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%                                  setting for  latex file
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%                               Realize the Double Spacing 
\renewcommand{\baselinestretch}{1.3}
\newcommand{\single}{}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%                                    % theorem envir.
\newtheorem{dfn}{Definition}
\newtheorem{clm}{Claim}
\newtheorem{exm}{Example}          
% not actually used 
%% % % % % % % % % % % % % % % % % % % % % % % % % % % 
%                                 Greek
\newcommand{\la}{\lambda}
\newcommand{\La}{\Lambda}
\newcommand{\sg}{\sigma}
\newcommand{\Sg}{\Sigma}
\newcommand{\Ga}{\Gamma}
\newcommand{\ep}{\epsilon}
\newcommand{\al}{\alpha}
\newcommand{\g}{\varphi}
%% % % % % % % % % % % % % % % % % % % % % % % % % % % 
%                                   bold-vector step 1
%- - - - - - - - - - - - - - - - - - - - - - - - - - 
\newcommand{\vecv}{{\boldmath $v$}}
\newcommand{\vecx}{{\boldmath $x$}}
\newcommand{\vecy}{{\boldmath $y$}}
\newcommand{\vecz}{{\boldmath $z$}}
\newcommand{\vece}{{\boldmath $e$}}
\newcommand{\vecr}{{\boldmath $r$}}
\newcommand{\vecm}{{\boldmath $m$}}
\newcommand{\veca}{{\boldmath $a$}}
\newcommand{\vecb}{{\boldmath $b$}}
\newcommand{\vecc}{{\boldmath $c$}}
\newcommand{\vecd}{{\boldmath $d$}}
\newcommand{\vecf}{{\boldmath $f$}}
\newcommand{\vecg}{{\boldmath $g$}}
\newcommand{\vech}{{\boldmath $h$}}
\newcommand{\veci}{{\boldmath $i$}}
\newcommand{\vecj}{{\boldmath $j$}}
\newcommand{\veck}{{\boldmath $k$}}
\newcommand{\vecn}{{\boldmath $n$}}
\newcommand{\vecl}{{\boldmath $l$}}
\newcommand{\vecp}{{\boldmath $p$}}
\newcommand{\vecq}{{\boldmath $q$}}
\newcommand{\vecs}{{\boldmath $s$}}
\newcommand{\vect}{{\boldmath $t$}}
\newcommand{\vecu}{{\boldmath $u$}}
\newcommand{\vecw}{{\boldmath $w$}}
\newcommand{\vecal}{{\boldmath $\al$}}
\newcommand{\vecbeta}{{\boldmath $\beta$}}
\newcommand{\vecgamma}{{\boldmath $\gamma$}}
\newcommand{\vecdelta}{{\boldmath $\delta$}}
\newcommand{\vecep}{{\boldmath $\ep$}}
\newcommand{\veczeta}{{\boldmath $\zeta$}}
\newcommand{\veceta}{{\boldmath $\eta$}}
\newcommand{\vectheta}{{\boldmath $\theta$}}
\newcommand{\veciota}{{\boldmath $\iota$}}
\newcommand{\veckappa}{{\boldmath $\kappa$}}
\newcommand{\vecla}{{\boldmath $\la$}}
\newcommand{\vecmu}{{\boldmath $\mu$}}
\newcommand{\vecnu}{{\boldmath $\nu$}}
\newcommand{\vecxi}{{\boldmath $\xi$}}
\newcommand{\vecpi}{{\boldmath $\pi$}}
\newcommand{\vecpho}{{\boldmath $\pho$}}
\newcommand{\vecsigma}{{\boldmath $\sigma$}}
\newcommand{\vectau}{{\boldmath $\tau$}}
\newcommand{\vecupsilon}{{\boldmath $\upsilon$}}
\newcommand{\vecphi}{{\boldmath $\phi$}}
\newcommand{\vecchi}{{\boldmath $\chi$}}
\newcommand{\vecpsi}{{\boldmath $\psi$}}
\newcommand{\vecomega}{{\boldmath $\omega$}}
%                                     % step 2
%- - - - - - - - - - - - - - - - - - - - - - - - - - 
\newcommand{\va}{\text{\veca}}
\newcommand{\vb}{\text{\vecb}}
\newcommand{\vc}{\text{\vecc}}
\newcommand{\vd}{\text{\vecd}}
\newcommand{\ve}{\text{\vece}}
\newcommand{\vf}{\text{\vecf}}
\newcommand{\vg}{\text{\vecg}}
\newcommand{\vh}{\text{\vech}}
\newcommand{\vi}{\text{\veci}}
\newcommand{\vj}{\text{\vecj}}
\newcommand{\vk}{\text{\veck}}
\newcommand{\vl}{\text{\vecl}}
\newcommand{\vm}{\text{\vecm}}
\newcommand{\vn}{\text{\vecn}}
\newcommand{\vo}{\text{\veco}}
\newcommand{\vp}{\text{\vecp}}
\newcommand{\vq}{\text{\vecq}}
\newcommand{\vr}{\text{\vecr}}
\newcommand{\vs}{\text{\vecs}}
\newcommand{\vt}{\text{\vect}}
\newcommand{\vu}{\text{\vecu}}
\newcommand{\vv}{\text{\vecv}}
\newcommand{\vw}{\text{\vecw}}
\newcommand{\vx}{\text{\vecx}}
\newcommand{\vy}{\text{\vecy}}
\newcommand{\vz}{\text{\vecz}}
\newcommand{\val}{\text{\vecal}}
\newcommand{\vbeta}{\text{\vecbeta}}
\newcommand{\vgamma}{\text{\vecgamma}}
\newcommand{\vdelta}{\text{\vecdelta}}
\newcommand{\vep}{\text{\vecep}}
\newcommand{\vzeta}{\text{\veczeta}}
\newcommand{\veta}{\text{\veceta}}
\newcommand{\vtheta}{\text{\vectheta}}
\newcommand{\viota}{\text{\veciota}}
\newcommand{\vkappa}{\text{\veckappa}}
\newcommand{\vla}{\text{\vecla}}
\newcommand{\vmu}{\text{\vecmu}}
\newcommand{\vnu}{\text{\vecnu}}
\newcommand{\vxi}{\text{\vecxi}}
\newcommand{\vpi}{\text{\vecpi}}
\newcommand{\vrho}{\text{\vecrho}}
\newcommand{\vsigma}{\text{\vecsigma}}
\newcommand{\vtau}{\text{\vectau}}
\newcommand{\vupsilon}{\text{\vecupsilon}}
\newcommand{\vphi}{\text{\vecphi}}
\newcommand{\vchi}{\text{\vecchi}}
\newcommand{\vpsi}{\text{\vecpsi}}
\newcommand{\vomega}{\text{\vecomega}}
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%                             % black boad bold
\newcommand{\C}{{\Bbb C}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\bP}{{\Bbb P}}
\newcommand{\N}{{\Bbb N}}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
% kahler moduli space
\newcommand{\kahler}{{\Bbb K}}
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%                              the D-brane moduli space
\newcommand{\Mod}{{\cal M}(\vr)}
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%                            GIT quotient symbol
\newcommand{\GIT}{/\negthickspace/}
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\newcommand{\CY}{{\C}^d/\Ga}  
% Calabi--Yau orbifold
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %  
%                 total number of facets
\newcommand{\nfacets}{|\La|}
%                 set of facets
\newcommand{\facetset}{\La}
%              a-th facet, b-th facet, \mu-th facet
\newcommand{\athfacet}{{\cal F}_{a}}
\newcommand{\bthfacet}{{\cal F}_{b}}
\newcommand{\muthfacet}{{\cal F}_{\mu}}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%           $\Ga$-orbit of the facets of $P$ 
\newcommand{\facetsetb}{\overline{\La}}
\newcommand{\ab}{\overline{a}}
\newcommand{\bb}{\overline{b}}
\newcommand{\cb}{\overline{c}}
\newcommand{\mub}{\overline{\mu}}
\newcommand{\oneb}{\overline{1}}
\newcommand{\twob}{\overline{2}}
\newcommand{\threeb}{\overline{3}}
\newcommand{\fourb}{\overline{4}}
\newcommand{\config}{configuration}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%            lattices for quotient toric
\newcommand{\Nb}{\overline{N}}
\newcommand{\Mb}{\overline{M}}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%  weight vector, primitive vector for quotient
\newcommand{\vwb}{\overline{\vw}}
\newcommand{\vvb}{\overline{\vv}}
\newcommand{\veb}{\overline{\ve}}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %%
%                                affine toric (n-1+d)-fold
\newcommand{\A}{{\cal A}}  
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%                                  complex torus
\newcommand{\ct}{({\C}^{*})}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%                                  Kahler
\newcommand{\Ka}{K\"ahler\ }
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%                                   regular representation
%\newcommand{\reg}{R}
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%
\newcommand{\what}{\widehat}
\newcommand{\Qr}{Q(\what{\vr})}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\newcommand{\amu}{a_{\mu}}
\newcommand{\anu}{a_{\nu}}         
\newcommand{\tM}{\widetilde{M}}
\newcommand{\ra}{\rightarrow}
\newcommand{\piQ}{\pi_{\Q}}
%\newcommand{\i*Q}{i^*_{\Q}}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%           various lattices associated with $\A$
\newcommand{\Mtot}{M^{(0)}}
\newcommand{\Msub}{M^{(1)}}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\newcommand{\cone}{C_{\text{basic}}}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\newcommand{\pos}{\mathop{\mathrm{cone}}}
\newcommand{\conv}{\mathop{\mathrm{conv}}}
\newcommand{\rec}{\mathop{\mathrm{rec}}}
\newcommand{\Spec}{\mathop{\mathrm{Spec}}}
\newcommand{\Proj}{\mathop{\mathrm{Proj}}}
\newcommand{\Ker}{\mathop{\mathrm{Ker}}}
\newcommand{\Hom}{\mathop{\mathrm{Hom}}}
\newcommand{\Aut}{\mathop{\mathrm{Aut}}}
%\newcommand{\dim}{\mathop{\mathrm{dim}}}
\newcommand{\codim}{\mathop{\mathrm{codim}}}
\newcommand{\Sym}{\mathop{\mathrm{Sym}}}
\newcommand{\hilb}{\mathrm{Hilb}}  
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\newcommand{\Vect}{{\Bbb W}}
\newcommand{\skima}{\thinspace}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
%                     $\Gamma$-Hilbert scheme
\newcommand{\Hilb}{\hilb^{\Ga}({\C}^{d})}
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
% Hilb. of length $n$ subscheme
\newcommand{\Hn}{\hilb^n({\C}^d)}  
%% % % % % % % % % % % % % % % % % % % % % % % % % % % 
\newcommand{\susy}{supersymmetry}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%    Kahler parameter for Hilb scheme
\newcommand{\kahilb}{r}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\numberwithin{equation}{section}   
%   (2.1)     section2, number 1 
%% % % % % % % % % % % % % % % % % % % % % % % % % % %
% footnote symbol    *, \dagger, . . .
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
%% % % % % % % % % % % % % % % % % % % % % % % % % % %
\begin{document}
%% % % % % % % % % % % % % % % % % % % % % % % % % % %
%\bibliographystyle{unsrt}
%% % % % % % % % % % % % % % % % % % % % % % % % % % %
\begin{center}
{\large
{\it Preliminary Version, \ June 1998}}
\end{center}
%
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%\begin{flushright}
%Preliminary version\\
% April 1998\\
%{ hep-th/9806052}
%\end{flushright}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\vspace{2cm}

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\begin{center}
%
{\LARGE {\bf  \Ka Moduli Space for a D-Brane 
\vspace{0.5cm}

at Orbifold Singularities}}
\vspace{0.5cm}

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\begin{center}
{\large {\boldmath $Dedicated$ $to$ $the$ $memory$ $of$ $a$ $cat$}}
%                   not a joke
\end{center}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\vspace{0.7cm}

{\large Kenji  \ Mohri}%\footnote[2]{mohri@@theory.kek.jp}
\vspace{0.5cm}

{\it Theory Group, Institute of Particle and Nuclear Studies},

{\it High Energy Accelerator Research Organization (KEK)},

{\it Oho~1-1 Tsukuba, Ibaraki 305-0801, Japan}

mohri@@theory.kek.jp

% phone~: 81-298-64-5400

% fax~: 81-298-64-5755

\end{center}
\vspace{4cm}

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\begin{abstract}
We develop a method to analyze systematically 
the configuration space of a D-brane 
localized at the orbifold singular point 
of a Calabi--Yau $d$-fold of the form ${\Bbb C}^d/\Gamma$
using the theory of toric quotients.
This approach elucidates 
the structure of the K\"ahler moduli space associated with
the problem. 
As an application, we compute the toric data of 
the $\Gamma$-Hilbert scheme.
\end{abstract}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\vfill
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\begin{flushright}
hep-th/9806052
\end{flushright}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\single

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\section{Introduction}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %



%   %   %   %   %   %   %   %   %   %   %   %   %   %   %   %
The \config\ space of a D-brane localized 
at the orbifold singularity of a Calabi--Yau $d$-fold
of the form $\CY$, 
where $\Ga$ is a finite subgroup of $\text{SU}(d)$, 
is an interesting object to study,
because it represents the ultra-short distance geometry
felt by the D-brane probe \cite{DKPS}, 
which may be different from the geometry of bulk string.
%   %   %   %   %   %   %   %   %   %   %   %   %   %   %   %   
On the mathematical side,
the D-brane \config\ space corresponds
to a generalization 
of the Kronheimer construction of 
the ADE type hyper-\Ka manifolds \cite{kronheimer}
to higher dimensions, which has been studied 
by Sardo Infirri \cite{sardo,infirri}. 
%
He has shown that the D-brane \config\ space 
is a blow-up of the orbifold $\CY$, 
the topology of which depends 
on the \Ka (or Fayet--Iliopoulos) moduli parameters~;
Moreover he has conjectured that for $d=3$,
the D-brane \config\ space is a smooth Calabi--Yau
three-fold for a generic choice of the \Ka moduli parameters.
%
The case in which $\Ga$ is Abelian is 
of particular importance,
because then the \config\ space is a toric variety,
which enables us to employ various methods 
of toric geometry to study it.
%   %   %   %   %   %   %   %   %   %   %   %   %   %   %   %
Using toric geometry,
several aspects of the D-brane \config\ space 
have been studied so far
\cite{DG,DGM,DM,greene,mohri,MR,muto,ray}.
%   %   %   %   %   %   %   %   %   %   %   %   %   %    %

%   %   %   %   %   %   %   %   %   %   %   %   %   %   %   % 
Our aim in this article is to give a method 
to analyze systematically 
the structure of the \Ka moduli space associated with
the D-brane \config\ space
which releases one from the previous brute force calculations, 
for example see \cite[(53--74)]{mohri}. 
It turns out that the theory of toric quotients 
developed by Thaddeus \cite{thaddeus} provides us with 
the most powerful tool to investigate 
the D-brane \config\ space.
This approach has already been taken in \cite{infirri},
where the analysis of the toric data
is reduced to the network flow problem  
on the McKay quiver defined by the orbifold.

%  %  %  %  %  %  %  %  
To save the notation, we consider
only cyclic groups for $\Ga$, 
but the generalization to an arbitrary Abelian group,
that is, a product of several cyclic groups,
should be straightforward.
\vspace{0.5cm}
%  %  %  %  %  %  %  %   %  %  %  %  %  %  %  %   

%   %   %   %   %   %   %   %   %   %   %   %   %   %   
The organization of this article is as follows~:

%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
In section 2, we explain in detail 
the construction by Thaddeus \cite{thaddeus}
of quasi-projective toric varieties and 
their quotients by subtori 
in terms of rational convex polyhedra.
This formulation gives us a clear picture of 
the \Ka moduli space 
associated with a toric quotient \cite{KSZ,thaddeus}.

%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
In section 3, we describe the \config\ space of a D-brane 
localized at the orbifold singularity as a toric variety
obtained by a toric quotient of an affine variety
closely following the treatment 
by Sardo Infirri \cite{infirri}.
Then we give typical examples
of phases of the D-brane \config\ spaces
for Calabi--Yau four-fold models. 

%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
Section 4 is devoted to 
an application of our construction
of the D-brane \config\ space 
to the $\Ga$-Hilbert scheme 
\cite{ito-nakajima,ito-nakamura,nakajima,nakamura,reid},
which is roughly the moduli space of $|\Ga|$ points
on $\C^d$ invariant under the action of $\Ga$,
in the hope that the investigation 
of various Hilbert schemes
sheds light on the geometrical aspect of D-branes 
on Calabi--Yau varieties \cite{BBMOOY,BVS,OOY}. 
\vspace{0.5cm} 
%-  -  -  -  -  -  -  -  -  -  -  -  -   -  -
%   %   %   %   %   %   %   %   %   %   %   %   %   %   

%   %   %   %   %   %   %   %   %   %   %   %   %   %   
For textbooks or monographs 
dealing with various aspects of toric varieties 
and related topics,
consult \cite{AGV,cox,ewald,fulton,GKZ,oda,sturmfels,ziegler},
as well as the physics articles \cite{AGM,MP,witten1}, 
which contain introductory materials intended for physicists.
%   %   %   %   %   %   %   %   %   %   %   %   %   %   


\single
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\section{Toric Varieties and Its Quotients}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\subsection{Polyhedra and Quasi-Projective Varieties}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
Let $N$ be a lattice of rank $p$ 
and $M=N^{*}$ be the dual lattice.
Let 
$T=\Hom(M,{\C}^*)
\cong N\otimes_{\Z}{\C}^*
\cong \ct^{p}$
be the associated torus.
Then we have the following identification~:
%- - - - - - - - - - - - - - - - - - - -
\begin{align}
M&=\Hom(T,{\C}^*),
\qquad \text{characters of } T,\\
\label{character}
%-  -  -  -  -  -  -  -  -  -  -  -  -  - 
N&=\Hom({\C}^*,T), 
\qquad \text{1-parameter subgroups of } T.
\label{1-parameter}
\end{align} 
%- - - - - - - - - - - - - - - - - - - - -
Let $P$ be a $p$-dimensional convex polyhedron 
%which is {\it bounded below}
in the vector space $M_{\Q}$.
We want to associate a quasi-projective toric variety
to the data $(M,P)$,
which we denote by $X(M,P)$ or simply by $X(P)$
if no confusion occurs.
%in this subsection.


$P$ can be represented 
as an intersection of half-spaces as follows~:
%- - - - - - - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
P=\left\{
\vm\in M_{\Q}\left|\ 
\langle \vm,\vv_a \rangle \geq t_a,\
\forall a\in \facetset 
%\{1,\dots ,A_1\}
\right.
\right\},
\label{convex-polyhedron}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - 
where ${\vv_a}\in N$ and  ${t_a}\in \Q$ and
$\facetset$ is an index set.

For technical reason, 
we put the following assumptions on $P$~:
%  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  
\begin{enumerate}
\item
Each $\vv_a$ is a {\it primitive} vector, that is,
for any integer $n>1$,
$(1/n)\skima \vv_a \not\in N$.
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
\item The expression of $P$ (\ref{convex-polyhedron})
is reduced in the sense that
the omission of the $a$~th inequality 
in (\ref{convex-polyhedron}) gives rise 
to a polyhedron strictly larger than $P$
for any $a\in \La$.
%  -   -  -  -  -  -  -  -  -  -  -  -  -  -  -
\item The vector space defined by  
$\{\vm\in M_{\Q}|\ \langle \vm,\vv_a\rangle=0,
\ \forall a\in \facetset\}$,
which is the maximal vector subspace in $P$,
is equal to  $\{\bold{0}\}$.
\end{enumerate}
%  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  

The $a$~th facet of $P$, 
which we denote by $\athfacet$,
is given by
%- - - - - - - - - - - - - - -
\begin{equation}
\athfacet:=
\left\{
\vm\in P\left|\ 
\langle \vm,\vv_a \rangle=t_a
\right.
\right\},         
\end{equation}
%- - - - - - - - - - - - - - - 
which shows that ${\vv_a}$ is an inner normal vector 
to $P$ at $\athfacet$.
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 


Here let us describe combinatorics of $P$ 
\cite[Lecture 2.2]{ziegler}.

