representation and 
$\tilde{\rho}$ a lifting of $\rho$ to a
representation of $\GL_2\ra \GL(V)$.
Let 
$$
\bG'':=\Stab_{gen}(\GL_2,V)\,\,\, {\rm  and}\,\,\, \bG:=\GL_2/\bG''.
$$ 
If the central $\sC_2\not\subset \bG''$ then
$$
\bP(V) \sim_{\bG} \bG\times S,
$$ 
where $S$ is a rational variety (with trivial $\bG$-action).

\no
If $ \sC_2\subset \bG''$ then
\begin{itemize}
\item
either the $\PGL_2$-action on  $\bP(V)$ has {\em no} slice 
and $\bG\ba \bP(V)$ is rational  
\item or 
$$
\bP(V) \sim_{\bG} \bG\times S,
$$
where the slice $S$ is a rational variety (with trivial $\bG$-action). 
\end{itemize}
\end{thm}


\


We now explain some general techniques 
in the study of rationality of quotient varieties. 

\

\begin{lemm}
\label{lemm:redu}
Let $E\ra X$ be a vector bundle of rank $r=\rk(E)$. 
Let $\bG$ be an (affine) reductive group acting on $E$ such that
the generic orbit of $\bG$ in $E$ projects isomorphically
onto a generic orbit of $\bG$ in $X$. Then  
$$
E\sim_{\bG} X\times \A^{r}
$$
with trivial $\bG$-action on the affine space $\A^{r}$. 
\end{lemm}

\begin{proof}
Denote by $O$ the $\bG$-orbit through a generic point in $E$.  
Shrinking (equivariantly)  
$X$, if necessary, we may assume that
the map
$$
\pi\,:\, H^0(X,E)\ra H^0(O,E|_{O})
$$
is surjective. With our assumptions, there exists a basis
$s_1,...,s_r$ such that for each $j$, the $\bG$-orbit of $s_j$  
projects isomorphically onto its image in $X$ and generates
a trivial 1-dimensional $\bG$-equivariant sub-bundle of
the restriction $E|_{O}$ of $E$ to the orbit $O$. 
It follows that $E|_{O}=\oplus_{j=1}^r \bG\cdot s_j$. In particular, 
$H^0(O,E)$ contains the trivial $\bG$-module $M$
generated by $s_1,...,s_r$. Moreover, 
$M$ generates $H^0(O,E|_{O})$ over every point of $O$. 
Since $\pi$ is a map of $\bG$-modules and $\bG$ is reductive
$H^0(X,E)$ contains a submodule $M'$ 
such that $\pi(M')=M$ (as $\bG$-modules).
The elements of $M'$ generate $E$ over a generic point of $X$. 
A basis $s_1',...,s_r'$ of $M'$ gives the 
desired splitting of the action.
\end{proof}


\


\begin{coro}
\label{coro:coro}
Let $\bG$ be a reductive group and 
$$
E''\ra E'\ra X
$$ 
a $\bG$-equivariant sequence of vector
bundles such that the generic $\bG$-orbit of $E'$
projects isomorphically onto its image.  
Choose a generic $\bG$-equivariant section $s'$ of $E'\ra X$
and denote by $E''_{s'}$ the restriction of $E''$ to this section.
Then  
$$
E''\sim_{\bG} E''_{s'}\times \A^{r'}
$$  
(where $r'=\rk E'$), with trivial $\bG$-action on $\A^{r'}$.
\end{coro}





\begin{prop}
\label{prop:modu}
Let $X$ be a variety with an action    
$\rho\,:\, \bG\ra X$ of a linear algebraic group $\bG$. Let 
$E\ra X$ be a vector bundle and $\tilde{\rho}\,:\, \tilde{\bG}\ra E$
a $\tilde{\bG}$-action lifting $\rho$.
Consider a generic orbit $\bG\cdot x\subset X$ and the linear
action of $\tilde{\bG}$ on the space of sections $H^0(X,E)$.
 
Assume that $\tilde{\bG}$ is
reductive and $V$ is a linear representation of $\tilde{\bG}$
which is contained in $H^0(X,E)$.
Then there exists an affine open $X'\subset X$ 
such that the vector bundle $E\ra X'$ admits a $\tilde{\bG}$-map 
onto a $\tilde{\bG}$-representation $V^*$. 
\end{prop}




If the action of $\bG$ on $X$ is almost free
we may think of $X$ as being (birational to) a principal fibration
over the quotient $\bG\ba X$  with fiber $\bG$.
If $\bG$ is {\em affine} we may assume that $X$ 
and $\bG\ba X$ are also affine. 
Let us also recall a 
standard general construction of $\bG$-maps:
if the ring $\C[X]$ is a direct sum of $\bG$-modules
then any $\bG$-submodule $V\subset \C[X]$ defines 
a $\bG$-map $X \ra \Spec(V)$.
We also have a vector bundle version of the above 
construction: let $E\ra X$ be a $\bG$-vector bundle
and $O$ a $\bG$-orbit through a generic point. 
Assume that $H^0(O,E)$  (the restriction 
of the space of sections to $O$) 
contains $V$ as a submodule.
We obtain a $\bG$-map 
$$
v\, :\, H^0(O,E)\ra V^*
$$
(the dual module, considered as a vector bundle over a point). 

\
 
\begin{lemm}
\label{lemm:mapp}
There exists a $\bG$-stable Zariski open $U\subset X$ and
a rational $\bG$-map of $H^0(U,E)\ra V^*$ extending
$v$.
\end{lemm}

\begin{proof}
A generic orbit $O$ has a $\bG$-equivariant  
neighborhood $U$, with $U/\bG$ affine,
such that 
$$
\xymatrix{H^0(U,E)\ar@{>>}[r] & H^0(O,E).}
$$
The module $H^0(U,E)$ is a direct sum of
finite dimensional irreducible $\bG$-modules. We can now take 
any submodule $V\subset H^0(U,E)$ which surjects isomorphically 
onto a submodule in $H^0(O,E)$.
\end{proof}

\


\begin{lemm}
\label{lemm:pgl2}
If $X$ has an $af$-action of $\PGL_2$ then
$$
X\times \bP(V_{2d})\sim_{\PGL_2} X\times \bP(V_{2d}),
$$
with diagonal $\PGL_2$-action on the left and trivial $\PGL_2$-action 
on $\P(V_{2d})$ on the right. 
\end{lemm}


\begin{proof}
We know that $\C[\PGL_2]$, as a $\PGL_2$-module,
is sum of all even modules $V_{2d}$. 
This gives a $\PGL_2$-map 
$s \,:\,X\ra \bP(V_{2d})$.  
The quotient 
$$
\PGL_2\ba X \times \bP(V_{2d})
$$ 
is a projective  bundle over the quotient 
$\PGL_2\ba X$, with a section obtained from $s$.
Therefore, it is birational to the  product
$(\PGL_2\ba X) \times \P(V_{2d})$, which  gives
the claimed $\PGL_2$-isomorphism.
\end{proof}

\

\begin{coro}
\label{coro:sect}
Let $X$ be a variety with an $af$-action 
of $\PGL_2$. Then $X$ is a $(\PGL_2,\bN_{\bT})$-slice in 
$$
X\times \P(V_2)
$$
(with diagonal $\PGL_2$-action). 
\end{coro}



\begin{lemm}
\label{lemm:y}
Assume that $X$ has an $af$-action $\rho$ of $\PGL_2$. 
Let $\cV\stackrel{V}{\longrightarrow} X$ be a vector bundle over $X$ with an action 
$\tilde{\rho}$ of $\GL_2$ lifting $\rho$. 
Assume that $X$ contains a $\PGL_2$-orbit $Y\sim \PGL_2$ 
such that the $\GL_2$-module $H^0(Y,\cV_Y)$ contains $V_{d}$, 
for some odd $d$.  
Then 
$$
\bP(\cV)\sim_{\PGL_2}\PGL_2\times  S,
$$ 
(with trivial $\PGL_2$-action on $S$). 
Otherwise, $\cV$ is induced from a 
$\GL_2$-vector bundle on $\PGL_2\ba X$.
\end{lemm}


\begin{proof}
Let $Y$ be an orbit such that 
$H^0(Y,\cV_Y)$ contains $V_{d}$, for some odd $d$.
Shrinking $X$, if necessary, gives a surjective map of 
$\GL_2$-modules
$$
\xymatrix{
H^0(X,\cV)\ar@{>>}[r] & H^0(Y,\cV_{Y}).} 
$$
Since  $H^0(Y,\cV_{Y})$ is an algebra over  
$H^0(Y,\cO_{Y})=\oplus_{d\ge 0} V_{2d}$, 
it contains $V_1$ as a submodule. 
We obtain a $\PGL_2$-equivariant surjective map 
$$
\bP(\cV)\ra \bP(V_1)=\bP^1.
$$ 
Since the stabilizer of a point in $\bP^1$ is 
solvable, we get a slice $S\subset \bP(\cV)$, as claimed.  

Assume that there is an orbit $Y\sim \PGL_2$ such that $\cV_Y$ contains only 
even weight $\GL_2$-submodules. Then the central  $\sC_2\subset \GL_2$  
acts trivially on $\cV_Y$. If follows that $\cV_Y$ is a trivial
$\PGL_2$-bundle, and $H^0(Y,\cV_Y)$ a trivial $\PGL_2$-module.
The semi-simplicity of the $\PGL_2$-action implies that
$H^0(X,\cV)$ contains $H^0(Y,\cV_Y)$ as a submodule. Shrinking $X$ if
necessary, we can find linearly independent $\PGL_2$-invariant sections, 
whose specializations to $Y$ generate $H^0(Y,\PGL_2)$. 
Therefore, $\cV$ is lifted from the quotient $\PGL_2\ba X$.
\end{proof}

\

\begin{lemm}
\label{lemm:sl}
Let $V$ be a representation of $\bG$ of dimension $\ge 2$  
(with $\bG$ acting on the left). Then $V\oplus V$ is  
a $\bG\times \GL_2$-space (with right $\GL_2$-action) and 
$$
V\oplus V\sim_{\bG\times \GL_2} 
\xymatrix{\cV \ar[d]^{\M2=V_1\oplus V_1}\\ \Gr(2,V),}
$$
a vector bundle with fibers $2\times 2$-matrices  
(with right $\GL_2$-action). 
\end{lemm}

\begin{proof} 
Consider the map 
$$
\begin{array}{ccc}
V\oplus V & \ra     &  \Gr(2,V)\\
(v,v')    & \mapsto &  \langle v,v'\rangle, 
\end{array}
$$
defined on the open, $\bG\times \GL_2$-invariant 
subset of noncollinear pairs $(v,v')\in V \oplus V$ 
(with fibers consisting of pairs spanning the same 
2-space). The $\GL_2$-action on the fibers is the right 
multiplication on matrices:
$$
(v,v')\mapsto (av+bv',cv+dv').
$$
\end{proof}

Assume that $\bG$ is reductive and denote by
$\bG'':=\Stab_{gen}(\bG,\Gr(2,V))$ and by
$\bG':= \bG/\bG''$ the quotient group 
of $\bG$ which acts effectively on $\Gr(2,V)$. 


