Remarks on abelian surfaces\\ in nonsingular toric Fano 4-folds
{\sc Hiroshi Sato}\thanks{Partly supported by the Grant-in-Aid for JSPS Fellows
The Ministry of Education
Science
Sports and Culture
Japan. \newline \hspace*{1.5em} {\em $2000$ Mathematics Subject Classification\/}. Primary 14M25; Secondary 14J45, 14K12.}
% Abstract In this paper, we investigate whether the 124 nonsingular toric Fano 4-folds admit totally nondegenerate embeddings from abelian surfaces or not. As a result, we determine the possibilities for such embeddings except for the remaining 21 nonsingular toric Fano 4-folds.
math0201257.tex
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\title{Remarks on abelian surfaces\\ in nonsingular toric Fano 4-folds}

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\author{{\sc Hiroshi Sato}\thanks{Partly supported by the Grant-in-Aid for JSPS Fellows, The Ministry of Education, Science, Sports and Culture, Japan.
\newline
\hspace*{1.5em} {\em $2000$ Mathematics Subject Classification\/}.
Primary 14M25;
Secondary 14J45, 14K12.}}

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\begin{document}
\maketitle


\begin{abstract}                                                   % Abstract

In this paper, we investigate whether the 124 nonsingular toric Fano 4-folds admit totally nondegenerate embeddings from abelian surfaces or not. As a result, we determine the possibilities for such embeddings except for the remaining 21 nonsingular toric Fano 4-folds.

\end{abstract}


%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{intro}
%%%%%%%%%%%%%%%%%%%%

\thispagestyle{empty}

\hspace{5mm} There exist no embeddings from abelian surfaces into nonsingular projective toric $3$-folds over ${\bf C}$ (see, e.g., Kajiwara \cite{kajiwara2} and \cite{kajiwara1}). So the next problem is to study which nonsingular projective toric $4$-folds admit embeddings from abelian surfaces. This problem was considered by many people (see Horrocks-Mumford \cite{horrocks1}, Hulek \cite{hulek1}, Kajiwara \cite{kajiwara2}, \cite{kajiwara1}, Lange \cite{lange1} and Sankaran \cite{sankaran1}). In this paper, we consider the following problem.

\begin{Prob}\label{mainfanoprob}
{\rm
Which nonsingular toric Fano $4$-fold admits a {\em totally nondegenerate embedding} from an abelian surface (see Definition \ref{defoftndabel})?
}

\end{Prob}

There exist exactly $124$ nonsingular toric Fano $4$-folds up to isomorphism (see Batyrev \cite{batyrev4} and Sato \cite{sato1}). We give a partial answer to Problem \ref{mainfanoprob} (see Theorem \ref{mt}).

The content of this paper is as follows: In Section \ref{prepare}, we recall the definition of a {\em totally nondegenerate embedding}. In Section \ref{somecri}, we describe criteria for the non-existence of totally nondegenerate finite morphisms, and using these criteria, we show the non-existence for some nonsingular projective toric $4$-folds. In Section \ref{main2blowup}, we consider the relationship between $2$-blow-ups of toric $4$-folds and totally nondegenerate finite morphisms. As a result, we can derive the main result in Section \ref{mainresult}. In Section \ref{secex}, we show the non-existence of totally nondegenerate finite morphisms for some nonsingular toric Fano $4$-folds. In Section \ref{mainresult}, we obtain the main result.

The author wishes to thank Professors Tadao Oda, Masa-Nori Ishida and Takeshi Kajiwara for advice and encouragement.





%%%%%%%%%%%%%%%%%%%%
\section{totally nondegenerate embedding}\label{prepare}
%%%%%%%%%%%%%%%%%%%% 

\hspace{5mm} The following notation is used throughout this paper. For fundamental properties of the toric geometry, see Oda \cite{oda2}.

Let $N:={\bf Z}^{4}$ and $M:=\Hom_{\bf Z}(N,{\bf Z})$ the dual group. For a finite complete nonsingular fan $\Sigma$ in $N$ and $0\leq i\leq 4$, we put $\Sigma(i):=\left\{ \sigma\in\Sigma\; |\;\dim\sigma=i\right\}$. Each $\tau\in\Sigma(1)$ determines a unique element $e(\tau)\in N$ which generates the semigroup $\tau\cap N$. We call
$$\G(\Sigma):=\left\{e(\tau)\in N\; |\;\tau\in\Sigma(1)\right\}$$
the set of primitive generators of $\Sigma$. Let $X$ be the complete nonsingular toric $4$-fold corresponding to $\Sigma$. Let $\G(\Sigma)=\left\{ x_1,\ldots,x_n\right\}$ and let $\left\{ D_1,\ldots,D_n\right\}$ be the corresponding $T_N$-invariant prime divisors on $X$. Especially, the Picard number of $X$ is $n-4$.

In this paper, we consider the finite morphisms from abelian surfaces to nonsingular complete toric $4$-folds satisfying the following condition.

\begin{Def}\label{defoftndabel}
{\rm
Let $X$ be a $4$-dimensional complete nonsingular toric variety and $A$ an abelian surface. A finite morphism $\varphi:A\rightarrow X$ is called a {\em totally nondegenerate finite morphism} if $D_i\cap \varphi(A)$ is non-empty on $\varphi(A)$ for any $T_N$-invariant prime divisor $D_i$ $(1\leq i\leq n)$. If $\varphi$ is an embedding, we call $\varphi$ a {\em totally nondegenerate embedding}.
}

\end{Def}

\begin{Not}

{\rm
We use the following notation throughout this paper.
\begin{enumerate}
\item Let $\varphi:A\rightarrow X$ be a totally nondegenerate finite morphism. For a $T_N$-invariant prime divisor $D_i$ on $X$, we put $C_i:=\varphi^{\ast}D_i$. $C_i$ is a divisor on $A$.
\item The types of nonsingular toric Fano $4$-folds are in the sense of Batyrev \cite{batyrev4} and Sato \cite{sato1}. We use characters $\mathcal{B},\mathcal{C},\mathcal{D}\ldots$ instead of $B,C,D\ldots$. See Table $1$ in Section \ref{tablefano}.
\end{enumerate}
}
\end{Not}


For a basis $\{x_1,x_2,x_3,x_4\}$ of $N$, by computing the divisors of the rational functions ${\bf e}(x^{\ast}_1),$ ${\bf e}(x^{\ast}_2),$ ${\bf e}(x^{\ast}_3),$ ${\bf e}(x^{\ast}_4)\in {\bf C}(X)$, where $\{x^{\ast}_1,x^{\ast}_2,x^{\ast}_3,x^{\ast}_4\}\subset M$ is the dual basis of $\{x_1,x_2,x_3,x_4\}$, we get four linear relations
$${\rm div}\left({\bf e}(x^{\ast}_1)\right)=0,\ {\rm div}\left({\bf e}(x^{\ast}_2)\right)=0,\ {\rm div}\left({\bf e}(x^{\ast}_3)\right)=0\mbox{ and }{\rm div}\left({\bf e}(x^{\ast}_4)\right)=0$$
among $T_N$-invariant prime divisors in ${\rm Pic}(X)$. We often use this argument in the following sections.


