A.C. Avram, M. Kreuzer, M. Mandelberg, H. Skarke

Searching for K3 Fibrations

21 pages, LaTeX2e, 4 eps figures, uses packages
  amssymb,latexsym,cite,epic

Abstract:
  We present two methods for studying fibrations of Calabi-Yau manifolds
embedded in toric varieties described by single weight systems. We analyse
184,026 such spaces and identify among them 124,701 which are K3 fibrations. As
some of the weights give rise to two or three distinct types of fibrations, the
total number we find is 167,406. With our methods one can also study elliptic
fibrations of 3-folds and K3 surfaces. We also calculate the Hodge numbers of
the 3-folds obtaining more than three times as many as were previously known.

\section{Introduction}

Ever since Calabi-Yau manifolds entered into the lexicon of string theory,
there have been various ways to classify them.
Initially, it was (and, of course, still is) of great interest to
calculate their Hodge numbers, thereby predicting the number of massless
families of matter fields.
This was of particular interest, since in the early days of the study of
Calabi-Yau manifolds, there were few known examples, and so it seemed that
it might be difficult to find one with Euler number $|\chi_E|=6$.
By now there are dozens of topologically distinct Calabi-Yau manifolds
with $|\chi_E|=6$, and in any case, these days we are happy to have
manifolds with other values of $\chi_E$.
They provide string theorists with a wide assortment of laboratories in
which to test their ideas.
\del [aa: i'd rather have this out. mm: why? hs: I agree with Alex.
What does this sentence tell us? ``Another'' as compared to what?]
Another {\it de facto} means of classifying Calabi-Yau manifolds was to
distinguish them by how they were constructed.
\enddel
In the beginning, there were  $\IP(4|5)$, $\IP(5|4~2)$, $\IP(5|3~3)$,
$\IP(6|3~2~2)$, $\IP(7|2~2~2~2)$ and the $\IZ_3$ orbifold\cite{chsw}.
Soon after followed complete intersections of polynomial hypersurfaces in
direct products of projective spaces\cite{{cdls,yt}}, and then transverse
polynomials in weighted projective spaces \cite{{nms,kl94}}.
Most recently Calabi-Yau manifolds have been constructed as embeddings in
toric varieties \cite{ba94}.
Using this approach, it was shown in \cite{{crp,wtc}} that there are
184,026 distinct sets of weights that define spaces in which Calabi-Yau
manifolds can be embedded.
These include both the weights that admit transverse polynomials, as well
as those that don't, but nonetheless
lead to reflexive polyhedra and thereby to varieties allowing
smooth Calabi-Yau hypersurfaces.
The purpose of this paper is to provide further means of
analysing these manifolds.

Among the dualities that have been recently studied is that between the
$E_8\times E_8$ heterotic string compactified on $K3\times T^2$ and the
Type IIA string compactified on a Calabi-Yau manifold.
In \cite{kv} specific examples for this Calabi-Yau manifold were proposed.
They rely on two pieces of evidence in support of the conjectured duality.
The first is that the massless spectra on both sides match at generic points
of their moduli spaces.
The second involves a close examination of certain regions of the moduli
space of the Calabi-Yau manifold.
This analysis suggests a natural identification of one of the K\"{a}hler
moduli with the heterotic dilaton.
Specifically, the authors analyze the heterotic string with the $T^2$ chosen
so that
classically there is a single extra $SU(2)$ at the self-dual point, $\tau=i$.
This suggests that when quantum effects are included, this singularity will
split into two points where hypermulitplets become massless, much as in
$N=2$ QFT \cite {sw}.

As it turns out, there is a Calabi-Yau manifold that has the correct Hodge
numbers.
The only known Calabi-Yau manifold with $h_{11}=2$ and $h_{21}=128$ is
{\cal M}=$\IP_4^{1,1,2,2,6}[12]$.
It is impressive that {\cal M} also has a moduli space that resembles the
Seiberg-Witten solution.
One of the keys to this correspondence comes from mirror symmetry.
By analysing the mirror map for {\cal M} $\leftrightarrow$ {\cal W}
\footnote{~   {\cal W} can be chosen as  $\IP_4^{12,13,18,25,28}[96]$.},
it can be shown that {\cal M} has a K\"{a}hler modulus that ``happens'' to
have
the modular transformation properties appropriate for a parameter describing
the moduli space of an $N=2$ theory with $SU(2)$ gauge symmetry.
It was suggested in \cite{klm} that this is a generic feature of Calabi-Yau
manifolds that are $K3$ fibrations.
Indeed it is not difficult to see that $\IP_4^{1,1,2,2,6}[12]$ is a $K3$
fibration with standard fiber given by $\IP_3^{1,1,1,3}$.
A short list of fibrations was provided in \cite{klm}.
Of these, 31 are transverse hypersurfaces in $\IP_4^\w$.%
\footnote{~
	Another 25 are intersections of two hypersurfaces in $\IP_5^\w$.
}
The authors proceed as follows.
Consider a $K3$ given by a hypersurface in $\IP_3^\w$.
If $\hbox{\bf k} = (1,k_2,k_3,k_4)$, then it is easy to see how to form a
$K3$ fibration in $\IP_4^{1,1,2k_2,2k_3,2k_4}$.
The $\IP_1$ base space is given by the ratio of the two coordinates with
weight one, while the $K3$ is as above.
Of the 95 $K3$'s that are given as a hypersurface in a $\IP_3^\w$,
41 have a coordinate with weight one.
Of these, 31 can generate a transverse polynomial in
$\IP_4^\w$.%
\footnote{~
	If we allow for the $\IP_4^\w$'s described in \cite{{crp,wtc}},
	then this approach ``saturates the bound'', and all 41
	of these $K3$'s produce fibrations.
}
These results were extended in \cite{hly}.
That article describes the general problem of determining whether a given
$\IP_4^\w$ is a $K3$ fibration.
The authors were able to identify 628 $K3$ fibrations among the 7,555
$\IP_4^\w$ that admit transverse polynomials.

The question of why $K3$ fibrations are so important to heterotic-Type II
duality has been largely laid to rest by Aspinwall and Louis in \cite{al}.
Their result is that, assuming the Type IIA dual of the weakly coupled
heterotic string is in the Calabi-Yau phase, then this Calabi-Yau manifold
must be a $K3$ fibration.
They achieve this result as a consequence of demanding that the
pre-potentials for both sides match.

Given this result, it has become useful to identify those Calabi-Yau
manifolds that are $K3$ fibrations.
We do this using the methods of toric geometry.
In Section 2 we briefly review some elements of toric geometry.
In Section 3 we describe in detail how a $K3$ fibration is manifested in a
polyhedron.
In \cite{cf} several such examples were analyzed.
It was noted that in each case, the
Newton polyhedron for the Calabi-Yau
manifold exhibited the fibration in a particularly simple manner, to wit,
there was a hyperplane through the origin whose intersection
with the polyhedron gave a reflexive three dimensional subpolyhedron
\footnote{ ~
        It was first conjectured in \cite{klm} that the polyhedron of the
        K3 should be a subpolyhedron of the polytope associated with the
        Calabi-Yau family.
}.
We will show that the general case is more complicated.
Any projection of the 4 dimensional polyhedron onto a 3 dimensional
sublattice that produces a reflexive polyhedron defines a $K3$ fibration.
This is explained and demonstrated by examples.
We also outline our approach for identifying fibrations.
Section 4 provides the technical details.
We explain how a certain kind of polyhedron known as a maximal
Newton polyhedron has the property that if it has a face that is a
reflexive 3 dimensional polyhedron, then there is always a
unique projection onto that face.
We also explain our second strategy, which is somewhat complementary to the
first, and
also
relies on some simple geometric properties of how faces behave
under projection. We summarize our results in Section 5.
Of the 184,026 polyhedra that we analyze, we identify 124,701 that are
$K3$ fibrations, 5,130 that are not, and 54,195 that do not yield to our
methods.
As an indication of the efficiency of our approaches, we note that
among the 7,555 transverse weights we find 5,370 models with one, two or
three fibrations
(to be compared with the 628 fibrations of \cite{hly}).
Finally, we have calculated the Hodge numbers for all 184,026 polyhedra.
We find 14,121 different pairs of Hodge numbers, thereby
increasing the number of known pairs by a factor of more than three.

\section{Calabi--Yau hypersurfaces in toric varieties}

In toric geometry algebraic varieties are described with
the help of a dual pair of lattices $N$ and $M$, each isomorphic
to $\IZ^n$, and a fan $\S$ defined on $N_\IR$ (the real
extension of the $N$ lattice).
Then there are various ways of constructing the variety $V_\S$ from
the toric data.

In the old ``gluing'' approach, affine varieties
$V_\s$ are associated with each cone $\s\in\S$, and $V_\S$
is obtained by gluing them together in a certain way (see, for example,
\cite{ful}).

In the holomorphic quotient approach \cite{cox} it is possible to assign
a single homogeneous coordinate system to $V_\S$ in a way
similar to the usual construction of $\IP^n$.
To this end one assigns a coordinate $z_k$, $k=1,\cdots,N$
(hopefully no confusion will arise from the two different
usages of the symbol $N$)
to each one dimensional cone in $\S$.
If the primitive generators $v_1,\cdots,v_N$ of these one
dimensional cones span $N_\IR$ (the real extension of $N$),
then there must be $N-n$ independent linear relations of
the type $\sum_k w^k_j v_k=0$.
These linear relations are used to define equivalence relations
of the type
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq (z_1,\cdots,z_N)\sim (\l^{w^1_j}z_1,\cdots,\l^{w^N_j}z_N),~~~~~~
j=1,\cdots,N-n  \eeql{er}
on the space $\IC^N-Z_\S$.
The set $Z_\S$ is determined by the fan $\S$ in the following way:
It is the union of spaces
$\{(z_1,\cdots,z_N):\; z_i=0\;\forall i\in I\}$,
where the index sets $I$ are those sets for which $\{v_i:\;i\in I\}$
does not belong to a cone in $\S$.
Thus $(\IC^*)^N\subset \IC^N \setminus Z_\S \subset \IC^N \setminus \{0\}$.
Then $V_\S=(\IC^N \setminus Z_\S)/(\IC^*)^{(N-n)}$, where the $N-n$
groups $\IC^*$
act by the equivalence relations given above.

A third approach is the symplectic quotient construction \cite{aud},
which is closely related to the holomorphic quotient construction.
Here each of the $\IC^*$'s is decomposed as $\IR_+\times U(1)$.
One first chooses representatives of the $\IR_+^{N-n}$
equivalence classes by imposing $N-n$ equations
$\sum_k w^k_j |z_k|^2=r_j$,
and then the resulting space is divided by the remaining $(U(1))^{N-n}$
invariance.