By the {\it face lattice} of $P$, we mean 
the set of all the faces of $P$ 
partially ordered by inclusion
relation, which is denoted by $L(P)$. 
We also denote the proper part of it by
$\overline{L}(P):=L(P)\backslash (\emptyset, P)$. 
For each $F\in \overline{L}(P)$,
we define a subset $I(F)$ of $\facetset$ by
%$\{1,\dots , A_1 \}$ by
%- - - - - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
I(F):=\{a\in \facetset
%\{1,\dots , A_1\}
|\ F\subset \athfacet \},
\end{equation}
%- - - n- - - - - - - - - - - - - - - - - - - - - -
where $\text{card}\ I(F)\geq \codim F$. 
Then each $F\in \overline{L}(P)$ can be represented as
an intersection of facets as follows~:
%- - - - - - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
F=\bigcap_{a\in I(F)}\athfacet.
\label{intersection}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - -
It is also convenient to set formally
$I(P)=\emptyset$,
$I(\emptyset)=\facetset$
%\{1,\dots , A_1\}$
and to regard (\ref{intersection})
valid even for $F=\emptyset, P$. 
Then the intersection $\cap$ of any two elements of 
$L(P)$ can be described in an obvious manner, that is,
%- - - - - - - - - - - - - - - - - - - - - - -- -
\begin{equation}
F_1\cap F_2=\bigcap_{a\in I(F_1)\, \cup \, I(F_2)}
\athfacet.
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - -
Again for $F_1,F_2\in L(P)$,
let $F_1\cup F_2\in L(P)$ be the smallest 
among those which contains both $F_1$ and $F_2$.
The operation $\cup$ is called {\it join}.
We see that for $F_1,F_2\in L(P)$,
%- - - - - - - - -
\begin{equation}
F_1\cup F_2=\bigcap_{a\in I(F_1)\,\cap \, I(F_2)}
\athfacet.
\end{equation}
%- - - - - - - - -








% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%                     Definition I 
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 

Define a rank $(q+1)$ lattice by 
$\tM:=\Z\times M$ 
and define a cone $C(P)$
in $\tM_{\Q}$,
which is called the homogenization of $P$
\cite[Lecture 1.5]{ziegler}, by
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{align}
C(P)&=\text{closure of }
\left\{ \la (1,\vm) \ \left| \ \la\in {\Q}_{\geq 0},
\ \vm\in P  \right.\right\}
\text{ in } \tM_{\Q}, \nonumber \\ 
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&=\left\{ \la (1,\vm)\ \left|\  \la \in {\Q}_{ > 0},
\ \vm \in P \right.
 \right\}
+\{ \bold{0} \}\times \rec P,
\end{align}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
where a Minkowski sum is used in the second line
and $\rec P$ is 
the {\it recession cone} of $P$ defined by
%- - - - - - definition  -  - - - - - - - - - - - - - - - - 
\begin{equation}
\rec P=\left\{\  \vm\in M_{\Q} 
\left|\ \vm^{'}+\la \vm\in P,\ 
\forall \vm^{'}\in P, \ 
\forall \la \in {\Q}_{>0} \right.\right\}.
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
In our case a more concrete expression is possible~:
%- - - - - - - - - - - - - -
\begin{equation}
\rec P
\cong
\left\{
\vm\in M_{\Q}
\left|\ 
\langle \vm,\vv_a\rangle \geq 0, 
\ \forall a\in \facetset
%\{1,\dots ,A_1\}
\right.
\right\}.
\label{alternate}
\end{equation}
%- - - - - - - - - - - - - - - -
%In fact any polyhedron $P$ can be
% represented as a Minkowski sum of
%a polytope and a cone $\rec P$.
%
$C(P)\cap \tM$ has a structure of 
a graded $\rec P$-algebra
graded by its first component, that is,
$(C(P)\cap \tM)_k:=C(P)\cap (\{k\}\times M)$ and
$(C(P)\cap \tM)_0=\rec P$,
which leads us to the following definition of 
$X(P)$ as a quasi-projective variety which is 
projective over an affine variety 
\cite[(2.9)]{thaddeus}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
X(P):=\Proj \left(C(P)\cap \tM\right)
{\longrightarrow}\ 
X_0(P):=\Spec \left(\rec P\right).
\label{Proj}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
Strictly speaking, every scheme $X$ in this article, 
either affine or projective,
should be replaced by the set of 
its $\C$-valued points 
$X(\C):=\Hom_{\C}(\Spec \C, X)$ 
\cite{oda}.  


% % % % % % %    Meaning of the Definition I   % % % % % % % % % %
To be more explicit, we construct $X(P)$
by the following procedure.
First let $(k_1,\vm_1)$, \dots , $(k_s,\vm_s)$
be the generators of 
$C(P)\cap \tM$.
Then we have an embedding %(or a closed immersion)
of $X(P)$ in the weighted projective space
$\bP(k_1,\dots ,k_s)$, where a degree $k_j$ may be 0~;
more precisely, the degree zero generators 
of $C(P)\cap \tM$ are those of $\rec P\cap M$.
The ambient space $\bP(k_1,\dots ,k_s)$ of $X(P)$ 
admits a following symplectic quotient realization~:
%which is also called the D-flatness equation
%in physics terminology~:
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\bP(k_1,\dots ,k_s)=
\left.
\left\{ (z_1,\dots ,z_s)\in  \C^s
\left|
\  \sum_{j=1}^s k_j\ |z_j|^2=1
\right.
\right\}
\right/
\text{U}(1).
\label{ambient}
\end{equation} 
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 

Second let $\psi$ be the lattice surjection {}from
$\Z^s$ to $C(P)\cap \tM$ defined by
%- - - - - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
\psi(\vc):=\sum_{j=1}^s  c_j (k_j,\vm_j).
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - 
Then $\Ker \psi$ is the lattice that represents
the relations between the generators of $C(P)\cap \tM$.
We convert them 
to equations for the homogeneous coordinates
$(z_j)$ of $\bP(k_1,\dots ,k_s)$,
which is called the F-flatness equations in physics terminology~:
%relation between the generators of $C(P)\cap \tM$
%of the form
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
%\begin{equation}
%\sum_{j=1}^{s}c_j\ (k_j,\vm_j)=\bold{0}, \quad c_j\in \Z,
%\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
\prod_{c_j>0}
 z_j^{c_j}
=\prod_{c_j<0}
z_j^{-c_j},
\quad \vc\in \Ker \psi,
\label{F-flat}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
where the degree of $z_j$ is $k_j$.

We now get a symplectic quotient realization of $X(P)$~:
%- - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
X(M,P):=
\left\{
(z_j) \in \C^s 
\left.
\left|
\begin{gathered}
\sum_{j=1}^s k_j\ |z_j|^2=1 \\ 
%\prod_{c_j>0}z_j^{c_j}=\prod_{c_j<0}z_j^{-c_j},
%\quad \vc\in \Ker \phi
\text{F-flatness equations}\  (\ref{F-flat})
\end{gathered}
\right\}
\right.
\right/ \text{U}(1).
\label{X(P)}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - -

% % % % % % % %  affine/projective  variety   % % % % % % % % % % %
If $P$ itself is a polyhedral cone in $M_{\Q}$,
%with an apex at $\bold{0}$,
then  $C(P)\cong {\Q}\times P$ so that
$\Proj \left(C(P)\cap \tM\right)$ is isomorphic to 
$\Spec \left(P\cap M\right)$, that is, 
$X(P)$ is an affine variety.

Another extreme case is when $P$ is a bounded polyhedron, 
that is, polytope. Then $X(P)$ is a projective variety.
%  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %
\begin{flushleft}
{\it Example.}
Let $M=\Z^2$ and 
$P=\pos \{\skima\bold{0},\ (1/2) \ve_1, \ (1/3) \ve_2\}
\subset M_{\Q}$.
Then $C(P)\cap \tM$ is freely generated by 
$(1,\bold{0})$, $(2,\ve_1)$ and $(3,\ve_2)$,
so that $X(P)=\bP(1,2,3)$.
\end{flushleft}
%  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %
\begin{flushleft}
{\it Example.}
Let $M=\Z^2$ and 
$P=\conv\skima \{\skima 3\ve_1,\ \ve_1+\ve_2,\  3\ve_2\}
+\pos \{\skima\ve_1, \ \ve_2\}$.
Then
%- - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
X(P)=
\left\{
\left(x_1,x_2;T_1,T_2,T_3\right)\in \C^2\times \bP^2
\left|\
\begin{gathered}
x_1T_3-x_2^2T_2=0,\ \ 
x_2T_1-x_1^2T_2=0 \\
T_1T_3-x_1x_2T_2^2=0   
% fu-you no generator deha nai!
\end{gathered}
\right.
\right\},           \nonumber
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - 
which is projective over the affine variety
$X(\rec  P)=\C^2$.
\end{flushleft}
%  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %
The $T$-action on the homogeneous coordinates is given by
\begin{equation}
z_j\ra \la^{\langle \pmb{m}_j, \pmb{n}\rangle}z_j,
\quad \vn\in N,\ \la\in \C^*,
\label{homogeneous}
\end{equation} 
where we regard $\vn\in N$ as a 1-parameter subgroup
of $T$ according to (\ref{1-parameter}).
In an evident way, (\ref{homogeneous})
induces a $T$-action on $C(P)\cap \tM$, 
which defines a linearization, that is, a lifting to
an ample line bundle, 
of the $T$-action on the base $X(P)$.
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\subsection{Toric Varieties {}from Fans}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
Now that we have given a variety $X(M,P)$ associated with 
a polyhedron $P\subset M_{\Q}$, 
it is natural to ask for the fan 
in $N_{\Q}$ that yields $X(M,P)$ as a toric variety.

To describe the fan associated with $X(M,P)$,
let us first define the following function~:
%- - - - - - - - - - - - - - -
\begin{equation}
h(\vn):=\text{min}\left\{ \langle \vm^{'}, \vn\rangle :\  
 \vm^{'}\in P \right\},
\label{def-h}
\end{equation}
%- - - - - - - - - - - - - - -
which is called the {\it support function} 
of $P\subset M_{\Q}$ 
\cite[Appendix]{oda}.
Note that the domain of definition of $h$, 
which we denote by $\text{dom }h$, is 
\begin{equation}
\text{dom }h
=\pos 
\left\{\skima
\vv_a\left|
\ a\in \facetset
%\{1,\dots , A_1\}
\right\}
\right.
\subset N_{\Q},
\end{equation}
which is $p$ dimensional owing to the third assumption
on $P$ that we put earlier.

Now define a cone $C(F)$ in $N_{\Q}$ for 
$F\in L(P)\backslash \emptyset$ by
%- - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
C(F):=\left\{\  
\vn\in \text{dom }h 
\left|\ 
\right.
 \langle \vm,\vn\rangle = h(\vn),
\ \ \forall \vm\in F 
\right\} \subset N_{\Q},
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - -
which is call the {\it normal cone} of $F$.

To be more explicit, for the $a$~th facet of $P$,
$C(\athfacet)
=\pos\{\skima\vv_a\}={\Q}_{\geq 0}\skima \vv_a$ 
%$a\in \facetset$ and 
%\{1,\dots,A\}$ and 
and for a lower dimensional face $F$,
%- - - - - - - - - - - - - - - - - -
\begin{equation}
C(F)=\pos \{\skima\vv_a |\ a\in I(F)\}
=\bigoplus_{a\in I(F)}\ {\Q}_{\geq 0}\skima \vv_a.
\end{equation}
%- - - - - - - - - - - - - - - - - -
We also  see that $C(P)=\{ \bold{0} \} \in N_{\Q}$ 
because we always assume that $\dim P=p$. 

Note that $\dim F+\dim C(F)=p$
and $F\in L(P)\backslash\emptyset$ can be recovered 
from $C(F)$ by
$$
F=\left\{
\vm\in P
\left|\ 
\langle \vm,\vn\rangle=h(\vn),\  \forall \vn\in C(F)
\right\}
\right..
$$



Moreover for $F_1, F_2\in L(P)\backslash \emptyset$, 
$C(F_1)$ is a face of $C(F_2)$ if and only if 
$F_2$ is a face of $F_1$,
and $C(F_1\cup F_2)=C(F_1)\cap C(F_2)$ is a common face of
$C(F_1)$ and $C(F_2)$. 
Thus we can define a fan in $N_{\Q}$ by
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
{\cal N}(P):=
\left\{
 C(F) 
\left|\  
F\in 
{L(P)\backslash \emptyset}
\right.
\right\},
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
which we call the {\it normal fan} of $P$, and
the support of which is $\text{dom }h$.




We denote by $X^*(N,{\cal N}(P))$ 
the toric variety associated with
the data $(N,{\cal N}(P))$.
%According to the general theory of toric varieties,
By definition,
$X^*(N,{\cal N}(P))$ has the following affine open covering~:
%- - - - - - - - - - - - - - - - - -
\begin{equation}
X^*(N,{\cal N}(P))
=\bigcup_{F\in L(P)\backslash \emptyset}
X(M,C(F)^{*}),            %=\Spec M\cap C(F)^{*},
\label{affine-open-cover}
\end{equation}
%- - - - - - - - - - - - - - - - - - - 
where 
$X(M,C(F)^{*})
=\Spec \left( M\cap C(F)^{*}\right)$, 
and for a cone $C\subset N_{\Q}$, its dual cone  
$C^{*}\subset M_{\Q}$ is defined by
\begin{equation}
C^{*}:=
\left\{
\vm\in M_{\Q}\left|\ 
\langle \vm,\vn\rangle\geq 0, \ \vn\in C
\right\}
\right..
\end{equation}

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\begin{flushleft}
{\it Proposition 2.1.}
{\sl
$X^*(N,{\cal N}(P))$ is isomorphic to $X(M,P)$.}
\end{flushleft}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %




This follows 
{}from the fact that the affine open covering of 
$X^*(N,{\cal N}(P))$ described  in (\ref{affine-open-cover}) is
identical with that of $X(M,P)$
given in \cite[Proposition (2.17)]{thaddeus}.
%- - - - - - - -- - - - 


The shape and the size of the polyhedron $P$ carry 
informations about the \Ka moduli parameters of $X(M,P)$, 
which are lost in converting $P$ into its normal fan 
${\cal N}(P)$.
Two polyhedra $P_1$, $P_2$ in $M_{\Q}$ 
are said to be {\it normally equivalent} 
if their normal fans  are isomorphic to each other,
that is, ${\cal N}(P_1)\cong {\cal N}(P_2)$.
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\begin{flushleft}
{\it Example.} Let us take $M=\Z^2$ and a pair of
normally equivalent polyhedra
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{align}
P_1&=\conv \skima
\{\bold{0},\ \ve_1,\ \ve_2,\ \ve_1+\ve_2 \},\nonumber \\
%   -   -   -   -   -   -   -   -   -   -   -   -   -   -
P_2&=\conv\skima
\{\bold{0},\ 4\ve_1,\ 3\ve_2,\ 4\ve_1+3\ve_2 \}.\nonumber
\end{align}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Both $X(M,P_1)\subset \bP^3$ and $X(M,P_2)\subset \bP^{19}$ 
are isomorphic to
$\bP^1\times \bP^1$~; 
the \Ka moduli of the former and the latter are
$(1,1)$ and $(4,3)$ respectively.
\end{flushleft}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
The use of the normal fan, however, is a far more  
efficient way to obtain the toric variety $X(M,P)$.



%Once the fan ${\cal N}(P)$ is known, we can realize
%the toric variety 
%Let us consider the lattice homomorphism
%$\eta:\Z^{|\La|}\ra N$
%defined by
%%- - - - - - - - - - - - - - - - - - - - -  
%\begin{equation}
%\eta(q_1,\dots,q_{|\La|})
%:=\sum_{a=1}^{|\La|} q_a \skima \vv_a,
%\end{equation}
%%- - - - - - - - - - - - - - - - - - - - - -
%and let $\vq^{(k)}$, $k=1,\dots, |\La|-p$, be the generators
%of the sublattice $\Ker \eta \subset \Z^{|\La|}$.
%We can realize $X^*(N,{\cal N}(P))$ as  
%a toric quotient of $\C^{|\La|}$
%by the action of $(\C^{*})^{|\La|-p}$, where
%the $k$~th charge on the $a$~th coordinate $x_a$
%is $q^{(k)}_a$.




















% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\subsection{Toric Quotient}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
Let $P\subset M_{\Q}$ be a polyhedron,
and  $X(M,P)$ be 
the associated  quasi-projective variety.
% with a $T$-linearization
%determined by the $T$-action on $M$.
Suppose that there is an exact sequence of lattices
%- - - - - - - - - - - - - - - - - -
\begin{equation}
0 \ra N^{'}\overset{\pi^*}{\ra} N 
\overset{i^*}{\ra} \Nb\ra 0, 
\label{exact-N}
\end{equation}
%- - - - - - - - - - - - - - - - - -
where $\text{rank}\  N^{'}=p-q$ and $\text{rank}\ 
 \Nb=q$,
then the dual sequence is also exact~:
%- - - - - - - - - - - - - - - - - - 
\begin{equation}
0 \ra \Mb\overset{i}{\ra} M 
\overset{\pi}{\ra} M^{'}\ra 0.
\label{exact-M}
\end{equation}
%- - - - - - - - - - - - - - - - - -
A sublattice $N^{'}\subset N$ defines 
a subtorus
$T^{'}=N^{'}\otimes\C^*=\Hom(M^{'},\C^*)$ 
of rank $p-q$, which acts on $X(M,P)$.

Now we want to define 
the geometric invariant theory (GIT)
quotient of $X(M,P)$ by the action of $T^{'}$.

The graded ring $C(P)\cap \tM$ admits a natural $T^{'}$-action
and the $T^{'}$-invariant part
$\left( C(P)\cap \tM \right)^{T^{'}}$ is also a graded ring.
Then we define the quotient variety by 
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
X(M,P)\GIT T^{'}:=\Proj \left( C(P)\cap \tM \right)^{T^{'}},
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
which is again projective 
over the affine variety defined by
the affine GIT quotient 
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
X_0(M,P)\GIT T^{'}:=\Spec \left(\rec P\cap M\right)^{T^{'}},
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
where $(\rec P\cap M)^{T^{'}}$ is the degree zero part
of $\left( C(P)\cap \tM \right)^{T^{'}}$.

We immediately see 
that the GIT quotient variety admits 
a following toric realization~:
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
X(M,P)\GIT T^{'}
=X\left(\Mb, P\cap \pi_{\Q}^{-1} (\bold{0})\right),
\label{GIT=0}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
where $\Mb= \Ker \pi=M\cap \pi_{\Q}^{-1}(\bold{0})$ is 
the sublattice of $M$ fixed by $T^{'}$.

The corresponding symplectic quotient construction 
can be done as follows~:
In addition to the D-flatness equation in (\ref{ambient}) 
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\sum_{j=1}^s k_j\ |z_j|^2=1
\label{1-ambient}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
for the ambient space $\bP(k_1,\dots ,k_s)$,
we put $p-q$ D-flatness equations associated with
$T^{'}$-action on $(z_j)$ with the \Ka 
(or Fayet--Iliopoulos) parameters  
$\vr=\bold{0}\in M^{'}_{\Q}$
followed by quotienting by $\text{U}(1)^{p-q}$.
More concretely, 
let $\vn^{'}_1,\dots ,\vn^{'}_{p-q}$ 
be the generators of $N^{'}$, 
each of which corresponds to a 1-parameter subgroup of 
$T^{'}\cong (\C^*)^{p-q}$.
Then the additional $p-q$ D-flatness equations 
can be written as
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\sum_{j=1}^s {\langle \pi(\vm_j),\vn^{'}_l \rangle}\ 
|z_j|^2=0, \qquad l=1,\dots ,p-q.
\label{additional}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
%followed by quotienting by $\text{U}(1)^p$.
A useful abbreviation of (\ref{additional}) is
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
\sum_{j=1}^s \pi(\vm_j)\ |z_j|^2 =\bold{0},
\label{0-vector}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
where we say that $z_j$ has $T^{'}$-charge $\pi(\vm_j)$.


% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%--------    deformation of Kahler parameter ------------------
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %            
Now we want to consider the toric quotient of $X(P)$ by $T^{'}$ 
with a nonzero \Ka moduli parameters
$\vr\in M^{'}_{\Q}$.
To this end let us take $\what{\vr}\in M_{\Q}$ such that
$\pi_{\Q}(\what{\vr})=\vr$ and consider the shifted polyhedron 
$P-\what{\vr}\subset M_{\Q}$.
The original generators $(k_j,\vm_j)$ of 
$C(P)\cap \tM$ are now shifted to
$(k_j,\vm_j-k_j \what{\vr})$ so that 
the $T^{'}$-charge of $z_j$ becomes
$(\pi(\vm_j)-k_j \vr)$.
This $T^{'}$ charge assignment 
for $(z_j)$ defines a new action of
$T^{'}$ on $(C(P)\cap \tM)$ 
which we denote by $T^{'}(\vr)$.
Then we can define 
the GIT quotient of $X(M,P)$ by $T^{'}(\vr)$ as
%- - - - - - - - - - - - - - - - - - - 
\begin{equation}
X(M,P)\GIT T^{'}(\vr):=\Proj  
\left(C(P)\cap \tM\right)^{T^{'}(\pmb{r})}
\label{quotient-def}
\end{equation}
%- - - - - - - - - - - - - - - - - - -
which is also projective over the affine variety
\begin{equation}
%- - - - - - - - - - - - - - - - - - - - - 
X_0(M,P)\GIT T^{'}(\vr):=
\Spec \left(\rec P\cap M\right)^{T^{'}(\pmb{r})}.
\label{quotient-def-affine}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - -
The ambiguity in the choice of $\what{\vr}$, which 
is isomorphic to $\Mb_{\Q}$, does not
affect the definitions (\ref{quotient-def}),
(\ref{quotient-def-affine}).
In fact it only affects 
the $\overline{T}:=T/T^{'}$-linearization
of the quotient variety, which is irrelevant to us.


To see that the definition of 
$X(M,P){\GIT} T^{'}(\vr)$ above corresponds to
the change of the \Ka parameters to $\vr\in M^{'}_{\Q}$,
we have only to describe the corresponding
symplectic quotient construction of $X(M,P)\GIT T^{'}(\vr)$.
The D-flatness equations associated with $T^{'}(\vr)$ are 
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\sum_{j=1}^s (\pi(\vm_j)-k_j \vr)\ |z_j|^2 =\bold{0}.
\label{modified}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
Combining (\ref{1-ambient}) and (\ref{modified}), 
we obtain the D-flatness equations associated with $T^{'}$
with the \Ka moduli parameters $\vr$~:
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\sum_{j=1}^s \pi(\vm_j)\ |z_j|^2 =\vr.
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
Thus we get $X(M,P)\GIT T^{'}(\vr)$ by
the following symplectic quotient of 
$X(P)$ by the $\text{U}(1)^{p-q}$-action 
with $\vr$ as a \Ka parameters~:
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{align}
X(M,P)\GIT T^{'}(\vr) &\cong
\left.
\left\{
\left[(z_j)\right]\in X(M,P)
\left|\
\sum_{j=1}^s \pi(\vm_j)\ |z_j|^2=\vr
\right.
\right\}\right/\text{U}(1)^{p-q}  \\
%\end{equation}
%- - - - - - - - - - - - - - -
%\begin{equation}
%X(M,P)\GIT T^{'}(\vr)
&\cong
\left\{
(z_j) \in \C^s 
\left.
\left|
\begin{gathered}
\sum_{j=1}^s k_j\ |z_j|^2=1,\ 
\sum_{j=1}^s \pi(\vm_j)\ |z_j|^2 =\vr\\
\text{F-flatness equations}\ (\ref{F-flat})
\end{gathered}
\right\}
\right.
\right/ \text{U}(1)^{p-q+1}.\nonumber
\end{align}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 

% % % % % %   GIT quotient as a toric variety  % % % % % % % % % %
In the following we argue that the GIT quotient 
$X(M,P){\GIT} T^{'}(\vr)$ defined above can be realized as a
quasi-projective toric variety~:


We will show that $X(M,P){\GIT} T^{'}(\vr)$ 
can be realized as a 
quasi-projective toric variety generalizing (\ref{GIT=0})~: 

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\begin{flushleft}
{\it Proposition 2.2.}
{\sl Fix $\what{\vr}\in M_{\Q}$ such that 
$\pi_{\Q}(\what{\vr})=\vr$ for $\vr\in {M}^{'}_{\Q}$, and
let $\Qr\subset \Mb_{\Q}$ be the polyhedron 
defined by $\Qr:=(P-\what{\vr})\cap \Mb_{\Q}$. 
Then  we have
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
X(M,P)\GIT T^{'}(\vr)
=X\left(\Mb, \Qr \right).
\label{GIT=r}
\end{equation}}
\end{flushleft}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %





%\begin{flushleft}
{\it Proof.}
%\end{flushleft}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

We see that (\ref{GIT=r}) holds when
$\vr\in M^{'}$ and $\what{\vr}\in M$ because upon the shift by 
$\what{\vr}$, each element of $C(P)\cap \tM$ turns to
one of $C(P-\what{\vr})\cap \tM$.