%%%%%%%%%%%%%%%%%%%%
\section{Criteria for non-existence}\label{somecri}
%%%%%%%%%%%%%%%%%%%%

\hspace{5mm} In this section, we present criteria for the non-existence of totally nondegenerate finite morphism from abelian surface $A$ to a projective nonsingular toric 4-fold. We reduce Kajiwara's method in \cite{kajiwara2} and \cite{kajiwara1} to a more convenient form. For fundamental properties of primitive collections and primitive relations, see Batyrev \cite{batyrev3}, \cite{batyrev4} and Sato \cite{sato1}.

\begin{Lem}\label{effonabel}
Let $A$ be an abelian surface and $D$ an effective divisor on $A$. Then we have $D^2\geq 0$.

\end{Lem}

\proof
We may assume that $D$ is an irreducible curve on $A$. For some point $x\in A$, we have $D(D+x)\geq 0$, where $D+x$ is the translation of $D$ by $x$. Since $D$ and $D+x$ are algebraically equivalent, we have $D^2=D(D+x)\geq 0$.\qed

\begin{Lem}\label{conntocomp}
Let $X$ be a complete nonsingular toric $4$-fold and $\varphi:A\rightarrow X$ a totally nondegenerate finite morphism. If $(\varphi^{\ast}D_i)(\varphi^{\ast}D_j)=0$ and $(\varphi^{\ast}D_j)(\varphi^{\ast}D_k)=0$ where $1\leq i,j,k\leq n$, then we have $(\varphi^{\ast}D_i)(\varphi^{\ast}D_k)=0$.

\end{Lem}

\proof
Suppose that $(\varphi^{\ast}D_i)(\varphi^{\ast}D_k)>0$. Since $\left( \varphi^{\ast}D_i+\varphi^{\ast}D_k\right)^{2}=(\varphi^{\ast}D_i)^{2}+(\varphi^{\ast}D_k)^{2}+2(\varphi^{\ast}D_i)(\varphi^{\ast}D_k)\geq 2(\varphi^{\ast}D_i)(\varphi^{\ast}D_k)>0$ by Lemma \ref{effonabel}, $\varphi^{\ast}D_i+\varphi^{\ast}D_k$ is an ample divisor on $A$. On the other hand, $\left( \varphi^{\ast}D_i+\varphi^{\ast}D_k\right)(\varphi^{\ast}D_j)=0$, by assumption. This contradicts the fact that $\varphi^{\ast}D_i+\varphi^{\ast}D_k$ is ample. Therefore, $(\varphi^{\ast}D_i)(\varphi^{\ast}D_k)=0$.\hfill q.e.d.

\bigskip


For a totally nondegenerate finite morphism $\varphi:A\rightarrow X$, we define a graph $\Gamma_{\varphi}$ as follows: The vertex set of $\Gamma_{\varphi}$ is $\{ 1,\ldots,n\}$, and $\{ i,j\}$ is an edge of $\Gamma_{\varphi}$ if $i\neq j$ and $(\varphi^{\ast}D_i)(\varphi^{\ast}D_j)=C_iC_j=0$.

\begin{Rem}\label{conntocomprem}
{\rm
Lemma \ref{conntocomp} implies that every connected component of $\Gamma_{\varphi}$ is complete, i.e., any pair of distinct vertices is connected by an edge. Especially, if $\Gamma_{\varphi}$ is connected, then $\Gamma_{\varphi}$ is a complete graph.
}

\end{Rem}

\begin{Lem}\label{projtononcomp}
Let $X$ be a {\em projective} nonsingular toric $4$-fold. If there exists a totally nondegenerate finite morphism $\varphi:A\rightarrow X$, then $(\varphi^{\ast}D_i)(\varphi^{\ast}D_j)>0$ for some $1\leq i<j\leq n$.

\end{Lem}

\proof
Since $X$ is projective, there exists an ample effective divisor $\sum_{k=1}^{n}a_kD_k$ on $X$. Since $\sum_{k=1}^{n}a_k\varphi^{\ast}D_k$ is also ample on $A$, we have $\left( \sum_{k=1}^{n}a_k\varphi^{\ast}D_k\right)^{2}>0$. Therefore, we have $(\varphi^{\ast}D_i)(\varphi^{\ast}D_j)>0$ for some $1\leq i\leq j\leq n$. If $i=j$, then there exists $1\leq l\leq n$ such that $l\neq i$ and $(\varphi^{\ast}D_i)(\varphi^{\ast}D_l)>0$ by Lemma \ref{conntocomp}.\hfill q.e.d.

\bigskip

\begin{Rem}\label{keypoint}
{\rm
Lemma \ref{projtononcomp} implies that $\Gamma_{\varphi}$ is not complete. Especially, by Remark \ref{conntocomprem}, $\Gamma_{\varphi}$ is not connected.
}

\end{Rem}

\begin{Rem}\label{keypoint2}
{\rm
For $n>5$, the assertion in Remark \ref{keypoint} is also true if we replace the vertex set of $\Gamma_{\varphi}$ by $S\subset\{1,\ldots,n\}$ such that $\left\{ D_i\right\}_{i\in S} \subset {\rm Pic}(X)$ generates ${\rm Pic}(X)$.
}

\end{Rem}


By using this incompleteness of $\Gamma_{\varphi}$, we can show the non-existence of totally nondegenerate finite morphisms for some projective nonsingular toric 4-folds. For example, the following holds.

\begin{Ex}\label{g1noexist}
{\rm
Let $X$ be the nonsingular projective toric $4$-fold corresponding to the fan $\Sigma$ whose primitive relations are
$$x_1+x_7=0,\ x_2+x_3+x_4=ax_1,\ x_4+x_5+x_6=(a+1)x_1,$$
$$x_5+x_6+x_7=x_2+x_3\ \mbox{and}\ x_1+x_2+x_3=x_5+x_6,$$
where $\G(\Sigma)=\{x_1,\ldots,x_7\}$ and $a$ is a positive integer. $D_1D_7=0$ on $X$ because $\{x_1,x_7\}$ is a primitive collection of $\Sigma$, and by a basis $\{x_1,x_2,x_4,x_5\}$ of $N$, we have
$$(1)\ D_1+aD_3+(a+1)D_6-D_7=0,\ (2)\ D_2-D_3=0,$$
$$(3)\ -D_3+D_4-D_6=0\ \mbox{and}\ (4)\ D_5-D_6=0$$
in ${\rm Pic}(X)$, respectively. Suppose that there exists a totally nondegenerate finite morphism $\varphi:A\rightarrow X$. By intersecting $D_1$ with both sides of $(1)$ and restricting the result to $A$, we have $C_1^{2}+aC_1C_3+(a+1)C_1C_6-C_1C_7=0$. So $C_1C_3=C_1C_6=0$, and $C_1C_2=C_1C_4=C_1C_5=0$ by $(2)$, $(3)$ and $(4)$. Hence $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}. Therefore, $X$ admits no totally nondegenerate finite morphism.
}

\end{Ex}

\begin{Rem}
{\rm
In Example \ref{g1noexist}, if $a=1$, then $X$ is the nonsingular toric Fano $4$-fold of type $\mathcal{G}_1$. So there exists no totally nondegenerate finite morphism to the nonsingular toric Fano $4$-fold of type $\mathcal{G}_1$.
}

\end{Rem}



\begin{Prop}\label{ainfdivcont}
If a nonsingular projective toric $4$-fold $X$ has an equivariant projective birational divisorial contraction to a possibly singular point, then there exist no totally nondegenerate finite morphism to $X$.