There are several reasons why we are mainly interested in the
holomorphic/symplectic quotient approach:
\begin{itemize}
\item The holomorphic quotient construction leads immediately to
the usual descriptions of projective spaces and is closely related
to weighted projective spaces.
\item Witten's gauged linear sigma model \cite{wit93} leads automatically
to the symplectic quotient construction.
\item Describing Calabi--Yau spaces with Batyrev's method \cite{ba94},
the generators of one dimensional cones are just the integer points
of the dual polyhedron $\D^*$ in the $N$ lattice.
\item The description of K3 fibrations of Calabi--Yau spaces is
far easier with these constructions.
\end{itemize}

Let us briefly outline the construction of a Calabi--Yau
hypersurface in a space described by a reflexive polyhedron:
We take $\D$ to be a reflexive polyhedron in $M_\IR$
(for example the Newton polyhedron of some weighted projective
space), $\D^*\subset N_\IR$ its dual, and $\S$ a fan defined by
a maximal triangulation of $\D^*$.
This means that the integer generators $v_1,\cdots,v_N$ of the one
dimensional cones are just the integer points (except the origin)
of $\D^*$.
The polynomial that determines the CY--hypersurface
takes the form
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq \sum_{x\in \D\cap M}a_x\prod_{k=1}^N z_k^{\<v_k,x\>+1}.  \eeql{polyn}
It is easily checked that it is quasihomogeneous with respect to all
$N-n$ relations of (\ref{er}) with degrees $d_j=\sum_{k=1}^N w_j^k$,
$j=1,\cdots N-n$.
Note how the reflexivity of the polyhedron ensures that the
exponents are nonnegative.

The Hodge numbers $h_{11}$ and $h_{21}$ for 3 dimensional
CY--hypersurfaces of this type are
\cite{ba94}
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq h_{11}=l(\D^*)-5-\sum_{{\rm codim }\th^* =1}l^*(\th^*)+
            \sum_{{\rm codim }\th^* =2}l^*(\th^*)l^*(\th)   \eeq
and
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq h_{21}=l(\D)-5-\sum_{{\rm codim }\th =1}l^*(\th)+
            \sum_{{\rm codim }\th =2}l^*(\th^*)l^*(\th),   \eeq
where $\th$ and $\th^*$ are faces of $\D$ and $\D^*$, respectively,
and $l(\cdot)$ and $l^*(\cdot)$ denote the numbers of integer points
and integer interior points of polytopes.
These formulas are invariant under the simultaneous
exchange of $\D$ with $\D^*$ and $h_{11}$ with $h_{21}$
	so that Batyrev's construction is manifestly mirror symmetric
	(at least at the level of spectra).

\section{Toric K3 fibrations}

In the previous section we have seen how the polyhedron $\D^*$
determines the fan in the $N$ lattice which is used to describe
the ambient space, whereas the polyhedron $\D$ describes the
polynomial whose vanishing locus is our CY hypersurface.
For the description of a fibration, we need the interplay
between both sides.
The following theorem relates properties of $\D$ and $\D^*$ which,
as we will soon see, are essential for the construction of
a fibration whose base space is $\IP^1$ and whose generic fiber is
an $n-1$ dimensional CY space.

\noindent
{\bf Theorem:} Let $\D$ be an n dimensional reflexive polyhedron
in $M_\IR$. Then the following statements are
equivalent:\\
(1) There exists a projection operator $P:~~M\to M_{n-1}$,
where $M_{n-1}$ is an $n-1$ dimensional sublattice, such that
$P\D$ is a reflexive polyhedron in $M_{n-1}$.\\
(2) There is a lattice hyperplane in $N_\IR$ through the origin whose
intersection with $\D^*$ is an $n-1$ dimensional reflexive polyhedron.\\
The respective $n-1$ dimensional polyhedra are dual to one another.
\\[4pt]
{\it Proof:}
For showing that (1) implies (2), we choose a lattice basis
$e_1,\cdots,e_n$ for $M$ such that $Pe_i=e_i$ for $i<n$, $Pe_n=0$
	(such a basis exists because $P$ projects onto a sublattice).
We denote coordinates with respect to this basis by $(x^i)$
and coordinates with respect to the dual basis $e^j$ of $N$ by $(y_j)$.
For (2) $\then$ (1) we start with a basis $e^j$ of $N$ such that
$\D^*\cap\{y_n=0\}$ is reflexive and define $e_i$ to be the dual basis.
In both cases we define
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq
\D_{n-1}=\{(x^1,\cdots,x^{n-1}):\exists~x^n \hbox{ with }
      (x^1,\cdots,x^{n-1},x^n)\in\D)\}
\eeq
and
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq
(\D^*)_{n-1}=\{(y_1,\cdots,y_{n-1}): (y_1,\cdots,y_{n-1},0)\in\D^*\}.
\eeq
Now we note that we can prove both directions if we manage to
show that $\D_{n-1}$ and $(\D^*)_{n-1}$ are dual.
This is the case, due to
\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\eeal#1{\label{#1}\eea
(\D_{n-1})^*&=&\{(y_1,\cdots,y_{n-1}):
            \<(y_1,\cdots,y_{n-1}),(x^1,\cdots,x^{n-1})\>\ge -1\;
             \forall   \;(x^1,\cdots,x^{n-1})\in \D_{n-1}\}  \nn\\
    &=&\{(y_1,\cdots,y_{n-1}):
            \<(y_1,\cdots,y_{n-1},0),(x^1,\cdots,x^n)\>\ge -1 \;
             \forall\; (x^1,\cdots,x^n)\in \D\}  \nn\\
    &=&(\D^*)_{n-1}.
\eea
\del
(1) $\then$ (2): Choose a lattice basis $e_1,\cdots,e_n$ for $M$
such that $Pe_i=e_i$ for $i<n$, $Pe_n=0$;
denote coordinates with respect to this basis by $(x^i)$.
Consider the prism consisting of points $(x^1,\cdots,x^{n-1},\l)$
($\l$ arbitrary)
such that there exists an $x^n$ with $(x^1,\cdots,x^{n-1},x^n)\in\D$.
Then $P$(prism)$=P\D=\D_{n-1}$.
Reflexivity of $\D_{n-1}$ implies that the $n-2$ dimensional
bounding hyperplanes of $\D_{n-1}$ are described by equations
of the type $\<y,x\>=-1$, where the $y\in N_{n-1}$ (the lattice dual
to $M_{n-1}$) are the vertices of $\D_{n-1}^*$.
The bounding hyperplanes of the prism are described by
$\<(y_1,\cdots,y_{n-1},0),x\>=-1$.
As duality inverts incidence relations, $\D\subset$ (prism)
implies that $\D^*$ contains all points of the form
$(y_1,\cdots,y_{n-1},0)$ with $(y_1,\cdots,y_{n-1})\in\D_{n-1}^*$.
If $\D^*\cap\{y_n=0\}$ contained points not in $\D_{n-1}^*$,
then $\D$ would have to lie in another prism (prism)' whose projection
would be a strict subset of the projection of the original prism, in
contradiction with $P\D=P$ (prism). Therefore   $\D^*\cap\{y_n=0\}$
must be isomorphic to $\D_{n-1}^*$.\\
(2) $\then$ (1):
By going through the arguments of the first part of the proof in the
inverse direction.
\enddel
\hfill$\Box$\\[4pt]
This theorem can be readily generalized to the case where
$P:~M\to M_{n-k}$ is a projection on an $n-k$ dimensional
sublattice. In particular for $n=4$ and $k=2$ we can use it to study
elliptic fibrations of 3-folds.
The duality of projections and intersections is most easily
visualised with the help of a simple example:
Fig. 1 shows a dual pair of reflexive triangles. The first triangle
has reflexive intersections along both coordinate axes, corresponding
to reflexive projections on the coordinate axes in the second triangle
(along the direction of the other axis).
The second triangle has only one reflexive intersection (along the
vertical axis), corresponding to the projection on the vertical
axis (along the horizontal axis) in the first triangle.

\begin{figure}		\ifundefined{epsfbox}

\begin{center}}  \def\EC{\end{center}	\unitlength=2.7pt
\BP(40,60)(-20,-30)	\thicklines\put(43,-30){\makebox(0,0)[t]
{{\bf
	Fig. 1:} A dual pair of reflexive polygons
}}
	\putlin(-18,-18,1,0,36)\putlin(-18,-18,3,4,18)\putlin(18,-18,-3,4,18)
	\putdot(-18,-18)\putdot(-9,-18)\putdot(0,-18)\putdot(9,-18)
	\putdot(18,-18)
	\putdot(-18,-6)\putdot(-9,-6)\putdot(0,-6)\putdot(9,-6)\putdot(18,-6)
	\putdot(-18,6)\putdot(-9,6)\putdot(0,6)\putdot(9,6)\putdot(18,6)
			\thinlines
	\putlin(-27,-6,1,0,54)
	\putlin(0,-27,0,1,42)
\EP\hspace{42mm}
\BP(40,60)(-20,-30)	\thicklines
	\putlin(-18,-18,1,0,36)\putlin(-18,-18,3,4,18)\putlin(18,-18,-3,4,18)
	\putdot(-18,-18)	\putdot(0,-18)	\putdot(18,-18)
	\putdot(0,6) \putdot(0,-6)\putdot(-18,6) \putdot(-18,-6)
	\putdot(18,6) \putdot(18,-6)
			\thinlines
	\putlin(-27,-6,1,0,54)
	\putlin(0,-27,0,1,42)
\EP
\EC
\else
	\epsfxsize=12cm
	\hfil\epsfbox{fig.eps}\hfill
	\caption{ A dual pair of reflexive polygons}
	\label{fig:pol}

\fi	\end{figure}

Let us now convince ourselves that reflexive polyhedra fulfilling
the criteria of the theorem correspond to spaces allowing
Calabi--Yau hypersurfaces that are fibrations whose generic
fibers are lower dimensional Calabi--Yau hypersurfaces.
Let us repeat our ingredients: In the $M$ lattice we have
a distinguished direction with integer generator $e_n$
(as in the proof of the theorem) defining a projection $P$
that projects our polyhedron $\D$ to a reflexive
polyhedron $\D_{n-1}$.
In the $N$ lattice we have a distinguished hyperplane
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq H=\{ y\in N: \<y,e_n\>=0\} \eeq
whose intersection with $\D^*$ is $\D_{n-1}^*$.