\


\begin{coro} 
\label{coro:1}
Assume that the action of $\bG'$ on $\Gr(2,V)$ has a slice
$S$ so that $\Gr(2,V) \sim S\times \bG'$.
Let $\cV_S$ be the restriction of $\cV$ to $S$ (this makes sense by 
Corollary~\ref{coro:coro}).
Then 
$$
\bG\ba \cV /\GL_2 \sim \bG'\ba \cV_S.
$$
\end{coro}  

\
  
\begin{rem}
\label{coro:2} 
The group $\bG''$ acts as scalars on $\cV=\cV_1\oplus \cV_1$ 
(it commutes with $\GL_2$).
\end{rem}
\

\begin{lemm} 
\label{lemm:inv}
Assume that we are in the situation of Corollary~\ref{coro:1}, 
$\bG = \GL_2$ and  $\bH\subset \GL_2$ has finite image in $\PGL_2$.
Then $\bG\ba \cV /\bH$ is rational.
\end{lemm}


\begin{proof} 
By Corollary~\ref{coro:2}, the   
slice $S$ is 3-stably rational, 
since 
$$
S\times \PGL_2 \sim \Gr(2,V)
$$
and $\Gr(2,V)$ is rational. 
The quotient of $\cV_S$
by a fiberwise linear action is birational to 
$(\M2 / \bH)\times S$  (every vector bundle admits 
an $\bH$-equivariant trivialization over an open subset of $S$).
There is a left  action of $\gm^2 \subset \GL_2$ on $\M2=V_1 \oplus V_1$ which
commutes with $\bH$. Thus $\M2 / \bH$ is (birationally) 
a three-dimensional variety with an $af$-action of $\gm$.
The quotient (a surface) is unirational, 
hence rational (by Theorem~\ref{thm:q}),
and 
$$
\bG\ba (V \oplus V)/\bH \sim S \times (V_1 \oplus V_1)/\bH \sim S\times\C^3.
$$ 
\end{proof}

\

The group $\PGL_2$ acts on $\P(\M2)$ on both sides. 
We will need an explicit description of the action for
some of its subgroups.

\

\begin{lemm}
\label{lemm:po(2)}
We have 
$$
\Stab_{gen}(\bN_{\bT}\times \bN_{\bT},\P(\M2))=\sC_2.
$$
\end{lemm}
  
\begin{proof}
Indeed $\bN_{\bT}$ contains 
$$
\begin{array}{ccccl}
\gm=\{ t\}  & :  &  (x,y) & \mapsto & (tx,t^{-1}y),\\
i           & :  &  (x,y) & \mapsto & (y,x).
\end{array}
$$ 
The corresponding actions on  $\P(\M2)$ are  
$$
(a,b,c,d) \mapsto (t_1t_2 a,t_1^{-1}t_2 b, t_1t_2^{-1}c, t_1^{-1}t_2^{-1} d)
$$ 
and   
$$
\begin{array}{ccc}
i_1 : &  a \ra c, &  b \ra d \\
i_2 : &  a \ra b, &  c \ra d,
\end{array}
$$
respectively.
A matrix $(a,b,c,d)\in \M2$ 
can be transformed to $(1,1,1,d)$ 
by a unique element of $\gm\times \gm$, the 
$\sS_2\times \sS_2$-orbit of which consists of 
two elements (for $d, d^{-1}$). 
\end{proof}

\

\begin{coro}
\label{coro:sopen}
The group $\bN_{\bT}\times \C^*$ acts almost freely on $\P(\M2)$.
There is an open, $\bN_{\bT}\times \bN_{\bT}$-stable subvariety $U\subset \P(\M2)$
such that
$$
\xymatrix{ U \ar[d]^{\C^*\times \C^*}\\ \C^*\subset \P^1,}
$$
with a transitive action of  
$\C^*\times \C^*\subset \bN_{\bT}\times \bN_{\bT}$ on the fibers. 
The diagonal subgroup 
$$
\sS_2^{\Delta}\subset \sS_2\times \sS_2 = (\bN_{\bT}\times \bN_{\bT})/(\C^*\times \C^*)
$$
acts on each fiber as an involution $x \mapsto x^{-1}$. The factor 
$\sS_2 = (\sS_2\times \sS_2)/\sS_2^{\Delta}$ acts on
the base $\C^*\subset \P^1$ as an involution without fixed points, 
on the first factor in the fiber as $x\to x^{-1}$, and as identity on 
the second factor. 
\end{coro}
 
\

\begin{coro}
\label{coro:conic}
Let $\sD\subset \bN_{\bT}$ be a dihedral subgroup such that
$\sD\ba \bN_{\bT} = \C^*$. Then the $\C^*$-bundle 
$$
\cC = \sD \ba \P(\M2)\ra \bN_{\bT}\ba \P(\M2)
$$ 
is induced from the $\C^*$-bundle 
$$
\sD\ba \P(\M2)/\bN_{\bT} \ra \bN_{\bT}\ba\P(\M2)/\bN_{\bT} = \P^1
$$ 
and is hence birationally trivial.
\end{coro}

\begin{proof}
Indeed, the left and the right actions of $\bN_{\bT}$ commute.
By Lemma~\ref{lemm:po(2)},  $\Stab_{gen}(\bN_{\bT}\times \bN_{\bT},\P(\M2))=\sC_2$, 
which implies that the bundle is induced. 
\end{proof}

\   

\begin{lemm}
\label{lemm:rh}
For every dihedral group 
$\sD$ and every $\bH\subset \bN_{\bT}$ the conic bundle 
$$
\cC_{\bH} = \sD\ba \P(\M2)/\bH \ra \bN_{\bT}\ba \P(\M2)/\bH,
$$
has a section.
\end{lemm}

\begin{proof}
The quotient 
$\sD\ba U /\bH $ from Corollary~\ref{coro:sopen}
admits a fibration
$$
\xymatrix{ \sD\ba U/\bH \ar[d]^{\C^*_{\sD}\times \C^*_{\bH}/\sS_2} \\ 
\P^1/\sS_2.}
$$
Here $\C^*_{\sD}\times \C^*_{\bH}$ is the 
quotient of the fiber $\C^*\times \C^*$ of $U\ra \C^*$ 
by the intersection of $\sD, \bH$ with the diagonal 
$\C^*_{\Delta}\subset \C^*\times \C^*$. 
Isomorphisms $\C^*_{\bH} \ra \C^*$ and $\C^*_{\sD}\ra \C^*$
induce a birational fiberwise isomorphism 
$$
\xymatrix{
\cC_{\bH} = &  \sD\ba \P(\M2)/\bH \ar[d] \\ 
            & \bN_{\bT}\ba \P(\M2)/\bH } \sim 
\xymatrix{
            \sS_2 \ba \P(\M2)/\sS_2 \ar[d] & = \cC_0\\
            \bN_{\bT}\ba \P(\M2)/\sS_2   &   }
$$  
and it suffices to consider $\sD = \sS_2, \bH= \sS_2$.
In this case, an alternative equivariant completion of $U$ is given by
$$
U\subset  \xymatrix{\P^1_1\times \P^1_2\times \P^1_3 \ar[d] \\  \P^1_3},
$$ 
with an action of $\sS_2\times \sS_2$, where 
the first $\sS_2$ acts as an involution on the first two factors and 
identity on the base while the complementary $\sS_2$ 
acts only on the base. 
Thus the quotient is a conic bundle over the complement in 
$$
\P^1\times \P^1/\sS_2\times \sS_2 = \P^1\times \P^1 
$$ 
to the branch locus of the quotient map. Here the left (resp. right) 
$\sS_2$ acts as an involution on the left (resp. right) $\P^1$ and the
branch locus is exactly the union of four lines.
By Lemma~\ref{lemm:cb}, 
this conic bundle has a section  
(it is nonsingular on a pencil of lines minus at most
two points).
\end{proof}
   

\

\begin{lemm}
\label{lemm:subgroup}
Let $\bG$ be a subgroup of $\SL_2$, 
not equal to $\tilde{\sA}_5$, and $V$ a linear representation of $\bG$.
Then $\bG\ba\P(V)$ is rational.
\end{lemm}

\begin{proof}
For  $\bG=\SL_2$ this is a theorem of Katsylo \cite{K}.
We now consider proper subgroups $\bG\subsetneq\SL_2$.
If $\bG$ is solvable and connected then rationality for 
the quotient follows from a theorem of Vinberg \cite{V}. 
For compact $\bG$ the proof is similar to the dihedral case
described below. 
Assume now that $\bG$ is finite and not equal to $\tilde{\sA}_5$. 
Then $\bG$ is either
\begin{enumerate}
\item a finite subgroup of $\C^*$,
\item a dihedral group or
\item $\tilde{\sA}_4, \tilde{\sS}_4$.
\end{enumerate}

The first case is easy.
For dihedral groups all irreducible representations of 
$\bG$ have dimension  $\le 2$ and the corresponding quotients
are rational by Theorem~\ref{thm:q}. 
Let $V$ be a faithful representation of a dihedral group $\sD$
(otherwise, we are reduced to a quotient group). 
Thus $V=W\oplus W'$, where $\dim W =2$ and $\dim W'\ge 1$.
Denote by  $\bG'=\bG/\sC'$ the quotient acting faithfully on $W'$
($\sC'$ is a cyclic group). 
We have $W\sim_{\sD} \C^*\times \P^1$, with trivial action of $\sD$
on $\C^*$ and trivial action of $\sC'$ on $\P^1$. 
By Lemma~\ref{lemm:redu}, 
$$
\C^*\times \P^1\times W' \sim_{\bG'} (\C^*\times \P^1) \times W',
$$ 
with trivial action of $\bG'$ on $\C^*\times \P^1$. 
Thus 
$$
\sD\ba V\sim (\sD'\ba W') \times (\C^*\times \P^1)
$$
and we can apply induction. 



We turn to the last case.
An irreducible representation of 
$\tilde{\sA}_4$ is either a character, 
or a faithful two-dimensional representation, 
or a three-dimensional
representation, 
trivial on the center 
(a faithful representation of $\sA_4$).
An irreducible representation of $\tilde{\sS}_4$ is either
a faithful two-dimensional representation, a faithful 
four-dimensional representation $W:=\Sym^3(V_1)$ 
or a representation of $\sS_4$ (of dimension $\le 3$). 