\end{Prop}

\proof
Suppose that there exists a totally nondegenerate finite morphsim $\varphi:A\rightarrow X$. By the Mori theory for projective toric varieties (see Reid \cite{reid1}), we may assume, without loss of generality, that we have a primitive relation $x_1+x_2+x_3+x_4=ax_5$, where $a$ is a positive integer. Obviously $D_5D_i=0$ for $6\leq i\leq n$. By a basis $\{x_1,x_2,x_3,x_5\}$ of $N$, we have $aD_4+D_5+b_6D_6+\cdots+b_nD_n=0$ in ${\rm Pic}(X)$. So $aC_4C_5+C_5^{2}=0$ on $A$. Therefore $C_4C_5=0$, and similarly $C_iC_5=0$ for $1\leq i\leq 4$. This means that $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}.\hfill q.e.d.


\begin{Rem}
{\rm
By Proposition \ref{ainfdivcont}, there exist no totally nondegenerate finite morphism to the nonsingular toric Fano $4$-folds of types $\mathcal{B}_1$, $\mathcal{B}_2$, $\mathcal{B}_3$ and $\mathcal{E}_3$.
}

\end{Rem}







We now consider the case where $X$ is decomposed into the product of ${\bf P}^{1}$ and a projective nonsingular toric $3$-fold. Let $X={\bf P}^{1}\times X'$ be a projective nonsingular toric $4$-fold, where $X'$ is a projective nonsingular toric $3$-fold. Suppose that $x_1$ and $x_2$ correspond to the class of fibers of the first projection $X\rightarrow {\bf P}^{1}$, where $\G(\Sigma)=\{ x_1,x_2,\ldots,x_n\}$. Then the following holds.

\begin{Prop}\label{subconn}
Suppose that there exists a totally nondegenerate finite morphism $\varphi:A\rightarrow X$. We define a subgraph $\Gamma'$ of $\Gamma_{\varphi}$ as follows$:$ The vertex set of $\Gamma'$ is $\{ 3,\ldots,n\}$, and $\{i,j\}$ $(3\leq i<j\leq n)$ is an edge of $\Gamma'$ if $\{i,j\}$ is an edge of $\Gamma_{\varphi}$. Then $\Gamma'$ is not complete.

\end{Prop}

\proof
Since each fiber of the second projection $p_2:X\rightarrow X'$ is ${\bf P}^1$, each fiber of $p_2$ is not contained in the abelian surface $A$. So $p_2\circ\varphi$ is a finite morphism. Therefore, for an ample divisor $E=\sum_{i=3}^{n}a_iE_i$ on the projective variety $X'$, where $E_3,\ldots,E_n$ are the toric divisors corresponding to $x_3,\ldots,x_n$, respectively, $(p_2\circ\varphi)^{\ast}(E)=\sum_{i=3}^{n}a_i(p_2\circ\varphi)^{\ast}D_i$ is also an ample divisor on $A$. So there exist $3\leq i<j\leq n$ such that $((p_2\circ\varphi)^{\ast}D_i)((p_2\circ\varphi)^{\ast}D_j)\neq 0$. Since $\{i,j\}$ is not a edge of $\Gamma'$, the graph $\Gamma'$ is not complete.\hfill q.e.d.

\bigskip

\begin{Ex}\label{l5noexist}
{\rm
Let $X$ be the nonsingular projective toric $4$-fold corresponding to the fan $\Sigma$ whose primitive relations are
$$x_1+x_8=0,\ x_2+x_3=0,\ x_4+x_5=ax_3\ \mbox{and}\ x_6+x_7=ax_3,$$
where $\G(\Sigma)=\{x_1,\ldots,x_8\}$ and $a$ is a positive integer. $D_1D_8=D_2D_3=D_4D_5=D_6D_7=0$ on $X$, and by a basis $\{x_1,x_2,x_4,x_6\}$ of $N$, we have
$$(1)\ D_1-D_8=0,\ (2)\ D_2-D_3-aD_5-aD_7=0,\ (3)\ D_4-D_5=0\ \mbox{and}\ (4)\ D_6-D_7=0$$
in ${\rm Pic}(X)$, respectively. $X$ is isomorphic to ${\bf P}^{1}\times X'$, where $X'$ is a toric $3$-fold, and $D_1$ and $D_8$ are fibers of the first projection $X\rightarrow {\bf P}^{1}$. Suppose that there exists a totally nondegenerate finite morphism $\varphi:A\rightarrow X$. By $(2)$, we have $C_3^{2}+aC_3C_5+aC_3C_7=C_2C_3=0$, hence $C_3C_5=C_3C_7=0$. Consequently, $C_3C_4=C_3C_6=0$ by $(3)$ and $(4)$. Hence the graph $\Gamma'$ as in Proposition \ref{subconn} is connected, a contradiction to  Proposition \ref{subconn}. Therefore, $X$ admits no totally nondegenerate finite morphism.
}

\end{Ex}

\begin{Rem}
{\rm
In Example \ref{l5noexist}, if $a=1$, then $X$ is the nonsingular toric Fano $4$-fold of type $\mathcal{L}_5$. So there exists no totally nondegenerate finite morphism to the nonsingular toric Fano $4$-fold of type $\mathcal{L}_5$.
}

\end{Rem}


For the main theorem of this paper, we show some results for the non-existence of totally nondegenerate finite morphisms, using Remark \ref{keypoint} and Proposition \ref{subconn}.

\begin{Prop}\label{faoverp2}
Let $X$ be an $F_a$-bundle over ${\bf P}^{2}$, where ${\rm F}_a$ is the Hirzebruch surface of degree $a$ $(a\geq 0)$, and $\G(\Sigma)=\{ x_1,\ldots x_7\}$. We introduce a coordinate in $N$ so that the coordinates of $x_1,x_2,x_3,x_4,x_5,x_6$ and $x_7$ are
\[
\pmatrix{
1 \cr
0 \cr
0 \cr
0 \cr},\ 
\pmatrix{
0 \cr
1 \cr
0 \cr
0 \cr},\ 
\pmatrix{
-1 \cr
-1 \cr
s \cr
t \cr},\ 
\pmatrix{
0 \cr
0 \cr
1 \cr
0 \cr},\ 
\pmatrix{
0 \cr
0 \cr
-1 \cr
a \cr},\ 
\pmatrix{
0 \cr
0 \cr
0 \cr
1 \cr}\mbox{ and }
\pmatrix{
0 \cr
0 \cr
0 \cr
-1 \cr},
\]
respectively, where $s$ and $t$ are integers. In this situation, the following hold:
\begin{enumerate}
\item If $s=t=0$, then $X$ is isomorphic to ${\bf P}^{2}\times{\rm F}_a$.
\item In the case $s\neq 0$ or $t\neq 0$, if one of the following conditions is satisfied, then $X$ admits no totally nondegenerate finite morphism.
\begin{enumerate}
\item $a=0$.
\item $a>0$ and $t\geq 0$.
\item $a>0$, $s>0$, $t<0$ and $as+t\geq 0$.
\end{enumerate}
\end{enumerate}


\end{Prop}

\proof
Suppose that there exists a totally nondegenerate finite morphism $\varphi:A\rightarrow X$. $($i$)$ is trivial, so let $s\neq 0$ or $t\neq 0$. By a basis $\{x_1,x_2,x_4,x_6\}$ of $N$, we have
$$(1)\ D_1-D_3=0,\ (2)\ D_2-D_3=0,\ (3)\ sD_3+D_4-D_5=0\mbox{ and }(4)\ tD_3+aD_5+D_6-D_7=0$$
in ${\rm Pic}(X)$, respectively. Moreover, we have $D_4D_5=D_6D_7=0$ on $X$, and $C_4^{2}=C_5^{2}=C_6^{2}=C_7^{2}=0$ on $A$.