In the gluing approach we have the following picture:
Depending on whether we consider $\D_{n-1}^*$ as a polytope in $N$
or in $N_{n-1}=N\cap H$, it defines $n$ or $n-1$ dimensional varieties
related by
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq V_{\D_{n-1}^*,N}\simeq V_{\D_{n-1}^*,N_{n-1}}\times \IC^*  \eeq
\del
In the gluing approach the toric variety defined by the subpolyhedron
of $\D^*$ corresponding
to $\D_{n-1}^*$ can be identified with $\IC^*$ times the ambient
space of the lower dimensional Calabi--Yau hypersurface
\enddel
(see, for example, pages 5,6 of \cite{ful}).
Therefore $V_\D$ can be considered as a compactification of
$V_{\D_{n-1}^*,N_{n-1}}\times \IC^*$, giving
a first hint in the desired direction.

A clearer picture emerges in the holomorphic quotient approach:
The integer generators $v_1,\cdots,v_N$ of the one dimensional cones
are just the integer points of $\D^*$.
We may split this set into $\{v_1,\cdots,v_{N'}\}$ (corresponding to
points of $\D_{n-1}^*$) and the remaining set $\{v_{N'+1},\cdots,v_N\}$.
The latter set may again be decomposed, namely into the set of
points `above' and the set of points `below' $\D_{n-1}^*$.
Then we may choose our linear relations such that we have
$N'-(n-1)$ relations involving only $v_1,\cdots,v_{N'}$
and $N-N'-1$ relations involving all of the $v_i$.
Similarly we may split our set $Z_{\Sigma_n}$ as
$Z_{\Sigma_n} \, = \, (Z_{\Sigma_{n-1}}\times \IC^{N-N'}) \cup Z_{\rm diff}$.
With a slight abuse of notation,
the embedding toric variety is then given by
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq \label{exsets}
\bigg\{ \bigg[ \Big[ \big( \IC^{N'} \setminus Z_{\Sigma_{n-1}} \big)
\big/
{\IC^*}^{N'+1-n} \Big] \times
\IC^{N-N'} \bigg] \setminus Z_{\rm diff}  \bigg\}
\bigg/ {\IC^*}^{N-N'-1}
\eeq
The direction of the projection $P$ determines a ray in the $M$ lattice
whose integer generator $e_n$
corresponds to a Laurent monomial
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq q(z_{N'+1},\cdots,z_N)=\prod_{j=1}^N z_j^{\<v_j,e_n\>}
     =\prod_{j=N'+1}^N z_j^{\<v_j,e_n\>}. \eeq
We will now argue that the value of $q$ determines a point in our
base space $\IP^1$.
A short glance at eq. (\ref{er}) shows us that $q$ is invariant
under the equivalence relations.
It takes the value $0$ if one of the $z_j$ corresponding
to points above $\D_{n-1}^*$ is $0$, and infinity if one of the
$z_j$ corresponding to points below $\D_{n-1}^*$ is $0$.
If we choose our maximal triangulation of $\D^*$ in such a way
that it contains a maximal triangulation of $\D_{n-1}^*$,
the rules for constructing $Z_\S$ tell us that these two cases
cannot occur simultaneously.
Thus we have indeed a well--defined (and obviously surjective)
map from $V_\S$ to $\IP^1$.
The quasihomogeneous polynomial of eq. (\ref{polyn})
may be seen as a polynomial in $z_1,\cdots,z_{N'}$
(quasihomogeneous with respect to the first $N'-n+1$ relations)
with coefficients that are polynomials in $z_{N'+1},\cdots,z_N$.
After fixing a point in our base space $\IP^1$ we may then proceed
to determine the structure of the fiber.
In particular, if the point in the base space lies in $\IC^*\subset\IP^1$,
we may use the second set of relations to eliminate all
coordinates $z_{N'+1},\cdots,z_N$, thus obtaining the fiber
as the zero locus of a quasihomogeneous polynomial in the space
determined by $\D_{n-1}$.
This shows us that the generic fiber in this construction is an
$n-2$ dimensional CY manifold.
A similar result is obtained at the level of embedding varieties.
Take a point where $q \neq 0$ is finite. Since $z_{N'+1},
\cdots,z_N$ are all different from zero there are no constraints
on the rest of the degrees of freedom coming from $Z_{\rm diff}$
which thus span the whole $V_{\Sigma_{n-1}}$.

How this works is best understood with the help of some examples.
The following two examples are based on 3 dimensional polyhedra
corresponding to elliptic fibrations of K3 manifolds,
because they can be visualised more easily.
Nevertheless they exhibit all of the features that are essential also
in the 4 dimensional case.

\noindent
{\bf Example 1:}
Take as $\D^*$ the pyramid with peak at $v_1=(-1,-1,2)$ and
base the square whose vertices are $v_2=(0,0,-1),~~v_3=(1,1,-1),~~
v_4=(0,1,-1)$ and $v_5=(1,0,-1)$.
\ifundefined{epsfbox}\else
\begin{figure}
\epsfxsize=5cm
\hfil\epsfbox{dualpoly1.eps}\hfill
\caption{The dual polytope $\D^*$ of Example 1}
\label{fig:dp1}
\end{figure}
\fi
\noindent
It is easily checked that there are no other integer points
(except the origin).
We have two linear relations: $v_1+v_2+v_3=0$ and $v_1+v_4+v_5=0$.
If we triangulate the base of the pyramid along the diagonal
$\overline{v_2v_3}$, we get $Z_\S=\{z_1=z_2=z_3=0\}\cup\{z_4=z_5=0\}$.
With these data we can describe a 2--dimensional Calabi--Yau space
(a K3 surface) as a codimension 1 surface in $(\IC^5 \setminus
Z_\S)/(\IC^*)^2$,
where the $(\IC^*)^2$ action is determined by
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq (z_1,z_2,z_3,z_4,z_5)\sim (\l\m z_1,\l z_2,\l z_3,\m z_4,\m z_5).
\eeq
The polynomial describing the surface is given by
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq z_1^3+z_1^2(z_2p_1^{(1)}+z_3p_1^{(2)})+
     z_1(z_2^2p_2^{(1)}+z_2z_3p_2^{(2)}+z_3^2p_2^{(3)})+
     z_2^3p_3^{(1)}+z_2^2z_3p_3^{(2)}+z_2z_3^2p_3^{(3)}+z_3^3p_3^{(4)},
\eeq
where $p_1^{(.)},p_2^{(.)},p_3^{(.)}$ are linear, quadratic and
cubic polynomials in $z_4$ and $z_5$, respectively.
Clearly the whole polynomial is homogeneous of degree 3 both in $\l$
and in $\m$.
Our base space is $\IP^1$ with homogeneous coordinates $(z_4:z_5)$.
In the coordinate patch $z_4\ne 0$ ($z_5\ne 0$) we may use $\m$
to set $z_4$ ($z_5$) to 1.
Fixing a point in the base space amounts to fixing the values of
the $p_i^{(.)}$. The generic fiber is an elliptic curve
determined by a cubic equation in $\IP^2$ with homogeneous coordinates
$(z_1:z_2:z_3)$.

\noindent
{\bf Example 2:}
Again we consider $N\simeq \BZ^3$. The polyhedron $\D^*_2$
corresponding to the fiber (the shaded area in Fig.\ref{fig:dp2})
is determined by the vertices
$v_1=(1,0,0),v_2=(0,1,0),v_3=(-1,-1,0)$, and the polyhedron
$\D^*_3$ has in addition two `upper' points $v_4=(0,0,1),v_5=(-1,-1,1)$
and one `lower'point $v_6=(0,0,-1)$.
\ifundefined{epsfbox}\else
\begin{figure}
\epsfxsize=5cm
\hfil\epsfbox{dualpoly2.eps}\hfill
\caption{The dual polytope $\D^*_3$ of Example 2}
\label{fig:dp2}
\end{figure}
\fi

\noindent

We have the linear relations $v_1+v_2+v_3=0$, $v_4+v_6=0$ and
$v_1+v_2+v_5+v_6=0$.
$\D^*_3$ has only triangular faces and
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq
Z_\S=\{z_1=z_2=z_3=0\}\cup\{z_1=z_2=z_5=0\}\cup\{z_3=z_4=0\}\cup
\{z_4=z_6=0\}\cup\{z_5=z_6=0\}.
\eeq
$V_\S$ is the space $(\IC^6 \setminus Z_\S)/(\IC^*)^3$ with the $(\IC^*)^3$
action given by
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq (z_1,z_2,z_3,z_4,z_5,z_6)\sim
(\l\n z_1,\l\n z_2,\l z_3,\m z_4,\n z_5,\m\n z_6).
\eeq
We can describe a K3 surface in this space by an equation of the
type
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq
z_1^3p_1^{(1)}+z_1^2z_2p_1^{(2)}+z_1z_2^2p_1^{(3)}+z_2^3p_1^{(4)}+
z_3(z_1^2p_2^{(1)}+z_1z_2p_2^{(2)}+z_2^2p_2^{(3)})+
z_3^2(z_1p_3^{(1)}+z_2p_3^{(2)})+z_3^3p_4=0
\eeq
where the $p_i^{(\cdot)}$ are polynomials in $z_4,z_5,z_6$ with bidegree
$(2,i)$ with respect to $(\m,\n)$.
The base space $\IP^1$ is $(z_4z_5:z_6)$.
In the patch where $z_4z_5\ne 0$ we may use $\m$ and $\n$
to set $z_4$ and $z_5$ to 1, thus taking $z_6$ as the variable
parametrizing the $\IP^1$.
Again the generic fiber is an elliptic curve given by a cubic
equation in $\IP^2$ with homogeneous coordinates $(z_1:z_2:z_3)$.
At the point $z_4z_5=0$, however, the fiber becomes degenerate:
If $z_4=0$ then $z_3\ne 0$ and $z_6\ne 0$.
With suitable redefinitions of $\l,\m,\n$ and partial fixing
of the variables we obtain the $z_4=0$ part of the fiber as the
zero locus of a quadratic equation in a $\IP^2$ described by
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq
(z_1,z_2,1,0,z_5,1)\sim (\n z_1,\n z_2,1,0,\n z_5,1),
\eeq
i.e. the $z_4=0$ part of the singular fiber is a double
cover of $\IP^1$.
At $z_5=0$ we have $(z_1,z_2)\ne (0,0)$.
After suitable redefinitions of $\l,\m,\n$ we get an equation of the type
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq
p_3(z_1,z_2)z_4+p_2(z_1,z_2)z_3=0
\eeq
in a space described by
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq
(z_1,z_2,z_3,z_4,0,1)\sim (\a z_1,\a z_2,\a\b z_3,\b z_4,0,1).
\eeq
This space may be projected to a $\IP^1$ described by $(z_1:z_2)$.
Then the above equation determines $z_3$ and $z_4$ uniquely up
to $\b$ equivalence at each point $(z_1:z_2)$, so the $z_5=0$ part
of the fiber over $z_4z_5=0$ is just $\IP^1$.
The two parts of the singular fiber intersect at the two points
corresponding to the zero locus of a quadratic equation in $\IP^1$:
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq
(z_1,z_2,1,0,0,1)\sim (\a z_1,\a z_2,1,0,0,1).
\eeq