Let $e$ be  the least positive integer %for $\what{\vr}$
such that $e\vr\in M^{'}$.
Without loss of generality, we can restrict 
$\what{\vr}\in \pi_{\Q}^{-1}(\vr)$ 
to those which satisfy $e\what{\vr}\in M$.
To deal with this case, % where $\what{\vr}$ is not quantized,
we use the dilatation invariance of the toric data~:
%- - - - - - - - - - - - - - 

For a graded ring $G:=\bigoplus_{k\geq 0}G_k$,
define its $e$~th Segre transform $G^{(e)}$ for $e\in \N$ by
$G^{(e)}_k=G_{ek}$ and
$G^{(e)}:=\bigoplus_{k\geq 0}G^{(e)}_k
=\bigoplus_{k\geq 0}G_{ek}$.
Then we have 
\begin{equation}
\Proj G \cong \Proj G^{(e)}. 
\end{equation}
%- - - - - - - -  - - - - - - - - - - - - - - - - - - - - - -
We easily see that the $e$~th Segre transform of $C(P)\cap \tM$
%$(C(P)\cap \tM)^{(e)}$ 
coincides with $C(eP)\cap \tM$, so that
\begin{equation}
X(M,P) \cong X(M,eP).
\label{scaling}
\end{equation}
%- - - - - - - -  - - - - - - - - - - - - - - - - - - - - - -

 Then we have
%- - - - - - - - - - 
\begin{align}
X\left( \Mb,(P-\what{\vr})\cap \Mb_{\Q}\right)
&\cong X\left( \Mb,(eP-e\what{\vr})
\cap \Mb_{\Q}\right)
\nonumber \\
&\cong \Proj  
\left(     
C(eP)\cap M
\right)^{T^{'}(e\pmb{r})}.
\label{tempo}
\end{align}
%- - - - - - - - - - -

To finish the proof of the Proposition, we have only to 
prove the following lemma~:

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\begin{flushleft}
{\it Lemma 2.2.1.}
{\sl The graded ring 
$\left(C(eP)\cap M\right)^{T^{'}(e\pmb{r})}$ 
coincides with 
$\left(C(P)\cap M\right)^{T^{'}(\pmb{r})}$.} 
\end{flushleft}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
{\it Proof of Lemma 2.2.1.}

For simplicity, we set temporally 
$G_k:=(C(P)\cap \tM)_k=kP\cap M$,
$G:=C(P)\cap \tM$, and $G^{(e)}:=C(eP)\cap \tM$.
Any element of $G^{T^{'}(\pmb{r})}$ can be written as
$\sum_{j=1}^L (k_j,\vm_j)$,
where the total $T^{'}(\vr)$ charge is
$\sum_{j=1}^L(\pi(\vm_j)-k_j\vr)=\bold{0}$, that is,
$(\sum_{j=1}^L k_j)\skima
\vr=\sum_{j=1}^L \pi(\vm_j)\in M^{'}$,
which implies that 
$\sum_{j=1}^N k_j$,
which is the degree of $\sum_{j=1}^L(k_j,\vm_j)$,
should be a multiple of $e$.
Thus we see that 
if we define a subring $H$ of $G$ by
\begin{align}
H_k&=G_k,\qquad \text{if}\ k\equiv 0 \mod{e}, \nonumber \\
H_k&=0,\qquad \ \ \text{otherwise},\nonumber
\end{align}
then we have $G^{T^{'}(\pmb{r})}=H^{T^{'}(\pmb{r})}$.

Now take an arbitrary element $(ek,\vm)\in H_{ek}=G^{(e)}_k$.
When regarded as an element of $H_{ek}$,
its $T^{'}(\vr)$ charge is $(\pi(\vm)-ek\vr)$, 
which is the same as its  $T^{'}(e\vr)$ charge 
regarded as an element of $G^{(e)}_k$. \hfill $\Box$

Then the combination of (\ref{tempo}) and Lemma~2.2.1
proves the Proposition~2.2.\hfill $\Box$


%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\subsection{\Ka Moduli Space}
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
We consider here the $\vr$-dependence 
of the topology, or the {\it phase} in physics terminology,
 of the quotient toric variety (\ref{GIT=r}).
%
The quotient variety is the toric variety associated with
the normal fan of the polyhedron $\Qr$,
which is given by the {\it slice} 
$P\cap \pi_{\Q}^{-1}(\vr)$ of $P$ translated by $-\what{\vr}$.
%
Therefore the topology of the quotient variety is determined 
virtually by the shape of the slice $P\cap \pi_{\Q}^{-1}(\vr)$,
which depends on the faces of $P$ that intersect with
the affine subspace $\piQ^{-1}(\vr)$ of $M_{\Q}$.

This observation leads us to define the following 
decomposition of the polyhedron $\piQ(P)$
induced by the $\piQ$-images 
of the faces of $P$ \cite{KSZ}. 
First for each $\vr\in \piQ(P)$,
let $L(\vr)$ be the subset of $\overline{L}(P)$, 
the proper faces of $P$, by
%- - - - - - - - - - - - - - -   Definition of L(r) - - -
%\begin{equation}
$
L(\vr):=
\left\{
F\in \overline{L}(P)
\left|\
\vr\in \piQ(F)
\right.
\right\}.
%\end{equation}
$
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Then define an equivalence relation $\sim$ 
in $\piQ(P)$ by
%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
%\begin{equation}
$\vr_1\sim \vr_2$ if and only if  
%\Longleftrightarrow 
$L(\vr_1)=L(\vr_2)$, 
%\qquad
%\text{for}\ 
for $\vr_1,\vr_2\in \piQ(P)$.
%\end{equation} 
%\qquad
%$\text{for}\ \vr_1,\vr_2\in \piQ(P).
%\end{equation} 
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -- 
We call an equivalence class $K^{0}$ in 
$\piQ(P){/\negthickspace\sim}$
a chamber. 
The polyhedron $\piQ(P)$ 
admits the decomposition
into the disjoint sum of these chambers~:
%- - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\piQ(P)=\coprod_{K^{0}\in \piQ(P)/\sim} K^{0},
\label{phase}
\end{equation} 
%- - - - - - - - - - - - - - - - - - - - - -  
and the topology of the quotient variety $X(\Mb,\Qr)$ 
is constant in each chamber \cite{KSZ}.
Therefore we see that
the decomposition (\ref{phase}) of the parameters space 
$\piQ(P)$ 
represents the phase structure of the toric quotient.

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
We also define a closed polyhedron $K$ 
to be the closure in $M^{'}_{\Q}$ of the chamber 
$K^0\in \piQ(P){/\negthickspace\sim}$.
Conversely $K^0$ is recovered as
the {\it relative interior} of $K$.

Then the collection of the polyhedra $\kahler$
defined by
%- - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}  
\kahler:=\{K|\ K^0\in \piQ(P){/\negthickspace\sim} \}
\label{polyhedral-complex}
\end{equation} 
%- - - - - - - - - - - - - - - - - - - - - - - - - - -
constitutes 
a {\it polyhedral complex}~\cite[Lecture 5.1]{ziegler}
in $M^{'}_{\Q}$,
which means that for each $K\in \kahler$, 
every face of $K$ belongs to $\kahler$ 
and the intersection $K_1\cap K_2$ 
of any two elements of $\kahler$ 
is the face of both $K_1$ and $K_2$~; in particular
$\kahler$ is a fan if it consists of polyhedral cones,
which is true if $P$ itself is a cone.
We call the polyhedral decomposition of $\piQ(P)$ 
defined by the complex (\ref{polyhedral-complex})
the \Ka moduli space 
associated with the toric quotient.

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
We define the \Ka walls to be 
the $\piQ$-image of the skeleton of $P$ 
consisting of all the faces of codimensions 
$q+1$.
The \Ka walls is the region 
where the toric quotient construction
{\it degenerates} in the sense that
for each $\vr$ in the \Ka walls,
there is a face $F$ of $P$ such that
$F$ and $\piQ^{-1}(\vr)$ intersect 
despite of the fact that
the sum of their codimensions in $M_{\Q}$
exceeds $p=\dim M_{\Q}$.

We are thus mainly interested 
in the \Ka moduli parameters
in the complement of the \Ka walls in $\piQ(P)$,
which is the disjoint union of the chambers
of the maximal dimensions \cite{thaddeus},
which we call the maximal chambers.
We also call the closure of a maximal chamber
in $M^{'}_{\Q}$ a ``maximal polyhedron''.




% % % % % % % % % % % % % % % %
 
The $\piQ$-image
of each  face of $P$ of codimensions less than
$q+1$ is a union of several maximal polyhedra.
%
Let $L(P)^{(k)}$ be the subset of $L(P)$
consisting of the faces of codimensions $k$. 
For each $F\in L(P)^{(k)}$, 
where $k\leq q$,
we can define a $k$-cone in $\Nb_{\Q}$ by
%- - - - - - - - - - - - - - - - -
\begin{equation}
\overline{C}(F):=i^*_{\Q}(C(F))
=\pos  
\left\{\skima
\vvb_a
\left|\
a\in I(F)
\right.
\right\},
\end{equation}
where $i^*$ is the lattice surjection
{}from $N$ to $\Nb$ (\ref{exact-N}).

%- - - - - - - - - - - - - - - - -
Then we see that for any 
$\vr\in \text{int}\ \piQ(F)$,
the normal fan ${\cal N}(\Qr)$ of the quotient 
has the $k$-cone $\overline{C}(F)$ defined above.
This is because if $\vr\in \text{int}\ \piQ(F)$,
then the slice $P\cap \piQ^{-1}(\vr)$ has the 
face $F\cap \piQ^{-1}(\vr)
=\bigcap_{a\in I(F)}\athfacet
\cap \piQ^{-1}(\vr)$ 
of codimensions $k$,
the normal cone of which is precisely $\overline{C}(F)$.
%

The following two cases are of particular importance~:
first for the $a$~th facet 
$\athfacet$ and 
for any $\vr\in \piQ(\athfacet)$,
the slice has the facet 
$\athfacet \cap \piQ^{-1}(\vr)$,
so that the normal fan ${\cal N}(\Qr)$ 
has the 1-cone $\pos \{\skima\vvb_a\}$,
which means that the quotient variety has 
the exceptional divisor corresponding to 
$\vvb_a$. 
%We call $\vvb_a$ the primitive vector
%of the quotient toric variety associated with
%$\athfacet$.
%
We say two vectors 
$\vvb_a$ and $\vvb_b$
in $\Nb$ to be {\it incompatible} 
if
$\text{int}\ \piQ(\athfacet)$ and 
$\text{int}\ \piQ(\bthfacet)$
have no common point~;
then the two vectors cannot appear 
simultaneously in the quotient fan 
outside the \Ka walls~;
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
second for $F\in L(P)^{(q)}$ and 
$\vr \in \text{int}\ \piQ(F)$,
the slice has the vertex
$\bigcap_{a\in I(F)}\athfacet
\cap \piQ^{-1}(\vr)$, 
which corresponds to 
the maximal cone 
$\overline{C}(F)\subset \Nb_{\Q}$
of the normal fan. 


% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%                Kahler chamber
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
Because the normal fan ${\cal N}(\Qr)$ 
is determined by listing
its maximal cones,
we obtain the following description
of the phase structure of the quotient variety
%when the moduli parameter $\vr$ does 
%not live in the \Ka walls~:
outside the \Ka walls.

% % % % % % % % % % % % % % % % % % % % % % % % 
Let us 
call a subset $S$ of $L(P)^{(q)}$ 
{\it coherent} if the collection 
of the cones in $\Nb_{\Q}$, 
\begin{equation}
\Sigma(S):=
\left\{
L\left(\overline{C}(F)\right) 
\left|\
 F\in S\right.
\right\},
\end{equation} 
defines a fan,
where $L(\overline{C}(F))$, 
the face lattice of $\overline{C}(F)$,
is the set of all the faces of $\overline{C}(F)$,
and if the subspace of $\piQ(P)$ defined by
%- - - - - - - - - - - - - - - - - - - - - --
\begin{equation}
K(S):=\bigcap_{F\in S}\piQ(F)=
\bigcap_{F\in S}
\piQ\left( \cap_{a\in I(F)} 
\athfacet \right)
\label{result}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - -
has an interior point, that is,
if $K(S)$ is a maximal polyhedron.

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\begin{flushleft}
{\it Proposition 2.3.}
{\sl The \Ka moduli space $\kahler$ associated with
the toric quotient is  
%- - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\kahler=\left\{
L\left(K(S)\right)
\left|\
S\subset L(P)^{(q)}:\  \mathrm{coherent}
\right.
\right\}.
\label{result-1}
\end{equation}}
%- - - - - - - - - - - - - - - - - - - - - - -
\end{flushleft}
\begin{flushleft}
{\it Proposition 2.4.}
{\sl For each coherent subset $S\subset L(P)^{(q)}$,
we have
%- - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
X(\Mb,\Qr)\cong X^*(\Nb,\Sigma(S)),
\quad \forall \vr\in \mathrm{int}\ K(S),
\label{result-2}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - -
where $X^*(\Nb,\Sigma(S))$ is the toric variety 
defined by the fan $\Sigma(S)$.}
\end{flushleft}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 

Note that (\ref{result-1}) and (\ref{result}) 
generalize the descriptions
of the GKZ secondary fan \cite{GKZ} and its maximal cones
given in \cite[(4.2)]{BFS},
where $M=\Z^p$,
$P=\pos \{\skima\ve_1,\dots, \ve_p\}
\cong ({\Q}_{\geq 0})^p$ is the basic simplicial cone,
and $X(M,P)\cong \C^{p}$,
which has been used in the investigation 
of the \Ka moduli space of {\it bulk string}
compactified on a Calabi--Yau manifold \cite{HLY}. 





\single
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\section{D-Brane Configuration Space}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 

\subsection{Calabi--Yau Orbifolds}
Let $\{a_1,\dots,a_d\}$ be a $d$-tuple of the integers,
and $\omega$ be a primitive $n$~th root of unity.
We define $\Ga$ to be a group
isomorphic to the cyclic group $\Z_n:=\Z/n\Z$ 
and define the action of the generator 
$g\in \Ga$ on $\C^d$ by
%- - - - - - - - - - - - - - - - - - - - -- 
\begin{equation}
g\cdot x_{\mu}=\omega^{\amu}x_{\mu},
\quad 1\leq \mu \leq d.
\end{equation} 
%- - - - - - - - - - - - - - - - - - - - - -
We denote the quotient space by $\CY$.
The followings are well-known~:
%- - - - - - - - - - - - - - -
\begin{itemize}
\item
$\CY$ has an isolated singularity at the origin 
if and only if \ $(\amu,n)=1,\quad  \forall \mu$.
%-  -  -  -  -  -  -  -  -  -  -  -  
\item
$\CY$ is a Calabi--Yau variety if and only if \ 
$\sum_{\mu=1}^d \amu\equiv 0\mod{n}$.
\end{itemize}
%- - - - - - - - - - - - - - - - - - -
We restrict ourselves to the models in which
the orbifold $\CY$ is a Calabi--Yau variety  
with an isolated singularity unless otherwise stated, 
because our main interest is 
the study of the \config\ space ${\cal M}$
of a D-brane localized 
at the singular point of the Calabi--Yau variety $\CY$.
We denote the model characterized by the integers
$(a_1,\dots,a_d;n)$ above by $1/n(a_1,\dots,a_d)$
for simplicity.

%%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  
Here we give some facts about the Calabi--Yau orbifolds.
An advanced introduction to this subject 
can be found in \cite{DH}.
First of all, $\CY$ is a toric variety.
A useful choice of the dual pair of the lattices 
to describe $\CY$ is the following~:
%- - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{align}
\Nb_0 &:=\Z^d+\frac{1}{n}(a_1,\dots,a_d)\ \Z,
\label{N-lattice}\\
%-    -   -   -   -   -   -   -   -   -   -   -   -   -
\Mb_0 &:=
\left\{
\vm\in \Z^d
\left|\
\vm \cdot \va \equiv 0 \mod{n}
\right.
\right\},\quad \va:=(a_1,\dots,a_d).
\label{M-lattice}
\end{align}
%- - - - - - - - - - - - - - - - - - - - - - - - - -
Let $\{\ve^*_1,\dots,\ve_d^*\}$ and
$\{\ve_1,\dots,\ve_d\}$ be the set of 
the fundamental vectors 
of $(\Nb_0)_{\Q}$ and $(\Mb_0)_{\Q}$ respectively, 
which generate the dual pair of simplicial cones~:
%- - - - - - - - - - - - - - - - - - - - - - - - - 
\begin{align}
C_0^*&=\pos \{\ve^*_1,\dots,\ve_d^*\}\cong
(\Q_{\geq 0})^d
\subset (\Nb_0)_{\Q}, 
\label{N-cone}\\
%-   -   -   -   -   -   -   -   -   -   -   -   -
C_0&=\pos \{
\ve_1,\dots,\ve_d\}\cong
(\Q_{\geq 0})^d
\subset (\Mb_0)_{\Q}.
\label{M-cone}
\end{align} 
%- - - - - - - - - - - - - - - - - - - - - - - - 
Then we have
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\CY
=X(\Mb_0,C_0)
=\Spec\skima (\Mb_0\cap C_0)
=X^*(\Nb_0,C^*_0).
\label{affine-orbifold}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
To see this, it suffices to note that
the affine coordinate ring of $\CY$ is
the $\Ga$-invariant part of $\C[x_1,\dots,x_d]$,
which is precisely $(\Mb_0\cap C_0)$.
%   %   %   %   %   %   %   %   %   %   %   %   %   %   %   
The simplicial cone $C^*_0\subset (\Nb_0)_{\Q}$
is the fan associated with $\CY$.
Thus a toric blow-up of $\CY$ 
corresponds to a subdivision of the cone $C^*_0$ 
by incorporating new 1-cones,
the primitive vectors of which 
correspond to {\it exceptional divisors}.
For simplicity, we will confuse the primitive vector
of a 1-cone with the exceptional divisor 
associated with it.
%
Let $T=\conv \{\ve^*_1,\dots,\ve_d^*\}$ 
be the fundamental simplex 
in $(\Nb_0)_{\Q}$ associated with the orbifold.
A primitive vector $\vvb\in \Nb_0$ is classified 
by its age,
which is defined to be the positive integer $k$ 
such that $\vvb\in kT$.
Incorporation of $\vvb\in \Nb_0$ 
in subdivision of the fan $C_0^*$ 
preserves the Calabi--Yau property  
if and only if its age is 1.
Thus a primitive vector of age 1 is said to be
{\it crepant}.
A crepant toric blow-up of $\CY$ corresponds 
to a subdivision of $T$ using lattice points in $T$.
%
We define the weight vector $\vwb$ 
associated with a primitive vector $\vvb\in \Nb$
by $\vwb:=n\vvb\in \Z^d$.


%- - - - - - - - - - 
We can read the physical Hodge numbers 
of bulk string $(h^{p,p})$
``compactified'' on $\CY$ from the Ehrhart series 
for $(\Nb_0,T)$ \cite{BD} as
%- - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\sum_{k\geq 0}l\left(kT\right)\skima y^k
=\frac{1}{(1-y)^d} \sum_{p=0}^{d-1} h^{p,p}\skima y^p,
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - -
where $l\left(kT\right)$ is the number of the lattice points 
in the dilated simplex $kT$, that is,
$l\left(kT\right)=\text{card}\skima (kT\cap \Nb_0)$.
In particular, the number of the crepant divisors
$h^{1,1}=l(T)-d$ equals to the dimensions 
of the \Ka moduli space of bulk string 
``compacitified'' on $\CY$.

%   %   %   %   %   %   %   %   %   %   %   %   %   %   %   
There is a striking difference between 
$d=4$ orbifolds from $d=2,3$ ones~: 
in general, incorporation of the crepant divisors only 
is not enough to resolve $\CY$ completely
into a smooth variety for $d=4$ as opposed to $d=2,3$ cases.