(a) Let $a=0$. In this case, $X$ is isomorphic to ${\bf P}^{1}\times X'$, where $X'$ is a toric $3$-fold.

If $s=0$ and $t\neq 0$, then $D_4$ and $D_5$ are in the class of fibers of the first projection $X\rightarrow{\bf P}^{1}$. Since $tD_3+D_6-D_7=0$ by $(4)$, we have $tC_3C_6=-C_6^{2}+C_6C_7=0$. Therefore $C_3C_6=0$, and $C_1C_6=C_2C_6=0$ by $(1)$ and $(2)$. This contradicts Proposition \ref{subconn}.

If $s\neq 0$ and $t=0$, then $D_6$ and $D_7$ are in the class of fibers of the first projection $X\rightarrow{\bf P}^{1}$. Since $sC_3C_4=-C_4^{2}+C_4C_5=0$ by $(3)$, we have $C_3C_4=0$. On the other hand, $C_1C_4=C_2C_4=0$ by $(1)$ and $(2)$. This contradicts Proposition \ref{subconn}.

(b) Let $a>0$ and $t\geq 0$. Since
$$(5)\ tC_3C_6+aC_5C_6=-C_6^{2}+C_6C_7=0$$
by $(4)$, we have $C_5C_6=0$.

If $t>0$, then $C_3C_6=0$ by $(5)$. Moreover $C_1C_6=C_2C_6=0$ by $(1)$ and $(2)$. So $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}.

Let $t=0$. Then $s\neq 0$ by assumption. So we have $C_3C_4=0$ as above, and $C_1C_4=C_2C_4=0$ by $(1)$ and $(2)$. So $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}.

(c) Let $a>0$, $s>0$, $t<0$ and $as+t\geq 0$. Then we have $C_3C_4=C_1C_4=C_2C_4=0$ as above. On the other hand, by $(3)$ and $(4)$, we have
$$(6)\ -tD_4+(as+t)D_5+sD_6-sD_7=0$$
in ${\rm Pic}(X)$. So $-tC_4C_6+(as+t)C_5C_6=-sC_6^{2}+sC_6C_7=0$, and we have $C_4C_6=0$ by the assumptions $t<0$ and $as+t\geq 0$. Therefore $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}.\hfill q.e.d.

\begin{Rem}
{\rm
By Proposition \ref{faoverp2}, there exists no totally nondegenerate finite morphism to the nonsingular toric Fano $4$-folds of types $\mathcal{D}_1$, $\mathcal{D}_2$, $\mathcal{D}_3$, $\mathcal{D}_5$, $\mathcal{D}_6$, $\mathcal{D}_8$, $\mathcal{D}_9$, $\mathcal{D}_{12}$ and $\mathcal{D}_{16}$. The corresponding $a$, $s$ and $t$ are as follows:

\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|}\hline
 & $\mathcal{D}_1$ & $\mathcal{D}_2$ & $\mathcal{D}_3$ & $\mathcal{D}_5$ & $\mathcal{D}_6$ & $\mathcal{D}_8$ & $\mathcal{D}_9$ & $\mathcal{D}_{12}$ & $\mathcal{D}_{16}$ \\ \hline
$a$ & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 \\ \hline
$s$ & 0 & 2 & 1 & 2 & 0 & 1 & 1 & 1 & 1 \\ \hline
$t$ & 2 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & -1 \\ \hline
\end{tabular}
\end{center}




}

\end{Rem}







%%%%%%%%%%%%%%%%%%%%
\section{2-blow-up}\label{main2blowup}
%%%%%%%%%%%%%%%%%%%%

\hspace{.5cm} The following is useful for deriving the main result in Section $6$.

\begin{Prop}\label{2blowupprop}
Let $X$ and $\widetilde{X}$ be nonsingular projective toric $4$-folds and $\psi:\widetilde{X}\rightarrow X$ a $2$-blow-up, where a ``$2$-blow-up'' means an equivariant blow-up along a $T_N$-invariant subvariety of codimension $2$. If $X$ admits no totally nondegenerate finite morphism, then $\widetilde{X}$ admits no totally nondegenerate finite morphism either.

\end{Prop}

\proof
If there exists a totally nondegenerate finite morphism $\varphi:A\rightarrow \widetilde{X}$, then $\psi\circ\varphi:A\rightarrow X$ is also a totally nondegenerate finite morphism (see Mumford \cite{mumford1}, p. 88). This is a contradiction.\qed

\bigskip

Especially, we have the following.

\begin{Cor}\label{dia2}
Let $X_1\leftarrow X_2\leftarrow\cdots\leftarrow X_{n-1}\leftarrow X_n$ be a sequence of $2$-blow-ups among nonsingular projective toric $4$-folds. If $X_1$ admits no totally nondegenerate finite morphism then $X_n$ admits no totally nondegenerate finite morphism either.

\end{Cor}



We end this section by proposing the following conjecture.

\begin{Conj}\label{2blowupconj}
Let $X$ and $\widetilde{X}$ be nonsingular projective toric $4$-folds and $\psi:\widetilde{X}\rightarrow X$ a $2$-blow-up. If $X$ admits no totally nondegenerate embedding, then $\widetilde{X}$ admits no totally nondegenerate embedding either.

\end{Conj}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Some examples}\label{secex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\hspace{.5cm} In this section, to describe the main result in Section \ref{mainresult}, we show the non-existence of totally nondegenerate embeddings for some other nonsingular toric Fano $4$-folds.