For irreducible representations of dimension $\le 3$ rationality for
the quotient follows from Theorem~\ref{thm:q}.
We turn to $W$. Recall that
$$
W=\Sym^3(V_1)=V_1^{\chi}\oplus V_1^{-\chi},
$$
as a $\tilde{\sA}_4$-representation, 
where 
$V_1^{\chi}=V_1\otimes \chi$, $V^{-\chi}=V_1\otimes \chi^{-1}$ 
and 
$$
\chi\,:\, \sA_4\ra \Z/3\subset \C^*
$$ 
is the cubic character. 
A pair of (generic) points 
$$
p_{\chi}\in \P^1= \P(V^{\chi}), 
p_{-\chi}\in \P^1=\P(V^{-\chi})
$$ 
defines a line $\P^1\subset \P(W)$. 
This shows that
$$
\P(W)\sim_{\tilde{\sS}_4} \xymatrix{ \ar[d]^{L} \\ \P^1\times \P^1 },
$$
where $\sS_4$ acts on the base, $\sA_4$ acts linearly on the fiber $L$
and $\sS_2=\tilde{\sS}_4 /\sS_4$ acts as an involution on the fiber $L$. 
Thus $\tilde{\sS}_4\ba \P(W)$ is a conic bundle over the rational surface
$\sS_4\ba (\P^1\times \P^1)$. 
We now analyze the geometry of this bundle in more detail.
Consider the action $\sD_2\subset \sS_4$ on
$\P^1\times \P^1$ and on $\P^2 = \Sym^2(\P^1)$.
Every involution $i\in \sD_2$ has two invariant
points $x_i,y_i$.  Consider the graphs $\P^1$ connecting 
the points $(x_i,y_i) - (y_i, x_i)$. Their set is equal to $\P^1$
and there is a graph: 
$$
l_i \,: \,(x_i,y_i) - (y_i, x_i)\subset \P^1\times \P^1 
$$ 
consisting of points $(x,i(x))$.  
The line $l_i$ is exactly the subset of $i$-invariant points in
$\P^1\times \P^1$. The action of $\sD_2$ is free 
outside the three lines $l_i, i\in \sD_2, i\neq 1$.
There are exactly $6$ points which 
are invariant under $\sD_2$. 
 
  
The corresponding action on $\P^2$ can be described as 
follows. There are three points corresponding to $(x_i,y_i)$
which are stable under $\sD_2$
and three lines (images of $l_i$) so that
the action is free on the torus $\C^*\times \C^*$
(the complement in $\P^2$ to the union of $l_i$).
The group $\sD_2$ acts on $\C^*\times \C^*$ as a translation
by the subgroup of points of order $2$.

The quotient $\P^2_q:=\sD_2\ba \P^2$ is a nonsingular 
variety isomorphic to $\P^2$
(indeed the only possible singularities
come from the three $\sD_2$-invariant points
in $\P^2$ but the quotient by the local action is 
nonsingular).  
The diagonal $\P^1_{\Delta}\subset  \P^1\times \P^1 $ projects onto
a conic $C\subset \P^2$, which is invariant under $\sD_2$.
The conic  $C$ intersects  the ``vertical'' and ``horizontal'' 
subgroups in $\C^*\times \C^*\subset \P^2$
in two points and does not intersect the line at infinity.
 
Thus in $\P^2_q = \sD_2\ba \P^2$, the image of
$\P^1_{\Delta}$ intersects $\C^*$ in one point.
Therefore, the images of $\P^1_{\Delta}$ 
and of $l_i$ are lines
(since pairwise intersections of the $l_i$ are equal to $1$)
and the $(\sC_2)^3$-covering $\P^1\times \P^1\to \P^2_q$  
is ramified exactly over a union of four lines. 
If suffices to observe that 
every conic bundle over $\P^2_q$ has a section.
Indeed, let $p$ be the intersection point of
two lines $l_i$ and $l_{i'}$ and consider the pencil of lines in $\P^2_q$
through $p$. Each line in this pencil
intersects the ramification locus 
in at most three points and we can 
apply Lemma~\ref{lemm:cb}.



\


Now we turn to reducible representations
$V=\oplus_{\al\in \cA} V^{\al}$ of $\tilde{\sA}_4$.
If $V$ is faithful for 
$\tilde{\sA}_4$ then there is an $\al_0\in \cA$ such that 
$V^{\al_0}$ is a three-dimensional irreducible {\em faithful}
representation of $\tilde{\sA}_4$ and 
$$
V\sim_{\tilde{\sA}_4} V^{\al_0} \times (\oplus_{\al\neq \al_0} V^{\al})
$$
with trivial action of $\tilde{\sA}_4$
on $\oplus_{\al\neq \al_0} V^{\al}$ (by Lemma~\ref{lemm:redu}).
If $V$ is faithful for $\sA_4$ then $V$ contains a 
faithful irreducible three-dimensional representation of $\sA_4$
and we can apply the same argument. 
In all other cases $V$ is a sum of one-dimensional 
representations and we are reduced to Case 1.


Finally, consider reducible representations $V$ of $\tilde{\sS}_4$. 
If $V$ is faithful then it contains either a faithful irreducible
two-dimensional representation or the faithful representation $W$. 
Again, we apply Lemma~\ref{lemm:redu} as before.
If $V$ is faithful for $\sS_4$ then it contains a faithful 
irreducible representation of dimension $\le 3$ and we conclude
as above. In all other cases $V$ is a sum of one-dimensional
representations.  
\end{proof}


\

\begin{lemm}
\label{lemm:grr}
Let $V$ be a representation of
$\bG\subsetneq\SL_2$, with $\bG\neq \tilde{\sA}_5$. Then 
$\bG\ba\Gr(2,V)$ is rational.
\end{lemm}

\begin{proof}
The relevant groups $\bG$ can be subdivided as follows:
\begin{enumerate}
\item $\bG$ is a subgroup of the normalizer of a maximal torus;  
\item $\bG$ an infinite subgroup of a Borel subgroup;
\item $\bG = \sA_4$, $\tilde{\sA}_4$; 
\item $\bG = \sS_4$, $\tilde{\sS}_4$. 
\end{enumerate}
Let $V=W\oplus W'$ be a reducible representation  of $\bG$.
Then (birationally)
$$
\xymatrix{\Gr(2,V)\ar[d]^{\Hom(\C^2_x, W)} \\
\Gr(2,W)}
$$
(where $x$ is a point on the base). 
In particular, if $\dim W\le 2$ then
$$
\Gr(2,V)\sim_{\bG} \Hom(W',W),
$$
with linear $\bG$-action on $\Hom(W', W)$. This reduction
suffices for the relevant infinite groups (for example, for
connected solvable $\bG$ we can apply Lemma~\ref{thm:qq}). 
Further,  
\begin{itemize}
\item if $\Stab_{gen}(\bG,\Gr(2,V))=1$ then (birationally) 
$$
 \bG\ba \Gr(2,V)\ra   \bG\ba\Gr(2,W),
$$
a vector bundle.
\item if $\Stab_{gen}(\bG,\Gr(2,V))=\sC\subset \bZ_{\bG}$ 
(a cyclic subgroup) then (birationally)
$$
\xymatrix{\bG\ba\Gr(2,V)  \ar[d]^{\sC\ba \Hom(\C^2_x, W)} \\      
\bG\ba \Gr(2,W).}
$$
\end{itemize} 


We now consider $\sA_4,\tilde{\sA}_4, \sS_4$. 
The rationality of $\bG\ba\Gr(2,V)$ for 
{\em irreducible} representations 
of these groups follows from the fact 
that all of them have dimension $\le 3$. 
Assume now that $V = W \oplus W' $, with $W$ irreducible of dimension $3$.
The classification of these representations implies that 
the action of the center must be trivial.
Then, birationally,
$$
\xymatrix{
\Gr(2,V)  \ar[d]^{\Hom(\C^2_x, W')} \\
\P^2=\P(W^*).}
$$
The $\bG$-action is equivalent to a 
$\bG$-action on a vector
bundle
$$
\xymatrix{\bG\ba \Gr(2,V) \ar[d]\\
\bG\ba \Gr(2,W) = \bG\ba\P^2.}
$$

\

Finally, let us consider the case of $\tilde{\sS}_4$.
Let $W$ be its unique { irreducible} 
representation of dimension four (as in Lemma~\ref{lemm:subgroup}).  
We claim that $\tilde{\sS}_4\ba \Gr(2,W)$ is rational.
Indeed, as $\tilde{\sA}_4$-modules, we have
$$
W=W^{\chi} \oplus W^{-\chi},
$$
where $W^{\chi},W^{-\chi}$ are two copies of 
the standard representation of $\tilde{\sA}_4$
of dimension $2$ and $\chi$ (resp. $-\chi$) 
indicates the eigenspace decomposition for the nontrivial character 
$$
\chi\, :\, \sA_4\to \Z/3\subset \C^*.
$$
Further,  
$$
\Gr(2,W) \sim \Hom(W^{\chi},W^{-\chi}),
$$
with a linear $\sA_4$-action (since the center acts trivially)
and a permutation $\sS_2$ 
inverting the map $w \in \Hom(W^{\chi},W^{-\chi})$.
More precisely, $W^{-\chi}= (W^{\chi})^*$ and  
$$
\Hom(W^{\chi},W^{-\chi})=\Sym^2(W^{-\chi}) \oplus C_1,
$$
where $C_1$ corresponds to skew symmetric maps and 
$\sA_4$ acts on $C_1$ by $\chi$.
The involution $\sS_2 = \sS_4/\sA_4$ acts on 
$C_1$ and on $\Sym^2(W^{-\chi})$ as $t \mapsto t^{-1}$.
In particular, if $\C^*\times \C^*$ is the diagonal group
acting on $\Sym^2(W^{-\chi}) \oplus C_1$ then $\sS_2$ acts as 
$$
X \ra s^{-1} X,
$$
where $s \in \C^*\times \C^*$ and $X\in \Sym^2(W^{-\chi}) \oplus C_1$.
There is an equivariant map 
$$
\begin{array}{ccccl}
f & : & \Hom(W^{\chi},W^{-\chi}) & \ra &   C_1,\\
  &   &   s                      & \mapsto &  (x,s(y)) - (s(x),y),
\end{array}
$$ 
with an effective action of $\sS_3 = \sS_4/\sD_2$ on the target $C_1$,
which to a subspace $s \in \C^2\subset W^{\chi} \oplus W^{-\chi}$
assigns the value of the $2$-form $(x,s(y)) - (s(x),y)$. 
The fiber of $f$ is $\sD_2$-birational to $\Sym^2(W^{\chi})=\P^2$. 
We have already seen in the proof of Lemma~\ref{lemm:subgroup} 
that $\sD_2\ba\P^2=\P^2$.
Thus $\tilde{\sS}_4\ba \Gr(2,W)$ is a $\C^*$- bundle over 
a $\P^2$-fibration over $\sS_2\ba C_1$.
It is clear that this $\P^2$-fibration is trivial.
The quotient conic bundle is nondegenerate
over a product of $\P^2$ with an open subvariety in $C_1/\sS_3$.
Hence it has a section. Rationality of 
$\tilde{\sS}_4\ba \Gr(2,W)$, and more generally, 
$\tilde{\sS}_4\ba \Gr(2,W\oplus \cdots \oplus W)$, follows (the latter is 
a vector bundle over the former). 