% % % % % % % % % % % % % % % % % % % % % % % % % % %
%     enumerate       (A),(B),(C),....
\renewcommand{\theenumi}{\Alph{enumi}}
\renewcommand{\labelenumi}{(\theenumi)}
%% % % % % % % % % % % % % % % % % % % % % % % % % % % 
In \cite{mohri}, we have divided the $d=4$ models 
into the following three classes~:
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
\begin{enumerate}
\item 
the models that admit a crepant resolution.
%   -    -   -   -   -   -   -   -   -   -   -   -
\item 
those that have no crepant divisors, the singularities of 
which are called {\it terminal},
consisting of the models of the form~:
$1/n(1,a,n-1,n-1)$
%$\dfrac{1}{n}(1,a,n-1,n-a)$, 
where $(n,a)=1$ \cite{MS}.
%   -    -   -   -   -   -   -   -   -   -   -   -
\item
those that have at least one crepant divisor, 
%( $h^{1,1}\geq 1$), 
but do not admit any crepant resolutions.
\end{enumerate}
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
The complete identification of the (A) class,
that is, the classification of the isolated 
cyclic quotient {\it Gorenstein} singularities
in four or higher dimensions 
for which crepant resolutions are possible 
is very interesting but unsolved 
mathematical problem \cite{reid}, 
the physical meaning of which is yet to be elucidated.
It is clear that the examples of the (A) class 
shown in \cite{mohri}, 
%- - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
1/(3m+1)(1,1,1,3m-2),\quad
%-   -   -   -   -   -   -   -   -   -   -   -
1/(4m)(1,1,2m-1,2m-1), \qquad m\in \N,
\label{trivial}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - 
are only the tip of the iceberg.
Recently, however, a considerable progress 
in this subject has been made in \cite{DHH,DH}. 
A remarkable new series in the (A) class,
the $m$~th member of which is called the 
{\it 4}-dimensional
{\it geometric progress singularity-series} of ratio $m$
$\left(\text{GPSS}(4;m)\right)$, 
is given in \cite[Conjecture 10.2]{DH}~: 
%  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
\begin{flushleft}
{\it Conjecture (Dais--Henk).}
{\sl $1/\{(1+m)(1+m^2)\}(1,m,m^2,m^3)$ model 
admits {\it a} crepant resolution for each $m\in \N$.}
\end{flushleft}
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
The {\it same} conjecture was also made by the author,
who have only checked that 
the {\it Delaunay triangulation} 
\cite{Del}, \cite[p.~146]{ziegler}, 
of $T$ by the lattice points in it
%incorporating the crepant divisors 
yields a crepant resolution, 
which is {\it not} unique for $m\geq 3$, up to $m=10$.
%
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\subsection{D-Brane Configuration Space}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
We consider the configuration space 
of a D1-brane localized at the singular point of $\CY$.
This can be realized as follows~:
%-   -   -   -   -   -   -   -   -   -   -   -   -   -   
First we consider $n=|\Ga|$ D1-branes 
localized at the origin of $\C^d$.
We assign the Chan--Paton indices $i\mod{n}$ to the D1-branes.
Then the world sheet theory on the D1-branes is 
$\text{U}(n)$ gauge theory with $(8,8)$ \susy.
The configuration of the D1-branes is described by the $d$-tuple
of the matrices  $\{ (X_{\mu})^i_j \}$ taking values 
in the adjoint representation of $\text{U}(n)$ \cite{witten2}~;
%   -   -   -   -   -   -   -   -   -   -   -   -   -   -
Second taking into account the $\Ga$-actions 
on the Lorentz indices $(\mu)$ and the Chan--Paton indices $(i)$,
on which $\Ga$ acts as cyclic permutations,
we define the configuration of a D1-brane on the orbifold $\CY$
to be that of $n$ D1-branes on $\C^d$ invariant under  
the simultaneous action of $\Ga$ on the Lorentz 
and the Chan--Paton indices \cite{DGM,DM}.
In the next section, we use a closely related idea 
in the definition of Hilbert schemes of $n$ points on $\C^d$. 

The world sheet \susy\  is reduced, at this point, 
to $(4,4)$, $(2,2)$ and $(0,2)$ for $d=2,3$ and for $d=4$ 
respectively, with the exception that
the \susy\  of $d=4$ (B) model is enhanced to $(0,4)$ 
\cite{mohri}.
We can also consider a model with 
$\sum_{\mu}a_{\mu}\not\equiv 0 \mod{n}$,
where the \susy\  is completely broken \cite{DGM}.
%   %   %   %   %   %   %   %   %   %   %   %   %    %   %
Let $R_a$ be the one dimensional representation of $\Ga$ 
over $\C$ on which the generator $g\in \Ga$ acts 
as multiplication by $\omega^a$.  
%   %   %   %   %   %   %   %   %   %   %   %   %    %    %
Then the D-brane matrices $(X_{\mu})$ take values in
$\left(Q\otimes \text{End}(R)\right)^{\Ga}
\cong \Hom_{\Ga}(R,R\otimes Q)$ 
\cite{kronheimer,sardo,infirri}, 
where the two $\Ga$-modules,
%- - - - - - - - - - - - - - - - - - - - - - - - -- -
\begin{equation}
R=\bigoplus_{i=1}^{n} R_i,\qquad
Q=\bigoplus_{\mu=1}^d R_{\amu},   \label{regular-rep}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - -- - - - - -
carry the Chan--Paton and the Lorentz $\Ga$-quantum numbers
of the matrices respectively. 
Note that we have done the discrete Fourier transformation
on the Chan--Paton indices, so that the $\Ga$-action on those
is diagonalized.

To be explicit, the matrix elements that can be nonzero are
%- - - - - - - - -
\begin{equation}
x_{\mu}^{(i)}:=(X_{\mu})^i_{i+\amu}, 
\label{as-before}
\end{equation}
%- - - - - - - - -
and the \config\ space of the D1-brane on $\CY$
is the solution space of the following equations~:
%- - - - - - - - - - - - - - - - -
\begin{alignat}{2}
&\left[X_{\mu},X_{\nu}\right]=O, &\quad
&\text{F-flatness equation},
\label{M-F-flat}\\
%-  -  -  -  -  -  -  -  -  -  -  -
\sum_{\mu=1}^d
&\left[X_{\mu},X_{\mu}^{\dagger}\right]
-\text{diag}(r_1,\dots,r_n)=O, &\quad
&\text{D-flatness equation},
\label{M-D-flat}
\end{alignat}
%- - - - - - - - - - - - - - - - -
divided by the action of 
$\text{U}(1)^n/\text{U}(1)_{\text{diag}}$,
where $x_{\mu}^{(i)}$ has the $i$~th $\text{U}(1)$ charge 1
and the $(i+\amu)$~th $\text{U}(1)$ charge $-1$, 
and the others 0 as seen from (\ref{M-D-flat}).

To have a solution to (\ref{M-D-flat}),
the Fayet--Iliopoulos (or \Ka) moduli parameters $\vr:=(r_i)$ 
must satisfy $\sum_{i=1}^n r_i=0$.
 
The F-flatness equation (\ref{M-F-flat})
can be solved as follows \cite{DGM}~:
We redefine the generator of $\Ga$ so that $a_{d}=-1 \mod{n}$.
Then the matrix elements (\ref{as-before}) can be represented 
by $x_d^{(i)}$, $i=1,\dots,n$ and 
$x_{\mu}^{(0)}$, $\mu=1,\dots,d-1$ as
%- - - - - - - - - - - - - - - - - -
\begin{equation}
x_{\mu}^{(i)}=x_{\mu}^{(0)}\ 
\frac{\prod_{j=1}^{i}x_d^{(j)}
\cdot\prod_{j=1}^{\amu}x_d^{(j)}}%
{\prod_{j=1}^{i+\amu}x_d^{(j)}}.
\label{solution-to-F}
\end{equation}
%- - - - - - - - - - - - - - - - - -
We see that the solution space 
of the F-flatness equation (\ref{M-F-flat}), 
which we denote by $\A$, is the $(n-1+d)$-dimensional
affine variety embedded in $\C^{nd}$ 
defined by the equations of monomial type
(\ref{solution-to-F}), 
which shows that $\A$ is a toric variety.
The \config\ space of the D1-brane, which we denote by $\Mod$,
is also toric because it is obtained as a toric quotient 
of $\A$ (\ref{M-D-flat}).

In the next subsection, we give a toric description of $\Mod$,
based on the formalism developed in the last section,
which elucidates the structure of the \Ka moduli space 
associated with the toric quotient 
$\A{\GIT}(\C^*)^{n-1}(\vr)$, 
as well as provides us with an efficient method
to compute the \config\ space $\Mod$
for any $\vr\in \Q^{n-1}$.








% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\subsection{Toric Description of the D-Brane Configuration Space}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
According to (\ref{solution-to-F}), we propose the following
toric description of $\A$ \cite{infirri}~:
Let $\Mtot$ be a lattice of rank $nd$
generated by $\ve_{\mu}^{(i)}$, 
$0\leq i \leq n-1$, $1\leq \mu\leq d$,
and $\Msub$ be the sublattice of rank $(n-1)(d-1)$
of $\Mtot$ generated by
% - - - - - - - - - - - - - - - -
\begin{equation}
\vf^{(i)}_{\mu}:=\ve_{\mu}^{(i)}-\ve_{\mu}^{(0)}
+\sum_{j=1}^{i+\amu}\ve_{d}^{(j)}
-\sum_{j=1}^{i}\ve_{d}^{(j)}
-\sum_{j=1}^{\amu}\ve_{d}^{(j)},
\quad  \mu\ne d,\ i\ne 0,
%1\leq \mu\leq d-1,
%\ 1\leq i\leq n-1,
\end{equation}
%- - - - - - - - - - - - - - - - -
with the injection $j:\Msub\ra \Mtot$.
Let $M=\Mtot/\Msub$
be the quotient lattice  
of rank $n-1+d$ and
$p:\Mtot\ra M$ be the projection.
If we define $\cone$ to be the basic simplicial cone 
in $\Mtot_{\Q}$, that is,
%   -   -   -   -   -   -   -   -
\begin{equation}
\left.
\cone=\pos 
\left\{\skima
\ve_{\mu}^{(i)}
\right|\ 
1\leq \mu\leq d,\ 
0\leq i\leq n-1
\right\},
\end{equation}
%   -   -   -   -   -   -   -   -
then its $p_{\Q}$-image
$P:=p_{\Q}(\cone)$ is a cone in $M_{\Q}$ 
and we have~\cite{infirri}
%-  -  -  -  -  -  -  - 
\begin{equation}
\A=X(M,P)=\Spec \left(P\cap M\right).
\end{equation}
%-  -  -  -  -  -  -  -
The D-brane \config\ space $\Mod$ 
can be realized as the toric quotient 
of $\A$ as follows \cite{infirri}~:
Let $M^{'}\subset \Z^n$ be 
the lattice of rank $n-1$ 
generated by $\ve_i-\ve_{i+1}$, 
$1\leq i \leq  n-1$,
where $(\ve_i)$ is the generators of $\Z^n$,
and $\pi^{'}:\Mtot\ra M^{'}$ the lattice projection
% - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\pi^{'}(\ve_{\mu}^{(i)}):=\ve_i-\ve_{i+\amu},
\end{equation}
% - - - - - - - - - - - - - - - - - - - - -
which is determined according to the $\text{U}(1)^n$
charge assignment of $x_{\mu}^{(i)}$.
It is easily seen that $\pi^{'}$ factors through
$p$, that is,
there is a projection 
$\pi:M\ra M^{'}$ such that
$\pi^{'}=\pi\circ p$. 
Finally we define a sublattice of rank $d$ 
of $M$ by
$\Mb:= \Ker \pi
\cong \Ker  \pi^{'}/\text{Im}\ j$. 
Note that $\pi_{\Q}(P)=M^{'}_{\Q}$.
Then we have
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\Mod:=X(M,P)\GIT T^{'}(\vr)=X(\Mb,\Qr),
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
where $T^{'}$ is the subtorus of $T$ associated with
the sublattice $N^{'}=(M^{'})^*\subset N=M^*$, 
and we regard the Fayet--Iliopoulos parameter 
$\vr$ as a point of $M^{'}_{\Q}$.
We can obtain the fan of $\Mod$ as the normal fan of
the $d$-polyhedron $\Qr$.

%- - - - - - - - - - - - - - - - - - - -
\begin{flushleft}
{\it Remark.} 
{\sl It may be confusing to have two lattices of rank $d$,
both of which are associated with the \config\ space $\Mod$~:
%  -  -  -  -  -  -  -  -  -  -  -  -  
$\Mb$ symbolically represents the lattice for a general
quotient toric variety (\ref{exact-M})~;
%  -  -  -  -  -  -  -  -  -  -  -  -  -  -  - 
on the other hand, $\Mb_0$, 
which was originally introduced 
as a useful lattice to describe the orbifold 
$\CY$ in (\ref{M-lattice}), 
is also suited for its blow-up $\Mod$.
%  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
Our intention is that  
we use $\Mb_0$ for the concrete descriptions 
of the toric data of $\Mod$ below.}     
\end{flushleft}
%- - - - - - - - - - - - - - - - - - - - 
\vspace{1cm}

%% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% 
%{\small
\setlength{\unitlength}{1mm}
\begin{picture}(115,95)(0,15)
\put(81,43){$\pi$}        
\put(85,55){$\pi^{'}$}       
\put(66,51){$p$}       
\put(67,73){$j$}           
\put(59,43){$i$}             
\put(26,40){0}
\put(31,41){\vector(1,0){15}}
\put(48,40){$\Mb$}            
\put(53,41){\vector(1,0){15}}
\put(70,40){$M$}                       
\put(75,41){\vector(1,0){15}}
\put(92,40){$M^{'}$}                  
\put(97,41){\vector(1,0){15}}
\put(114,40){0}
\put(72,38){\vector(0,-1){15}}
\put(71,18){0}
\put(70,62){$\Mtot$}                     
\put(72,60){\vector(0,-1){15}}
\put(70,84){$\Msub$}
\put(72,82){\vector(0,-1){15}}
\put(72,105){\vector(0,-1){15}}
\put(71,107){0}
\put(77,60){\vector(1,-1){15}}
\end{picture}
%}    
% the tail of {\small.... 
%
\begin{center}
{\bf Figure 1.} \ Sequences of Lattices
\end{center}
%\caption{Sequenses of lattices}
%\label{SES}
%\end{figure}
%% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% 
%

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\subsection{Some Properties of the \Ka Moduli Space}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
We define the action of a generator of $\Ga$ 
on the Chan--Paton indices by
$\g(i):=i+1\pmod{n}$, 
which can be extended to an action of $\Ga$ 
as an automorphism on each lattice shown in Figure 1
in such a manner that any lattice 
homomorphism in Figure 1 becomes $\Ga$-equivariant,
which we denote by
%   -   -   -   -   -   -   -   -   -   -   -
$\g^{(1)},\g^{(0)},\g,\g^{'},\overline{\g}$
for 
$\Msub,\Mtot,M,M^{'},\Mb$ 
%   -   -   -   -   -   -   -   -   -   -   -
respectively. 
For example, the action on the generators of $\Mtot$ 
reads as follows~:
%- - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
\g^{(0)}(\ve_{\mu}^{(i)})=\ve_{\mu}^{(i+1)},
\label{Ga-action}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - -
while the action on those of $\Msub$ is
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\g^{(1)}(\vf_{\mu}^{(i)})
=\vf_{\mu}^{(i+1)}-\vf_{\mu}^{(1)}.
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
The \Ka moduli space of $1/n(a_1,\dots,a_d)$,
which we denote by
%     -       -      -     -      -     -
$\kahler^n(a_1,\dots,a_d)$,
%
is the complete fan in $M^{'}_{\Q}$
obtained as the subdivision of $M^{'}_{\Q}$ induced by 
the $\pi_{\Q}$-images of the faces 
of the cone $P$ in $M_{\Q}$.

The Propositions 3.1--3 stated below 
are immediate consequences of our definitions~:
%%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  
\begin{flushleft}
{\it Proposition 3.1.}
{\sl If $\{a_1,\dots, a_d\}
=\{b_1,\dots, b_e \}$
as {\it sets}, 
then the two models
$1/n(a_1,\dots,a_d)$ and 
$1/n(b_1,\dots, b_e)$
have the {\it same} \Ka moduli space, that is,
%- - - - - - - - - - - - - - - - - - - - 
\begin{equation}
\kahler^n(a_1,\dots,a_d)
=\kahler^n(b_1,\dots, b_e),
\label{reduction}
\end{equation}
%- - - - - - - - - - - - - - - - - - - -
where the two models above need not 
necessarily satisfy the Calabi--Yau condition.}
\end{flushleft}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 

We say that the $d$-fold model
$1/n(a_1,\dots,a_d)$ can be reduced to $e$~dimensions,
when (\ref{reduction}) occurs with $d>e$.

\begin{flushleft}
{\it Example.}
We have the reductions of the Calabi--Yau four-fold models
to two~dimensions according to the following identifications~:
%- - - - - - - - - - - - - - - - - - - - - 
\begin{align}
\kahler^m(1,1,m-1,m-1)&=
\kahler^m(1,m-1),     \label{4-1}      \\
%-   -   -   -   -   -   -   -   -   -   -
\kahler^{3m+1}(1,1,1,3m-2)&=
\kahler^{3m+1}(1,3m-2), \label{4-2}         \\
%-   -   -   -   -   -   -   -   -   -   -
\kahler^{4m}(1,1,2m-1,2m-1)&=
\kahler^{4m}(1,2m-1). \label{4-3}
\end{align}
%- - - - - - - - - - - - - - - - - - - - - 
\end{flushleft}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % %

A toric two-fold has the virtue
that the listing of the 1-cones alone 
determines its fan.
The four-fold models entering in (\ref{4-1}--\ref{4-3}) 
inherit this property
{}from the corresponding two-fold models,
which considerably simplifies 
the analysis of the \Ka moduli space 
of these four-fold models. 

Let us take $1/m(1,m-1)$ model.
The maximal chambers of the \Ka moduli space 
$\kahler^m(1,m-1)$ can identified 
with the Weyl chambers of $\text{SU}(m)$ \cite{kronheimer},
in which  the D-brane \config\ space
$\Mod$ is in the minimal blow-up phase.  
Correspondingly, the phase of the four-fold
$1/m(1,1,m-1,m-1)$ 
in the Weyl chambers turns out to be 
the non-Calabi--Yau smooth phase 
with Euler number $4(m-1)$,
the fan of which is given 
by the following collection of the maximal cones~:
%- - - - - - - - - - - - - - - - - - - - - - - -
\begin{align}
&\langle 1,3,4,5 \rangle, \ 
\langle 2,3, 4,5 \rangle, \  
\langle 1,2, 3,m+3 \rangle, \  
\langle 1,2, 4,m+3 \rangle,
 \\ 
%  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 1,3,l,l+1 \rangle,  \  
\langle 1,4,l,l+1 \rangle,   \ 
\langle 2,3, l,l+1 \rangle,   \  
\langle 2,4,l,l+1 \rangle,
\ \  5\leq l\leq m+2,\nonumber
\end{align}
%- - - - - - - - - - - - - - - - - - - - - - - -
where the weight vectors above are given by
%- - - - - - - - - - - - - - - - - - - - - - - - -
\begin{align}
&\vwb_1=(m,0,0,0), \ 
\vwb_2=(0,m,0,0), \ 
\vwb_3=(0,0,m,0), \ 
\vwb_4=(0,0,0,m), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\vwb_l=(l-4,l-4,m+4-l,m+4-l), \qquad 5\leq l \leq m+3.
\end{align}
%- - - - - - - - - - - - - - - - - - - - - - - - - -
Here our convention for the expression  
of the maximal cone \cite{HLY} is~: 
%- - - - - - - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
\langle s_1,\dots,s_k\rangle:=
\pos \{\skima
\vwb_{s_1},\dots,\vwb_{s_k}\}\subset (\Nb_0)_{\Q}.
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - - -
Brute force calculations for $m=2,3$ cases 
can be found in \cite[Section 6.2]{mohri}.

As for $1/(4m)(1,1,2m-1,2m-1)$ model and
$1/(3m+1)(1,1,1,3m-2)$ model,
the D-brane \config\ space $\Mod$
of the four-fold model is 
in the smooth Calabi--Yau phase
if and only if 
that of the corresponding two-fold model is 
in the minimal blow-up phase~;
the former is
in the non-Calabi--Yau smooth phases 
if and only if the latter is
in the non-minimal blow-up phases. 
%-  -  -  -  -  -  -  -  -  -  -  -

In the same way, a {\it two-parameter model}~: 
$1/n(1,\dots,1,a,b)$ treated in \cite{DHH}
is one which can be reduced to three dimensions.




% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\begin{flushleft}
{\it Proposition 3.2.}
{\sl The polyhedron $P$ admits an action of $\Ga$, that is,
$\g_{\Q}(P)=P$.}

{\it Corollary 3.2.1.} 
{\sl The set of the facets of $P$,
which we previously denoted by 
$L(P)^{(1)}=\{ \athfacet|\ a\in \facetset \}$,
are decomposed into $\Ga$-orbits.}
\end{flushleft}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
Within each $\Ga$-orbit, the facets share 
a common weight vector for the quotient toric variety.
Each model has the following $d$ $\Ga$-singlets
%- - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
{\cal F}_{\mu}:=
\pos
\left.
\left\{
\skima \vp_{\nu}^{(i)}
\right|\ 
\nu\ne \mu
\right\},
\quad 1\leq \mu \leq d,
\end{equation}
%- - - - - - - - - - - - - - - - - - - - -
where we set $p(\ve_{\mu}^{(i)})=\vp_{\mu}^{(i)}\in M$ 
for simplicity. 
The weight vector associated with ${\cal F}_{\mu}$ is
$\vwb_{\mu}=n\ve_{\mu}$ for $1\leq \mu\leq d$.