\bigskip

(a) ``type $\mathcal{I}$'s''\quad Let $X$ be the nonsingular projective toric $4$-fold corresponding to the fan $\Sigma$ defined as follows: Let $\G(\Sigma)=\{x_1,\ldots,x_8\}\subset N$ such that the coordinates of $x_1,\ldots,x_8$ are
\[
\pmatrix{
1 \cr
0 \cr
0 \cr
0 \cr},\ 
\pmatrix{
-1 \cr
b \cr
0 \cr
c \cr},\ 
\pmatrix{
0 \cr
1 \cr
0 \cr
0 \cr},\ 
\pmatrix{
0 \cr
0 \cr
1 \cr
0 \cr},\ 
\pmatrix{
0 \cr
-1 \cr
-1 \cr
a+1 \cr},\ 
\pmatrix{
0 \cr
0 \cr
0 \cr
-1 \cr},\ 
\pmatrix{
0 \cr
1 \cr
0 \cr
-1 \cr},\ 
\pmatrix{
0 \cr
0 \cr
0 \cr
1 \cr},
\]
respectively, and that the primitive collections of $\Sigma$ are $\{ x_3,x_4,x_5\}$, $\{ x_4,x_5,x_7\}$, $\{ x_7,x_8\}$, $\{ x_3,x_6\}$, $\{ x_6,x_8\}$ and $\{ x_1,x_2\}$. For some values of $a$, $b$ and $c$, $X$ becomes the nonsingular toric Fano $4$-fold of type $\mathcal{I}$. The corresponding $a$, $b$ and $c$ are as follows:
\[
\begin{tabular}{|c||c|c|c|c|}\hline
 & $\mathcal{I}_4$ & $\mathcal{I}_6$ & $\mathcal{I}_{12}$ & $\mathcal{I}_{15}$ \\ \hline
$a$ & 1 & 0 & 0 & 1 \\ \hline
$b$ & 1 & 1 & 0 & 0 \\ \hline
$c$ & -1 & 0 & -1 & -1 \\ \hline
\end{tabular}
\]
$D_7D_8=D_3D_6=D_6D_8=D_1D_2=0$ on $X$, and by a basis $\{x_1,x_3,x_4,x_8\}$ of $N$, we have
$$(1)\ D_1-D_2=0,\ (2)\ bD_2+D_3-D_5+D_7=0,$$
$$(3)\ D_4-D_5=0\ \mbox{and}\ (4)\ cD_2+(a+1)D_5-D_6-D_7+D_8=0$$
in ${\rm Pic}(X)$, respectively. Moreover, we have
$$(5)\ (b+c)D_2+D_3+aD_5-D_6+D_8=0$$
by $(2)$ and $(4)$. Suppose that there exists a totally nondegenerate finite morphism $\varphi:A\rightarrow X$.

If $b=0$ and $c=-1$, then $C_5C_6=C_3C_6+C_6C_7=0$ by $(2)$. On the other hand, by $(4)$, we have $C_2C_5=C_5C_8+(a+1)C_5^{2}-C_5C_6-C_5C_7=0$. Hence $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}. Therefore, $X$ admits no totally nondegenerate finite morphism. Especially, the nonsingular toric Fano $4$-folds of types $\mathcal{I}_{12}$ and $\mathcal{I}_{15}$ admit no totally nondegenerate finite morphism.

In the case $b=1$, if $a=1$ and $b+c=0$, then $C_3C_5=-C_3^{2}+C_3C_6-C_3C_8=0$ by $(5)$, and $C_2C_3=-C_3^{2}+C_3C_5$ $-C_3C_7=0$ by $(2)$. On the other hand, if $a=0$ and $b+c=1$, then $C_2C_3=0$ by $(5)$, and $C_3C_5=0$ by $(2)$. In any case, $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}. Therefore, $X$ admits no totally nondegenerate finite morphism. Especially, the nonsingular toric Fano $4$-folds of types $\mathcal{I}_{4}$ and $\mathcal{I}_{6}$ admit no totally nondegenerate finite morphism.

\bigskip


(b) ``type $\mathcal{J}_2$''\quad Let $X$ be the nonsingular projective toric $4$-fold corresponding to the fan $\Sigma$ whose primitive relations are
$$x_3+x_6=x_7,\ x_1+x_2+x_8=x_4+x_5,\ x_4+x_5+x_6=x_1+x_2,\ x_7+x_8=x_3,\ x_6+x_8=0,$$
$$x_3+x_4+x_5=x_8,\ x_4+x_5+x_7=0,\ x_1+x_2+x_3=0\ \mbox{and}\ x_1+x_2+x_7=x_6,$$
where $\G(\Sigma)=\{x_1,\ldots,x_8\}$. $X$ is the nonsingular toric Fano $4$-fold of type $\mathcal{J}_2$. $D_3D_6=D_7D_8=D_6D_8=0$ on $X$, and by a basis $\{x_1,x_2,x_4,x_5\}$ of $N$, we have
$$(1)\ D_1-D_3+D_6-D_8=0,\ (2)\ D_2-D_3+D_6-D_8=0,$$
$$(3)\ D_4-D_6-D_7+D_8=0\ \mbox{and}\ (4)\ D_5-D_6-D_7+D_8=0$$
in ${\rm Pic}(X)$, respectively. So we have $D_1=D_2$ and $D_4=D_5$. Suppose that there exists a totally nondegenerate finite morphism $\varphi:A\rightarrow X$. By $(1)$, we have $C_1C_3=C_3^{2}-C_3C_6+C_3C_8=0$. On the other hand, by $(3)$, we have $C_3C_4=C_3C_6+C_3C_7-C_3C_8=0$. Hence $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}. Therefore, $X$ admits no totally nondegenerate finite morphism.




\bigskip

(c) ``type $\mathcal{L}$'s''\quad Let $X$ be the nonsingular projective toric $4$-fold corresponding to the fan $\Sigma$ defined as follows: Let $\G(\Sigma)=\{x_1,\ldots,x_8\}\subset N$ such that the coordinates of $x_1,\ldots,x_8$ are
\[
\pmatrix{
0 \cr
1 \cr
0 \cr
0 \cr},\ 
\pmatrix{
-1 \cr
1 \cr
0 \cr
0 \cr},\ 
\pmatrix{
1 \cr
0 \cr
0 \cr
0 \cr},\ 
\pmatrix{
0 \cr
0 \cr
1 \cr
0 \cr},\ 
\pmatrix{
a \cr
b \cr
-1 \cr
0 \cr},\ 
\pmatrix{
0 \cr
0 \cr
0 \cr
1 \cr},\ 
\pmatrix{
c \cr
d \cr
0 \cr
-1 \cr},\ 
\pmatrix{
0 \cr
-1 \cr
0 \cr
0 \cr},
\]
respectively, and that the primitive collections of $\Sigma$ are $\{ x_1,x_8\}$, $\{ x_2,x_3\}$, $\{ x_4,x_5\}$ and $\{ x_6,x_7\}$. For some values of $a$, $b$, $c$ and $d$, $X$ becomes the nonsingular toric Fano $4$-fold of type $\mathcal{L}$. The corresponding $a$, $b$, $c$ and $d$ are as follows:
\[
\begin{tabular}{|c||c|c|c|}\hline
 & $\mathcal{L}_1$ & $\mathcal{L}_{2}$ & $\mathcal{L}_{10}$ \\ \hline
$a$ & 0 & 1 & 1 \\ \hline
$b$ & 1 & 0 & 0 \\ \hline
$c$ & 0 & 1 & -1 \\ \hline
$d$ & 1 & 0 & 1 \\ \hline
\end{tabular}
\]
$D_1D_8=D_2D_3=D_4D_5=D_6D_7=0$ on $X$, and by a basis $\{x_1,x_3,x_4,x_6\}$ of $N$, we have
$$(1)\ D_1+D_2+bD_5+dD_7-D_8=0,\ (2)\ -D_2+D_3+aD_5+cD_7=0,$$
$$(3)\ D_4-D_5=0\ \mbox{and}\ (4)\ D_6-D_7=0$$
in ${\rm Pic}(X)$, respectively. Suppose that there exists a totally nondegenerate finite morphism $\varphi:A\rightarrow X$. By $(1)$ and $(2)$, we have $C_1C_2+bC_1C_5+dC_1C_7=-C_1^{2}+C_1C_8=0$ and $aC_3C_5+cC_3C_7=-C_3^{2}+C_2C_3=0$, respectively. Especially, $C_1C_2=0$.