\

Assume now that $V = nW \oplus V'$, 
where $\dim V'\ge 1$, and $n\in \N$. 
Since the $\sS_4$-action on $\Gr(2,nW)$ is $af$
there is a $\tilde{\sS}_4$-equivariant homogeneous 
rational  map  $f\, :\, \Gr(2, nW )\ra V'$
sending the generic  $\tilde{\sS}_4$-orbit in $W$ to
the generic $\tilde{\sS}_4$-orbit in $V'$.
Notice that the center $\sC_2$ acts as a scalar on
$\Hom (W, V')$.
We have (birationally)
\begin{equation}
\label{eqn:bbd}
\xymatrix{
\tilde{\sS}_4\ba\Gr(2,V)  \ar[d]^{\sS_2\ba\Hom(\C^2_x ,V')} & \sim 
& \C^* \times &
 \ar[d]^{ \P(\Hom (\C^2_x,V'))} \\
\tilde{\sS}_4\ba \Gr(2,nW) &   &  & \sS_4\ba  \Gr(2,nW)
} 
\end{equation}
(with rational bases).
The projective bundle on the right has a section.
Indeed, 
\begin{equation}
\label{eqn:bb}
\xymatrix{ \ar[d]^{\Hom(\C^2_x, V')} \\ \Gr(2,nW) }
\end{equation}
is an equivariant quotient
bundle of the trivial bundle with fiber $\Hom(W, V')$.
The map $f$ defines an $\sS_4$-equivariant section $s(f)$
in the projective bundle in (\ref{eqn:bbd}).
The (equivariant) linear projection 
$$
\Hom(W, V')\ra \Hom(\C^2_x, V')
$$
maps $s(f)$ to an equivariant section of the
bundle in (\ref{eqn:bb}).
Thus $s(f)$ projects onto a section of the bundle
on the right in (\ref{eqn:bbd}), making it birationally trivial.
\end{proof}

\

We proceed to describe possible $\SL_2$, resp. $\PGL_2$-actions on 
Grassmannians. (If all weights in $V$ are of the same parity then
$\Gr(2,V)$ carries the $\PGL_2$-action, otherwise the $\SL_2$-action.) 

\

\begin{lemm}
\label{lemm:free}
%\label{coro:stabb}
Let $V$ be a faithful $\SL_2$-representation of dimension $\ge 3$.
Then 
$$
\begin{array}{c|c}
V             &     \Stab_{gen} \\
\hline
\dim \ge 5    &  1     \\
V_4           &  \sC_2 \\
V_3           &  \sD_2  \\
V_2\oplus V_0 &  \sC_2  \\
V_2           &  \bN_{\bT} \\
V_1\oplus V_1 & \C^*   \\
V_1\oplus V_0 &  \tilde{\bB} 
\end{array}
$$

Further, 
\begin{itemize}
\item 
$\Gr(2,V_4)$ has a $(\PGL_2, \bN_{\bT})$-slice 
$S= \Sym^2(\P^2)$ with an $af$-action of $\bN_{\bT}/\sC_2$,
(where $\sC_2$ is the center of $\bN_{\bT}$);
\item 
$\Gr(2,V_3)$ has a $(\PGL_2, \sA_4)$-slice birational to $\P^1$, 
with $\sA_4$ acting on $\P^1$ as $\sC_3$.
\end{itemize}
\end{lemm}

\begin{proof}
Consider first irreducible representations $V=V_{d}=\Sym^d(V_1)$ 
and assume that the stabilizer of a 
generic line $\P^1\subset \P(V)$ contains a nontrivial 
cyclic group $\sC$. 
Then $\sC$ fixes at least two points in this $\P^1$. 
Any orbit of $\sC$ on $\P^1$ is a union of a zero-cycle 
$\sC \cdot x$ and a zero-cycle supported in the fixed
points. In particular, the subvariety of points in $\P^(V_d)$
which are fixed by $\sC$ has dimension
$\le d/|\sC|$. The dimension 
of the variety of $\sC$-fixed lines in $\P(V)$ 
is therefore $\leq 2 d/ |\sC|$.
The subvariety of distinct cyclic subgroups 
$\sC\subset \PGL_2$ has dimension $2$
and $\dim \Gr(2,V_d) = 2d - 2$.
Since $d/|\sC| \leq d/2$ the inequalities 
$$
2d - 4 > 2d/2 \,\,\, {\rm and} \,\,\,  d - 4 > 0
$$ 
imply the result. 

Assume that $V=\oplus_{j\in J} V_{d_j}$, $|J|\ge 2$
and that $\Stab_{gen}\neq 1$. 
Then $d_j\le 2$, for all $j\in J$. Indeed, the stabilizer
of a generic $\P^1$ through a generic point $p\in \P(V_d)$
is a subgroup of the stabilizer of $p$, which stabilizes 
some generic line in the tangent space
at $p$. This group is trivial for $d > 2$ and equal to $\sC_2$ for $d=2$. 

If $V = V_2 \oplus V'$, with $\dim V' > 2$, then 
$\Gr(2,V)$ is (birationally) a fibration  over 
$\Gr(2,V_2)$, with fibers $\Hom(\C^2, V')$ so that 
$\Stab_{gen}=1$ if $\dim V' > 3$.
If $V = V_2 \oplus V_1$ then 
$\Stab_{gen}$ is the same as 
the (generic) stabilizer of the $\bN_{\tilde{\bT}}$-action on
$\Hom(V', V_1), V' \in \Gr(2,V_2) =\P^2$, hence trivial.
For $V=V_2 \oplus V_0$, $\Stab_{gen}=\sC_2$.  


In the remaining cases $d_j=0$ or $1$, for all $j\in J$. 
If $V$ contains at least three copies of $V_1$ then the argument above
shows that the action is $af$.
Similarly, if $V = V_1 \oplus V_1 $ then
$\Stab_{gen}=\C^*$ and if $V=V_1 \oplus V_1 \oplus V_0$ 
then $\Stab_{gen}=1$.
For $V_1 \oplus 3 V_0$, the generic stabilizer is the same
as for three linear functionals - which is zero.
\end{proof}


\

\begin{lemm}
\label{lemm:gra}
The quotient $\PGL_2\ba \Gr(2,V)$ is 2-stably rational. 
\end{lemm}

\begin{rem}
For {\em even} $d\ge 10$, $\PGL_2\ba \Gr(2,V_d)$ is rational by \cite{sh-2}.
\end{rem}


\begin{proof}
By Lemma~\ref{lemm:free}, if $\dim V\ge 5$ then
the $\Stab_{gen}=1$ and we
can apply Lemma~\ref{lemm:pgl2} and Corollary~\ref{coro:sect} 
to conclude that
$$
\PGL_2\ba \Gr(2,V)\times \C^2\sim_{\bG} \bN_{\bT}\ba\Gr(2,V).
$$
The claim follows from Lemma~\ref{lemm:grr}.
It remains to consider: 
\begin{enumerate}
\item $\Gr(2,V_4)$, 
\item $\Gr(2,V_3)$,
\item reducible $V$.
\end{enumerate}

In the first case, $\Stab_{gen}(\PGL_2,\Gr(2,V_4))=\sS_2$, with
normalizer $\bN_{\bT}\subset \PGL_2$. 
We claim that the subset  $X\subset \Gr(2,V_4)$ of $\sS_2$-invariant points
is a $(\PGL_2, \bN_{\bT})$-slice. Indeed, 
there is a Zariski open subset $ U\subset X$
such that the stabilizer of each point in $U$ 
is exactly $\sS_2$. In particular, $g\cdot U $ intersects $U$ only
if $g\in \bN_{\bT}$. Consider the 
$\P^2\subset \P(V_4)$ consisting of $\sS_2$-invariant subschemes 
containing $4$ points. Any line in $U$ joins a pair of points 
in this $\P^2$. Therefore, we have a (birational) 
$\bN_{\bT}$-isomorphism of
$U$ and $\Sym^2(\P^2)$. The stabilizer
of a generic point in $X$ is a central subgroup in
$\bN_{\bT}$ whose action on $\P^2$ is equivalent to a linear 
action on $\C^2$. 
(Indeed, $\Sym^2(V_1)=\C \oplus W_2$, where $\C$ is the trivial
representation - the invariant symmetric form - 
and $W_2$ is a faithful two-dimensional representation  of 
$\bN_{\bT}/\sS_2$).
Thus instead of $X$ with the $\bN_{\bT}$-action we can consider 
$\C^2\times \C^2$ with the $(\bN_{\bT}/\sS_2)\times \sS_2$-action 
(where the second $\sS_2$ interchanges the factors).
In particular, (by linearity)
$$
\bN_{\bT}\ba X \sim \C^*\times \bN_{\bT} \ba \P^3,
$$ 
and is hence rational.


In the second case, $\Gr(2,V_3)$ has a 
surjection of degree 2 onto $\P(V_4)$.
The connected component of the
preimage of the $(\PGL_2,\sS_4)$-slice $\P^1$ in $\P(V_4)$
is a $(\PGL_2,\sA_4)$-slice, isomorphic to $\P^1$.
The quotient is rational.


If $V$ is reducible and the $\PGL_2$-action on the 
Grassmannian has nontrivial stabilizer then $\dim V < 5$. 
Rationality follows since $\dim \Gr(2,V) \leq 4$ and the 
generic orbit has dimension at least $2$.
\end{proof}
  


\begin{prop}
\label{prop:dihedr}
Let $\bG, \bH $ be finite solvable subgroups of $\PGL_2$. Then 
$$
\bG\ba \PGL_2/\bH
$$ 
is rational. 
\end{prop}

\begin{proof}
The action is birational to the (projective) action of 
$\bG\times \bH$ on $\P(\M2)$,
where $\bG$ acts on the right and $\bH$ on the left.
The groups $\bG,\bH$ are either:
\begin{itemize}
\item cyclic;
\item dihedral or
\item $\sA_4$, $\sS_4$.
\end{itemize}
The case of {\em primitive} 
solvable groups is covered by Theorem~\ref{thm:kolp}, 
\cite{km}.
If $V$ is reducible then there is a nontrivial action
of $\C^*$ on $\bG\ba \P(V)/\bH$, leading 
to rationality. This covers the case when either 
$\bG$ or $\bH$ is cyclic.  

\

We claim that if $V$ is irreducible and 
imprimitive (for the $\bG\times \bH$-action) 
then either $\bG$ or $\bH$ is dihedral.  
By definition, 
$V:=\M2=\oplus_{\al} V^{\al}$, such that $\tilde{g}V^{\al}=V^{\al'}$
for all $\tilde{g}\in \bG\times \bH$.
Moreover, by irreducibility, all $V^{\al}$ must have the 
same dimension, $=1$ or $2$.
Notice that imprimitivity for an action of a group $\bG'$ implies 
imprimitivity for the induced action of every subgroup $\bG''\subset \bG'$
(with the same decomposition of $V$). 
We now claim that the actions of 
$\sA_4\times \sA_4$, and consequently of 
$\sA_4\times \sS_4$ and $\sS_4\times \sS_4$ are 
primitive. Indeed, $\sA_4\times \sA_4$ contains 
$\sD_2\times \sD_2$ as a normal subgroup, for which the imprimitive
structure is either a sum of two subspaces
of dimension $2$ or four subspaces of dimension $1$, corresponding to 
the choice of a subgroup $\sS_2\subset \sD_2$.  
The first possible imprimitive structure for $\sD_2\times \sD_2$ does not 
extend to one for $\sA_4\times \sA_4$
(which has no index $2$ subgroups). 
The second structure is also impossible: 
$\sA_4$ rotates the subgroups $\sS_2\subset \sD_2$,
hence there is no $\sA_4$-invariant imprimitive
structures for $\sD_2\times \sD_2$.