The remaining $\Ga$-orbits are denoted by
%- - - - - - - - - - - - - - - - -
\begin{equation}
\left.
\left\{ 
{\cal F}_k^{(j)}:=\g_{\Q}^j
\left(
{\cal F}_{k}^{(0)}
\right)
\right|\ 
0\leq j \leq m_k-1
\right\},
\quad k\geq d+1,
\label{Ga-orbit}
\end{equation} 
%- - - - - - - - - - - - - - - - - 
where $m_k$ is the length of the $k$~th $\Ga$-orbit,
%that is,
%$m_k$ is the least positive integer among those 
%that satisfy
%$\g_{\Q}^{m}\left({\cal F}_{k}^{(0)}\right)
%={\cal F}_k^{(0)}$,
and we denote the weight vector associated with
the $k$~th orbit by $\vwb_k$, 
which we call the $k$~th exceptional divisor.
% % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\begin{flushleft}
{\it Example.}
%
For $1/5(1,2,3,4)$ model, the exceptional divisors are
%- - - - - - - - - - - - - - - - - -  - -
\begin{alignat}{2}
\vwb_5   &=(1,2,3,4), &\qquad
\vwb_6   &=(2,4,1,3)\nonumber \\
% -  -  -  -  -  -  -  -  -  -  
\vwb_7   &=(3,1,4,2), &\qquad
\vwb_8   &=(4,3,2,1)\nonumber \\
%-  -  -  -  -  -  -  -  -  -  -
\vwb_9   &=(4,3,2,6), &\qquad
\vwb_{10}&=(2,4,6,3)\nonumber \\
%-  -  -  -  -  -  -  -  -  -  -
\vwb_{11}&=(3,6,4,2), &\qquad
\vwb_{12}&=(6,2,3,4).
\end{alignat} 
%- - - - - - - - - - - - - - - - - - - - - - 
To describe $k$~th $\Ga$-orbit, 
it suffices to show its 0~th member
${\cal F}_k^{(0)}$ as in (\ref{Ga-orbit}).
Then we have for the age 2 exceptional divisors
%- - - - - - - - - - - - -  - - - - - - 
\begin{align}
{\cal F}_{5}^{(0)}&=
\pos
\left\{
\skima
\vp_{1}^{(1)},
\vp_{1}^{(2)},
\vp_{1}^{(3)},
\vp_{1}^{(4)},
\vp_{2}^{(1)},
\vp_{2}^{(2)},
\vp_{2}^{(3)},
\vp_{3}^{(1)},
\vp_{3}^{(2)},
\vp_{4}^{(1)}
\right\},\\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
{\cal F}_{6}^{(0)}&=
\pos
\left\{
\skima
\vp_{1}^{(1)},
\vp_{1}^{(3)},
\vp_{1}^{(4)},
\vp_{2}^{(3)},
\vp_{3}^{(1)},
\vp_{3}^{(2)},
\vp_{3}^{(3)},
\vp_{3}^{(4)},
\vp_{4}^{(1)},
\vp_{4}^{(3)}
\right\},\\
%-  -  -  -  -  -  -  -  -  -  -  -
{\cal F}_{7}^{(0)}&=
\pos
\left\{
\skima
\vp_{1}^{(0)},
\vp_{1}^{(2)},
\vp_{2}^{(0)},
\vp_{2}^{(1)},
\vp_{2}^{(2)},
\vp_{2}^{(4)},
\vp_{3}^{(0)},
\vp_{4}^{(0)},
\vp_{4}^{(2)},
\vp_{4}^{(4)}
\right\},\\
%-  -  -  -  -  -  -  -  -  -  -
{\cal F}_{8}^{(0)}&=
\pos
\left\{
\skima
\vp_{1}^{(0)},
\vp_{2}^{(0)},
\vp_{2}^{(4)},
\vp_{3}^{(0)},
\vp_{3}^{(3)},
\vp_{3}^{(4)},
\vp_{4}^{(0)},
\vp_{4}^{(2)},
\vp_{4}^{(3)},
\vp_{4}^{(4)}
\right\},
\end{align}
%- - - - - - - - - - - - - - 
and for the age 3 exceptional divisors
\begin{align}
{\cal F}_{9}^{(0)}&=
\pos
\left\{
\skima
\vp_{1}^{(2)},
\vp_{1}^{(4)},
\vp_{2}^{(1)},
\vp_{2}^{(4)},
\vp_{3}^{(0)},
\vp_{3}^{(2)},
\vp_{3}^{(4)},
\vp_{4}^{(4)}
\right\},\\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
{\cal F}_{10}^{(0)}&=
\pos
\left\{
\skima
\vp_{1}^{(2)},
\vp_{1}^{(3)},
\vp_{1}^{(4)},
\vp_{2}^{(2)},
\vp_{2}^{(3)},
\vp_{3}^{(2)},
\vp_{4}^{(1)},
\vp_{4}^{(2)}
\right\},\\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
{\cal F}_{11}^{(0)}&=
\pos
\left\{
\skima
\vp_{1}^{(0)},
\vp_{1}^{(1)},
\vp_{2}^{(0)},
\vp_{3}^{(0)},
\vp_{3}^{(4)},
\vp_{4}^{(0)},
\vp_{4}^{(3)},
\vp_{4}^{(4)}
\right\},\\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
{\cal F}_{12}^{(0)}&=
\pos
\left\{
\skima
\vp_{1}^{(0)},
\vp_{2}^{(0)},
\vp_{2}^{(2)},
\vp_{2}^{(4)},
\vp_{3}^{(0)},
\vp_{3}^{(3)},
\vp_{4}^{(0)},
\vp_{4}^{(2)}
\right\}.
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
\end{align}
%- - - - - - - - - - - - - - - - - - -
The action of $\Ga$ on a facet is as follows~:
%- - - - - - - - - - - - - - - -
\begin{equation}
\g_{\Q}\left({\cal F}_{5}^{(0)}\right)
%={\cal F}_{5}^{(1)}
=\pos
\left\{
\skima
\vp_{1}^{(2)},
\vp_{1}^{(3)},
\vp_{1}^{(4)},
\vp_{1}^{(0)},
\vp_{2}^{(2)},
\vp_{2}^{(3)},
\vp_{2}^{(4)},
\vp_{3}^{(2)},
\vp_{3}^{(3)},
\vp_{4}^{(2)}
\right\}.
\end{equation}
%- - - - - - - - - - - - - - - - 
We see that the length of each $\Ga$-orbits above is {\tt 5}.
\end{flushleft}
\begin{flushleft}
{\it Example.} For the case of $(1/12)(1,1,5,5)$ model,
the exceptional divisors and 
the lengths of the $\Ga$-orbits are as follows~:
%- - - - - - - - - - - - - -
\begin{alignat}{2}
&\vwb_5=(1,1,5,5): &\quad &\text{\tt 12}\nonumber \\
&\vwb_6=(3,3,3,3): &\quad &\text{\tt 12}+\text{\tt 12}
               +\text{\tt 12}+\text{\tt 4}\nonumber \\
&\vwb_7=(5,5,1,1): &\quad &\text{\tt 12} \\
&\vwb_8=(4,4,8,8): &\quad 
&\text{\tt 6}+\text{\tt 6}+\text{\tt 3}\nonumber \\
&\vwb_9=(8,8,4,4): &\quad 
&\text{\tt 6}+\text{\tt 6}+\text{\tt 3}.\nonumber
\end{alignat}
%- - - - - - - - - - - - - -
The representatives of $\Ga$-orbits 
for the crepant divisors are
%- - - - - - - - - - - - - - -  -
\begin{align}
{\cal F}_{5}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(1)},
\vp_{\mu}^{(2)},
\vp_{\mu}^{(3)},
\vp_{\mu}^{(4)},
\vp_{\mu}^{(5)},
\vp_{\mu}^{(6)},
\vp_{\mu}^{(7)},
\vp_{\mu}^{(8)},
\vp_{\mu}^{(9)},
\vp_{\mu}^{(10)},
\vp_{\mu}^{(11)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&\vp_{\nu}^{(1)},
\vp_{\nu}^{(2)},
\vp_{\nu}^{(4)},
\vp_{\nu}^{(6)},
\vp_{\nu}^{(7)},
\vp_{\nu}^{(9)},
\vp_{\nu}^{(11)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\}\nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -
{}^1{\cal F}_{6}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(0)},
\vp_{\mu}^{(1)},
\vp_{\mu}^{(2)},
\vp_{\mu}^{(5)},
\vp_{\mu}^{(6)},
\vp_{\mu}^{(8)},
\vp_{\mu}^{(9)},
\vp_{\mu}^{(10)},
\vp_{\mu}^{(11)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&\vp_{\nu}^{(1)},
\vp_{\nu}^{(2)},
\vp_{\nu}^{(4)},
\vp_{\nu}^{(5)},
\vp_{\nu}^{(6)},
\vp_{\nu}^{(8)},
\vp_{\nu}^{(9)},
\vp_{\nu}^{(10)},
\vp_{\nu}^{(11)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\}\nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -
{}^2{\cal F}_{6}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(3)},
\vp_{\mu}^{(4)},
\vp_{\mu}^{(5)},
\vp_{\mu}^{(6)},
\vp_{\mu}^{(7)},
\vp_{\mu}^{(8)},
\vp_{\mu}^{(9)},
\vp_{\mu}^{(10)},
\vp_{\mu}^{(11)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&\vp_{\nu}^{(1)},
\vp_{\nu}^{(2)},
\vp_{\nu}^{(3)},
\vp_{\nu}^{(4)},
\vp_{\nu}^{(6)},
\vp_{\nu}^{(7)},
\vp_{\nu}^{(8)},
\vp_{\nu}^{(9)},
\vp_{\nu}^{(11)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\}\nonumber  \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -
{}^3{\cal F}_{6}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(0)},
\vp_{\mu}^{(2)},
\vp_{\mu}^{(3)},
\vp_{\mu}^{(6)},
\vp_{\mu}^{(7)},
\vp_{\mu}^{(8)},
\vp_{\mu}^{(9)},
\vp_{\mu}^{(10)},
\vp_{\mu}^{(11)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&\vp_{\nu}^{(0)},
\vp_{\nu}^{(2)},
\vp_{\nu}^{(3)},
\vp_{\nu}^{(5)},
\vp_{\nu}^{(6)},
\vp_{\nu}^{(7)},
\vp_{\nu}^{(8)},
\vp_{\nu}^{(10)},
\vp_{\nu}^{(11)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\} \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -
{}^4{\cal F}_{6}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(0)},
\vp_{\mu}^{(1)},
\vp_{\mu}^{(2)},
\vp_{\mu}^{(4)},
\vp_{\mu}^{(5)},
\vp_{\mu}^{(6)},
\vp_{\mu}^{(8)},
\vp_{\mu}^{(9)},
\vp_{\mu}^{(10)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&\vp_{\nu}^{(0)},
\vp_{\nu}^{(1)},
\vp_{\nu}^{(2)},
\vp_{\nu}^{(4)},
\vp_{\nu}^{(5)},
\vp_{\nu}^{(6)},
\vp_{\nu}^{(8)},
\vp_{\nu}^{(9)},
\vp_{\nu}^{(10)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\}\nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -
{\cal F}_{7}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(5)},
\vp_{\mu}^{(6)},
\vp_{\mu}^{(7)},
\vp_{\mu}^{(8)},
\vp_{\mu}^{(9)},
\vp_{\mu}^{(10)},
\vp_{\mu}^{(11)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&
\vp_{\nu}^{(1)},
\vp_{\nu}^{(2)},
\vp_{\nu}^{(3)},
\vp_{\nu}^{(4)},
\vp_{\nu}^{(5)},
\vp_{\nu}^{(6)},
\vp_{\nu}^{(7)},
\vp_{\nu}^{(8)},
\vp_{\nu}^{(9)},
\vp_{\nu}^{(10)},
\vp_{\nu}^{(11)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\},\nonumber 
%-  -  -  -  -  -  -  -  -  -  -  -  -  -
\end{align}
%- - - - - - - - - - - - - - - - - 
while those for the age 2 divisors are
%- - - - - - - - - - - - - - -  -
\begin{align}
{}^1{\cal F}_{8}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(0)},
\vp_{\mu}^{(1)},
\vp_{\mu}^{(2)},
\vp_{\mu}^{(4)},
\vp_{\mu}^{(6)},
\vp_{\mu}^{(7)},
\vp_{\mu}^{(8)},
\vp_{\mu}^{(10)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&\vp_{\nu}^{(0)},
\vp_{\nu}^{(4)},
\vp_{\nu}^{(6)},
\vp_{\nu}^{(10)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\}\nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -
{}^2{\cal F}_{8}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(0)},
\vp_{\mu}^{(3)},
\vp_{\mu}^{(4)},
\vp_{\mu}^{(5)},
\vp_{\mu}^{(6)},
\vp_{\mu}^{(9)},
\vp_{\mu}^{(10)},
\vp_{\mu}^{(11)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&\vp_{\nu}^{(2)},
\vp_{\nu}^{(3)},
\vp_{\nu}^{(8)},
\vp_{\nu}^{(9)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\}\nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -
{}^3{\cal F}_{8}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(0)},
\vp_{\mu}^{(1)},
\vp_{\mu}^{(3)},
\vp_{\mu}^{(4)},
\vp_{\mu}^{(6)},
\vp_{\mu}^{(7)},
\vp_{\mu}^{(9)},
\vp_{\mu}^{(10)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&\vp_{\nu}^{(0)},
\vp_{\nu}^{(3)},
\vp_{\nu}^{(6)},
\vp_{\nu}^{(9)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\} \\
%-  -  -  -  -  -  -  -  -  -  -  -
{}^1{\cal F}_{9}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(3)},
\vp_{\mu}^{(4)},
\vp_{\mu}^{(9)},
\vp_{\mu}^{(10)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&\vp_{\nu}^{(0)},
\vp_{\nu}^{(1)},
\vp_{\nu}^{(2)},
\vp_{\nu}^{(3)},
\vp_{\nu}^{(6)},
\vp_{\nu}^{(7)},
\vp_{\nu}^{(8)},
\vp_{\nu}^{(9)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\}\nonumber  \\
%-  -  -  -  -  -  -  -  -  -  -  -
{}^2{\cal F}_{9}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(0)},
\vp_{\mu}^{(2)},
\vp_{\mu}^{(6)},
\vp_{\mu}^{(8)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&\vp_{\nu}^{(0)},
\vp_{\nu}^{(2)},
\vp_{\nu}^{(4)},
\vp_{\nu}^{(5)},
\vp_{\nu}^{(6)},
\vp_{\nu}^{(8)},
\vp_{\nu}^{(10)},
\vp_{\nu}^{(11)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\}\nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -
{}^3{\cal F}_{9}^{(0)}&=
\pos
\left.
\left\{ \skima
\begin{aligned}
&\vp_{\mu}^{(2)},
\vp_{\mu}^{(5)},
\vp_{\mu}^{(8)},
\vp_{\mu}^{(11)}\\
%   -   -   -   -   -   -   -   -   -   -   -   -
&\vp_{\nu}^{(1)},
\vp_{\nu}^{(2)},
\vp_{\nu}^{(4)},
\vp_{\nu}^{(5)},
\vp_{\nu}^{(7)},
\vp_{\nu}^{(8)},
\vp_{\nu}^{(10)},
\vp_{\nu}^{(11)}
\end{aligned} 
\right|
\begin{gathered}
 \mu=1,2\\
\nu=3,4
\end{gathered}
\right\}. \nonumber
%-  -  -  -  -  -  -  -  -  -  -  -
\end{align}
%- - - - - - - - - - - - - - - - - - - - -
Note that the reducibility of this model
to two dimensions (\ref{4-3})
is reflected in the structure of the facets of $P$.
\end{flushleft}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %













% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\begin{flushleft}
{\it Proposition 3.3.}
{\sl $\Ga$ acts on $M^{'}_{\Q}$ 
as an symmetry of the toric quotient~:
\begin{equation}
{\cal M}(\vr)\cong {\cal M}(\g^{'}_{\Q}(\vr)), 
\quad \vr\in M^{'}_{\Q}.
\end{equation}}
\end{flushleft}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 






%\single
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\subsection{Phases of Calabi--Yau Four-Fold Models}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
Here we describe some typical phases of
Calabi--Yau four-fold  models, 
leaving the cases of $d=3$ for the reader's exercise.

In this subsection, we identify 
$M^{'}
=\{\skima \vr\in \Z^n |\sum_{i=0}^{n-1}\skima r_i=0 \}$ 
with $\Z^{n-1}$ by discarding its zeroth component $r_0$.
%  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  
\subsubsection{(1/12)(1,1,5,5) model}
First of all, we need to choose the representative
of the facets for each weight vector $\vwb_k$ 
for $k=5,\dots,9$, which we denote by ${\cal F}_k$, 
so that they are compatible, that is, 
%- - - - - - - - - -
\begin{equation}
\bigcap_{k=5}^9 \pi_{\Q}\left({\cal F}_k\right)
%\subset M^{'}_{\Q}
\end{equation}
%- - - - - - - - - -
is a $4$ dimensional cone in $M^{'}_{\Q}$. 
Our choice is as follows~:
%- - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
{\cal F}_5={\cal F}_5^{(3)},\quad
{\cal F}_6={}^4{\cal F}_6^{(0)},\quad
{\cal F}_7={\cal F}_7^{(1)},\quad   
{\cal F}_8={}^1{\cal F}_8^{(0)},\quad
{\cal F}_9={}^2{\cal F}_9^{(0)}.
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - -
Consider the following candidates of the phases 
realized in maximal cones of the \Ka moduli space
$\kahler^{12}(1,1,5,5)$~:
\vspace{0.5cm}

phase I ($\Sigma_{\text{I}}$)
%- - - - - - - - - - - - - - - 
\begin{alignat}{4}
&\langle 1,2,3,7 \rangle, &\quad
&\langle 1,2,4,7 \rangle, &\quad
&\langle 1,3,4,5  \rangle, &\quad
&\langle 2,3,4,5  \rangle, \nonumber \\
%-   -   -   -   -   -   -   -   -
&\langle 2,4,5,6 \rangle, &\quad
&\langle 2,4,6,7  \rangle, &\quad
&\langle 1,4,6,7  \rangle, &\quad
&\langle 2,3,6,7  \rangle,           \\
%-   -   -   -   -   -   -   -   -
&\langle 1,4,5,6  \rangle, &\quad
&\langle 1,3,6,7  \rangle, &\quad
&\langle 2,3,5,6  \rangle, &\quad
&\langle 1,3,5,6  \rangle. \nonumber 
\end{alignat} 
%- - - - - - - - - - - - - - - - - - -

phase II ($\Sigma_{\text{II}}$)
%- - - - - - - - - - - - - - - 
\begin{alignat}{4}
&\langle 1,2,3,7 \rangle, &\quad
&\langle 1,2,4,7 \rangle, &\quad
&\langle 1,3,4,5  \rangle, &\quad
&\langle 2,3,4,5  \rangle, \nonumber \\
%-   -   -   -   -   -   -   -   -
&\langle 2,4,6,8 \rangle, &\quad
&\langle 1,4,6,7  \rangle, &\quad
&\langle 2,4,5,8  \rangle, &\quad
&\langle 1,4,5,8  \rangle,      \\
%-   -   -   -   -   -   -   -   -
&\langle 1,4,6,8  \rangle, &\quad
&\langle 1,3,5,8  \rangle, &\quad
&\langle 2,3,6,8  \rangle, &\quad
&\langle 1,3,6,8  \rangle, \nonumber \\ 
%-   -   -   -   -   -   -   -   -
&\langle 2,4,6,7  \rangle, &\quad
&\langle 2,3,5,8  \rangle, &\quad
&\langle 2,3,6,7  \rangle, &\quad
&\langle 1,3,6,7  \rangle. \nonumber 
\end{alignat} 
%- - - - - - - - - - - - - - - - - - -



phase III ($\Sigma_{\text{III}}$)
%- - - - - - - - - - - - - - - 
\begin{alignat}{4}
&\langle 1,2,3,7 \rangle, &\quad
&\langle 1,2,4,7 \rangle, &\quad
&\langle 1,3,4,5  \rangle, &\quad
&\langle 2,3,4,5  \rangle, \nonumber \\
%-   -   -   -   -   -   -   -   -
&\langle 1,4,6,9 \rangle, &\quad
&\langle 2,4,5,6  \rangle, &\quad
&\langle 2,4,6,9  \rangle, &\quad
&\langle 2,4,7,9  \rangle,      \\
%-   -   -   -   -   -   -   -   -
&\langle 1,4,7,9  \rangle, &\quad
&\langle 1,3,6,9  \rangle, &\quad
&\langle 1,3,7,9  \rangle, &\quad
&\langle 1,4,5,6  \rangle, \nonumber \\ 
%-   -   -   -   -   -   -   -   -
&\langle 2,3,7,9  \rangle, &\quad
&\langle 2,3,6,9  \rangle, &\quad
&\langle 2,3,5,6  \rangle, &\quad
&\langle 1,3,5,6  \rangle. \nonumber 
\end{alignat} 
%- - - - - - - - - - - - - - - - - - -

phase IV ($\Sigma_{\text{IV}}$)
%- - - - - - - - - - - - - - - 
\begin{alignat}{4}
&\langle 1,2,3,7 \rangle, &\quad
&\langle 1,2,4,7 \rangle, &\quad
&\langle 1,3,4,5  \rangle, &\quad
&\langle 2,3,4,5  \rangle, \nonumber \\
%-   -   -   -   -   -   -   -   -
&\langle 2,4,6,8 \rangle, &\quad
&\langle 2,4,5,8  \rangle, &\quad
&\langle 2,4,6,9  \rangle, &\quad
&\langle 1,4,6,9  \rangle,  \nonumber \\
%-   -   -   -   -   -   -   -   -
&\langle 2,4,7,9  \rangle, &\quad
&\langle 1,4,6,8  \rangle, &\quad
&\langle 1,4,7,9  \rangle, &\quad
&\langle 1,3,6,9  \rangle,           \\
%-   -   -   -   -   -   -   -   -
&\langle 2,3,7,9 \rangle, &\quad
&\langle 1,3,7,9  \rangle, &\quad
&\langle 1,4,5,8  \rangle, &\quad
&\langle 2,3,6,9  \rangle, \nonumber \\
%-   -   -   -   -   -   -   -   -
&\langle 2,3,6,8  \rangle, &\quad
&\langle 2,3,5,8  \rangle, &\quad
&\langle 1,3,5,8  \rangle, &\quad
&\langle 1,3,6,8   \rangle. \nonumber 
\end{alignat} 
%- - - - - - - - - - - - - - - - - - -
Here $\Sigma_{\text{I--IV}}$ means 
the corresponding fan.
The phase I is the smooth Calabi--Yau phase~;
the phase II--IV are non-Calabi--Yau smooth phases,
which are blow-ups of the phase I.