If either $a>0$ and $c>0$ or $b>0$ and $d>0$, then $C_3C_5=C_3C_5=0$ or $C_1C_5=C_1C_7=0$, respectively. In any case, $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}. Therefore, $X$ admits no totally nondegenerate finite morphism. Especially, the nonsingular toric Fano $4$-folds of types $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ admit no totally nondegenerate finite morphism.

If $X$ is of type $\mathcal{L}_{10}$, then $C_1C_7=0$ because $b\geq 0$ and $d>1$. Moreover, by $(2)$, we have $C_1C_5=-C_1C_3+C_2C_3+C_1C_7=0$. Hence $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}. Therefore, the nonsingular toric Fano $4$-fold of type $\mathcal{L}_{10}$ admits no totally nondegenerate finite morphism.





\bigskip



(d) ``type $\mathcal{L}_{12}$''\quad This case is special. Let $X$ be the nonsingular projective toric $4$-fold corresponding to the fan $\Sigma$ whose primitive relations are
$$x_1+x_8=0,\ x_2+x_3=x_1,\ x_4+x_5=x_8\ \mbox{and}\ x_6+x_7=x_4,$$
where $\G(\Sigma)=\{x_1,\ldots,x_8\}$. $X$ is the nonsingular toric Fano $4$-fold of type $\mathcal{L}_{12}$. $D_1D_8=D_2D_3=D_4D_5=D_6D_7=0$ on $X$, and by a basis $\{x_1,x_2,x_4,x_6\}$ of $N$, we have
$$(1)\ D_1+D_3-D_5-D_8=0,\ (2)\ D_2-D_3=0,$$
$$(3)\ D_4-D_5+D_7=0\ \mbox{and}\ (4)\ D_6-D_7=0$$
in ${\rm Pic}(X)$, respectively. So we have $D^{2}_2=D^{2}_3=D^{2}_6=D^{2}_7=0$,
$$(5)\ D^{2}_{8}=D_3D_8-D_5D_8\ \mbox{and}\ (6)\ D^{2}_5=D_5D_7$$
on $X$. Suppose that there exists a totally nondegenerate embedding $\varphi:A\hookrightarrow X$. Then by $(3)$, we have $D_4D_7A=-(D_4A)^{2}+D_4D_5A=0$. Moreover, since $D_1A$ is an effective divisor on $A$ and $D_3D_8A-D_5D_8A=0$ by $(5)$, we have $(D_1A)^{2}=(-D_3A+D_5A+D_8A)^{2}=-2D_3D_5A-2D_3D_8A+2D_5D_8A=-2D_3D_5A\geq 0$. Therefore, we have $D_3D_5A=0$. On the other hand, $\{ D_3D_5,D_3D_7,D_3D_8,D_5D_7,D_5D_8,D_7D_8\}$ generates ${\rm A}^{2}(X)$ by the equalities $(5)$ and $(6)$. So we can express the class of $A$ in ${\rm A}^{2}(X)$ as
$$A=a_1D_3D_5+a_2D_3D_7+a_3D_3D_8+a_4D_5D_7+a_5D_5D_8+a_6D_7D_8\in {\rm A}^{2}(X).$$
Since $D_3D^{2}_5D_7=D_3(D_4+D_7)D_5D_7=0$, $D_3D^{2}_5D_8=D_3(D_4+D_7)D_5D_8=1$ and $D^{2}_5D_7D_8=(D_4+D_7)D_5D_7D_8=0$, we have the following:
\[
(7)\ \left\{
\begin{array}{c}
D_3D_5A=a_4D_3D^{2}_5D_7+a_5D_3D^{2}_5D_8+a_6D_3D_5D_7D_8=a_5+a_6=0, \\
D_3D_7A=a_5D_3D_5D_7D_8=a_5=0\mbox{ and} \\
D_5D_7A=a_1D_3D^{2}_5D_7+a_3D_3D_5D_7D_8+a_5D^{2}_5D_7D_8=a_3=0. \\
\end{array}
\right.
\]
By these equalities, $a_3=a_5=a_6=0$. So $A^{2}=a_1D_3D_5A+a_2D_3D_7A+a_4D_5D_7A=0$. Therefore, by the following, we have $c_2(X)A=0$.

\begin{Lem}[Van de Ven \cite{vandeven1}, Proposition 3]\label{selfintersection}
Let $A\hookrightarrow X$ be an embedding from an abelian surface to a $4$-dimensional nonsingular projective toric variety. Then we have
$c_2(X)A=A^{2}$ in the group ${\rm A}^{4}(X)$ of codimension four cycles on $X$ modulo rational equivalence.

\end{Lem}

Since
$$c_2(X)=\sum_{1\leq i<j\leq n}D_iD_j,$$
$\Gamma_{\varphi}$ is connected. This contradicts Remark \ref{keypoint}. So $X$ admits no totally nondegenerate embedding.


\bigskip

(e) ``type $\mathcal{M}$'s''\quad Let $X$ be the nonsingular projective toric $4$-fold corresponding to the fan $\Sigma$ defined as follows: Let $\G(\Sigma)=\{x_1,\ldots,x_7\}\subset N$ such that the coordinates of $x_1,\ldots,x_7$ are
\[
\pmatrix{
1 \cr
0 \cr
0 \cr
0 \cr},\ 
\pmatrix{
0 \cr
1 \cr
0 \cr
0 \cr},\ 
\pmatrix{
-1 \cr
-1 \cr
1 \cr
1 \cr},\ 
\pmatrix{
0 \cr
0 \cr
1 \cr
0 \cr},\ 
\pmatrix{
a \cr
0 \cr
-1 \cr
0 \cr},\ 
\pmatrix{
0 \cr
0 \cr
0 \cr
1 \cr},\ 
\pmatrix{
ac+b \cr
0 \cr
-c \cr
-1 \cr},\ 
\pmatrix{
-1 \cr
0 \cr
0 \cr
0 \cr},
\]
respectively, and that the primitive collections of $\Sigma$ are $\{ x_1,x_8\}$, $\{ x_1,x_2,x_3\}$, $\{ x_4,x_6,x_8\}$, $\{ x_4,x_5\}$, $\{ x_6,x_7\}$, $\{ x_2,x_3,x_5\}$ and $\{ x_2,x_3,x_7\}$. For some values of $a$, $b$ and $c$, $X$ becomes the nonsingular toric Fano $4$-fold of type $\mathcal{M}$. The corresponding $a$, $b$ and $c$ are as follows:
\[
\begin{tabular}{|c||c|c|c|c|}\hline
 & $\mathcal{M}_1$ & $\mathcal{M}_2$ & $\mathcal{M}_3$ & $\mathcal{M}_4$ \\ \hline
$a$ & 0 & 1 & 1 & 1 \\ \hline
$b$ & 0 & 1 & 0 & 0 \\ \hline
$c$ & 0 & 0 & 1 & 0 \\ \hline
\end{tabular}
\]
$D_1D_8=D_4D_5=D_6D_7=0$ on $X$, and by a basis $\{x_1,x_2,x_4,x_6\}$ of $N$, we have
$$(1)\ D_1-D_3+aD_5+(ac+b)D_7-D_8=0,\ (2)\ D_2-D_3=0,$$
$$(3)\ D_3+D_4-D_5-cD_7=0\ \mbox{and}\ (4)\ D_3+D_6-D_7=0$$
in ${\rm Pic}(X)$, respectively. Suppose that there exists a totally nondegenerate finite morphism $\varphi:A\rightarrow X$. By $(4)$, we have $C_3C_6=-C_6^{2}+C_6C_7=0$. By $(1)$, $(3)$ and $(4)$, we have
$$(5)\ D_1+(a-1)D_3+aD_4+bD_7-D_8=0\ \mbox{and}\ (6)\ D_1+aD_5+D_6+(ac+b-1)D_7-D_8=0.$$
Let $a\geq 0$, $b\geq 0$ and $c\geq 0$.