\

It remains to consider the case when both $\bG$ and $\bH$ are
dihedral.
On $V_1$ there is a unique imprimitive structure,
corresponding to the eigenspaces $C_1,C_2$ of the
elements of $\bG$.
In particular, there is an imprimitive structure on 
$$
\M2 =V_1 \oplus V_1' = (C_1 \oplus C_1') \oplus (C_2 \oplus C_2)'.
$$
We claim that (birationally)
$$
\xymatrix{
\bG\ba\P(\M2)/\bH  \ar[d] \\
\P^2 = \bG\ba \Sym^2(\P^1)}
$$
is a conic bundle degenerating precisely
over the image of the diagonal and the subvarieties 
in $\P^2$ with nontrivial stabilizers.

Indeed, since $\bH\subset \bN_{\bT}$  (a $\sC_2$-extension 
of $\C^*$), (birationally)
$$
\xymatrix{\bG\ba\P(\M2)/\bH \ar[d]^{\C^* = \bN_{\bT}/\bH} \\ 
\bN_{\bT}\ba\P(\M2)/\bH.}
$$
The quotient $\C^*\ba\P(\M2)$ is (birationally)  a fibration
over $\P^1\times \P^1$, with $\sS_2$
acting by permutation, where the coordinate $\P^1$s
are the projectivizations of the two-dimensional
eigenspaces for the $\C^*$-action on $\M2$.
Thus 
$$
\xymatrix{
\P(\M2)/\bH  \ar[d] \\
\P^2 = \P^1\times \P^1/\sS_2}
$$
is a conic bundle nondegenerate
outside a conic (the image of the diagonal in
$\P^1\times \P^1$).
The $\bG$-action commutes with the
$\bN_{\bT}$-action and is effective on the base. This proves the claim.

\

We have  $\bG\subset \bN_{\bT}$ and
$$
\bG\ba \P^2 \ra \bN_{\bT}\ba \P^2 
$$ 
is a conic bundle. Since the left and right actions
of $\bN_{\bT}$ commute, 
$\bG\ba\P^2$ contains an open subvariety 
$ U\times \C^*$ where the restriction of the conic bundle 
is nondegenerate. Here $\C^* = \bG\ba\bN_{\bT}$ and 
$U$ is a subset of $\P^1 =\bN_{\bT}\ba \P^2$.
Therefore the conic bundle has at most $2$ singular fibers 
on any completion of the fiber $\C^*\subset U\times \C^*$.
Rationality follows.


We can now describe some open subvariety in the quotient 
$\bG\ba\P(\M2)/\bH$ explicitly.
Consider the action of $\C^*\subset \bN_{\bT}$ on both sides 
$\C^*\ba\P(\M2)/ \C^*$. With respect to this 
action $\P(\M2)$ is birationally equivalent to a trivial 
$\C^*\times \C^*$-fibration over $\P^1$.
Now we add the action of $\sS_2$ on both sides.
The product $\sS_2\times \sS_2$ acts on the base
$\P^1$. The group $\sS_4$ contains
a normal subgroup $\sD_2\subset \bN_{\bT}$
and the action of each $\sS_2\subset \sD_2$ inverts  the respective 
$\C^*$ action. Thus (birationally)
$$
\xymatrix{
\bN_{\bT}\ba \P(\M2)/\bN{\bT} \ar[d]^{\bN_{\bT}\times \bN_{\bT}} \\ \P^1 -3\, pts,}
$$
where the deleted points are the ramification points
of the map $\P^1 \ra \P^1/\sD_2$.
In particular, there is an open $U$ such that 
$$
\xymatrix{
\bG\ba\P(\M2)/\bH \ar[d]^{\C^*} 
\\ 
U \ar[d]^{\C^*}
\\
\P^1 - 3\, pts. }
$$
By Lemma~\ref{lemm:cb}, the conic bundles are trivial.
 
\


Finally, the  conic bundles 
on $\P^2/\sS_4$ and $\P^2/\sA_4$ have sections.
Indeed, both $\sA_4$ and $\sS_4$ contain dihedral subgroups
of index 3 ($\sD_2$, resp. $\sD_4$).
The image of the section in the conic bundle over
$\sD_2\ba \P^2$ (resp. $\sD_4\ba \P^2$), has 
odd degree in the conic bundles over 
$\sA_4\ba \P^2$ and $\sS_4\ba \P^2$, respectively. We  
apply Lemma~\ref{lemm:conic}. 
\end{proof}
  
\

\begin{prop}
\label{prop:even} 
Let $V$ be an irreducible $\GL_2$-representation 
and $\bH\subset \SL_2$ a finite group, not equal to $\tilde{\sA}_5$.  
Then 
$$
\GL_2\backslash (V\oplus V)/\bH
$$ 
is rational. 
\end{prop}

\begin{proof}
First of all, $V_1\oplus V_1/\bH$ is rational. 
Next, by Lemma~\ref{lemm:sl}, 
$$
V\oplus V\sim_{\GL_2\times \GL_2} \xymatrix{\cV \ar[d]^{\M2=V_1\oplus V_1} 
\\ \Gr(2,V).}
$$

First we assume that $V$ has odd weight.
The Grassmannian $\Gr(2,V)$ carries the action of $\PGL_2$. 
If we restrict the bundle $\cV$ to a generic 
$\PGL_2$-orbit $O$ in $\Gr(2,V)$ then the corresponding module
$H^0(O, \cV_{O})$ contains $V_1$ as a submodule.
By Lemma~\ref{lemm:mapp}, this gives an equivariant map
$$
\cV\ra V_1 \oplus V_1
$$ 
with a $1$-transitive action
of $\GL_2$ on the target.
Thus 
\begin{equation}
\label{eqn:vy}
\GL_2\ba \cV/\bH \sim \bH\ba \Gr(2,V)
\end{equation}
(with the {\em same} subgroup 
$\bH\subset \GL_2$ appearing on the left).
Indeed, $\GL_2\subset (V_1 \oplus V_1) = \M2$ and multiplication
by $\bH$ on the right gives an orbit $x\cdot \bH$.
This orbit is a  $(\GL_2\times \bH, \bH^x\times \bH)$-slice 
(with $\bH^x = x \bH x^{-1}$) and 
it is stabilized exactly by $\bH^x\times \bH$, acting 
doubly transitively on the set $\bH^x\cdot x$. 
It follows that every point $x' \in x\cdot \bH$ is
a $(\bH^x\times \bH,\bH^x)$-slice of the orbit $x\cdot \bH$.
The quotient $\bH\ba \Gr(2,V)$ is rational by \ref{lemm:grr}. 


\


Assume that $V$ has even weight.
If the $\PGL_2$-action is $af$ then 
$$
\GL_2\ba \cV /\bH \sim 
(\PGL_2\ba \Gr(2,V))\times (\C^*\ba (V_1\oplus V_1)/\bH).
$$ 
If it is not $af$, then, by Lemma~\ref{lemm:free}, 
$V = V_4 $ or $V_2$.



For $V=V_4$ we have the $(\PGL_2,\bN_{\bT})$-slice 
$X = \Sym^2(\P^2)$ with
the $\bN_{\bT}$-action which we can replace by 
$\C^2\times \C^2$ with a $(\bN_{\bT}/\sC_2)\times \sC_2$-linear action.
In particular, we identify the quotient with a quotient of 
$\C^2 \oplus \C^2 \oplus V_1 \oplus V_1$ by a linear action of 
$\bN_{\tilde{\bT}}\times \sS_2\times \bH$ (where
$\bN_{\tilde{\bT}}\subset \GL_2$).
The action of $\bN_{\tilde{\bT}}\times \bH$ on $V_1$ 
is transitive with stabilizer $\sC_2\times \bH$. Hence
it is equivalent to the action of $\sD_2\times \bH$ on
$\C^2 \oplus \C^2 \oplus V_1$, which 
is a $\C^2$-vector bundle (permutation of the anti-invariant part of
$\sS_2$-action) over $\C^2\times V_1$, with $\sD_2\times \bH$ action.
The latter quotient is rational.
For $V = V_2$ the action is transitive on $\Gr(2,V)= \P^2$ and
the quotient has dimension $2$ - rationality follows.
\end{proof}


We will also need a more general result for $\bH= \sS_2$.


\begin{prop}
\label{prop:yz}
Let 
$$
X\stackrel{L}{\lra} Y=\prod_{j\in J} \P(V_{d_j})
$$
be a $\GL_2$-homogeneous line bundle. 
If at least one $d_j\neq 2$ then 
$\GL_2 \backslash X\times X/\sS_2$ is rational.  
\end{prop}

\begin{proof}
{\bf Case 1.} $|J|=1$.  If $d=d_1$ is even or 
if $d$ is odd and the line bundle has odd degree on $\P(V_d)$ 
then 
$$
X\times X \sim_{\GL_2\times \sS_2} V_d \oplus V_d
$$ 
and we apply Proposition~\ref{prop:even}.
If the line bundle has even degree then it
is trivial and $\GL_2$ acts as $\PGL_2\times \C^*$.
If the $\PGL_2$-action on $\P(V_d)$ is $af$ 
we have 
$$
\P(V_d)\sim_{\PGL_2} S\times \PGL_2,
$$
for a rational slice $S$ (with trivial $\PGL_2$-action).
We have a $\PGL_2\times \C^*\times \sS_2$-action on 
$$
\C \times \PGL_2\times  S\times \C \times \PGL_2\times S.
$$ 
The quotient variety is a vector
bundle over $\PGL_2 \ba \PGL_2\times \PGL_2/ \sS_2$
(rational by Lemma~\ref{lemm:qqq}). The claim follows.
If the $\PGL_2$-action is not $af$, then $V=V_3$ or
$V_1$. For $V_1$ the quotient is rational by dimensional reasons.
For $V_3$ we have a projection
$$
\xymatrix{
\C\times \P^3\times \C\times  \P^3 \ar[d]^{\P^1\times \P^1}\\
\Gr(2,V_3)}
$$ 
commuting with both actions.
Recall that $\Gr(2,V_3)$ has $\P^1$ as a $(\PGL_2,\sA_4)$-slice, 
with $\sA_4$ effectively acting as a cyclic group $\sC_3= \sA_4/\sD_2$
on $\P^1$ (the group $\sD_2$ acts trivially on the
$(\PGL_2,\sA_4)$-slice $\P^1\subset \P^4$ and similarly for
$\Gr(2,V_3)$, see Lemma~\ref{lemm:free}). 
Thus the quotient is the same as for the bundle
$$
\xymatrix{  \ar[d]^{\P^1\times \C\times \P^1\times \C}\\ \P^1}
$$
under the action of $\sA_4\times \sS_2$.
In particular, it is a vector bundle
over a $\P^2 =\sD_2\ba \P^1\times \P^1/\sS_2$-fibration over 
$\P^1 = \P^1/\sC_3$, hence is rational.