%  blow-downs 
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\setlength{\unitlength}{1mm}
\begin{picture}(80,40)(5,5)
\put(9,19){IV}
\put(29,10){III}
\put(29,28){II}
\put(50,19){I}
\put(14,23){\vector(2,1){13}}
\put(14,19){\vector(2,-1){13}}
\put(35,29){\vector(2,-1){13}}
\put(35,12){\vector(2,1){13}}
\end{picture}

%\begin{center}
{\bf Figure 2.}  Blow-Down Diagram
%\end{center}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\vspace{0.5cm}

Each of the fans $\Sigma_{\text{I--IV}}$ 
defines a coherent subset $S_{\text{I--IV}}$ of 
$L(P)^{(4)}$, the set of the codimension four faces of $P$.
Then according to (\ref{result}),
we can construct the maximal cones
$K_{\text{I--IV}}:=K(S_{\text{I--IV}})$ 
of the \Ka moduli space $\kahler^{12}(1,1,5,5)$.
The result is as follows~:
%- - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
K_{\text{I}}=\pos 
\left\{
\begin{gathered}
-\ve_3+\ve_4-\ve_7+\ve_8-\ve_{11},\ 
-\ve_4+\ve_9,\
-\ve_8+\ve_9,\
-\ve_7    \\
%-   -   -   -   -   -   -
-\ve_3+\ve_8,\ 
-\ve_3+\ve_4,\
-\ve_1+\ve_2,\
-\ve_4+\ve_5,\
-\ve_5+\ve_{10}\\
%-   -   -   -   -   -   -   -
-\ve_{11},\ 
-\ve_7+\ve_9-\ve_{11},\ 
-\ve_3+\ve_6,\
-\ve_4+\ve_6-\ve_8+\ve_9 \\
%-   -   -   -   -   -   -   
-\ve_7+\ve_{10},\
-\ve_1+\ve_2-\ve_4+\ve_6,\
-\ve_5+\ve_6-\ve_8+\ve_{10}\\
%-   -   -   -   -   -   -
-\ve_1+\ve_2-\ve_5+\ve_6-\ve_9+\ve_{10},\ 
-\ve_3+\ve_5-\ve_7+\ve_8\\
%-   -   -   -   -   -   -   -   -
-\ve_1+\ve_6,\
-\ve_4+\ve_5-\ve_7+\ve_9
%-   -   -   -    -   -
\end{gathered}
\right\}.
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
K_{\text{II}}=\pos 
\left\{
\begin{gathered}
-\ve_7,\
-\ve_3+\ve_4-\ve_5+\ve_6-\ve_9+\ve_{10}-\ve_{11},\
-\ve_3+\ve_6 \\
%-   -    -   -   -   -   -   -   -   -   - 
-\ve_5+\ve_6-\ve_8+\ve_{10},\
-\ve_1+\ve_2-\ve_5+\ve_6-\ve_9+\ve_{10} \\
%-   -   -   -   -   -    -   -   -   -   -
-\ve_3+\ve_8,\
-\ve_3+\ve_4,\
-\ve_5+\ve_{10},\
-\ve_9+\ve_{10},\
-\ve_1+\ve_6    \\
%-   -   -   -   -   -   -   -   -   -   -
-\ve_{11},\
-\ve_3+\ve_4-\ve_7+\ve_8-\ve_{11}
\end{gathered}
\right\}.
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
K_{\text{III}}=\pos 
\left\{
\begin{gathered}
-\ve_{11},\
-\ve_1+\ve_2,\
-\ve_1+\ve_2-\ve_3+\ve_6-\ve_7+\ve_8-\ve_9 \\
%-   -   -   -   -   -   -   -   -   -   -   -
-\ve_1+\ve_2-\ve_4+\ve_6,\
-\ve_1+\ve_2-\ve_5+\ve_6-\ve_9+\ve_{10} \\
%-   -   -   -   -   -   -   -   -   -   -   -
-\ve_4+\ve_5,\
-\ve_3+\ve_8,\
-\ve_3+\ve_4,\
-\ve_1+\ve_6,\
-\ve_3+\ve_6   \\
%-   -   -   -   -   -   -   -
-\ve_7,\
-\ve_3+\ve_5-\ve_7+\ve_8,\
-\ve_3+\ve_4-\ve_7+\ve_8-\ve_{11}
%-   -   -   -   -   -   -   -   -   -
\end{gathered}
\right\}.
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
K_{\text{IV}}=\pos 
\left\{
\begin{gathered}
-\ve_{11},\
-\ve_3+\ve_8,\
-\ve_3+\ve_4-\ve_5+\ve_6-\ve_9+\ve_{10}-\ve_{11}\\
%-   -   -   -   -   -   -   -   -   -   -   -   -
-\ve_7,\
-\ve_3+\ve_4,\
-\ve_1+\ve_2-\ve_3+\ve_6-\ve_7+\ve_8-\ve_9\\
%-   -   -   -   -   -   -   -   -   -   -   -   - 
-\ve_1+\ve_2-\ve_5+\ve_6-\ve_9+\ve_{10},\
-\ve_1+\ve_6,\
-\ve_3+\ve_6  \\
%-   -   -   -   -   -   -   -   -   -   -   -
-\ve_9+\ve_{10},\
-\ve_3+\ve_4-\ve_7+\ve_8-\ve_{11}
\end{gathered}
\right\}.
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - -

%  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  %  
\subsubsection{1/5(1,2,3,4) model}
Let us first choose the following
representatives for the $\Ga$-orbits~: 
%- - - - - - - - - - - - - - - - - - - -
\begin{alignat}{4}
{\cal F}_{5}&={\cal F}_{5}^{(4)},  &\quad
{\cal F}_{6}&={\cal F}_{6}^{(2)},  &\quad
{\cal F}_{7}&={\cal F}_{7}^{(0)},  &\quad
{\cal F}_{8}&={\cal F}_{8}^{(0)}, \nonumber \\
%-   -   -   -   -   -   -   -   -   -
{\cal F}_{9} &={\cal F}_{9}^{(1)},   &\quad
{\cal F}_{10}&={\cal F}_{10}^{(3)},  &\quad
{\cal F}_{11}&={\cal F}_{11}^{(0)},  &\quad
{\cal F}_{12}&={\cal F}_{12}^{(0)},
\label{choice-1}
\end{alignat}
%- - - - - - - - - - - - - - - -
which satisfy the compatibility condition~:
%- - - - - - - - - - - - - - - - - - - -
\begin{equation}
\bigcap_{k=5}^{12}\pi_{\Q}
\left({\cal F}_{k}\right)
=\pos
\{\skima 
-\ve_1,\ 
-\ve_2,\
-\ve_3,\
-\ve_4
\}
\label{zentaide}
\end{equation}
%- - - - - - - - - - - - - - - - - -
If we define the phase I by
\vspace{0.5cm}

phase I ($\Sigma_{\text{I}}$)
%- - - - - - - - - - - - - - - - - - 
\begin{align}
&\langle 2,3,10,11  \rangle,  \quad
\langle 1,4,7,12  \rangle,   \quad
\langle 3,5,7,10  \rangle,   \quad
\langle 2,3,8,11  \rangle,   \quad
\langle 1,7,8,12  \rangle,   \nonumber \\
%-   -   -   -   -   -   -
&\langle 2,6,8,11  \rangle,   \quad
\langle 2,3,4,5  \rangle,    \quad
\langle 1,3,7,8  \rangle,    \quad
\langle 2,4,5,6  \rangle,    \quad
\langle 1,2,6,8  \rangle,\nonumber \\
%-   -   -   -   -   -   -
&\langle 1,4,6,9  \rangle,   \quad
\langle 1,2,4,6  \rangle,   \quad
\langle 1,3,4,7  \rangle,   \quad
\langle 1,2,3,8  \rangle,   \quad
\langle 1,4,9,12 \rangle,\nonumber \\
%-   -   -   -   -   -   -
&\langle 3,4,5,7   \rangle,  \quad
\langle 2,3,5,10  \rangle,  \quad
\langle 4,5,6,9   \rangle,   \\
%-   -   -   -   -   -   -   -   -   -
&\langle 3,7,8,10,11 \rangle,   \quad
\langle 2,5,6,10,11  \rangle,  \quad
\langle 4,5,7,9,12  \rangle,   \quad
\langle 1,6,8,9,12  \rangle,\nonumber \\
%-   -   -   -   -   -   -   -   -   -   -
&\langle 5,6,7,8,9,10,11,12  \rangle, \nonumber
\label{hilb-1234-5}
\end{align}
%- - - - - - - - - - - - - - - - - - - - - - -
the associated cone 
$K_{\text{I}}:=K(S_{\text{I}})$ 
coincides with (\ref{zentaide}).
Therefore the phase I is the only possible phase
under the choice of the representatives 
for the exceptional divisors (\ref{choice-1}).
\vspace{0.5cm}

%  %  %  %  %  %  %%  %  %  %  %  %  %%  %  %  %  %  %  %
The second choice of the compatible representatives
for the exceptional divisors is~:
%- - - - - - - - - - - - - - - - - - - -
\begin{alignat}{4}
{\cal F}_{5}&={\cal F}_{5}^{(4)}, &\quad
{\cal F}_{6}&={\cal F}_{6}^{(2)}, &\quad
{\cal F}_{7}&={\cal F}_{7}^{(0)}, &\quad
{\cal F}_{8}&={\cal F}_{8}^{(1)}, \nonumber \\
%-   -   -   -   -   -   -   -   -   -
{\cal F}_{9} &={\cal F}_{9}^{(1)},  &\quad
{\cal F}_{10}&={\cal F}_{10}^{(3)}, &\quad
{\cal F}_{11}&={\cal F}_{11}^{(1)}. &\quad
\phantom{{\cal F}_{12}}&\phantom{={\cal F}_{12}^{(0)}.}
\label{choice-2}
\end{alignat}
%- - - - - - - - - - - - - - - -
Note the absence of the facet associated with
$\vwb_{12}$ in (\ref{choice-2}).


Consider the following two phases
\vspace{0.5cm}

phase II ($\Sigma_{\text{II}}$)
%- - - - - - - - - - - - - -  - - - - - 
\begin{align}
&\langle 5,8,10,11   \rangle,\quad
\langle 2,3,4,5  \rangle,\quad
\langle 3,4,5,7   \rangle,\quad
\langle 4,5,6,8   \rangle, \nonumber \\
%-   -   -   -   -   -   -  
&\langle 1,2,3,8   \rangle,\quad
\langle 2,3,8,11   \rangle,\quad
\langle 1,4,5,8   \rangle,\quad
\langle 3,8,10,11   \rangle,\nonumber   \\
%-   -   -   -   -   -   -  
&\langle 2,5,10,11   \rangle,\quad
\langle 1,4,6,8   \rangle,\quad
\langle 1,3,4,7   \rangle,\quad
\langle 2,3,10,11   \rangle,  \\
%-   -   -   -   -   -   -  
&\langle 1,2,6,8   \rangle,\quad
\langle 2,3,5,10   \rangle,\quad
\langle 2,4,5,6   \rangle,\quad
\langle 3,5,7,10   \rangle, \nonumber   \\
%-   -   -   -   -   -   -  
&\langle 1,4,5,7   \rangle,\quad
\langle 1,2,4,6   \rangle, \nonumber \\
%-   -   -   -   -   -   -  
&\langle 1,3,7,8,10   \rangle,\quad
\langle 1,5,7,8,10   \rangle,\quad
\langle 2,5,6,8,11   \rangle.\nonumber
%-   -   -   -   -   -   -  
\end{align} 
%- - - - - - - - - - - - - - - -  - - - - - -

phase III ($\Sigma_{\text{III}}$)
%- - - - - - - - - - - - - - - - - - - - -
\begin{align}
&\langle 3,8,10,11   \rangle,\quad
\langle 3,5,7,10   \rangle,\quad
\langle 1,2,3,8   \rangle,\quad
\langle 2,3,4,5   \rangle,  \nonumber \\
%-   -   -   -   -   -   -  
&\langle 2,3,8,11   \rangle,\quad
\langle 1,2,6,8   \rangle,\quad
\langle 5,8,10,11   \rangle,\quad
\langle 2,3,10,11   \rangle, \nonumber \\
%-   -   -   -   -   -   -  
&\langle 2,5,10,11   \rangle,\quad
\langle 1,2,4,6   \rangle,\quad
\langle 3,4,5,7   \rangle,\quad
\langle 4,5,6,9   \rangle,  \\
%-   -   -   -   -   -   -  
&\langle 1,4,5,9   \rangle,\quad
\langle 2,3,5,10   \rangle,\quad
\langle 2,4,5,6   \rangle,\quad
\langle 1,3,4,7   \rangle,\nonumber  \\
%-   -   -   -   -   -   -  
&\langle 1,4,6,9   \rangle,\quad
\langle 1,4,5,7   \rangle, \nonumber \\
%-   -   -   -   -   -   -  
&\langle 2,5,6,8,11  \rangle,\quad
\langle 1,3,7,8,10   \rangle,\quad
\langle 1,5,7,8,10   \rangle,\quad
\langle 1,5,6,8,9   \rangle.\nonumber
%-   -   -   -   -   -   -  
\end{align}
%- - - - - - - - - - - - - - - - - - - - - - - 
Using (\ref{result}), we see that 
these two phases can be realized in the maximal cones
$K_{\text{II}}$ and $K_{\text{III}}$ defined by
%- - - - - - - - - - - - - - -  - - - - - - -
\begin{gather}
K_{\text{II}}=\pos
\left\{
-\ve_3,\ 
\ve_1-\ve_3-\ve_4,\
\ve_1-\ve_2-\ve_3-\ve_4,\
\ve_1-\ve_2-\ve_3
\right\},\\
%-   -   -   -   -   -   -   -   -   -
K_{\text{III}}=\pos
\left\{
-\ve_3,\ 
\ve_1-\ve_3-\ve_4,\
\ve_1-\ve_2-\ve_3-\ve_4,\
-\ve_4
\right\}.\phantom{-\ve_2-\ve_3}
\end{gather}
%- - - - - - - - - - - - - - - - - - - - - -
So far we have seen only the phases the fan of which 
is not simplicial, which means that the singularity of 
$\Mod$ is worse than orbifold ones in these phases.

In fact, combinatorics of the facets of $P$
admits neither 
the smooth phases incorporating 
two age 3 divisors, for example,
%- - - - - - - - - -
\begin{alignat}{5}
&\langle 3,7,8,10  \rangle,   &\quad
&\langle 5,6,8,11  \rangle,   &\quad
&\langle 3,8,10,11  \rangle,   &\quad
&\langle 2,5,10,11  \rangle,   &\quad
&\langle 2,3,4,5    \rangle,   \nonumber \\
%-   -   -   -   -   -   -   -   -   -
&\langle 1,4,7,8  \rangle,   &\quad
&\langle 2,4,5,6  \rangle,   &\quad
&\langle 1,2,4,6  \rangle,   &\quad
&\langle 3,4,5,7  \rangle,   &\quad
&\langle 4,5,7,8  \rangle,   \nonumber \\
%-   -   -   -   -   -   -   -   -   -
&\langle 5,8,10,11  \rangle,   &\quad
&\langle 5,7,8,10  \rangle,   &\quad
&\langle 4,5,6,8  \rangle,   &\quad
&\langle 2,5,6,11  \rangle,   &\quad
&\langle 2,3,8,11  \rangle,   \\
%-   -   -   -   -   -   -   -   -   -
&\langle 1,4,6,8  \rangle,   &\quad
&\langle 2,6,8,11  \rangle,   &\quad
&\langle 1,3,4,7  \rangle,   &\quad
&\langle 2,3,10,11  \rangle,   &\quad
&\langle 2,3,5,10  \rangle,   \nonumber \\
%-   -   -   -   -   -   -   -   -   -
&\langle 3,5,7,10  \rangle,   &\quad
&\langle 1,3,7,8  \rangle,   &\quad
&\langle 1,2,6,8  \rangle,   &\quad
&\langle 1,2,3,8  \rangle,   &\quad
&\phantom{\langle 1,1,1,1\rangle,}\nonumber
%-   -   -   -   -   -   -   -   -   -
\end{alignat}
%  %  %  %  %  %  %  %  %  %  %  %  %  %  %  % 
nor the phase with the simplicial fan incorporating
all the exceptional divisors~:
%- - - - - - - -- 
\begin{alignat}{5}
&\langle 1,4,7,12  \rangle,   &\quad
&\langle 1,4,9,12  \rangle,   &\quad
&\langle 8,9,10,11  \rangle,   &\quad
&\langle 5,6,9,11  \rangle,   &\quad
&\langle 4,7,9,12  \rangle,   \nonumber \\
%-   -   -   -   -   -   -   -   -   -
&\langle 2,5,10,11  \rangle,   &\quad
&\langle 1,7,8,12  \rangle,   &\quad
&\langle 7,8,9,12  \rangle,   &\quad
&\langle 1,8,9,12  \rangle,   &\quad
&\langle 2,3,4,5  \rangle,   \nonumber \\
%-   -   -   -   -   -   -   -   -   -
&\langle 1,3,4,5  \rangle,   &\quad
&\langle 1,3,4,7  \rangle,   &\quad
&\langle 1,2,4,6  \rangle,   &\quad
&\langle 3,4,5,7  \rangle,   &\quad
&\langle 2,3,8,11  \rangle,   \nonumber \\
%-   -   -   -   -   -   -   -   -   -
&\langle 4,5,7,9  \rangle,   &\quad
&\langle 2,6,8,11  \rangle,   &\quad
&\langle 1,4,6,9  \rangle,   &\quad
&\langle 6,8,9,11  \rangle,   &\quad
&\langle 4,5,6,9  \rangle,   \\
%-   -   -   -   -   -   -   -   -   -
&\langle 2,3,10,11  \rangle,   &\quad
&\langle 3,8,10,11  \rangle,   &\quad
&\langle 5,9,10,11  \rangle,   &\quad
&\langle 3,7,8,10  \rangle,   &\quad
&\langle 7,8,9,10  \rangle,   \nonumber \\
%-   -   -   -   -   -   -   -   -   -
&\langle 5,7,9,10  \rangle,   &\quad
&\langle 2,5,6,11  \rangle,   &\quad
&\langle 2,3,5,10  \rangle,   &\quad
&\langle 3,5,7,10  \rangle,   &\quad
&\langle 1,6,8,9  \rangle,   \nonumber \\
%-   -   -   -   -   -   -   -   -   -
&\langle 1,2,6,8  \rangle,   &\quad
&\langle 1,2,3,8  \rangle,   &\quad
&\langle 1,3,7,8  \rangle,   &\quad
&\phantom{\langle 1,1,1,1  \rangle,}   &\quad
&\phantom{\langle 1,1,1,1  \rangle,}\nonumber
%-   -   -   -   -   -   -   -   -   -
\end{alignat}
%- - - - - - - - - - - - - - - - - - - - - - - 
the fans of which are found by the Delaunay triangulations 
\cite{Del}, \cite[p.~146]{ziegler} of $6T$. 













\single
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\section{$\boldsymbol\Ga$-Hilbert Scheme}
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
%     enumerate       (i),(ii),(iii),....
\renewcommand{\theenumi}{\roman{enumi}}
\renewcommand{\labelenumi}{(\theenumi)}
%% % % % % % % % % % % % % % % % % % % % % % % % % % % 


\subsection{Symplectic Quotient Construction}
%
% % % %  rough algebraic definition of Hilb  % % % % % % % % % % % 
%
Let $X$ be a quasi-projective variety with a fixed embedding
in a projective space.
The Hilbert scheme $\text{Hilb}^P(X)$ is 
the moduli space that parametrizes all the closed subschemes
of $X$ with a fixed Poincar\'e polynomial $P(z)$,
where $P(l)\in {\Z}$, for all $l\in {\Z}$.
See \cite{hilbert},\  \cite[Chapter I]{kollar}
for more detailed informations.

Let us take the following pair~:
$X=\C^d$, $P(z)=n$ (constant),
and consider the moduli space of
zero-dimensional closed subschemes
of length $n$ in $\C^d$,
which we denote by $\Hn$
\cite{ito-nakajima,ito-nakamura,%ito-nakamura-2,ito-nakamura-3,
nakajima,reid}.
A point $Z\in \Hn$ corresponds 
to a ideal $I\subset A$
of colength $n$, where
%-  -  -  -  -  -  -  -  -  -  - 
$A=:{\C}[x_1,\dots ,x_d]$
%-  -  -  -  -  -  -  -  -  -  -
is the coordinate ring of $\C^d$.
Therefore we have
\begin{equation}
\Hn=\left\{\skima \mathrm{ideal}\ I\subset A \left|\
\dim_{\C} A/I = n  \right. \right\}.     
\label{Hilbn}
\end{equation}
For $d,n \geq 3$, $\Hn$ is a singular variety.

The action of $\Ga$ on ${\C}^d$ 
is naturally extended to
that on $\Hn$.
Let $(\Hn)^{\Ga}$ be the subset of 
$\Hn$ which is fixed by the action of  
$\Ga$.
Each  point of $(\Hn)^{\Ga}$ corresponds
to a $\Ga$-invariant ideal $I$ of $A$. 
Consequently, for  $I\in (\Hn)^{\Ga}$,
$A/I=H^0(Z,{\cal O}_Z)$ becomes 
a $\Ga$-module of rank $n$.

For example, $\Ga$-orbit of a point 
$p\in \C^d\setminus\bold{0}$ 
is a point of $(\Hn)^{\Ga}$,  and it
constitutes the regular representation
$R$ (\ref{regular-rep}) of $\Ga$.

Now we give a definition of the $\Ga$-Hilbert scheme
following \cite{ito-nakajima}~:
%- - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
\Hilb:=
\left.
\left\{
I\in (\Hn)^{\Ga} 
\right|\ 
 A/I\cong R\ 
\right\},
 \label{Gamma-Hilb}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - 
which means that the $\Ga$-Hilbert scheme $\Hilb$
parametrizes all the zero-dimensional closed 
subschemes $Z\subset {\C}^d$ such that
$H^0(Z,{\cal O}_Z)$ is isomorphic to the
regular representation $R$ of $\Ga$.
%
The mathematical aspect  
of the $\Ga$-Hilbert scheme $\Hilb$ 
for $d=2,3$ has been largely uncovered~:
For $d=2$, $\hilb^{\Ga}(\C^2)$ is a minimal resolution 
of the singularity $1/m(1,m-1)$ \cite{ito-nakamura}~;
What is more, it has been shown that even for $d=3$, 
$\hilb^{\Ga}(\C^3)$ is a crepant resolution 
of the Calabi--Yau three-fold singularity $\C^3/\Ga$
by I.~Nakamura in \cite{nakamura}, despite of the fact 
that $\hilb^n(\C^3)$ itself is {\it singular}.