Returning to our general discussion, we may ask whether our projection
$P\D$ in the $M$ lattice is isomorphic to the intersection of $\D$
with some lattice hyperplane $H$. If $H$ intersects the interior of
$\D$, then the only possibility for $\D\cap H$ to have an interior
point is that this point is the interior point of $\D$.
If $H$ does not intersect the interior of $\D$, then the requirement
that $\D\cap H$ is an $n-1$ dimensional polyhedron tells us
that $H$ must in fact be a bounding hyperplane, i.e. that $\D\cap H$
is a facet of $\D$.
Summing up, there are three possibilities:
\begin{enumerate}
\item There is no hyperplane $H$ such that $P\D$ is isomorphic to $H\cap \D$;
\item $P\D$ is isomorphic to $H\cap \D$, where $H$ is a hyperplane through
the origin;
\item $P\D$ is isomorphic to a facet of $\D$.
\end{enumerate}
Of course cases 2 and 3 are not mutually exclusive.
An example is provided by the first triangle of Fig.1, with $P$
the projection along the horizontal axis.
There are also examples of polyhedra allowing different
projections to lower dimensional polyhedra, corresponding to different
cases.
For the second triangle of Fig.1, the projection along the
horizontal axis corresponds to case 2 whereas the projection along the
vertical axis corresponds to case 3.
Case 2 is particularly pretty because there not only
the CY--manifold we are considering but also its mirror have
the structure of a fibration.
It is realised by the projections along the horizontal axes in Fig. 1.
Case 1 is realised, for example, by the triangle with vertices
(1,0), (0,1) and (-1,-1) (the dual of the Newton polyhedron of $\IP^2$).
Here the projections along both coordinate directions and along the
(1,1)--direction are reflexive, but no intersection is.

Let us now consider possible strategies for looking for
torically realised K3 fibrations.

Of course a complete analysis is possible by considering
all 3 dimensional hyperplanes in the $N$ lattice that are
linearly spanned by points of $\D^*$, one of which is the interior point.
To this end, we would have to analyse ${l-1}\choose 3$ hyperplanes
for each polyhedron, where $l$ is the number of points in $\D$.
We did not attempt to do that.

A more modest approach is to search for projections corresponding
to case 3.
Hosono, Lian and Yau \cite{hly} have analysed the 7,555 CY--hypersurfaces
corresponding to transverse polynomials in weighted projective spaces
with respect to a specialisation of this case, where the reflexivity
of the facet can be read off from the weights.
In the present work we extend their results in two directions:
On the one hand we consider not only the 7,555 old models,
but the 184,026 models of \cite{wtc}, and on the other hand we analyse
the geometry of the faces instead of properties of weights,
thereby finding {\it all} fibrations corresponding to facets.

In fact, there is even more that we can learn by considering
facets of $\D$:
Unless the projection $P$ is parallel to some facet $\th$,
any interior point of $\th$ will be mapped to an interior point
of $P\D$.
In particular, this implies the following:
If $\th$ has more than one interior point, $P\D$ can only be reflexive
if $P$ is parallel to $\th$, and if $\th$ has exactly one
interior point and $P\D$ is reflexive, then either $P$ is parallel
to $\th$ or the interior point of $\th$ is mapped to the interior
point of $P\D$.
If $\D$ has enough facets with interior points, these considerations
are sufficient to determine all reflexive projections.

In the following section we will consider maximal Newton polyhedra
\cite{crp} corresponding to single weight systems (classified in
\cite{wtc}) and show how special properties of these polyhedra make the
application of these strategies particularly simple.

\section{Maximal Newton polyhedra}

In ref. \cite{crp}, an algorithm for the classification  of
reflexive polyhedra was presented. The key to this algorithm is the
fact that any reflexive polyhedron is a subpolyhedron of a polytope
defined by a weight system or a combination of weight systems, in the
following way:
A weight system is just a collection of positive rational numbers
$q_i=w_i/d$ with $\sum q_i=1$ ($\sum w_i=d$).
In the present work we consider only the case of a single
weight system with 5 weights.
$(w_1,\cdots,w_5)$ defines a sublattice $\G^4$ of the lattice
$\G^5\simeq\BZ^5$ by
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq \G^4=\{(x^i)\in\G^5:\sum w_ix^i=d\}.    \eeql{g4}
If we also consider the real extensions $\G^5_\IR$ and $\G^4_\IR$,
we may define the simplex $Q$ (whose vertices are generically rational)
as the
intersection of $\G^4_\IR$ with the positive hyperoctant in $\G^5_\IR$:
\begin{equation}} \def\eeq{\end{equation}} \def\eeql#1{\label{#1}\eeq Q=\{(x^i)\in\G^4_\IR: x^i\ge 0~~~ \forall i\}, \eeql{Q}
and the maximal Newton polyhedron (MNP) $\D_{\rm max}$ as the set of
integer points in $Q$: $\D_{\rm max}=Q\cap \G^4$.
It is easy to see that the point $\ip=(1,\cdots,1)$ is the only
integer point in the interior of $Q$, and that its integer distance
to any of the hyperplanes $x^i=0$ bounding $Q$ is 1.
If $\ip$ is also in the interior of $\D_{\rm max}$ (which we assume
henceforth), we say that the weight system $(w_1,\cdots,w_5)$ has the
`interior point property'.
These weight systems, which play a crucial role
for the algorithm of \cite{crp}, were classified in \cite{wtc}.
In particular, in the case of 4 dimensions considered here,
$\D_{\rm max}$ is reflexive by another result of \cite{wtc}.
The $M$ lattice is identified with $\G_4$, with $\hbox{\bf 0}}		\def\w{{\bf k}_M$ corresponding
to $\ip$.

It is tempting to identify spaces defined by an MNP
with the weighted projective spaces defined by the weight system.
A weighted projective space, however, {\it is} a toric variety,
determined by a fan with n+1 one-dimensional cones.
In our language, these are the cones over the vertices
of the integer (but not reflexive) simplex $Q^*$.
One can define a weighted projective space with any set of weights,
but the resulting variety is usually very ill behaved, whereas
the singularities of spaces defined by maximal triangulations of
reflexive polyhedra have codimension $\ge 4$ \cite{ba94}.
What we consider are the varieties defined by the corresponding MNP's,
which are blowups of weighted projective spaces,
because $\D_{\rm max}\subset Q$ implies $Q^*\subset \D_{\rm max}^*$.
A special case occurs when the {\it vertices} of $Q^*$ are not just points,
but also {\it vertices} of $\D^*$.
Then, by duality, the hyperplanes dual to these points
(which are just the coordinate hyperplanes $x^i=0$ which carry the
facets of the simplex $Q$) must also carry facets of $\D$.
Hence vertices of $Q^*$ are vertices of $\D^*$ if and only if
the hyperplanes $x^i=0$ are affinely spanned by points of $\D$.
In this case we say that a weight system has the `span property'.
The weight systems with both the `interior point' and the `span property'
are the ones that are relevant for the classification scheme of \cite{crp}.

As we have seen in the previous section, interior points of
facets are the key ingredients for our strategies for the search
for reflexive projections of reflexive polyhedra.
The following lemmata will provide useful tools for identifying
interior points of facets of MNP's.
They all refer to MNP's defined by a single weight system.

\noindent
\del
{\bf Lemma 1:}
Facets of $\D_{\rm max}$ that do not correspond to hyperplanes of the
type $x^k=0$ have no interior points.
\\[4pt]
{\it Proof:}
Such a point would have to be in the interior of $Q$, but the integer
interior point of $Q$ is unique and not on a facet.
\hfill$\Box$\\[4pt]
\enddel
{\bf Lemma 1:}
An integer interior point of a facet $x^k=0$ of $Q$ defines a projection
$P:\;\G^4_\IR\to \G^4_\IR\cap\{x^k=0\}$ such that
$P\D_{\rm max}=\D_{\rm max}\cap\{x^k=0\}$.
\\[4pt]
{\it Proof:}
An integer interior point of a facet $x^k=0$ of $Q$ must have coordinates
$x^k=0$ and $x^i=1+y^i,\;y^i\ge 0$
for $i\ne k$ (if one of the $x^i$ with $i\ne k$ were zero, our point would
be at the boundary of the facet $x^k=0$).
By comparison with the interior point
$(1,\cdots,1)$ of $\D_{\rm max}$ we see that $\sum_{i\ne k}w_iy^i=w_k$.
Then, for $(x^1,\cdots,x^5)\in \G^4_\IR$, $P$ is defined by $x^k\to 0$,
$x^i\to x^i+y^ix^k$ for $i\ne k$.
Obviously $\D_{\rm max}\cap\{x^k=0\}\subseteq P\D_{\rm max}$, and the
maximality assumption on $\D_{\rm max}$ also ensures
$P\D_{\rm max}\subseteq\D_{\rm max}\cap\{x^k=0\}$.
\hfill$\Box$\\[4pt]
{\bf Lemma 2:}
An integer interior point of $Q\cap\{x^k=0\}$ is also an interior point of
$\D_{\rm max}\cap\{x^k=0\}$.
\\[4pt]
{\it Proof:}
An integer interior point of $Q\cap\{x^k=0\}$ is the image of $(1,\cdots,1)$
under the projection $P$ of Lemma 1.
If it were at the boundary of $\D_{\rm max}\cap\{x^k=0\}=P\D_{\rm max}$,
then $(1,\cdots,1)$ would be at the boundary of $\D_{\rm max}$.
\hfill$\Box$\\[4pt]
{\bf Lemma 3:}
(1) Facets of $\D_{\rm max}$ that do not correspond to hyperplanes of the
type $x^k=0$ have no interior points.\\
(2) Interior points of a facet of $\D_{\rm max}$ defined by $x^k=0$
are in one-to-one correspondence with partitions of $w_k$ by $\{w_i:i\ne k\}$.
\\[4pt]
{\it Proof:}
(1) Such a point would have to be in the interior of $Q$, but the integer
interior point of $Q$ is unique and not on a facet.\\
(2) Partitions of $w_k$ by $\{w_i:i\ne k\}$ define projections as in the
proof of Lemma 2. Projections of $(1,\cdots,1)$ are integer interior
points of $Q\cap\{x^k=0\}$ and (by Lemma 1) of $\D_{\rm max}\cap\{x^k=0\}$.
There is a one-to-one correspondence between partitions of $w_k$ and
interior points of $\D_{\rm max}\cap\{x^k=0\}$ defined by the corresponding
projections.
\hfill$\Box$

Lemma 3 is the key to our strategies for identifying reflexive projections:

Searching for reflexive facets, we simply look for weights that have
unique partitions by the other weights.
By Lemma 1 we can be sure that such a facet corresponds to a projection.
For 3 dimensional polyhedra (weight systems with 4 weights)
Lemma 3 would even guarantee reflexivity of the facet, because
any polygon with a single interior point is reflexive.
In the present case we still have to check for reflexivity
of the facet, and indeed it turns out that there are many cases
where a facet has a single interior point without being
reflexive.
If, for example, $w_1$ has the unique partition $w_1=\sum_{i>1}w_iy^i$,
then the vector $e_n$ determining our projection $P$ is given
by $(-1,y_2,\cdots,y_5)$.