{\bf Case 2.} $|J|\ge 2$. 
If at least one $d_j$ is odd and $>1$ or if all $d_j=1$ and 
$|J|>2$, then there is a slice $S$ and the 
$\PGL_2$-action is $af$. 
We can write $Y$ as (the total space of the) line bundle:
$$
\xymatrix{ X\ar[d]^{L}\\ 
S\times \PGL_2}
$$ 
and, using Lemma~\ref{lemm:y}, reduce to either
a vector bundle over 
$$
\PGL_2\ba \PGL_2\times \PGL_2/\sS_2,
$$
when $L$ is trivial on $\PGL_2$,
or to 
$$
\GL_2\backslash \GL_2\times \GL_2/\sS_2
$$ 
otherwise. In both cases the base is rational 
by Lemma~\ref{lemm:qqq}.



If $d_j=1$ for every $j\in J$ and $|J| = 2$ 
then the there is a map  
$$
(\P^1)^4 \ra \P(V_4)= \Sym^4(\P^1) = \P^4
$$
(of degree $24$, mapping $4$ points to a form of degree $4$).
The preimage in $(\P^1)^4$ 
of the $(\PGL_2,\sS_4)$-slice $\P^1_s=\P^1$ of
$\P^4$, will be a set of six lines $\P^1_{g,h}$, labeled by
a pair of generators $g,h \in \sD_2$ (which act trivially on 
$\P^1_s\subset \P^4$). 
More precisely, the line $\P^1_{g,h}$ is the set given by
$(x:gx:hx:ghx)\in (\P^1)^4$, for $x\in \P^1$.
The map $\P^1_{g,h}\ra \P^1_s=\P^1_{t,s}/\sD_2$ has 
degree $4$. Thus $\P^1_{g,h}$ is a $(\PGL_2, \sD_2)$-slice
of $(\P^1)^4$ 
and the quotient of a vector bundle  
$\stackrel{L\oplus L}{\longrightarrow}\P^1$ 
by a linear action of $\sD_2$ is rational.

   
\


Assume that all $d_i$ are even. 
Then $L$ is (birationally) trivial. 
Unless $|J|=2$ and $d_1=d_2 = 2$, there is a decomposition of
$$
Y\times Y =\P(V_d)\times Y'\times \P(V_d)\times Y'
$$
such that the $\PGL_2$-action is $af$ and 
$$
\P(V_d)\times Y'\times \P(V_d)\times Y'
\sim_{\PGL_2\times \sS_2} (Y'\times Y') \times 
(\P(V_d)\times\P(V_d))
$$
(with trivial $\PGL_2$-action on $\P(V_d)$), by Lemma~\ref{lemm:pgl2}.
The quotient is birational to a vector
bundle over $\PGL_2\times \C^* \ba X'\times X'/\sS_2$,  
where $X'$ is the trivial line bundle over $Y'$.

We have reduced to $|J|=1$ treated in Case 1 
or to $|J|=2$ and $d_1=d_2=2$, treated in Lemma~\ref{lemm:p2}. 
\end{proof}


\begin{lemm}
\label{lemm:p2}
The quotient
$$
X:=\PGL_2\ba (\P_1(V_2)\times \P_2(V_2) \times \P_1(V_2)\times \P_2(V_2))/\sS_2
$$
is rational, where $\P_1(V_2)$ and $\P_2(V_2)$ 
are different copies of $\P^2=\P(V_2)$ and 
$\sS_2$ acts by permutation.
\end{lemm}



\begin{proof}
Consider the projection 
$$
X\ra \PGL_2\ba \P_1(V_2)\times \P_1(V_2)/\sS_2
$$
and the $\PGL_2\times \sS_2$-equivariant map of degree $6$
$$
\begin{array}{cccc}
\pr\,: &  \P(V_2)\times \P(V_2)       & \ra      &    \P(V_4)\\
       &     (Q_1,Q_2)          & \mapsto  & Q_1\cdot Q_2.
\end{array}
$$
The space $\P(V_4)$ has a $(\PGL_2,\sS_4)$-slice $\P^1_s$
(the $\sD_2$-invariant polynomials). 
The zeroes of a (polynomial) $p\in \P^1_s$ form an orbit under $\sD_2$.
The preimage $\pr^{-1}(\P^1_s)\subset \P^2\times \P^2$
consists of $3$ lines, each invariant under $\sD_2$.
Indeed, the ordered pair $(Q_1,Q_2)$ corresponds to a
choice of a generator $g\in \sD_2$ such that $x, g(x)$ are zeroes 
of $Q_1$ and $h(x), hg(x)$ are zeroes of $Q_2$.
Thus the line $\P^1_g\subset \P^2\times \P^2$ consists
of tupels $\{(x,gx),(hx,ghx)\}$, where $x$  is an arbitrary point
in $\P^1$ and $(x,gx) =Q_1, (hx,ghx)= Q_2$.
The map $\P^1_g\ra \P^1_s$ has degree two and its fibers
coincide with orbits of $h$ (since $g$ acts trivially
on $\P^1_g$). The action of $h$ is given by
$$
h\,:\, \{(x,gx),(hx,ghx)\} \mapsto \{ (hx,ghx),(x, gx)\}.
$$
Thus $h(Q_1,Q_2) = (Q_2,Q_1)$ and the action 
of $h$ coincides with the restriction of the permutation action
on $\P^2\times \P^2$ to $\P^1_g$.
The line $\P^1_g$ is invariant under
${\sD}_4\times \sS_2$ (considered as a subgroup of $(\PGL_2\times \sS_2)$).
The group ${\sS}_4$ permutes the lines in $\pr^{-1}(\P^1_s)$.
Each $\P^1_g$ is a $(\PGL_2\times \sS_2, {\sD}_4\times \sS_2)$-slice of  
$\P^2\times \P^2$.
Therefore, 
$$
X \sim  {\sD}_4\ba \P^1\times \P^2\times \P^2  / \sS_2.
$$ 
The space $\P^2\times \P^2$ contains a subspace
$\C^2\times \C^2$ with a {\em linear} action of $\sD_4\times \sS_2$.
Indeed, the action of $\sD_4$ on $\P^1$ corresponds to the
irreducible representation of
$\tilde{\sD}_4$ on $\C^2= V$. 
Under the $\sD_4$-action, one has
a decomposition 
$\Sym^2(V)=V'\oplus V''$, where $\dim V'=2$, $\dim V''=1$ and the 
action of $\sD_4$ on $\P^2$ is equivalent 
to the linear action on $V'$.
The additional $\sS_2$ permutes the $\P^2$ and hence
acts by permutation on $V' \oplus V'$.
Thus  
$$
\P^1\times \P^2\times \P^2\sim_{\sD_4\times \sS_2}
\xymatrix{ \ar[d]^{V'\oplus V'}  \\  \P^1}
$$
(a vector bundle).


Consider the effective action of (the nonabelian group) 
$\sD_4\times \sS_2$ on $\P^1$.
It has a normal subgroup $\sD_2\times \sS_2$ with generators
$g,h,k$ and an element $i, i^2 = 1$  which commutes 
with $g,k$ and acts on $h$ as $ihi = gh$.  
The stabilizer of a generic point  on $\P^1_g$ 
is a normal abelian subgroup generated by $g, hk$. 
Thus $\sD_4\times \sS_2$ acts on $\P^1$ effectively through the
quotient $\sD_4/\langle g,hk\rangle = \sD_2$. 
The action of this $\sD_2$ on $\P^1$ is
almost free. Indeed, the action of $k$ coincides with the action 
of $h$ and permutes $Q_1,Q_2$. Thus the orbits
of $h$ and $k$ on $\P^1_g$ coincide with fibers of the map
$\P^1_g\to \P^1_s$. On the other hand, $i$ acts nontrivially on $\P^1_s$. 
We claim that 
$$
\xymatrix{\sD_4\ba (V'\oplus V')\times \bP^1/\sS_2 \ar[d]\\
          \sD_4\ba (V'\times \P^1)}
$$
is a vector bundle.
Indeed, consider the subspace $V'_{inv} \subset V'\oplus V'$ of invariant
vectors (under the permutation).
The action of $\sD_4\times \sS_2 $ on  
$((V'\oplus V')/V'_{inv}) \times \bP^1$ is almost free. Hence
$$
\xymatrix{\sD_4\ba (V'\oplus V')\times \P^1/\sS_2\ar[d] \\
\sD_4\ba ((V'\oplus V')/V'_{inv}) \times \P^1/\sS_2}
$$
is a vector bundle with base a quotient of the 
vector bundle $ (V'\oplus V'/V'_{inv})\ra \P^1_g$ by 
$\sD_4\times \sS_2$. The variety 
$(V'\oplus V'/V'_{inv})\times \P^1$ has a fiberwise (scalar)
$\C^*$-action commuting with the $\sD_4\times \sS_2$-action.
Since every $\C^*$-action has a slice,  
$$
X':=\sD_4\ba ((V'\oplus V')/V'_{inv}) \times \P^1/\sS_2,
$$
is rational by dimensional reasons: 
$X'/\C^*$ is a unirational, therefore, rational surface and   
$$
X'\sim (X'/\C^*)\times \C^*.
$$ 
\end{proof}

\


\begin{prop}
\label{prop:syml}
Let $X$ be $V\oplus V$, where $V=V_d$ is an 
irreducible $\GL_2$-representation,
$\ell>0$ and $\bH\subset \SL_2$ with $\bH\neq \tilde{\sA}_5$. 
Then 
$$
\GL_2\ba X\times \P(V_{\ell})/\bH
$$
is rational (where $\bH$ acts trivially on $\P(V_{\ell})$). 
\end{prop}

\begin{proof}
If $\ell$ is even and the action of 
$\GL_2$ or a quotient of $\GL_2$ by a central subgroup is $af$
then we apply Lemma~\ref{lemm:pgl2} combined with
Proposition~\ref{prop:even}, 
resp. \ref{prop:yz}. 

If $\ell$ is odd and the action is $af$ then there exists a slice,
which is a rational variety, 
by Lemma~\ref{lemm:grr} resp. \ref{lemm:subgroup}. 
Rationality follows. 


Now we assume that the action is not $af$. 
This means that $d\le 4$. 
The subcases with $d\le 2$ are trivial since
the action on the corresponding 
Grassmannian is transitive. 
If $\ell$ is odd, then
the $\PGL_2$-action on $\Gr(2,V)\times \P(V_{\ell})$ 
has a rational slice and our claim follows. 