If $a=1$ and $c=0$, then by $(3)$, we have $C_3C_4=-C_4^{2}+C_4C_5=0$. On the other hand, by $(5)$, we have $C_1C_4+bC_1C_7=-C_1^{2}+C_1C_8=0$, and $C_1C_4=0$. Hence $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}. Therefore, $X$ admits no totally nondegenerate finite morphism. Especially, the nonsingular toric Fano $4$-folds of types $\mathcal{M}_2$ and $\mathcal{M}_4$ admit no totally nondegenerate finite morphism.

If $a=b=c=0$, then $C_3C_4=0$ as above. On the other hand, by $(5)$, we have $C_1C_3=C_1^{2}-C_1C_8=0$, and $C_1C_3=0$. Hence $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}. Therefore, $X$ admits no totally nondegenerate finite morphism. Especially, the nonsingular toric Fano $4$-fold of type $\mathcal{M}_1$ admits no totally nondegenerate finite morphism.

If $a=c=1$, then $C_1C_4=0$ as above. On the other hand, by $(6)$, we have $C_1C_5+C_1C_6+bC_1C_7=-C_1^{2}+C_1C_8=0$, and $C_1C_6=0$. Hence $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}. Therefore, $X$ admits no totally nondegenerate finite morphism. Especially, the nonsingular toric Fano $4$-fold of type $\mathcal{M}_3$ admits no totally nondegenerate finite morphism.





\bigskip

(f) ``type $\mathcal{M}_5$''\quad Let $X$ be the nonsingular projective toric $4$-fold corresponding to the fan $\Sigma$ whose primitive relations are
$$x_1+x_8=x_5,\ x_4+x_5=x_7,\ x_6+x_7=x_1,\ x_1+x_2+x_3=x_6,\ x_2+x_3+x_5=x_6+x_8$$
$$x_2+x_3+x_7=0\ \mbox{and}\ x_4+x_6+x_8=0,$$
where $\G(\Sigma)=\{x_1,\ldots,x_8\}$. $X$ is the nonsingular toric Fano $4$-fold of type $\mathcal{M}_5$. $D_1D_8=D_4D_5=D_6D_7=0$ on $X$, and by a basis $\{x_1,x_2,x_4,x_6\}$ of $N$, we have
$$(1)\ D_1-D_3+D_5+D_7=0,\ (2)\ D_2-D_3=0,$$
$$(3)\ D_4-D_5-D_8=0\ \mbox{and}\ (4)\ D_3-D_5+D_6-D_7-D_8=0$$
in ${\rm Pic}(X)$, respectively. Suppose that there exists a totally nondegenerate finite morphism $\varphi:A\rightarrow X$. By $(1)$, $(3)$ and $(4)$, we have $C_5C_8=-C_4C_5+C_5^{2}=0$ and $C_6C_8=-C_1C_8+C_8^{2}=0$, respectively. On the other hand, by Lemma \ref{conntocomp}, $C_1C_5=C_1C_7=0$. So by $(1)$, we have $C_1C_3=C_1^{2}+C_1C_5+C_1C_7=0$. Hence $\Gamma_{\varphi}$ is connected, a contradiction to Remark \ref{keypoint}. Therefore, $X$ admits no totally nondegenerate finite morphism.



\begin{Rem}\label{vksymmetric}
{\rm
Kajiwara \cite{kajiwara2}, \cite{kajiwara1} showed that the pseudo del Pezzo $4$-fold $\widetilde{V}^{4}$ admits no totally nondegenerate finite morphism similarly as above.
}

\end{Rem}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The main results}\label{mainresult}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\hspace{.5cm} By Examples \ref{g1noexist} and \ref{l5noexist}, Propositions \ref{ainfdivcont} and \ref{faoverp2}, the results (a), (b), (c), (d), (e) and (f), and Remark \ref{vksymmetric}, the nonsingular toric Fano $4$-folds of types $\mathcal{B}_1$, $\mathcal{B}_2$, $\mathcal{B}_3$, $\mathcal{D}_1$, $\mathcal{D}_2$, $\mathcal{D}_3$, $\mathcal{D}_5$, $\mathcal{D}_6$, $\mathcal{D}_8$, $\mathcal{D}_9$, $\mathcal{D}_{12}$, $\mathcal{D}_{16}$, $\mathcal{G}_1$, $\mathcal{I}_4$, $\mathcal{I}_6$, $\mathcal{I}_{12}$, $\mathcal{I}_{15}$, $\mathcal{J}_2$, $\mathcal{L}_1$, $\mathcal{L}_2$, $\mathcal{L}_5$, $\mathcal{L}_{10}$, $\mathcal{L}_{12}$, $\mathcal{M}_1$, $\mathcal{M}_2$, $\mathcal{M}_3$, $\mathcal{M}_4$, $\mathcal{M}_5$ and $\widetilde{V}^{4}$ admit no totally nondegenerate embedding. Moreover, by Corollary \ref{dia2}, any nonsingular projective toric $4$-fold admits no totally nondegenerate embedding, if it is obtained by finite successions of $2$-blow-ups from one of them.

\begin{Rem}
{\rm
Kajiwara \cite{kajiwara1} and Sankaran \cite{sankaran1} showed that the nonsingular toric Fano $4$-folds of types $\mathcal{B}_5$, $\mathcal{D}_{19}$, $\mathcal{G}_{2}$ and $\mathcal{G}_6$ admit no totally nondegenerate embedding using more complicated methods (see Kajiwara \cite{kajiwara1} for types $\mathcal{D}_{19}$, $\mathcal{G}_{2}$ and $\mathcal{G}_6$, Sankaran \cite{sankaran1} for type $\mathcal{B}_5$). Since their method differs from ours, we cannot determine whether nonsingular projective toric $4$-folds obtained by finite successions of $2$-blow-ups from one of these types admit a totally nondegenerate embedding or not.
}

\end{Rem}


To describe the main result, we need the following proposition.

\begin{Prop}\label{dptimesdp}
If $X$ is a nonsingular toric Fano $4$-fold such that $X\cong X_{1}\times X_{2}$, where $X_{1}$ and $X_{2}$ are nonsingular toric del Pezzo surfaces, then there exists a totally nondegenerate embedding.