On the other hand, if we have three facets (say, $x^1=0$, $x^2=0$
and $x^3=0$) with interior points, then any projection that does
not correspond to one of these facets must be parallel to each
of them.
Thus it must be parallel to the edge $x^1=x^2=x^3=0$ of $Q$,
connecting the vertices $(0,0,0,d/w_4,0)$ and $(0,0,0,0,d/w_5)$
of $Q$.
Denoting by $g$ the greatest common divisor of $w_4$ and $w_5$,
$e_n=(0,0,0,w_5/g,-w_4/g)$.
With four or five facets with interior points (this happens, in
particular, if we have one or two weights equal to 1),
projections not corresponding to facets are excluded.

\section{Results}

	\def\#$\P$}	\def\nFa{\#$F${\#$\P$}	\def\nFa{\#$F$}
	\def\VR#1#2{\vrule height #1mm depth #2mm width 0pt}
	\def\TVR#1#2{@{~~\VR{#1}{#2}}}
	\def\\}	\def\N{--{\\\hline}
	\defT\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}{T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}}

In order to find K3 fibrations we pursued the two different strategies
described in the previous sections. In the search for K3 facets we first
searched for weights that have a unique partition in the remaining weights,
which is a necessary condition for a unique interior point and hence for
reflexivity. The results are listed in Table 1 for the complete
list of reflexive weights and for the much smaller list of spanning weights.

\begin{center}					\defUP weight{UP weight}
\smallskip
\begin{tabular}{||\TVR{4.5}2 c||c|c|c|c||c||} \hline\hline
weights		& no UP weight  & 1 UP weight  & 2 UP weight s & 3 UP weight s &
		total\\\hline\hline
reflexive	& 43,988 & 133,386 & 6,571 & 81 & 184,026\\\hline
spanning	& 9,939  & 26,400  & 2,361 & 27 & 38,727  \\\hline
\hline\end{tabular}
\\[4mm]
	{\bf Table 1:} Numbers of weights with unique partitions
\end{center}

If we keep only those weights whose corresponding facets actually are
reflexive then the numbers reduce to the numbers of K3 fibrations that arise
as projections onto facets as given in Table 2.
It turns out that all of the 81 polytopes where 3 weights have
a unique partition have at most 1 reflexive facet.
An example of this type is $(3,4,6,7,8;28)$, where 6, 7 and
8 have unique partitions while only the
facet corresponding to the weight 7 is reflexive.

\del
7 3 4 6 8  28		7 3 4 8 9  31		10 3 4 8 9  34
8 3 5 9 10  35		11 3 5 9 10  38		9 4 5 12 15  45
13 4 5 12 15  49	14 4 5 12 15  50	18 4 5 12 15  54
11 5 6 18 20  60	21 5 6 18 25  75	21 4 9 20 27  81
27 5 6 18 25  81	21 4 9 22 28  84	30 4 9 20 27  90
21 4 9 28 31  93	51 4 15 42 56  168
\enddel

\vbox{
\begin{center}
\smallskip
\begin{tabular}{||\TVR{4.5}2 c||c|c|c||c||} \hline\hline
weights		& no K3 facet & 1 K3 facet & 2 K3 facets  &
		total\\\hline\hline
reflexive	& 76,460 & 104,036 & 3,530 & 184,026\\\hline
spanning	& 19,410 & 18,356  & 961  & 38,727 \\\hline
\hline\end{tabular}
\\[4mm]
	{\bf Table 2:} Numbers of K3 facets
\end{center}}

In both tables partitions of equal weights are counted only once because
the corresponding projections onto facets are along the same
direction $e_n=\pm(1,-1,0,\ldots)$ and thus define the same fibration
(regardless of the fact that there are two reflexive facets for each such pair
of equal weights).
This is always the case, for example, if 2 and only 2 weights are equal to 1
(this type of example has been used extensively in the literature).
A nice case with 2 pairs of equal weights is $(2,2,3,3,4;14)$, where
2 and 3 have unique partitions, so there are 4 reflexive facets but
only 2 inequivalent K3 fibrations (see Table 6 below).

Our alternative search method, which is based on the constraints on directions
of projection due to facets with interior points, is not applicable in 55\%
of all cases although about half of these even have projections onto K3
facets. In the remaining cases it allows, however, for a complete analysis,
providing many examples with up to 3 different K3 projections and a powerful
consistency check. In particular, for 5,130 weights we have shown that the
corresponding Newton polytopes do not admit any reflexive projection.
It is interesting to compare the results of our two approaches in some detail,
and this is done by the
compilations in the following tables, where we first give the
overall statistics in Table 3, and then list the respective numbers for the
weights that do (T) or do not (--) admit transverse polynomials and for which
the Newton polytopes do (S) or do not (--) span all coordinate hyperplanes
in Table 4.
It turns out that the number of toric fibrations goes up to 3. Actually that
number can go up to at least 5, as one can see from the example of the
mirror of the quintic threefold for which any projection along a line from
the origin to one of the 5 vertices is reflexive (since none of the facets has
an interior point, however, our algorithms do not find these fibrations).

\begin{figure}
\begin{center}
\begin{tabular}{||c||c|c|c||c||} \hline\hline
	& \nFa=0& \nFa=1& \nFa=2& total	\\\hline\hline
\#$\P$}	\def\nFa{\#$F$=? & 54,195 & 45,792 & 1,508 & 101,495 \\\hline
\#$\P$}	\def\nFa{\#$F$=0 & 5,130 &  &  & 5,130 \\\hline
\#$\P$}	\def\nFa{\#$F$=1 & 17,135 & 20,080 &  & 37,215 \\\hline
\#$\P$}	\def\nFa{\#$F$=2 & 0 & 38,164 & 1,011 & 39,175 \\\hline
\#$\P$}	\def\nFa{\#$F$=3 & 0 & 0 & 1,011 & 1,011 \\\hline
\hline total & 76,460 & 104,036 & 3,530 & 184,026 \\\hline
\hline
\end{tabular}
\\[4mm]{\bf Table 3:} Numbers of K3 projections (\#$\P$}	\def\nFa{\#$F$) and K3 facets (\nFa)
\end{center}\end{figure}

\begin{figure}
\begin{center}	\footnotesize	\small
\begin{tabular}{||c||c|c|c||c||} \hline\hline
TS	& \nFa=0& \nFa=1& \nFa=2& total	\\\hline\hline
\#$\P$}	\def\nFa{\#$F$=? & 703 & 299 & 12 & 1014 \\\hline
\#$\P$}	\def\nFa{\#$F$=0 & 507 &  &  & 507 \\\hline
\#$\P$}	\def\nFa{\#$F$=1 & 697 & 1,471 &  & 2,168 \\\hline
\#$\P$}	\def\nFa{\#$F$=2 & 0 & 383 & 88 & 471 \\\hline
\#$\P$}	\def\nFa{\#$F$=3 & 0 & 0 & 9 & 9 \\\hline
\hline total & 1,907 & 2,153 & 109 & 4,169 \\\hline
\hline
\end{tabular}
~
~
~
~
\begin{tabular}{||c||c|c|c||c||} \hline\hline
T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}	& \nFa=0& \nFa=1& \nFa=2& total	\\\hline\hline
\#$\P$}	\def\nFa{\#$F$=? & 859 & 622 & 39 & 1,520 \\\hline
\#$\P$}	\def\nFa{\#$F$=0 & 116 &  &  & 116 \\\hline
\#$\P$}	\def\nFa{\#$F$=1 & 536 & 586 &  & 1,122 \\\hline
\#$\P$}	\def\nFa{\#$F$=2 & 0 & 577 & 27 & 604 \\\hline
\#$\P$}	\def\nFa{\#$F$=3 & 0 & 0 & 24 & 24 \\\hline
\hline total & 1,511 & 1,785 & 90 & 3,386 \\\hline
\hline
\end{tabular}
\\[5mm]	~~~~~{\bf Table 4a:} Transverse spanning weights
\hfill	{\bf Table 4b:} Transverse non-spanning weights ~ ~
\end{center}

\begin{center}\small
\begin{tabular}{||c||c|c|c||c||} \hline\hline
\RS	& \nFa=0& \nFa=1& \nFa=2& total	\\\hline\hline
\#$\P$}	\def\nFa{\#$F$=? & 10,222 & 3,804 & 297 & 14,323 \\\hline
\#$\P$}	\def\nFa{\#$F$=0 & 2,258 &  &  & 2,258 \\\hline
\#$\P$}	\def\nFa{\#$F$=1 & 5,023 & 6,692 &  & 11,715 \\\hline
\#$\P$}	\def\nFa{\#$F$=2 & 0 & 5,707 & 356 & 6,063 \\\hline
\#$\P$}	\def\nFa{\#$F$=3 & 0 & 0 & 199 & 199 \\\hline
\hline total & 17,503 & 16,203 & 852 & 34,558 \\\hline
\hline
\end{tabular}
~
~
~
\begin{tabular}{||c||c|c|c||c||} \hline\hline
\RN	& \nFa=0& \nFa=1& \nFa=2& total	\\\hline\hline
\#$\P$}	\def\nFa{\#$F$=? & 42,411 & 41,067 & 1160 & 84,638 \\\hline
\#$\P$}	\def\nFa{\#$F$=0 & 2,249 &  &  & 2,249 \\\hline
\#$\P$}	\def\nFa{\#$F$=1 & 10,879 & 11,331 &  & 22,210 \\\hline
\#$\P$}	\def\nFa{\#$F$=2 & 0 & 31,497 & 540 & 32,037 \\\hline
\#$\P$}	\def\nFa{\#$F$=3 & 0 & 0 & 779 & 779 \\\hline
\hline total & 55,539 & 83,895 & 2,479 & 141,913 \\\hline
\hline
\end{tabular}
\\[5mm]	~~~{\bf Table 4c:} Non-transverse spanning weights
\hfill		{\bf Table 4d:} Non-transverse non-spanning weights$\!$
\end{center}
\end{figure}