If $d=3$, the action of $\PGL_2$ on $\Gr(2,V_3)$
has a $(\PGL_2, \sA_4)$-slice $\P^1$.
For even $\ell>0$ the action of $\sA_4$ on $\P^{\ell}$ is faithful and it 
lifts to a linear representation of
of $\sA_4$. Further, $\sA_4$-acts on $\P^1$ is through
a cyclic quotient. 
Thus 
$$
(\P^1\times \P(V_\ell)\sim_{\sA_4} \P^1\times \P(V_\ell)
$$
with trivial $\sA_4$-action on the $\P^1$ on the right.
This implies that the quotient is 
equivalent to
$$
\P^1\times (\P^{\ell}/\sA_4) \times (V_1\oplus V_1)/ \C^*\times \bH,
$$   
a product of rational varieties.


If $d=4$, the action of $\PGL_2$ on $\Gr(2,V_4)$
has a $(\PGL_2, \bN_{\bT})$-slice
$X'$. The action of $\bN_{\bT}$ on $\P(V_{\ell})$ 
is linear and the quotient
of $X\times \P^{\ell}$ is a vector bundle over the quotient of 
$X$, which is rational.
\end{proof}


\begin{prop}
\label{prop:xxx}
Let $X=(\stackrel{L}\longrightarrow Y)^2 $, where $Y=
\prod_{j\in J} \P(V_{d_j})$ and  $\ell >0$. 
Then 
$$
\GL_2\ba X\times \P(V_{\ell})/\sS_2
$$
is rational (where $\sS_2$ acts trivially on $\P(V_{\ell})$ and
by permutation on $X$).
\end{prop}


\begin{proof}
The same argument as in the proof of Proposition~\ref{prop:syml}
shows that it suffices to assume that the action on  
$X$ is not $af$. This happens only if $Y=\P^2$ or $\P^1$.
The case $Y=\P^2$ reduces to Proposition~\ref{prop:syml}
(Grassmannian). 
If $Y=\P^1$ then the action of $\PGL_2$ on $\P^1\times \P^1$ is
transitive and 
$$
\GL_2\ba X\times \P(V_{\ell})/\sS_2 
\sim  (\C^*\ba \P(V_{\ell}))\times (\C^2/\C^*\times \sS_2),
$$
a rational variety. 
\end{proof}






\section{Special rationality results}
\label{sect:rat-sp}


In this section we collect rationality results
for  spaces of rational maps $\Pr^1\ra \Pr^1$ with prescribed 
(special) ramification over exactly three distinguished
points $(0,1,\infty)$ and unspecified ramifications over other points.  

Let $\cR(r_0,r_1,r_{\infty})$
be the space of rational maps $f\,:\, \Pr^1\ra \Pr^1$
with local ramification data (vectors)
$r_0,r_1,r_{\infty}$ over the points $0,1,\infty$. 


\begin{prop}
\label{prop:rrr}
Assume that $(r_0,r_1,r_{\infty})$ satisfies one of the following:
\begin{itemize}
\item all entries of the vectors $r_0,r_\infty$ are even 
and some fixed number of entries of $r_1$ is even; 
\item all entries of the vectors $r_0,r_{\infty}$ are even
and a fixed number of entries of $r_1$ is divisible by $3 ;$ 
\item all entries of the vectors $r_0,r_{\infty}$ are divisible by $3$
and all entries of $r_1$ are even.
\end{itemize}
Then $\cR(r_0,r_1,r_{\infty})$ is a finite union of 
irreducible rational varieties.
\end{prop}

\begin{proof}
In these cases the map $f=f_0/f_\infty$ is given by coprime polynomials
satisfying the equations:
\begin{itemize}
\item $f_0^2-f_{\infty}^2=g_1^2g_1'$;  
\item $f_0^2-f_{\infty}^2=g_1^3g_1'$; 
\item $f_0^3-f_{\infty}^3=g_1^2g_1'$, 
\end{itemize}
where $g_1'$ is an arbitrary polynomial. 
The first equation leads to 
$$
(f_0-f_{\infty})(f_0+f_{\infty})=g_1^2g_1'
$$
and, by coprimality, to  
$$
\begin{array}{ccc}
f_0-f_{\infty} & = & g_{11}^2g_{11}', \\
f_0+f_{\infty} & = & g_{12}^2g_{12}',
\end{array}
$$
with arbitrary $g_{11},g_{11}',g_{12},g_{12}'$
(satisfying the obvious degree conditions) --- a union of rational
varieties. 

The second case is analogous. 
Consider the third case:
since $f_0^3-f_\infty^3$ is a square we obtain 
$$
\begin{array}{ccc}
f_0- f_\infty          & =  &  g_1^2\\
f_0-\zeta f_\infty     & =  &  g_2^2\\
f_0-\zeta^2 f_{\infty} & =  &  g_3^2
\end{array}
$$ 
(where $\zeta^3=1$)
and  we need to solve 
$$
\frac{2\zeta}{1+\zeta} g_1^2 + \frac{1-\zeta}{1+\zeta} g_2^2 = g_3^2.
$$ 
Now we apply the parametrization 
as above. 
\end{proof}

\begin{coro}
\label{coro:rr-r}
Let $\cR(r_0,r_1,r_{\infty})$ be as in \ref{prop:rrr}. 
Then 
$$
\PGL_2\backslash \cR(r_0,r_1,r_{\infty})
$$
is rational.
\end{coro}

\begin{proof}
We have established an explicit parametrization
of $\cR(r_0,r_1,r_{\infty})$ 
as a direct sum of spaces of polynomials (with different weights
as irreducible $\GL_2$-representations). 
By the theorem of Katsylo \ref{thm:katsylo}, the corresponding quotients are rational.
\end{proof}

\begin{rem}
\label{rem:8.3}
Only the first case with 
$g_1'=1$ can admit a nontrivial action of $H_{\Ga}$ 
(which necessarily is $\Z/3$). 
But even in this case the action of $\Z/3$ is linear
and it commutes with the action of $\GL_2$ on pairs of
polynomials. Lemma~\ref{lemm:grr} implies 
rationality.
\end{rem}


\begin{lemm}\label{thG}  
Every irreducible component of the variety  $\cR$ of rational maps 
$f\,:\, {\Pr}^1\to {\Pr}^1$ of degree $5$ and prescribed
global ramification datum
$$
\RD(f)=[(2,2,1)_0,(2,2,1)_1,(2,2,1)_{\infty},(2),(2)]
$$
is rational.
\end{lemm}

\begin{proof} 
Changing the variables 
(fixing two ramification points over $1\in{\Pr}^1$ as $0,\infty$),
we can write  $f=F_1/F_2$ where 
$$
\begin{array}{ccc}
F_1(x) & = & \hat{f}_1(x)^2\hat{a}_1(x)^2 \hat{b}_1(x)\\
F_2(x) & = & \hat{f}_2(x)^2\hat{a}_2(x)^2 \hat{b}_2(x)
\end{array}
$$
where 
$\hat{f}_1,\hat{f}_2,\hat{a}_1,\hat{a}_2,\hat{b}_1,\hat{b}_2$ 
are linear forms in $x$.
Since the leading coefficients of $F_1$ and $F_2$ are equal 
we can assume that they are both equal to $1$ and write
$\hat{f}_1(x)=x+f_1,\dots,\hat{b}_2(x)=x+b_2$,
with some nonzero constants $f_1,\dots,b_2$. 
Since we have one free parameter (under the action of $\PGL_2$)
we can assume that $b_1=1$.  
Thus 
$$
\hat{f}_1(x)^2\hat{a}_1(x)^2\hat{a}_2(x)-
\hat{f}_2(x)^2\hat{b}_1(x)^2 \hat{b}_2(x) =\sum_i g_ix^i= c_1 x^2(x+c_2)
$$ 
with arbitrary constants $c_1,c_2$. 
We get a system of equations on the coefficients $g_j$: 
$$
g_4=0, g_1=0, g_0=0.
$$ 
Remark that the coefficients of $g$ are
symmetric functions on pairs $(f_1,a_1)$ and $(f_2,a_2)$. 
To parametrize $\cR$ we introduce  the following variables: 
$$
X_1=a_1+f_1,\, Y_1=a_1f_1,\, X_2=f_2+a_2,\, Y_2=f_2a_2,\, b_1,\,b_2.
$$ 
Write the equations on the coefficients $g_j$ as
$$
\begin{array}{rcl}
2X_1+b_1         & = & 2X_2+b_2\\
Y_1^2b_1         & = & Y_2^2b_2\\
Y_1^2+2X_1Y_1b_1 & = & Y_2^2+2X_2Y_2b_2.
\end{array}
$$
Since  $b_1=1$, for a fixed $b_2$  we get 
$$
\begin{array}{rcl}
2X_1+1                 & = & 2X_2+b_2\\
Y_1                    & = & \pm \sqrt{b_2}Y_2\\ 
b_2Y_2 +2\sqrt{b_2}X_1 & = & Y_2 +2X_2b_2.
\end{array}
$$
This is a union of two (affine) lines.  
After a rational covering $(\sqrt{b_2})$ our surface
is (rationally) a $\Pr^1$-bundle over $\Pr^1$, a rational
surface. 
\end{proof}





\begin{lemm}
\label{lemm:4}
Every irreducible component of the variety $\cR$ of rational maps
$f\,:\, \Pr^1\to \Pr^1$ of degree $4$ and ramification datum
$$
\RD(f)=[(2,2)_0,(2,1,1)_1,(2,1,1)_{\infty}]
$$ 
is a rational surface.
\end{lemm}

\begin{proof}
Using the $\PGL_2$-action on the preimage 
$\Pr^1$ we can assume that the
points $(2,2)$ are $+1,-1$, respectively,  and
that the point of degree $2$
(in the local ramification datum $(2,1,1)$) over $0$ is $\infty$. 
Thus we can write
$$
(x^2-1)^2 - c(x+c_1)(x+c_2)(x+c_3)^2=g_2(x),
$$
where $g_2$ is an arbitrary  
polynomial of degree 2 and $c$ is some constant. 
We get two equations
$$
c=1,
$$
$$
c_1+c_2+2c_3=0.
$$
Thus we have a (rational) surjection of $\Pr^2$ onto $\cR$.
\end{proof}



\begin{lemm}
\label{lemm:5}
Every irreducible component of the 
variety  $\cR$ of  rational maps $f\,:\, \Pr^1\to \Pr^1$ 
of degree $4$ with ramification datum 
$$
\RD(f)=[(2,2)_0,(3,1)_{1},(2,1,1)_{\infty},(2),(2)]
$$ 
is a rational curve.
\end{lemm} 

\begin{proof}
A generic map with this ramification datum 
is given by the equation $f=f_1/f_2$, 
where
$$
f_1=(x^2-1)^2,\,\, f_2=(x+c_1)(x+c_2)(x+c_3)^2
$$ 
and 
$$
f_1-f_2=(x^2-1)^2-c(x+c_1)(x+c_2)(x+c_3)^2=g_1(x),
$$
where $g_1(x)$ is linear. Thus $c=1$ and 
$$
c_1+c_2+2c_3=0,
$$
$$
c_1c_2+2c_1c_3 + 2c_2c_3 + c_3^2 =0,
$$
clearly rational.
\end{proof}



\begin{lemm}
\label{lemm:6}
The irreducible component of the 
variety  $\cR$ of  rational maps $f\,:\, \Pr^1\to \Pr^1$ 
of degree $3$ with ramification datum 
$$
\RD(f)=[(2,1)_0,(2,1)_{1},(2,1)_{\infty},(2)]
$$ 
is a rational curve.
\end{lemm} 

\begin{proof}
Reduces easily to the rationality of a cuspidal cubic curve.
\end{proof}