Thus our interest here is also
concentrated on the $d=4$ case.
We will show later that $\hilb^{\Ga}(\C^4)$ 
is singular in general.

The definition of Hilbert schemes $\Hn$ and $\Hilb$ 
given above may seem abstract.
However Y.~Ito and H.~Nakajima has shown that
they can be realized as holomorphic (GIT)/symplectic 
quotients of flat spaces associated with the gauge group 
$\mathrm{U}(n)$ and $\mathrm{U}(1)^{n}$ 
respectively \cite{ito-nakajima,nakajima},
that is, we can identify $\Hn$ and $\Hilb$ 
as the classical Higgs moduli spaces 
of supersymmetric gauge theories \cite{MP,witten1}.
In particular $\Hilb$ can be described as a toric variety.
Furthermore, it is isomorphic to the D-brane \config\ space 
$\Mod$ with a particular choice 
of the Fayet--Iliopoulos parameter $\vr\in M^{'}_{\Q}$
\cite{ito-nakajima},
which is the main point of this subsection.

% % % % % % %   Construction of Hilb^n    % % % % % % % % % % % 
Let us first explain a holomorphic quotient construction of
$\Hn$. 
Fix $I\in \Hn$ and let $V=A/I$ be the associated
$n$ dimensional vector space.
The multiplication of $x_{\mu}$ on $V$ 
defines $d$-tuple of the elements of $\text{End}(V)$
which we denote by $X_{\mu}$.
%
To be more explicit, we choose an arbitrary basis  
$\vep_i$, $i=1,\dots ,n$ of $V$,
%as a basis of $A/I$ %over $\C$ 
and we define the matrix elements of $X_{\mu}$ by
$(x_{\mu}+I)\cdot \vep_i=\sum_{j=1}^{n}(X_{\mu})^j_i\ \vep_j$.
If we also define a basis of $\C^d$ by $\vbeta_{\mu}$,
$\mu=1,\dots ,d$,
then we define an element $X$ 
of $\Hom(V,\C^d\otimes V)$ 
by
%- - - - - - - - - - - - - - - - - - - 
\begin{equation}
X(\vep_i):=
\sum_{\mu=1}^d \sum_{j=1}^n 
\vbeta_{\mu}\otimes \vep_j\ 
(X_{\mu})^j_i.
\end{equation}
%- - - - - - - - - - - - - - - - - - - 
Similarly the image of the map 
$1\hookrightarrow A\ra A/I$
defines a non-zero element of $V$ which we denote by
$p(1)=\sum_{i=1}^np^i\ \vep_i$,
where we mean by $p$
the associated element of $\Hom(\C,V)$, that is,\ 
$p(\la):=\la\ p(1)$\  for $\la \in \C$.

It is clear by construction that $(X,p)$ satisfies 
the following two conditions~:
%- - - - - - - - - - - - - - - - -
\begin{enumerate}
\item $\left[X_{\mu}, X_{\nu}\right]=O$    \qquad
(F-flatness).
%-   -  -  -  -  -  -  -  -  -  -  -  -  -  
\item 
$p(1)$ is a cyclic vector, that is,
$V$ is generated by $X_{\mu}$ over $p(1)$ 
\ \ (stability). 
\end{enumerate}
%- - - - - - - - - - - - - - - - - - -

Conversely let $\Vect$ be the vector space
$\Hom(\C^n,\C^d\otimes \C^n)\oplus \Hom(\C,\C^n)$,
and take such an element  $(X,p)\in \Vect$
that satisfies the above two conditions (i), (ii) with $V=\C^n$.
Then $(X,p)$ defines a point $I\in\Hn$ as follows~: 
First we define a surjective homomorphism 
$\kappa: A \ra {\C}^n$ 
of vector spaces over $\C$ by
%- - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\kappa(x_{\mu_1}\cdots x_{\mu_s}):=
X_{\mu_1}\cdots X_{\mu_s}\cdot p(1) %\nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
=\sum_{i_1,\dots ,i_s,j}
\vep_{i_1}\skima (X_{\mu_1})^{i_1}_{i_2}\skima 
(X_{\mu_2})^{i_2}_{i_3}\skima \cdots  \skima
(X_{\mu_s})^{i_s}_{j}
\medspace p^j, %\in \C^n,
\end{equation}
%- - - - -  - - - - - - - - - - - - - - - - - - - - 
where we define
%  -  -  -  -  -  -  -
$X_{\mu}(\vep_i):=\sum_{j=1}^n \vep_j\ (X_{\mu})^j_i$,
%
and 
%  -  -  -  -  -  -  -
$p(1):=\sum_{i=1}^n p^i\skima \vep_i$ 
%  -  -  -  -  -  -  -
for a basis $\vep_i$ of $\C^n$~; 
Second  let  $I:=\Ker \kappa \subset A$
be an ideal of $A$,
then $A/I\cong {\C}^n$ as a vector space, 
which implies $I\in\Hn$ according to (\ref{Hilbn})~;
Third, it is clear that two elements 
 $(X,p)$ and  $(X^{'},p^{'})$ of $\Vect$
define the same point of $\Hn$
if and only if 
%- - - - - - - - - - - - - - - -
\begin{equation*}
(X^{'},p^{'})=(g X g^{-1},gp),\quad
\exists g\in \text{GL}(n,{\C}).
\end{equation*}
%- - - - - - - - - - - - - - -- 
Thus we have arrived at the following
holomorphic quotient construction  of $\Hn$~:
%- - - - - - - - - - - - - - - - - - - - - - - - - 
\begin{equation}
\Hn\cong 
\left\{
(X,p) \in \Vect\ 
\left|\ 
% -  -  -  -  -  -  -  -  -  -  -  -  -
\begin{aligned}
&\text{condition (i) : F-flatness}\\
%-   -   -   -   -   -   -   -   -   -   -
&\text{condition (ii) : stability}
\end{aligned}
% -   -  -  -  -  -  - -  -  -  -  -  -
\right\}
\right/
\text{GL}(n,\C).
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - 
We can also obtain 
the corresponding 
symplectic quotient construction of $\Hn$
by replacing the stability condition (ii) and
the quotient by $\text{GL}(n,\C)$ above
by the D-flatness condition
% % % % % % % % % %  D-flatness  % % % % % % % % % % % % % % % %
\begin{equation}
D_{\kahilb}:=\sum_{\mu=1}^{d}
\left[X_{\mu},X_{\mu}^{\dagger}\right]
+p\cdot p^{\dagger}
-\kahilb\skima \text{diag}(1,\dots ,1)
%----------------------
%\begin{pmatrix}
% 1   &  {}    &   {} \\
% {}  & \ddots &   {} \\
% {}  &  {}    &    1
%\end{pmatrix}
%----------------------
=O,
\label{D-flat}
\end{equation}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
and the quotient by $\text{U}(n)$,
where $\kahilb>0$
is a unique Fayet--Illiopoulos parameter
associated with the $\text{U}(1)$ factor of
$\text{U}(n)$.
%Needless to say, 
%If we stick to obtain a quasi-projective variety,
%$\xi$ must be quantized, that is, $\xi\in \N$.

If we set $\kahilb=0$,
we obtain the symmetric product  
$(\C^d)^n/{\frak S}_n$ as a quotient variety 
reflecting the existence of the Hilbert-Chow morphism
$\Hn\ra (\C^d)^n/{\frak S}_n$.
\vspace{1cm}
% % % % % % % %  $\Ga$-Hilbert scheme  % % % % % % % % % % % % 

Let us turn to the holomorphic quotient construction
of $\Hilb$ based on that of $\Hn$ given above.
The only difference {}from the previous treatment 
of $\Hn$ is that this time 
we must assign the $\Ga$-quantum numbers
to the objects : $x_{\mu}$, 
$\vep_i$, $\vbeta_{\mu}$ and $p$.
However we would not mind repeating 
almost the same argument for convenience.

Let us first redefine the generator $g$ of $\Ga$ 
so that the action of which on $x_{\mu}$ becomes
$g\cdot x_{\mu}= \omega^{-\amu}x_{\mu}$
for consistency.
Second take a point $I \in \Hilb$ and 
define  $V=A/I$, which is now isomorphic to
the regular representation
$R$ as a $\Ga$-module by definition.
%
Let $\vep_i \in V$ be a generator of
$R_i$ for $i=1,\dots ,n$, that is,
$g\cdot \vep_i=\omega^i \vep_i$ and 
$V=\bigoplus_{i=0}^{n-1}\C \ \vep_i$ 
is the irreducible decomposition of $\Ga$-modules.

We also introduce somewhat abstractly 
$\vbeta_{\mu}$
as a generator of $R_{\amu}$  
for $\mu=1,\dots ,d$ 
and let
$Q=\bigoplus_{\mu=1}^d \C\  \vbeta_{\mu}$ 
be a $\Ga$-module.

Then we define a $\Ga$-equivariant homomorphism
{}from $V$ to $Q\otimes V$, which we call $X$, by
%- - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
X : f\in V \ra 
\sum_{\mu=1}^d 
\vbeta_{\mu}\otimes (f\cdot x_{\mu})\in Q\otimes V,
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - -
where the product of polynomials $f\cdot x_{\mu}$
is evaluated modulo $I$.
In particular $\mu$~th component 
of $X(\vep_{i+\amu})$ becomes
%- - - - - - - - - - - - - - - - - -
\begin{equation}
x_{\mu}\cdot \vep_{i+\amu} =(X_{\mu})^{i}_{i+\amu} \ 
 \vep_{i}, \quad \exists (X_{\mu})^{i}_{i+\amu} \in \C.
\end{equation}
%- - - - - - - - - - - - - - - - - - - 
Thus we get the matrices $(X_{\mu})$ 
of the same content as
those for a D-brane at the orbifold singularity.
%
The map $1\hookrightarrow A\ra A/I$ now induces
an element $0\ne p\in \Hom_{\Ga}(\C,V)$, where
$p(1)=p^0\skima \vep_0\in V$.
Thus an element $I\in \Hilb$ defines an element
$(X,p)\in
\Hom_{\Ga}(V,Q\otimes V)
\oplus \Hom_{\Ga}(\C,V)$ 
and it clear  by construction that $(X,p)$ satisfies
the conditions (i) (F-flatness) and (ii) (stability) above.

Conversely %let
%$Q=\bigoplus_{\mu=1}^d \C\ \vbeta_{\mu}$ %and
%$R=\bigoplus_{i=1}^n \C\ \vep_i$ 
%be an irreducible decomposition of a $\Ga$-module
%and 
take an element $(X,p)$ of
$\Vect^{\skima\Ga}:=\Hom_{\Ga}(R, Q\otimes R)
\oplus \Hom_{\Ga}(\C,R)$
such that $(n,n)$ matrices $(X_{\mu})$ 
and $(n,1)$ matrix $p(1)$ 
defined by
\begin{equation*}
X(\vep_i)=\sum_{\mu=1}^d %\bigoplus_{i=1}^n 
(X_{\mu})^{i-\amu}_i\ 
\vbeta_{\mu}\otimes \vep_{i-\amu},\qquad 
p(1)=p^0 \ \vep_0,
\end{equation*}
satisfies the conditions (i), (ii) with $V=R$.
Then we can define a $\Ga$-equivariant
surjective homomorphism $\kappa :A\ra R$ by
%- - - - - - - - - - - - - - - - - - - -
\begin{equation}
\kappa(x_{\mu_1}\cdots x_{\mu_s})
:=X_{\mu_1}\cdots X_{\mu_s}\cdot p(1)
%\nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -
=\sum_{i_1,\dots ,i_s}
\skima \vep_{i_1} \skima (X_{\mu_1})^{i_1}_{i_2} 
\skima 
(X_{\mu_2})^{i_2}_{i_3}\skima \cdots \skima
(X_{\mu_s})^{i_s}_{0} \medspace p^0,
\end{equation}
%- - - - - - - - - - - - - - - - - - - - -
so that the ideal 
$I:=\Ker \kappa$ 
is $\Ga$-invariant
and we obtain the $\Ga$-module isomorphism
$A/I\cong R$, that is, $(X,p)$ defines 
an element $I$ of $\Hilb$.
With the basis $\vbeta_{\mu}$ of $Q$ fixed, 
two elements 
$(X,p)$ and $(X^{'},p^{'})$ 
of
$\Vect^{\skima\Ga}$ 
which satisfy the conditions (i) and (ii)
define the same point $I\in \Hilb$ if and only if
they are related  as
$(X^{'},p^{'})=(uX u^{-1},up)$
by an element 
%-  -  -  -  -  -  -  -  -  -  -  -  -
$u:=(u_i) \in \Aut_{\Ga}(R)
\cong \prod_{i=1}^n\Aut(R_i)
\cong (\C^*)^n$,
%-  -  -  -  -  -  -  -  -  -  -  -  -
where $u_i$ acts on $\vep_i$ by
$\vep_i\ra u_i^{-1}\ \vep_i$.

Thus  we get the holomorphic quotient construction
of $\Hilb$~:
% % % %   Holomorphic Quotient of $\Hilb$   % % % % % % % % % % % %
\begin{equation}
\Hilb\cong 
\left\{ 
(X,p)\in \Vect^{\skima\Ga}
\left|\ 
%- - - - - - - - - - - -
\begin{aligned}
&\text{condition (i) :  F-flatness}\\
%-   -   -   -   -   -   -
&\text{condition (ii) :  stability}
\end{aligned}
%- - - - - - - - - - - -
\right\}
\right/
\prod_{i=1}^n\Aut(R_i),
\end{equation}
% % % % % %  % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
which in particular shows that $\Hilb$ is toric.
The associated symplectic quotient can be obtained  
by replacing the stability condition (ii) 
and the quotient by $\prod_{i}\Aut(R_i)$
by the D-flatness condition 
that takes the {\em same form} as (\ref{D-flat}) 
followed by the quotient 
by $\prod_{i}\text{U}(R_i)\cong \text{U}(1)^n$.
%
Consequently, the Fayet--Iliopoulos parameters 
associated with $\text{U}(1)^n$ is $\kahilb (1,\dots ,1)$. 


                                                      
To sum up, 
we have the symplectic quotient realization of $\Hilb$~: 
% % % % %           symplectic quotient   % % % % % % % % % % % % % %
\begin{equation}
\Hilb\cong
\left\{
(X,p)\in \Vect^{\skima\Ga}
\left|\ 
%- - - - - - - - - - - - - - - - - - - - -
\begin{aligned}
&\text{F-flatness :}\quad
\left[X_{\mu},X_{\nu}\right]=O \\
%-   -   -   -   -   -   -   -
&\text{D-flatness :}
\quad  D_{\kahilb}=O 
\end{aligned}
%- - - - - - - - - - - - - - - - - - - - -
\right\}
\right/
\text{U}(1)^n. 
\label{symplectic}
\end{equation}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
The relation between $\Hilb$ and $\Mod$ can be easily seen
if we write down the D-flatness equations
for $\Hilb$ in components~:
%- - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
\sum_{\mu=1}^{d}
\left(
|x_{\mu}^{(i)}|^2-|x_{\mu}^{(i-\amu)}|^2
\right)+|p^0|^2\  \delta^{i,0}
 \ = \ \kahilb, \quad i=0,1,\dots ,n-1,
\end{equation}
%- - - - - - - - - - - - - - - - - - - - - - - - - -
where we set
$x_{\mu}^{(i)}:=(X_{\mu})^{i}_{i+\amu}$ 
as before (\ref{as-before}).
We can delete $p^0$ and the diagonal $\text{U}(1)$
{}from the symplectic quotient construction
owing to the Higgs mechanism \cite{ito-nakajima}~: 
%- - - - - - - - - - - - - - -
\begin{equation}
|p^0|^2=n\skima \kahilb.
\end{equation}
%- - - - - - - - - - - - - - -
Then we are left with
the matrices $(X_{\mu})$,
which satisfy the same equations
as those of D-brane matrices 
(\ref{M-F-flat},\skima\ref{M-D-flat})
with the Fayet--Illiopulos parameter
%- - - - - - - - - - - - - - - - 
\begin{equation}
(r_0,r_1,\dots ,r_{n-1})
=\kahilb\skima (-(n-1),1,\dots ,1)\in M^{'}_{\Q}.
\end{equation}
%- - - - - - - - - - - - - - - -
Thus we come to the conclusion~:
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\begin{equation}
\Hilb\cong {\cal M}(\kahilb\skima\bold{1}),\quad
%\text{where}\
\bold{1}:=(\overbrace{1,\dots ,1}^{n-1}),
\label{conc1}
\end{equation}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
where we have identified 
$M^{'}$ with $\Z^{n-1}$ by neglecting the zeroth component.
We also note the existence of 
the Hilbert-Chow morphism 
$\Hilb\ra \CY$,
which comes {}from the isomorphism~:
${\cal M}(\bold{0})\cong \CY$ \cite{sardo}.


% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\subsection{Another Algorithm for Computation}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
The aim of this subsection is to translate 
the algorithm to compute the $\Ga$-Hilbert scheme 
given by Reid in \cite{reid}, which seems quite different
from the one given in the previous subsection,
into the language of convex polyhedra.
Closely related topics can be found in \cite{AGV,sturmfels}. 

%   %   %   %   %   %   %   %   %   %   %   %   %   %   %   %   %   %  
Let $A=\C[x_1,\dots,x_d]$ the coordinate ring
of $\C^d$, where $g\in \Ga$ acts on $x_{\mu}$ as
multiplication by $\omega^{\amu}$, which defines 
the action of $\Ga$ on $\C^d$ from the {\it right}.
For $i=0,\dots, n-1$, we define ${\cal L}_i$ 
to be the ``orbifold line bundle'' on $\CY$
associated with the irreducible representation $R_{-i}$ 
of $\Ga$ where $g\in \Ga$ acts as multiplication 
by $\omega^{-i}$.
The global section of ${\cal L}_i$  is 
$\left(R_{-i}\otimes A\right)^{\Ga}$, that is,
the weight $i$ subspace of $A$.
Note that as a $\Ga$-module,
$A\cong \Sym Q=\oplus_{n=0}^{\infty}\skima S^n Q$.
The set of the monomial generators over $\C$ 
of $\left(R_{-i}\otimes A\right)^{\Ga}$,
which we denote by $M_i$, is given by
% - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
M_i=
\left\{ 
\vm\in ({\Z}_{\geq 0})^d 
\left| \  
\vm\cdot \va \equiv i \mod{n}
\right.
\right\},
%\subset {\N}^{d},
\end{equation}
% - - - - - - - - - - - - - - - - - - - - - - - - - - - 
where $\vm =(m_1,\dots ,m_d)$ and $\va =(a_1,\dots ,a_d)$.
%
$M_0$ coincides with the coordinate ring 
of $\CY$, and each $M_i$ has a structure 
of a finitely generated $M_0$-module, 
the set of the generators of which
we denote by $B_i$.
%
Let $P_i=\conv  M_i$ 
be the {\it Newton polyhedron}
of the global monomial sections of ${\cal L}_i$,
which can be regarded as a polyhedron in $(\Mb_0)_{\Q}$,
where the lattice $\Mb_0$ is defined in (\ref{M-lattice}).
Then the toric variety $X(\Mb_0,P_i)$
defines the blow-up of $\CY=X(\Mb_0,P_0)$ 
by ${\cal L}_i$, 
which is denoted by $\text{Bl}_i(\CY)$. 
The normal fan ${\cal N}(P_i)$ in 
$(\Nb_0)_{\Q}$ (\ref{N-lattice})
is the fan associated with $\text{Bl}_i(\CY)$.
%
Evidently, $P_i$ can be 
expressed as the Minkowski sum
of the polytope $\conv  B_i$ 
and the cone $C_0=P_0$ (\ref{M-cone}).
%
On the other hand,
a celebrated theorem of E.~Noether adapted to 
$1/n(a_1,\dots ,a_d,n-i)$ model,
which is not Calabi--Yau, tells us
that all the members of $B_i$ can be found  
among those in $M_i$ of degree $\leq n$,
which implies the following way 
to construct the Newton polyhedron $P_i$ 
without any knowledge of $B_i$~: 
%- - - - - - - - - - - - - - - - - - -
\begin{equation}
P_i\cong \conv  B_i^{'}+C_0,
\qquad
B^{'}_i:=
\big\{
\vm \in M_i
\big| \ 
\sum_{\mu=1}^d m_{\mu}\leq n
\big\}
%\right.
\supset B_i.
\end{equation}
%- - - - - - - - - - - - - - - - - -
%%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  
\begin{flushleft}
{\it Example.}
We take $1/5(1,2,3,4)$ model.
The four convex polyhedra are given by
\begin{align}
P_1&=\conv\skima \{\ve_1,\  3\ve_2,\  2\ve_3,\ 4\ve_4,\ 
 \ve_2+\ve_4\} +C_0,\nonumber \\
%-   -   -   -   -   -   -   -   -   -
P_2&=\conv\skima \{2\ve_1, \ \ve_2, \ 4\ve_3,\ 
3\ve_4,\  \ve_3+\ve_4\}+C_0,\nonumber \\
%-   -   -   -   -   -   -   -   -   -   - 
P_3&=\conv\skima \{3\ve_1, \ 4\ve_2, \ \ve_3,\ 
2\ve_4, \ \ve_1+\ve_2\}+C_0,\nonumber \\
%-   -   -   -   -   -   -   -   -   -   - 
P_4&=\conv\skima \{4\ve_1, \ 2\ve_2, \ 3\ve_3,\
\ve_4, \ \ve_1+\ve_3\}+C_0.
\end{align}
\end{flushleft}
%%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  %%  
According to \cite{reid}, $\Hilb$ is the toric variety
associated with the fan in $(\Nb_0)_{\Q}$ 
that is the {\it coarsest common refinement} 
of the normal fans
${\cal N}(P_i)$, $i=1,\dots, n-1$,
which we denote by
${\cal N}( P_1)\ \cap\  \cdots \ \cap \ {\cal N}(P_{n-1})$.
%
To put differently, $\Hilb$ is the toric variety 
associated with the polyhedron $P_{\text{Hilb}}$ 
defined by
%- - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
P_{\text{Hilb}}:=P_1+ \cdots + P_{n-1}
=\conv \left( B_1+\cdots +\ B_{n-1}\right)+C_0,
\label{conc2}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - -
because of the formula \cite[Proposition 7.12]{ziegler}~:
%- - - - - - - - - - - - - - - - - - - - - - - - - - - -
\begin{equation}
{\cal N}( P_1)\ \cap\  \cdots \ \cap \ {\cal N}(P_{n-1}) 
={\cal N}( P_1+\cdots + P_{n-1}). \label{special}
\end{equation}
%- - - - - - - - - - - - - - - - - - - - -  - - - - - - -
It is clear by construction that 
$\Hilb$ is projective over $\CY=X(\Mb_0,C_0)$,
and that each $P_i$ defines a line bundle 
generated by global sections,
and $P_{\text{Hilb}}$ an ample one on $\Hilb$.