In Table 5 we give the beginning and the end of our complete list of
results which has 184,026 lines and is available via WWW
\footnote{~
	The URL is http://tph.tuwien.ac.at/{\tiny$^\thicksim$}kreuzer/CY}
or by e-mail from the authors.
Weights with unique partitions in terms of the other weights such that
the facets carried by the corresponding coordinate hyperplanes are reflexive
are typed in bold face (if a weight occurs twice it is only marked once since
the pair defines only a single fibration; see above). We also list the
Hodge numbers $h_{11}$ and $h_{21}$ and the numbers $P$ and $V$ of points and
vertices for the MNP and its dual, and also indicate the transversality and
spanning properties of the weights.

\begin{figure}
\begin{center}
\begin{tabular}{||\TVR{3.5}1 c|ccccc|c|cc|cc|cc|cc||} \hline\hline	\VR41
$d$	&$w_1$	& $w_2$	& $w_3$ & $w_4$ & $w_5$ &  TS	& $h_{11}$ & $h_{12}$
& $\#P$	& $\#V$	& $\#\5P$	& $\#\5V$	& $\P$	& F \\\hline\hline
5 &1 & 1 & 1 & 1 & 1 & TS& 1&101 & 126&5 & 6&5 & 0&0\\}	\def\N{--
6 &1 & 1 & 1 & 1 & 2 & TS& 1&103 & 130&5 & 6&5 & 0&0\\}	\def\N{--
7 &1 & 1 & 1 & 1 & 3 & TS& 2&122 & 159&8 & 7&6 & 0&0\\}	\def\N{--
7 &1 & 1 & 1 & 2 & 2 & TS& 2&95 & 120&9 & 7&6 & 0&0\\}	\def\N{--
8 &1 & 1 & 1 & 1 & 4 & TS& 1&149 & 201&5 & 7&5 & 0&0\\}	\def\N{--
8 &1 & 1 & 1 & 2 & 3 & TS& 2&106 & 136&8 & 7&6 & 0&0\\}	\def\N{--
8 &\bf1 & 1 & 2 & 2 & 2 & TS& 2&86 & 105&5 & 7&5 & 1&1\\}	\def\N{--
9 &1 & 1 & 1 & 2 & 4 & TS& 3&123 & 162&9 & 8&6 & 0&0\\}	\def\N{--
9 &1 & 1 & 1 & 3 & 3 & TS& 4&112 & 145&5 & 7&5 & 0&0\\}	\def\N{--
9 &1 & 1 & 2 & 2 & 3 & TS& 2&86 & 109&9 & 7&6 & 0&0\\}	\def\N{--
10 &1 & 1 & 1 & 2 & 5 & TS& 1&145 & 196&5 & 7&5 & 0&0\\}	\def\N{--
10 &1 & 1 & 1 & 3 & 4 & \RS& 4&126 & 165&10 & 9&7 & 0&0\\}	\def\N{--
10 &\bf1 & 1 & 2 & 2 & 4 & TS& 3&99 & 126&8 & 8&6 & 1&1\\}	\def\N{--
10 &1 & 1 & 2 & 3 & 3 & TS& 3&87 & 111&9 & 8&6 & 0&0\\}	\def\N{--
10 &1 & 2 & 2 & 2 & 3 & TS& 3&75 & 87&8 & 8&6 & 0&0\\}	\def\N{--
11 &1 & 1 & 1 & 3 & 5 & TS& 4&144 & 192&10 & 9&7 & 0&0\\}	\def\N{--
11 &\bf1 & 1 & 2 & 2 & 5 & TS& 4&109 & 144&10 & 9&7 & 1&1\\}	\def\N{--
11 &1 & 1 & 2 & 3 & 4 & TS& 4&94 & 121&13 & 9&8 & 0&0\\}	\def\N{--
11 &1 & 2 & 2 & 3 & 3 & TS& 4&64 & 81&13 & 9&7 & 0&0\\}	\def\N{--
12 &1 & 1 & 1 & 3 & 6 & TS& 3&165 & 225&5 & 8&5 & 0&0\\}	\def\N{--
12 &1 & 1 & 1 & 4 & 5 & \RS& 4&154 & 204&7 & 10&6 & 0&0\\}	\def\N{--
12 &\bf1 & 1 & 2 & 2 & 6 & TS& 2&128 & 171&5 & 8&5 & 1&1\\}	\def\N{--
12 &1 & 1 & 2 & 3 & 5 & TS& 3&105 & 137&7 & 8&6 & 0&0\\}	\def\N{--
12 &\bf1 & 1 & 2 & 4 & 4 & TS& 5&101 & 130&5 & 8&5 & 1&1\\}	\def\N{--
12 &1 & 1 & 3 & 3 & 4 & TS& 5&89 & 115&5 & 7&5 & 0&0\\}	\def\N{--
12 &1 & 2 & 2 & 2 & 5 & TS& 4&94 & 108&8 & 9&6 & 0&0\\}	\def\N{--
12 &1 & 2 & 2 & 3 & 4 & TS& 2&74 & 89&5 & 7&5 & 0&0\\}	\def\N{--
12 &1 & \bf2 & 3 & 3 & 3 & TS& 3&69 & 81&5 & 8&5 & 1&1\\}	\def\N{--
12 &2 & 2 & 2 & \bf3 & 3 & TS& 6&60 & 63&5 & 8&5 & 1&1\\}	\def\N{--
	   $\cdots$ & $\cdots$ &$\cdots$ &$\cdots$ &$\cdots$ &$\cdots$ &
$\cdots$ & $\cdots$ & $\cdots$ &$\cdots$ &$\cdots$ &$\cdots$ &$\cdots$ &
						    $\cdots$ &$\cdots$ \\}	\def\N{--
3192 &37 & 39 & \bf456 & 1064 & 1596 & \RN& 491&11 & 26&5 & 680&5 & 2&1\\}	\def\N{--
3234 &36 & 41 & \bf462 & 1078 & 1617 &T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}& 462&12 & 27&6 & 639&6 & 2&1\\}	\def\N{--
3234 &37 & 40 & \bf462 & 1078 & 1617 & \RN& 491&11 & 26&5 & 680&5 & 2&1\\}	\def\N{--
3234 &38 & 39 & \bf462 & 1078 & 1617 & \RN& 491&11 & 26&5 & 680&5 & 2&1\\}	\def\N{--
3276 &37 & 41 & \bf468 & 1092 & 1638 & \RN& 491&11 & 26&5 & 680&5 & 2&1\\}	\def\N{--
3318 &37 & 42 & \bf474 & 1106 & 1659 & \RN& 491&11 & 26&5 & 680&5 & 2&1\\}	\def\N{--
3318 &38 & 41 & \bf474 & 1106 & 1659 & \RN& 491&11 & 26&5 & 680&5 & 2&1\\}	\def\N{--
3318 &39 & 40 & \bf474 & 1106 & 1659 & \RN& 491&11 & 26&5 & 680&5 & 2&1\\}	\def\N{--
3360 &39 & 41 & \bf480 & 1120 & 1680 & \RN& 491&11 & 26&5 & 680&5 & 2&1\\}	\def\N{--
3402 &40 & 41 & \bf486 & 1134 & 1701 & \RN& 491&11 & 26&5 & 680&5 & 2&1\\}	\def\N{--
3486 &41 & 42 & \bf498 & 1162 & 1743 &T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}& 491&11 & 26&5 & 680&5 & 2&1\\}	\def\N{--
\hline\end{tabular}
\\[4mm]
	{\bf Table 5:} Results from the complete list of reflexive weights
	(available via WWW)
\end{center}
\end{figure}

In Table 6 we give the same information for a number of examples that
illustrate the phenomena that can occur. The first line is an example
of a weight combination whose MNP is the mirror of the quintic, i.e. a
simplex with 6 points such that all vertices have distance 5 from the
opposite facet. This polytope has 5 reflexive projections, but none is onto
a facet and it also cannot be analysed with our second approach since none
of the facets has an interior point. The second example features the maximal
number of vertices of $\D$, which is 18; $\D^*$ can have up to 21 vertices,
which occurs twice. The number of points goes up to 680 both for $\D$
(this number occurs for the Fermat weights in the next line of Table 6)
and $\D^*$ (apparently the dual of the Fermat simplex, represented many
times by different weight systems, for example at the end of Table 5).
The remaining lines
give examples (of minimal degree) for all combinations of transversality and
spanning properties, general projections and projections onto facets.

	It is straightforward to apply our methods also to the well known
	95 K3 weights, this time producing elliptic fibrations. Since
	all 2D polytopes with one interior point are automatically reflexive,
	no reflexivity check is required
	and the search for `elliptic' facets can be done by hand.
	The results are listed in Table~7, where the (up to 2) weights with
	unique partition are again in bold face. With our alternative search
	algorithm we again find up to 3 fibrations per Newton polytope,
	so that we find a total of 110 ellipitic fibrations.%
\footnote{~
	The algorithm of \cite{hly} yields 18,
	with 1 double fibration.
}
	The statistics are given in Table~8.

We also computed the Hodge numbers and compared and combined them with
the complete results that exist for weighted projective spaces \cite{nms,kl94}
and abelian orbifolds thereof \cite{ao} in Table~9.
A pronounced feature is the absence of mirror symmetry, which is already
familiar from the list of transversal weights.
	The well known plot Fig.4 becomes much denser,
	but otherwise does not change shape.
In our context
it does not make sense any longer to omit the mirror spectra since they are
produced by the dual polytopes (which, however, cannot be MNPs for a
single weight system whenever the spectrum does not occur in our original
list).
Counting all spectra of MNPs and their duals we thus get an increase in the
known Calabi--Yau spectra by more than a factor of 3. Inclusion of the
abelian orbifold spectra for the transversal weights,
which have been analysed completely in \cite{ao},
only adds 173 new spectra to those for reflexive
MNPs of single weight systems.
If we mirror-symmetrize by hand, as is appropriate in the
toric framework (all abelian orbifolds correspond to MNPs on sublattices,
which are reflexive because of the results of \cite{wtc}), this number
even goes down to 95.
This can be interpreted as an indication that our list of spectra already
might be fairly complete.