\section{Rationality of moduli}
\label{sect:ratio}

\begin{thm}
\label{thE}
Any connected component of a moduli space of rational or 
K3 elliptic surfaces with fixed monodromy group is rational.
\end{thm}

\begin{proof}
In Proposition~\ref{lemm:struct}
we have identified (Zariski open subsets of) the 
corresponding moduli spaces
$\cF_{r,\tilde{\Ga}}$ 
as quotients 
(by the left $\PGL_2$ and right $H_{\Ga}$-action) 
$$
\PGL_2\ba \cU_{r,\tilde{\Ga},\ell}'/H_{\Ga}.
$$
Here 
$$
\cU_{r,\tilde{\Ga},\ell}'\sim_{\PGL_2\times H_{\Ga}} 
\Sym^{\ell}(\P^1)\times \cR_{\Ga}
$$ 
and 
$$
\cR_{\Ga}=\{ f\,:\, \P^1\ra \P^1\}
$$
is the space of rational maps 
(with prescribed ramification).  
For elliptic rational or K3 surfaces $\ell\le 3$
and $H_{\Ga}$ is either trivial, cyclic, dihedral or
a subgroup of $\sS_4$ (see Corollary~\ref{coro:hgamma}).
The actions if $\PGL_2$ and $H_{\Ga}$  commute and $H_{\Ga}$
acts only on $\cR_{\Ga}$.  


\

First we consider {\em general families}:
$$
\ET(\E)-12\ell =\deg(j_{\E})\ET(\Ga).
$$
For ${\bf d}=(d_1,...,d_k)\in \N^k$ we put
$$
\P^{\bf d}:= \prod_{j=1}^k \P(V_{d_j}).
$$
Recall that $\cR_{\Ga}$ is (birationally) 
the total space of a line bundle over the space
$$
\P^{\bf d}\times \P^{\bf d'},
$$
where $\sum_{j=1}^k d_j=\sum_{j=1}^{k'} d_j'$.

\

{\bf Case 1.} ${\bf d}\neq {\bf d}'$. Then, by 
\ref{coro:hgamma}, 
$H_{\Ga}=1$ and rationality of $\PGL_2\ba \cR_{\Ga}$ 
(in all cases) follows from the rationality of
$$
\PGL_2\ba \P^{\bf d}\times \P^{\bf d'},
$$
which is the theorem of Katsylo \ref{thm:katsylo}. 



\

{\bf Case 2.} ${\bf d}={\bf d}'$ and $k\ge 2$.
By Corollary~\ref{coro:hgamma}, 
$H_{\Ga}=\sS_2$ (permutation of the factors).   
This case is covered by Proposition~\ref{prop:yz}.

\

{\bf Case 3.} ${\bf d}={\bf d}'=(d)$. This case is covered by 
Proposition~\ref{prop:even}.


\




Now we discuss the {\em special families}:
$$
\ET(\E)-12\ell <\deg(j_{\E})\ET(\Ga).
$$
We use the classification of these families established in 
Section~\ref{sect:ellmono}.
All families listed in Lemma~\ref{lemm:24-sp} are covered by 
Propositions~\ref{prop:yz} and the Theorem~\ref{thm:katsylo}.
Consider the families listed in Lemma~\ref{lemm:table-tree}:
Lemma~\ref{lemm:subgroup} covers the cases 
$j_1,j_4,j_5,j_6,j_{13}$. 
The case $j_2,j_8$ and $j_{12}$ are covered by Proposition~\ref{prop:rrr}, 
$j_3$ by Lemma~\ref{lemm:5}, $j_7,j_9,j_{10}$ by \ref{prop:rrr} and
\ref{rem:8.3}, $j_{11}$ by Lemma~\ref{lemm:4}. The case $j_{14}$ is
covered by Lemma~\ref{lemm:6}.
Finally, the families $j_{15}$ and $j_{16}$ 
(listed in Lemma~\ref{lemm:table-rat}) are
covered by Proposition~\ref{prop:yz} and the remaining
families $j_{17}-j_{20}$ by Theorem~\ref{thm:katsylo}.
\end{proof}


\begin{rem}
Our methods extend to some moduli spaces of 
elliptic surfaces with higher Euler characteristic. 
In particular, the results of Section~\ref{sect:rat-sp} 
imply that any moduli space of Jacobian elliptic surfaces
over $\Pr^1$ such that a generic surface in this space 
has only singular fibers of multiplicative type is rational. 
However, we expect that there are nonrational 
moduli spaces already for 
Euler characteristic 36. 
\end{rem}


\section{Pictures}
\label{sect:pictures}

In this section we give a combinatorial description of
monodromy groups of elliptic K$3$ surfaces. 
More precisely, we describe a simple procedure which allows
to enumerate all the possible graphs $\Ga$ 
with given $\ET(\Ga)$.
Let $\E\ra \Pr^1$ be an elliptic K$3$ surface. 
We have shown in Section~\ref{sect:comb}
that 
$$
48=\ET(\E)\geq\ET(\Ga)
$$
and that $\ET(\Ga)$ is divisible by 12.
Thus  $\ET(\Ga)$ equals $12, 24, 36$ or $48$
and all possible $\Ga\subset \PSL_2(\Z)$ 
are described by connected trivalent graphs
$T_{\Ga}$ with $\leq 8$ edges embedded into 
${\mathbb S}^2$, with an arbitrary bicoloring of the ends.  

\

\no
{\bf Case} $\ET(\Ga)=12$ :
There is only one tree $T_{12}$ with $ \ET(T_{12})=12$
\begin{figure}[htb]
\centerline{\includegraphics[width=.15\textwidth]{tree12.eps}}
\caption{The tree $T_{12}$.}
\end{figure}

The ends of $T_{12}$ can be either $A$ or $B$-vertices. 
To obtain all possible graphs $T_{\Ga}$ 
with $\ET(\Ga)=12$ we just need to attach 
to $T_{12}$ a single loop $L$.
\begin{figure}[htb]
\centerline{\includegraphics[width=.2\textwidth]{loop.eps}}
\caption{The loop $L$.}
\end{figure}

This gives the  following list of graphs:
\begin{figure}[htb]
\centerline{\includegraphics[width=.5\textwidth]{et12_1.eps}}
\caption{The case $\ET(\Ga)=12$.}
\end{figure}

%\pagebreak
There is only one saturated graph from the list above which has no
outer loops (Figure $4$).
\begin{figure}[htb]\label{fig:sat}
\centerline{\includegraphics[width=.15\textwidth]{sat12.eps}}
\caption{}
\end{figure}

This graph will be a basic building block 
in the construction of graphs with $\ET(\Ga)>12$ - 
we will attach trees and loops to its edges. 
The edges are numbered to simplify the
count of all possible outcomes.


\pagebreak

 
\no
{\bf Case} $\ET(\Ga)=24$: 
Again, we have only one topological tree $T_{24}$ with
$\ET(T)=24$:
\begin{figure}[htb]
\centerline{\includegraphics[width=.1\textwidth]{tree24.eps}}
\caption{The tree $T_{24}$.}
\end{figure}
\

\no
{\bf Case} $\ET(\Ga)=36$: 
There are only $3$ saturated graphs without end-loops (modulo
equivalent embedding into the sphere):
\begin{figure}[htb]
\centerline{\includegraphics[width=.6\textwidth]{36.eps}}
\caption{The case $\ET(\Ga)=36$.}
\end{figure}

Any other graph is either a 
tree or a sum of a saturated graph $T'$
with $\ET(T')=0,12,24$ with trees (with complementary
$\ET$). 
There is only one topological tree $T_{36}$ with
$\ET(T_{36})=36$.
\begin{figure}[htb]
\centerline{\includegraphics[width=.2\textwidth]{tree36.eps}}
\caption{}
\end{figure}  

The number of possible markings of the tree or 
loops at the ends is $81$ but due to the 
symmetry of the graph the actual number of graphs
$T_{\Ga}$ corresponding to different placement 
of loops at the end and markings is smaller: 
there are $34$ different $T_{\Ga}$ of this type.

The number of markings of $T_{36}$ 
is $16$ but due to its symmetry 
the number of different graphs $T_{\Ga}$ is $7$. 
(Recall that two graphs $T_{\Ga}$ give the same $\Ga$ 
modulo conjugation if they are isotopic in 
a ${\mathbb S}^2$).

The graphs of tree type with one end loop are topologically
equivalent to:

%\begin{figure}[htb]
\centerline{
\includegraphics[width=.2\textwidth]{tree36loop.eps}}

%\caption{}
%\end{figure}  

\

There are $8$ possible markings of the above graph and they all
give different $T_{\Ga}$ with $\ET(\Ga)=36$. We have $12$
different $T_{\Ga}$ with $2$ end-loops, $6$ with $3$ 
end-loops and one with $4$ end-loops.

All topological graphs which 
are sums of a loop and a tree can be
obtained by placing a loop into a tree. 
Thus there are two types:
\begin{figure}[htb]
\centerline{\includegraphics[width=.35\textwidth]{tree36top.eps}}
\caption{}
\end{figure}  

This gives $8$ graphs $T_{\Ga}$ in the 
first case and $4$ in the second case. 

\


\no
{\bf Case} $\ET(\Ga)=48$:
We have one tree $T_{48}$ with $\ET(\Ga)=48$:
\begin{figure}[htb]
\centerline{\includegraphics[width=.2\textwidth]{tree48.eps}}
\caption{The tree $T_{48}$.}
\end{figure}

Here is the list of all saturated graphs with $\ET(\Ga)=48$.
\begin{figure}[htb]
\centerline{\includegraphics[width=.6\textwidth]{48.eps}}
\caption{Saturated graphs in the case $\ET(\Ga)=48$.}
\end{figure}



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\end{document}



math0205124.extracted_bib
\begin{thebibliography}
\frenchspacing
  
\bibitem{Alex} J. Alexander, {\em Note on Riemann spaces}, Bull.  AMS
  {\bf 26}, 370--372, (1920).


\bibitem{am} M. Artin, D. Mumford, {\em Some elementary examples of 
unirational varieties which are nor rational}, Proc. Lond. Math. Soc.
{\bf 25}, 75--95, (1972). 
  
\bibitem{Bart} W. Barth, C. Peters, A. Van de Ven, {\em Compact
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