Note that $P_{\text{Hilb}}$ defined in (\ref{special})
is by no means a unique candidate for a polyhedron 
yielding the $\Ga$-Hilbert scheme~:
indeed any polyhedron of the form 
$\sum_{i=1}^{n-1}k_i\skima P_i$, where $k_i >0$,
for all $i$ fits for the job.
%
A distinguished feature of $P_{\text{Hilb}}$ (\ref{special})
among the family $\sum_{i=1}^{n-1}k_i\skima  P_i$  
is the following~:
 
% % % % %     conjecture  % % % % % % % % % % % % % % % % % % % % % % %
\begin{flushleft}
{\it Conjecture.} 
{\sl Two polyhedra $(\Mb,Q(\widehat{\bold{1}}))$ 
and $(\Mb_0,P_{\mathrm{Hilb}})$ are isomorphic to each other 
modulo translation.}
% (modulo translation).
\end{flushleft}
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
Recall that
$\widehat{\bold{1}}$ is an element of $M_{\Q}$ which satisfies
$\piQ(\widehat{\bold{1}})=\bold{1}$.




               


\single
\subsection{Computations}
Here we compute the $\Ga$-Hilbert schemes of some 
Calabi--Yau four-fold models to show 
the {\em power} of the formula (\ref{GIT=r}) 
of the toric quotient combined with (\ref{conc1}).
Another method (\ref{conc2}), though less effective,
serves as a consistency check 
of the result of (\ref{conc1}).

% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\begin{flushleft}
\subsubsection{(1/17)(1,1,6,9) model}
\end{flushleft}

The fan of the $\Ga$-Hilbert scheme is given by the following
collection of the maximal cones~:
%- - - - - - - - - - - - - - - - 
\begin{alignat}{5}
&\langle 2,3,4,5 \rangle, &\quad 
&\langle 1,3,5,6 \rangle, &\quad 
&\langle 1,2,7,9 \rangle, &\quad 
&\langle 1,2,3,6 \rangle, &\quad 
&\langle 2,5,6,8 \rangle,         \nonumber \\
% -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 1,2,4,7 \rangle, &\quad 
&\langle 2,3,5,6 \rangle, &\quad 
&\langle 1,2,8,9 \rangle, &\quad 
&\langle 1,7,8,9 \rangle, &\quad 
&\langle 2,7,8,9 \rangle,         \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 2,5,7,8 \rangle, &\quad 
&\langle 2,4,5,7 \rangle, &\quad 
&\langle 1,2,6,8 \rangle, &\quad 
&\langle 1,4,5,7 \rangle, &\quad 
&\langle 1,3,4,5 \rangle,         \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  
&\langle 1,5,6,8 \rangle, &\quad 
&\langle 1,5,7,8 \rangle, &\quad 
&\phantom{\langle 2,3,4,5 \rangle,} &\quad 
&\phantom{\langle 2,3,4,5 \rangle,} &\quad 
&\phantom{\langle 2,3,4,5 \rangle,}
\label{hilb-1169}
\end{alignat} 
%- - - - - - - - - - - - - - - - - - - - - - 
where the weight vectors are
%- - - - - - - - - - - - - -  - -
\begin{alignat}{3}
&\vwb_5=( 1, 1, 6, 9), &\quad
&\vwb_6=( 2, 1,12, 1), &\quad
&\vwb_7=( 3, 3, 1,10), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\vwb_8=( 4, 4, 7, 2), &\quad
&\vwb_9=( 6, 6, 2, 3). &\quad
&\phantom{\vwb_1=(17, 0, 0, 0)}
\end{alignat}
%- - - - - - - - - - - - - - - - - - - -
The fan (\ref{hilb-1169}) defines
one of the five crepant resolutions of
$(1/17)(1,1,6,9)$ model.

For other Calabi--Yau four-fold models
which admit crepant resolutions,
we only give the following conjecture.
\vspace{0.3cm}

%   %   %   %   %   %   %   %   %   %   %   
\begin{flushleft}
{\it Conjecture.}
{\sl The $\Ga$-Hilbert schemes of
$1/(3m+1)(1,1,1,3m-2)$ and $1/(4m)(1,1,2m-1,2m-1)$ models
(\ref{trivial}) are the crepant resolutions of 
the corresponding orbifolds described in \cite{mohri}.}
\end{flushleft}
%   %   %   %   %   %   %   %   %   %   %  



% % % % % % % % % % % % % % % % % % % % % % % % % % %
\subsubsection{1/5(1,2,3,4) model}
The $\Ga$-Hilbert scheme coincides with the phase I
found in the previous section (\ref{hilb-1234-5}).
% % % % % % % % % % % % % % % % % % % % % % % % % 
\subsubsection{1/7(1,2,5,6) model}
The exceptional divisors appearing 
in the $\Ga$-Hilbert scheme are as follows~:
%- - - - - - - - - - - - - - - - - 
\begin{alignat}{4}
\vwb_5&=(1,2,5,6), &\quad
\vwb_6&=(2,4,3,5), &\quad
\vwb_7&=(3,6,1,4), &\quad
\vwb_8&=(4,1,6,3), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -
\vwb_9&=(5,3,4,2), &\quad
\vwb_{10}&=(6,5,2,1), &\quad
\vwb_{11}&=(2,4,10,5), &\quad
\vwb_{12}&=(3,6,8,4), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -
\vwb_{13}&=(4,8,6,3), &\quad
\vwb_{14}&=(5,3,4,9), &\quad
\vwb_{15}&=(5,10,4,2), &\quad
\vwb_{16}&=(6,5,2,8),  \\
%-  -  -  -  -  -  -  -  -  -  -
\vwb_{17}&=(8,2,5,6), &\quad
\vwb_{18}&=(9,4,3,5), &\quad
\vwb_{19}&=(9,4,3,12), &\quad
\vwb_{20}&=(12,3,4,9).\nonumber
%-  -  -  -  -  -  -  -  -  -  -
\end{alignat}
%- - - - - - - - - - - - - - - - - -
The fan of the $\Ga$-Hilbert scheme is given by
%- - - - - - - - - - - - - - - - - - - - - - 
\begin{align} 
&\langle 1,2,3,10   \rangle, \quad
\langle 2,4,6,7      \rangle, \quad
\langle 1,2,7,10      \rangle, \quad
\langle 2,3,13,15      \rangle, \quad
\langle 2,6,12,13      \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 2,3,12,13      \rangle, \quad
\langle 3,4,5,8      \rangle, \quad
\langle 2,3,10,15      \rangle, \quad
\langle 1,2,4,7      \rangle, \quad
\langle 2,3,11,12      \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 2,3,5,11      \rangle, \quad
\langle 2,3,4,5     \rangle, \quad
\langle 1,3,8,9      \rangle, \quad
\langle 1,9,10,18      \rangle, \quad
\langle 4,5,6,14      \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 1,4,19,20      \rangle, \quad
\langle 1,3,9,10      \rangle, \quad
\langle 2,4,5,6      \rangle, \quad
\langle 3,9,12,13      \rangle, \quad
\langle 6,9,12,13      \rangle,  \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 1,4,17,20      \rangle, \quad
\langle 1,3,4,8      \rangle, \quad
\langle 1,4,16,19      \rangle, \quad
\langle 1,4,8,17      \rangle, \quad
\langle 1,4,7,16      \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 2,7,10,15       \rangle, \quad
\langle 4,6,7,16      \rangle, \quad
\langle 3,5,8,11      \rangle, \quad
\langle 1,8,9,17  \rangle, \nonumber \\
% - - - - - - - - - - - - - - - - - - - - - - -
&\langle 3,8,9,11,12      \rangle, \quad
\langle 2,6,7,13,15      \rangle, \quad
\langle 1,9,17,18,20      \rangle, \quad
\langle 4,6,14,16,19      \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  - 
&\langle 3,9,10,13,15  \rangle, \quad
\langle 2,5,6,11,12 \rangle, \quad
\langle 1,16,18,19,20 \rangle, \quad
\langle 4,14,17,19,20  \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  - 
&\langle 1,7,10,16,18      \rangle, \quad
\langle 4,5,8,14,17   \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  
&\langle 6,7,9,10,13,15,16,18 \rangle, \quad
\langle 4,5,8,9,11,12,14,17  \rangle, \quad
\langle 6,9,14,16,17,18,19,20 \rangle.\nonumber
\end{align}
%- - - - - - - - - - - - - - - - - - - - - - - - -
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 
\subsubsection{1/7(1,1,2,3) model}
This model has seven weight vectors~:
%- - - - - - - - - - - - -  - - -
\begin{alignat}{4}
\vwb_{5} &=(1,1,2,3),&\
\vwb_{6} &=(3,3,6,2),&\
\vwb_{7} &=(4,4,1,5),&\
\vwb_{8} &=(5,5,3,1),\\
%-  -  -  -  -  -  -  -  -
\vwb_{9} &=(6,6,5,4),&\
\vwb_{10} &=(8,8,2,3), &\
\vwb_{11} &=(9,9,4,6).&\
\phantom{\vwb_{13}} &\phantom{=(2,2,2,2),}\nonumber
\end{alignat}
%- - - - - - - - - - - - - - - -
The fan of the $\Ga$-Hilbert scheme,
which is a smooth non-Calabi--Yau four-fold,
has only five of them~:
%- - - - - - - - - - - - - - - - 
\begin{alignat}{5}
&\langle 2,4,5,7 \rangle, &\quad
&\langle 1,2,7,10 \rangle, &\quad
&\langle 1,4,5,7 \rangle, &\quad
&\langle 1,2,3,8 \rangle, &\quad
&\langle 1,4,5,8 \rangle, \nonumber \\
%-   -   -   -   -   -   -   -   -
&\langle 2,4,5,8 \rangle, &\quad
&\langle 1,3,5,6 \rangle, &\quad
&\langle 2,3,5,6 \rangle, &\quad
&\langle 1,2,4,7 \rangle, &\quad
&\langle 1,3,4,5 \rangle, \\
%-   -   -   -   -   -   -   -   -
&\langle 1,5,7,10 \rangle, &\quad
&\langle 1,2,8,10 \rangle, &\quad
&\langle 2,3,6,8 \rangle, &\quad
&\langle 1,5,8,10 \rangle, &\quad
&\langle 2,5,8,10 \rangle, \nonumber \\
%-   -   -   -   -   -   -   -   -
&\langle 1,3,6,8 \rangle, &\quad
&\langle 2,3,4,5 \rangle, &\quad
&\langle 2,5,7,10 \rangle. &\quad
&\phantom{\langle 1,1,1,1 \rangle,} &\quad
&\phantom{\langle 2,2,2,2 \rangle,} \nonumber
%-   -   -   -   -   -   -   -   -
\end{alignat}
%- - - - - - - - - - - - - - - - - - - - - -




% % % % % % % % % % % % % % % % % % % % % % % % % % % 
\subsubsection{(1/16)(1,3,5,7) model}
%The polyhedron $P$ has 85018 facets.
The weight vectors appearing in the $\Ga$-Hilbert scheme 
are given by
%- - - - - - - - - - - - - - - - -
\begin{alignat}{3}
\vwb_5&=(17,3,21,7), &\quad
\vwb_6&=(18,6,10,14), &\quad
\vwb_7&=(14,42,6,18), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -
\vwb_8&=(5,15,9,3), &\quad
\vwb_9&=(11,33,7,13), &\quad
\vwb_{10}&=(12,4,12,20), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -
\vwb_{11}&=(12,4,12,4), &\quad
\vwb_{12}&=(8,24,8,8), &\quad
\vwb_{13}&=(20,12,4,12), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -
\vwb_{14}&=(13,7,33,11), &\quad
\vwb_{15}&=(3,9,15,5), &\quad
\vwb_{16}&=(6,2,14,10), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -
\vwb_{17}&=(8,8,24,8), &\quad
\vwb_{18}&=(7,5,3,1), &\quad
\vwb_{19}&=(22,2,14,10),  \\
%-  -  -  -  -  -  -  -  -  -  -  -  -
\vwb_{20}&=(18,6,42,14), &\quad
\vwb_{21}&=(30,10,6,18), &\quad
\vwb_{22}&=(18,6,10,30), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -
\vwb_{23}&=(17,3,5,7), &\quad
\vwb_{24}&=(1,3,5,7), &\quad
\vwb_{25}&=(7,21,3,17), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -
\vwb_{26}&=(13,7,1,11), &\quad
\vwb_{27}&=(23,5,3,17), &\quad
\vwb_{28}&=(10,14,2,6), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -
\vwb_{29}&=(14,10,6,18), &\quad
\vwb_{30}&=(7,5,3,17), &\quad
\vwb_{31}&=(11,1,7,13), \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -
\vwb_{32}&=(4,12,4,12), &\quad
\vwb_{33}&=(17,3,5,23), &\quad
\vwb_{34}&=(10,14,2,22).\nonumber
%-  -  -  -  -  -  -  -  -  -  -  -  -
\end{alignat}
%- - - - - - - - - - - - - - - - - - - - 
The fan of the $\Ga$-Hilbert scheme is 
defined by the following 104 maximal cones~:
%- - - - - - - - - - -  - - - - - - - - - 
\begin{align}
&\langle 1,3,11,18 \rangle,  \quad
\langle 1,4,31,33 \rangle,  \quad
\langle 2,4,24,32 \rangle,  \quad
\langle 1,18,21,23 \rangle, \quad
\langle 2,9,12,32 \rangle,  \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 3,11,17,18 \rangle,   \quad
\langle 3,4,16,24  \rangle,   \quad
\langle 18,24,29,32 \rangle,  \quad
\langle 6,18,24,29  \rangle,  \quad
\langle 1,13,18,21  \rangle,  \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 1,11,18,23 \rangle, \quad
\langle 2,3,8,18   \rangle, \quad
\langle 2,25,28,34 \rangle, \quad
\langle 3,8,15,18  \rangle, \quad
\langle 2,26,28,34 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 2,4,26,34  \rangle, \quad
\langle 2,3,15,24  \rangle, \quad
\langle 2,8,12,18 \rangle, \quad
\langle 2,3,4,24  \rangle, \quad
\langle 2,8,15,24 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 6,11,16,24 \rangle, \quad
\langle 2, 9,12,18 \rangle, \quad
\langle  2,8,12,24 \rangle, \quad
\langle  8,15,18,24 \rangle, \quad
\langle  3,14,17,24 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 4,22,30,33  \rangle, \quad
\langle 11,17,18,24 \rangle, \quad
\langle 3,15,17,24 \rangle, \quad
\langle 4,26,30,34 \rangle, \quad
\langle 15,17,18,24 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 1,2,4,26   \rangle, \quad
\langle 1,21,23,27 \rangle, \quad
\langle 4,10,22,24 \rangle, \quad
\langle 6,10,22,24 \rangle, \quad
\langle 4,10,16,24 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 6,11,18,24 \rangle, \quad
\langle 6,10,16,24 \rangle, \quad
\langle 18,28,29,32 \rangle, \quad
\langle 1,4,26,27 \rangle, \quad
\langle 1,3,5,11 \rangle,  \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 1,2,26,28  \rangle, \quad
\langle 2,7,25,28 \rangle, \quad
\langle 1,13,26,28 \rangle, \quad
\langle 1,2,18,28 \rangle, \quad
\langle 1,13,18,28 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 13,18,28,29 \rangle, \quad
\langle 6,11,18,23 \rangle, \quad
\langle 6,23,29,30 \rangle, \quad
\langle 8,12,18,24  \rangle, \quad
\langle 13,18,21,29 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 1,3,5,19 \rangle, \quad
\langle 2,3,8,15 \rangle, \quad
\langle 3,11,14,17 \rangle, \quad
\langle 6,18,23,29 \rangle, \quad
\langle 1,2,3,18   \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 2,4,25,34 \rangle, \quad
\langle 18,21,23,29 \rangle, \quad
\langle 3,5,16,19 \rangle, \quad
\langle 4,22,24,30 \rangle, \quad
\langle 6,22,24,30 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 4,26,27,30 \rangle, \quad
\langle 1,19,23,31 \rangle, \quad
\langle 1,3,4,31 \rangle, \quad
\langle 1,3,19,31 \rangle, \quad
\langle 2,7,9,32 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 3,5,11,20 \rangle, \quad
\langle 3,16,19,31 \rangle, \quad
\langle 3,4,16,31 \rangle, \quad
\langle 6,24,29,30 \rangle, \quad
\langle 4,10,16,31 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 9,12,18,32 \rangle, \quad
\langle 3,11,14,20 \rangle, \quad
\langle 2,7,25,32 \rangle, \quad
\langle 12,18,24,32 \rangle, \quad
\langle 2,12,24,32 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 2,4,25,32 \rangle, \quad
\langle 7,25,28,32 \rangle, \quad
\langle 1,23,27,33 \rangle, \quad
\langle 24,29,30,32 \rangle, \quad
\langle 4,24,30,32 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 1,4,27,33 \rangle, \quad
\langle 3,15,17,18 \rangle, \quad
\langle 23,27,30,33 \rangle, \quad
\langle 4,27,30,33 \rangle, \quad
\langle 1,23,31,33 \rangle, \nonumber \\
%\end{align}
%
% %% %% %%   Emergency Break        %% %% %% %% %% %% %% %% 
%
%\begin{align}
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 3,5,16,20  \rangle, \quad
\langle 11,14,17,24 \rangle, \quad
\langle 5,11,16,20 \rangle, \nonumber  \\
%-  -  -  -  -  -  -  -  -  -
&\langle 1,13,21,26,27 \rangle, \quad
\langle 2,7,9,18,28   \rangle, \quad
\langle 21,23,27,29,30 \rangle, \quad
\langle 6,22,23,30,33 \rangle, \nonumber \\
% -  -  -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 4,10,22,31,33 \rangle, \quad
\langle 7,9,18,28,32 \rangle, \quad
\langle 1,5,11,19,23 \rangle, \quad
\langle 4,25,30,32,34 \rangle, \nonumber \\
%-  -  -  -  -  -  -  -  -  -  -  -  -  -  - 
&\langle 3,14,16,20,24 \rangle, \quad
\langle 11,14,16,20,24 \rangle,    \nonumber \\
%-  -  -   -  -  -  -  -  -  -  -  -  -  -  -  -
&\langle 6,10,22,23,31,33 \rangle, \quad
\langle 5,6,11,16,19,23   \rangle, \quad
\langle 13,21,26,27,29,30   \rangle,\nonumber \\
%-  -  -  -  -  -  -  -  -  
&\langle 25,28,29,30,32,34 \rangle, \quad
\langle 13,26,28,29,30,34   \rangle, \quad
\langle  6,10,16,19,23,31  \rangle.   
\end{align}
%- - - - - - - - - - - - - - - - - - - - - - -
We see that in general 
the $\Ga$-Hilbert scheme of a Calabi--Yau orbifold 
for $d=4$ is neither smooth nor Calabi--Yau
in contrast with the cases of $d=2,3$.
%
%
\vspace{0.5cm}

%% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %%
\begin{flushleft}
{\it Acknowledgement.}
%\vspace{0.3cm}

I would like to thank Mitsuko Abe (Tokyo Inst. of Technology) 
for many discussions.
\end{flushleft}
%% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %% %%







%%%   %%%   %%%   %%%   %%%   %%%   %%%   
\single
\begin{thebibliography}{99}
%----------------------------------------------
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%------------------------------------------------------
\bibitem{AGM}P.S.~Aspinwall, B.R.~Greene and D.R.~Morrison,
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%-------------------------------------------------------
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\bibitem{BBMOOY}K.~Becker, M.~Becker, D.R.~Morrison,
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%------------------------------------------------------
\bibitem{BVS}M.~Bershadsky, C.~Vafa and V.~Sadov,
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%--------------------------------------------------------
\bibitem{BFS}L.J.~Billera, P.~Fillman and B.~Sturmfels,
Constructions and Complexity of Secondary Polytopes,
{\it Adv. Math.} {\bf 83} (1990) pp.~155--179. 
%-----------------------------------------------------
\bibitem{cox}D.A.~Cox,
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{\it J. Alg. Geom.} {\bf 4} (1995) pp.~17--50,
alg-geom/9210008~; 
%- - - - - - - - - - - - - - - - - - - - - - - - - - - 
D.A.~Cox, 
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alg-geom/9606016.
%----------------------------------------------------
\bibitem{DHH}D.I.~Dais, U.-U.~Hause and M.~Henk,
On Crepant Resolutions of 2-Parameter Series 
of Gorenstein Cyclic Quotient Singularities, 
math/9803096.
%-----------------------------------------------------
\bibitem{DH} D.I.~Dais and M.~Henk,
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