\begin{figure}
\begin{center}
\begin{tabular}{||\TVR{3.5}1 c|ccccc|c|cc|cc|cc|cc||} \hline\hline	\VR41
$d$	&$w_1$	& $w_2$	& $w_3$ & $w_4$ & $w_5$ &  TS	& $h_{11}$ & $h_{12}$
& $\#P$	& $\#V$	& $\#\5P$	& $\#\5V$	& $\P$	& F \\\hline\hline
95 &15 & 17 & 18 & 19 & 26 & \RN& 101&1 & 6&5 & 126&5 & ?&0\\\hline
47 &3 & 4 & 5 & 14 & 21 & \RS& 26&39 & 54&\bf18 & 35&15 & ?&0\\\hline
69 &7 & 8 & 10 & 19 & 25 & \RS& 59&10 & 16&13 & 75&\bf21 & ?&0\\
97 &7 & 8 & 11 & \bf26 & 45 & \RS& 63&15 & 24&15 & 71&\bf21 & ?&1\\\hline
84 &\bf1 & 1 & 12 & 28 & 42 & TS& 11&491 & 680&5 & 26&5 & 1&1\\\hline
24 &3 & 4 & 5 & 6 & 6 & TS& 10&34 & 36&8 & 12&7 & ?&0\\
26 &3 & 4 & 5 & 7 & 7 & \RS& 22&22 & 31&13 & 21&10 & ?&0\\
33 &3 & 6 & 6 & 7 & 11 & \RN& 19&37 & 34&7 & 22&6 & ?&0\\
36 &3 & 6 & 6 & 10 & 11 & T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}& 19&49 & 38&7 & 22&6 & ?&0\\\hline
26 &3 & 4 & 5 & \bf6 & 8 & \RS& 14&24 & 32&14 & 19&10 & ?&1\\
36 &5 & \bf7 & 7 & 8 & 9 & \RN& 30&12 & 19&10 & 28&9 & ?&1\\
39 &3 & \bf6 & 9 & 10 & 11 & TS& 17&41 & 33&12 & 22&13 & ?&1\\
52 &4 & 6 & \bf8 & 11 & 23 & T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}& 29&33 & 34&9 & 36&8 & ?&1\\\hline
34 &3 & \bf6 & 7 & 8 & \bf10 & \RS& 18&20 & 27&13 & 23&12 & ?&2\\
44 &4 & \bf8 & 9 & 10 & \bf13 & \RN& 29&17 & 22&9 & 31&9 & ?&2\\
55 &3 & 10 & \bf13 & 14 & \bf15 & TS& 28&16 & 23&12 & 35&14 & ?&2\\
63 &7 & 9 & \bf14 & 15 & \bf18 & T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}& 44&8 & 15&6 & 37&6 & ?&2\\\hline
5 &1 & 1 & 1 & 1 & 1 & TS& 1&101 & 126&5 & 6&5 & 0&0\\
10 &1 & 1 & 1 & 3 & 4 & \RS& 4&126 & 165&10 & 9&7 & 0&0\\
25 &1 & 5 & 5 & 6 & 8 & T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}& 17&49 & 65&7 & 15&7 & 0&0\\
26 &1 & 5 & 5 & 7 & 8 & \RN& 19&49 & 65&9 & 19&7 & 0&0\\\hline
20 &2 & 3 & 4 & 4 & 7 & \RS& 13&45 & 51&10 & 14&8 & 1&0\\
20 &2 & 3 & 5 & 5 & 5 & TS& 6&48 & 50&8 & 11&6 & 1&0\\
30 &2 & 5 & 6 & 6 & 11 & \RN& 27&39 & 45&7 & 25&6 & 1&0\\
36 &2 & 5 & 6 & 6 & 17 & T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}& 24&54 & 60&7 & 25&6 & 1&0\\\hline
8 &\bf1 & 1 & 2 & 2 & 2 & TS& 2&86 & 105&5 & 7&5 & 1&1\\
13 &\bf1 & 1 & 2 & 4 & 5 & \RS& 6&108 & 141&12 & 11&8 & 1&1\\
35 &2 & 7 & \bf8 & 9 & 9 & \RN& 35&23 & 33&11 & 30&8 & 1&1\\
40 &4 & 5 & \bf9 & 10 & 12 & T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}& 22&18 & 25&7 & 20&7 & 1&1\\\hline
19 &2 & 3 & \bf4 & 5 & 5 & \RS& 11&33 & 43&14 & 14&9 & 2&1\\
27 &2 & 3 & \bf4 & 9 & 9 & TS& 14&44 & 56&9 & 13&7 & 2&1\\
36 &\bf4 & 4 & 6 & 9 & 13 & \RN& 31&31 & 33&6 & 29&6 & 2&1\\
40 &\bf4 & 4 & 6 & 9 & 17 & T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}& 26&38 & 39&7 & 29&6 & 2&1\\\hline
14 &\bf2 & 2 & \bf3 & 3 & 4 & TS& 5&51 & 57&10 & 10&7 & 2&2\\
19 &2 & \bf3 & 3 & \bf4 & 7 & \RS& 11&39 & 51&14 & 16&9 & 2&2\\
30 &3 & \bf5 & 5 & \bf6 & 11 & \RN& 33&21 & 33&6 & 25&6 & 2&2\\
35 &3 & \bf5 & 5 & \bf6 & 16 & T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}& 26&28 & 42&7 & 25&6 & 2&2\\\hline
28 &4 & \bf5 & 5 & 6 & \bf8 & \RS& 18&20 & 27&10 & 18&8 & 3&2\\
36 &4 & 6 & \bf8 & \bf9 & 9 & TS& 23&23 & 26&6 & 16&6 & 3&2\\
40 &4 & \bf7 & 7 & 10 & \bf12 & \RN& 28&16 & 23&7 & 25&6 & 3&2\\
42 &6 & \bf7 & 7 & 10 & \bf12 & T\,\bf{--}} \def\RS{{\bf{--}}\,S} \def\RN{\bf{--}\,\bf{--}& 35&11 & 19&6 & 23&6 & 3&2\\
\hline\end{tabular}
\\[4mm]
	{\bf Table 6:} 	Examples from the long list of weights with different
			features
\end{center}
\end{figure}

\begin{figure}	\vspace{-9mm}
\begin{center}	{	\footnotesize	\def\\}	\def\N{--{\\}	\def\N{--}
\def\tabtop
{\begin{tabular}{||\TVR{3.5}1
		c|c@{~}c@{~}c@{~}c|c|c@{~}c@{~}|c@{~}c@{~}|c@{~~~}c||}
	\hline\hline	\VR41
$d$	&$w_1$	& $w_2$	& $w_3$ & $w_4$ &  S
	& $\#P$	& $\#V$	& $\#\5P$ & $\#\5V$ & $\P$
	& F \\\hline\hline
}							\hspace*{-6mm}\tabtop
4 & 1 &1 &1 &1 & S& 35 & 4 & 5 & 4 & 0 & 0\\}	\def\N{--
5 & 1 &1 &1 &2 & S& 34 & 6 & 6 & 5 & 0 & 0\\}	\def\N{--
6 & 1 &1 &1 &3 & S& 39 & 4 & 6 & 4 & 0 & 0\\}	\def\N{--
6 & \bf1 &1 &2 &2 & S& 30 & 4 & 6 & 4 & 1 & 1\\}	\def\N{--
7 & \bf1 &1 &2 &3 & S& 31 & 7 & 8 & 6 & 1 & 1\\}	\def\N{--
8 & 1 &2 &2 &3 & S& 24 & 6 & 8 & 5 & 0 & 0\\}	\def\N{--
8 & \bf1 &1 &2 &4 & S& 35 & 4 & 7 & 4 & 1 & 1\\}	\def\N{--
9 & 1 &\bf2 &3 &3 & S& 23 & 6 & 8 & 5 & 1 & 1\\}	\def\N{--
9 & \bf1 &1 &3 &4 & S& 33 & 5 & 9 & 5 & 1 & 1\\}	\def\N{--
10 & 1 &2 &2 &5 & S& 28 & 4 & 8 & 4 & 0 & 0\\}	\def\N{--
10 & 1 &\bf2 &3 &4 & S& 23 & 7 & 11 & 6 & 1 & 1\\}	\def\N{--
10 & \bf1 &1 &3 &5 & S& 36 & 5 & 9 & 5 & 1 & 1\\}	\def\N{--
11 & 1 &\bf2 &3 &5 & S& 24 & 8 & 13 & 7 & 1 & 1\\}	\def\N{--
12 & 1 &\bf2 &3 &6 & S& 27 & 4 & 9 & 4 & 1 & 1\\}	\def\N{--
12 & 1 &\bf2 &4 &5 & S& 24 & 5 & 12 & 5 & 1 & 1\\}	\def\N{--
12 & 1 &\bf3 &4 &4 & S& 21 & 4 & 9 & 4 & 1 & 1\\}	\def\N{--
12 & 2 &\bf3 &3 &\bf4 & S& 15 & 4 & 9 & 4 & 2 & 2\\}	\def\N{--
12 & \bf1 &1 &4 &6 & S& 39 & 4 & 9 & 4 & 1 & 1\\}	\def\N{--
12 & \bf2 &2 &3 &5 & S& 17 & 5 & 11 & 5 & 1 & 1\\}	\def\N{--
13 & 1 &\bf3 &4 &5 & S& 20 & 7 & 15 & 7 & 1 & 1\\}	\def\N{--
14 & 1 &\bf2 &4 &7 & S& 27 & 5 & 12 & 5 & 1 & 1\\}	\def\N{--
14 & 2 &3 &\bf4 &\bf5 & S& 13 & 7 & 16 & 7 & 3 & 2\\}	\def\N{--
14 & \bf2 &2 &3 &7 & S& 19 & 5 & 11 & 5 & 1 & 1\\}	\def\N{--
15 & 1 &\bf2 &5 &7 & S& 26 & 6 & 17 & 6 & 1 & 1\\}	\def\N{--
15 & 1 &\bf3 &4 &7 & S& 22 & 6 & 17 & 6 & 1 & 1\\}	\def\N{--
15 & 1 &\bf3 &5 &6 & S& 21 & 5 & 15 & 5 & 1 & 1\\}	\def\N{--
15 & 2 &3 &5 &5 & S& 14 & 6 & 11 & 5 & 1 & 0\\}	\def\N{--
15 & \bf3 &3 &4 &5 & S& 12 & 5 & 12 & 5 & 1 & 1\\}	\def\N{--
16 & 1 &\bf2 &5 &8 & S& 28 & 5 & 14 & 5 & 1 & 1\\}	\def\N{--
16 & 1 &\bf3 &4 &8 & S& 24 & 5 & 12 & 5 & 1 & 1\\}	\def\N{--
16 & 1 &\bf4 &5 &6 & S& 19 & 6 & 17 & 6 & 1 & 1\\}	\def\N{--
16 & 2 &3 &\bf4 &7 & S& 14 & 6 & 18 & 6 & 2 & 1\\}	\def\N{--
17 & 2 &3 &\bf5 &7 & S& 13 & 8 & 20 & 8 & 2 & 1\\}	\def\N{--
18 & 1 &\bf2 &6 &9 & S& 30 & 4 & 12 & 4 & 1 & 1\\}	\def\N{--
18 & 1 &\bf3 &5 &9 & S& 24 & 5 & 15 & 5 & 1 & 1\\}	\def\N{--
18 & 1 &\bf4 &6 &7 & S& 19 & 6 & 20 & 6 & 1 & 1\\}	\def\N{--
18 & 2 &3 &\bf4 &9 & S& 16 & 5 & 14 & 5 & 2 & 1\\}	\def\N{--
18 & 2 &3 &\bf5 &8 & S& 14 & 6 & 20 & 6 & 2 & 1\\}	\def\N{--
18 & 3 &4 &5 &\bf6 & S& 10 & 6 & 17 & 6 & ? & 1\\}	\def\N{--
19 & 3 &4 &5 &\bf7 & S& 9 & 7 & 24 & 8 & ? & 1\\}	\def\N{--
20 & 1 &\bf4 &5 &10 & S& 23 & 4 & 13 & 4 & 1 & 1\\}	\def\N{--
20 & 2 &3 &\bf5 &10 & S& 16 & 5 & 14 & 5 & 2 & 1\\}	\def\N{--
20 & 2 &5 &\bf6 &\bf7 &\N& 11 & 5 & 23 & 5 & 3 & 2\\}	\def\N{--
20 & 2 &\bf4 &5 &9 &\N& 13 & 4 & 23 & 4 & 1 & 1\\}	\def\N{--
20 & 3 &4 &5 &8 & S& 10 & 6 & 22 & 6 & ? & 0\\}	\def\N{--
21 & 1 &\bf3 &7 &10 &\N& 24 & 4 & 24 & 4 & 1 & 1\\}	\def\N{--
21 & 1 &\bf5 &7 &8 &\N& 18 & 5 & 24 & 5 & 1 & 1\\}	\def\N{--
21 & 2 &3 &\bf7 &9 & S& 14 & 6 & 23 & 6 & 2 & 1\\}	\def\N{--
\hline\end{tabular}	\hspace{9mm}
	\tabtop
21 & 3 &5 &\bf6 &7 &\N& 9 & 5 & 21 & 5 & ? & 1\\}	\def\N{--
22 & 1 &\bf3 &7 &11 &\N& 25 & 5 & 20 & 5 & 1 & 1\\}	\def\N{--
22 & 1 &\bf4 &6 &11 & S& 22 & 6 & 20 & 6 & 1 & 1\\}	\def\N{--
22 & 2 &\bf4 &5 &11 &\N& 14 & 5 & 19 & 5 & 1 & 1\\}	\def\N{--
24 & 1 &\bf3 &8 &12 & S& 27 & 4 & 15 & 4 & 1 & 1\\}	\def\N{--
24 & 1 &\bf6 &8 &9 & S& 18 & 5 & 24 & 5 & 1 & 1\\}	\def\N{--
24 & 2 &3 &8 &11 &\N& 15 & 4 & 27 & 4 & 1 & 0\\}	\def\N{--
24 & 2 &3 &\bf7 &12 & S& 16 & 5 & 20 & 5 & 2 & 1\\}	\def\N{--
24 & 3 &4 &5 &12 & S& 12 & 5 & 18 & 5 & ? & 0\\}	\def\N{--
24 & 3 &4 &\bf7 &10 & S& 10 & 5 & 26 & 6 & 2 & 1\\}	\def\N{--
24 & 3 &\bf6 &7 &8 &\N& 9 & 4 & 21 & 4 & ? & 1\\}	\def\N{--
24 & 4 &5 &6 &\bf9 & S& 8 & 5 & 26 & 6 & ? & 1\\}	\def\N{--
25 & 4 &5 &7 &\bf9 &\N& 7 & 5 & 32 & 6 & ? & 1\\}	\def\N{--
26 & 1 &\bf5 &7 &13 &\N& 21 & 5 & 24 & 5 & 1 & 1\\}	\def\N{--
26 & 2 &3 &8 &13 &\N& 16 & 5 & 23 & 5 & 1 & 0\\}	\def\N{--
26 & 2 &5 &\bf6 &13 &\N& 13 & 5 & 23 & 5 & 2 & 1\\}	\def\N{--
27 & 2 &5 &\bf9 &11 &\N& 11 & 6 & 32 & 6 & 2 & 1\\}	\def\N{--
27 & 5 &6 &7 &9 &\N& 6 & 5 & 30 & 6 & ? & 0\\}	\def\N{--
28 & 1 &\bf4 &9 &14 &\N& 24 & 4 & 24 & 4 & 1 & 1\\}	\def\N{--
28 & 3 &4 &\bf7 &14 & S& 12 & 5 & 18 & 5 & 2 & 1\\}	\def\N{--
28 & 4 &6 &7 &\bf11 &\N& 7 & 4 & 35 & 4 & ? & 1\\}	\def\N{--
30 & 1 &\bf4 &10 &15 & S& 25 & 5 & 20 & 5 & 1 & 1\\}	\def\N{--
30 & 1 &\bf6 &8 &15 & S& 21 & 5 & 24 & 5 & 1 & 1\\}	\def\N{--
30 & 2 &3 &10 &15 & S& 18 & 4 & 18 & 4 & 1 & 0\\}	\def\N{--
30 & 2 &\bf6 &7 &15 &\N& 13 & 4 & 23 & 4 & 1 & 1\\}	\def\N{--
30 & 3 &4 &\bf10 &13 &\N& 10 & 5 & 35 & 5 & 2 & 1\\}	\def\N{--
30 & 4 &5 &6 &15 & S& 10 & 5 & 20 & 5 & ? & 0\\}	\def\N{--
30 & 5 &6 &8 &\bf11 &\N& 6 & 4 & 39 & 4 & ? & 1\\}	\def\N{--
32 & 2 &5 &\bf9 &16 &\N& 13 & 5 & 29 & 5 & 2 & 1\\}	\def\N{--
32 & 4 &5 &7 &16 &\N& 9 & 5 & 27 & 5 & ? & 0\\}	\def\N{--
33 & 3 &5 &\bf11 &14 &\N& 9 & 4 & 39 & 4 & 2 & 1\\}	\def\N{--
34 & 3 &4 &\bf10 &17 &\N& 11 & 6 & 31 & 6 & 2 & 1\\}	\def\N{--
34 & 4 &6 &7 &\bf17 &\N& 8 & 5 & 31 & 5 & ? & 1\\}	\def\N{--
36 & 1 &\bf5 &12 &18 &\N& 24 & 4 & 24 & 4 & 1 & 1\\}	\def\N{--
36 & 3 &4 &\bf11 &18 &\N& 12 & 4 & 30 & 4 & 2 & 1\\}	\def\N{--
36 & 7 &8 &9 &12 &\N& 5 & 4 & 35 & 4 & ? & 0\\}	\def\N{--
38 & 3 &5 &\bf11 &19 &\N& 10 & 5 & 35 & 5 & 2 & 1\\}	\def\N{--
38 & 5 &6 &8 &\bf19 &\N& 7 & 5 & 35 & 5 & ? & 1\\}	\def\N{--
40 & 5 &7 &8 &20 &\N& 8 & 4 & 28 & 4 & ? & 0\\}	\def\N{--
42 & 1 &\bf6 &14 &21 & S& 24 & 4 & 24 & 4 & 1 & 1\\}	\def\N{--
42 & 2 &5 &14 &21 &\N& 15 & 4 & 27 & 4 & 1 & 0\\}	\def\N{--
42 & 3 &4 &\bf14 &21 & S& 13 & 5 & 26 & 5 & 2 & 1\\}	\def\N{--
44 & 4 &5 &\bf13 &22 &\N& 9 & 4 & 39 & 4 & 2 & 1\\}	\def\N{--
48 & 3 &5 &\bf16 &24 &\N& 12 & 4 & 30 & 4 & 2 & 1\\}	\def\N{--
50 & 7 &8 &10 &\bf25 &\N& 6 & 4 & 39 & 4 & ? & 1\\}	\def\N{--
54 & 4 &5 &\bf18 &27 &\N& 10 & 5 & 35 & 5 & 2 & 1\\}	\def\N{--
66 & 5 &6 &\bf22 &33 &\N& 9 & 4 & 39 & 4 & 2 & 1\\}	\def\N{--
				&&&&&&&&&&&	\\}	\def\N{--
\hline\end{tabular}\hspace*{-6mm}
}
\\[4mm]
	{\bf Table 7:} 	Elliptic fibration data for the 95 K3 weights
			(58 are spanning; all are transversal)
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{tabular}{||c||c|c|c||c||} \hline\hline
	& \nFa=0& \nFa=1& \nFa=2& total	\\\hline\hline
\#$\P$}	\def\nFa{\#$F$=? & 7	&	11	& 0 &	18 \\\hline
\#$\P$}	\def\nFa{\#$F$=0 & 5 &  &  & 5 \\\hline
\#$\P$}	\def\nFa{\#$F$=1 & 5 & 42 &  & 47 \\\hline
\#$\P$}	\def\nFa{\#$F$=2 & 0 & 22 & 1 & 23 \\\hline
\#$\P$}	\def\nFa{\#$F$=3 & 0 & 0 & 2 & 2 \\\hline	\hline
total & 17 & 75 & 3 & 95 \\\hline
\hline
\end{tabular}
\\[4mm]
{\bf Table 8:} 	Numbers of reflexive projections (\#$\P$}	\def\nFa{\#$F$) and facets (\nFa)
		for the K3 weights
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{tabular}{||c||c|c|c||c||} \hline\hline
Spectra	& 	MNPs	& no mirror 	& together	& $\c=0$\\\hline\hline
$WP^4$	&   	2,780	& 669		& 3,449		& 63	\\\hline
orbifolds & 	3,537	& 781		& 4,318		& 72	\\\hline
MNPs	&   	10,237	& 3,884		& 14,121	& 103	\\\hline\hline
total	&	10,410	& 3,806		& 14,216	& 104	\\\hline
\hline\end{tabular}
\\[4mm]
	{\bf Table 9:} Hodge numbers for transversal weights, orbifolds and
		maximal Newton polyhedra
\end{center}
\end{figure}

\ifundefined{epsfbox}\else

\begin{figure}
\epsfxsize=15cm
\hfil\epsfbox{hodge.eps}\hfill
\caption{$b_{11}+b_{21}\; vs\; \chi_E \;$ for single weight systems}
\label{fig:hdg}
\end{figure}
\newpage
\fi

\vfill\break
\bigskip	{\it Acknowledgements.}

A.C.A. is supported by the Robert A. Welch Foundation and by the NSF under
grant PHY/9511632. M.K. is supported by the {\it Austrian Research Fund}
under grant number P10641-PHY. M.M. is supported by the United  States
National Science Foundation
under grants PHY-9404057 and PHY-9457916.
H.S. is supported by the {\it Austrian National Bank} under grant number 5674.
It is a pleasure to acknowledge useful conversations with
Per Berglund, Philip Candelas and Anamaria Font. Most of this work was done
while M.M. was at the University of Texas at Austin.

\del
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\.klun	A.Klemm, unpublished	\>
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\enddel

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