\end{Prop}

\proof
A smooth element $E_1$ in $|-K_{X_1}|$ (resp. $E_2$ in $|-K_{X_2}|$) is an elliptic curve. By an easy calculation of intersection numbers, $E_1 \times E_2\hookrightarrow X$ is obviously a totally nondegenerate embedding.\hfill q.e.d.

\begin{Rem}
{\rm
In Proposition \ref{dptimesdp}, if there exists an abelian surface embedding $A\hookrightarrow X$, then $A$ is isomorphic to the direct product of two elliptic curves as stated in the proof of Proposition \ref{dptimesdp}, by the results of Kajiwara \cite{kajiwara2} and \cite{kajiwara1}.
}

\end{Rem}

By these results and Table $1$ in Sato \cite{sato1}, we get the following:

\begin{Thm}\label{mt}
Let $X$ be a nonsingular toric Fano $4$-fold. Then, one of the following holds.

\begin{enumerate}

\item $X$ admits no totally nondegenerate embedding.

\item $X\cong {\bf P}^{4}$ or $X\cong {\bf P}^{1}\times{\bf P}^{3}$. There exists a totally nondegenerate embedding in this case $($see {\rm Horrocks-Mumford \cite{horrocks1}} and {\rm Lange \cite{lange1}}$)$.

\item $X\cong X_{1}\times X_{2}$, where $X_{1}$ and $X_{2}$ are nonsingular toric del Pezzo surfaces. There exists a totally nondegenerate embedding in this case $($see Proposition $\ref{dptimesdp})$.
\item $X$ is of one of the types $\mathcal{C}_1$, $\mathcal{C}_2$, $\mathcal{C}_3$, $\mathcal{D}_7$, $\mathcal{D}_{10}$, $\mathcal{D}_{11}$, $\mathcal{D}_{14}$, $\mathcal{D}_{17}$, $\mathcal{D}_{18}$, $\mathcal{G}_3$, $\mathcal{G}_4$, $\mathcal{G}_5$, $\mathcal{L}_{11}$, $\mathcal{L}_{13}$, $\mathcal{I}_9$, $\mathcal{Q}_{16}$, $\mathcal{U}_8$, $V^4$, $\mathcal{Z}_1$, $\mathcal{Z}_2$ and $\mathcal{W}$.

\end{enumerate}

\end{Thm}

\begin{Rem}
{\rm
For the nonsingular toric Fano $4$-fold $X$ of type $\mathcal{C}_1$, Sankaran \cite{sankaran1} showed that there exists a totally nondegenerate embedding $A\hookrightarrow X$. However, his paper seems to contain gaps unfortunately. So we do not yet know whether $X$ admits a totally nondegenerate embedding or not.
}

\end{Rem}



\section{Table of nonsingular toric Fano $4$-folds}\label{tablefano}

\hspace{.5cm} In this section, we give the table of nonsingular toric Fano $4$-folds classified in Batyrev \cite{batyrev4} and Sato \cite{sato1} with $2$-blow-up relations among them. We describe the results about totally nondegenerate embeddings obtained in the previous sections. In the third column, we show whether the nonsingular toric Fano $4$-fold admits a totally nondegenerate embedding or not. The symbol ``$\exists$'' means that there exists a totally nondegenerate embedding, while the symbol ``$\times$'' means that there does not exist a totally nondegenerate embedding. We omit a reference in the case where the noningular toric Fano $4$-fold is obtained by finite successions of $2$-blow-ups from one of the nonsingular toric Fano $4$-folds of types $\mathcal{B}_1$, $\mathcal{B}_2$, $\mathcal{B}_3$, $\mathcal{D}_1$, $\mathcal{D}_2$, $\mathcal{D}_3$, $\mathcal{D}_5$, $\mathcal{D}_6$, $\mathcal{D}_8$, $\mathcal{D}_9$, $\mathcal{D}_{12}$, $\mathcal{D}_{16}$, $\mathcal{G}_1$, $\mathcal{I}_4$, $\mathcal{I}_6$, $\mathcal{I}_{12}$, $\mathcal{I}_{15}$, $\mathcal{J}_2$, $\mathcal{L}_1$, $\mathcal{L}_2$, $\mathcal{L}_5$, $\mathcal{L}_{10}$, $\mathcal{L}_{12}$, $\mathcal{M}_1$, $\mathcal{M}_2$, $\mathcal{M}_3$, $\mathcal{M}_4$, $\mathcal{M}_5$ and $\widetilde{V}^{4}$ (see Corollary \ref{dia2}).
\setlongtables
\bigskip

\bigskip

\begin{center}
Table 1: nonsingular toric Fano $4$-folds
\end{center}



\begin{longtable}{|c||l|l||c|}

\hline
\endhead
\hline
\endfoot

\hspace{1cm} &  $2$-blow-up of\hspace{2.0cm} & embedding &notation\hspace{0.5cm} \\ \hline
(1) & none & $\exists$ (See Horrocks-Mumford \cite{horrocks1}) & ${\bf P}^{4}$ \\ \hline
(2) & none & $\times$ (See Proposition \ref{ainfdivcont}) & $\mathcal{B}_1$ \\ \hline
(3) & none & $\times$ (See Proposition \ref{ainfdivcont}) & $\mathcal{B}_2$ \\ \hline
(4) & none & $\times$ (See Proposition \ref{ainfdivcont}) & $\mathcal{B}_3$ \\ \hline
(5) & none & $\exists$ (See Lange \cite{lange1}) & $\mathcal{B}_4$ \\ \hline
(6) & ${\bf P}^{4}$ & $\times$ (See Sankaran \cite{sankaran1}) & $\mathcal{B}_5$ \\ \hline
(7) & none & unknown & $\mathcal{C}_1$ \\ \hline
(8) & none & unknown & $\mathcal{C}_2$ \\ \hline
(9) & none & unknown & $\mathcal{C}_3$ \\ \hline
(10) & none & $\exists$ (See Proposition \ref{dptimesdp}) & $\mathcal{C}_4$ \\ \hline
(11) & $\mathcal{B}_1,\ \mathcal{B}_2$ & $\times$ & $\mathcal{E}_1$ \\ \hline
(12) & $\mathcal{B}_2,\ \mathcal{B}_3$ & $\times$ & $\mathcal{E}_2$ \\ \hline
(13) & $\mathcal{B}_3,\ \mathcal{B}_4$ & $\times$ & $\mathcal{E}_3$ \\ \hline
(14) & none & $\times$ (See Proposition \ref{faoverp2}) & $\mathcal{D}_1$ \\ \hline
(15) & $\mathcal{C}_1$ & $\times$ (See Proposition \ref{faoverp2}) & $\mathcal{D}_2$ \\ \hline
(16) & none & $\times$ (See Proposition \ref{faoverp2}) & $\mathcal{D}_3$ \\ \hline
(17) & $\mathcal{B}_2$ & $\times$ & $\mathcal{D}_4$ \\ \hline
(18) & none & $\times$ (See Proposition \ref{faoverp2}) & $\mathcal{D}_5$ \\ \hline
(19) & $\mathcal{C}_3$ & $\times$ (See Proposition \ref{faoverp2}) & $\mathcal{D}_6$ \\ \hline
(20) & none & unknown & $\mathcal{D}_7$ \\ \hline
(21) & $\mathcal{C}_2$ & $\times$ (See Proposition \ref{faoverp2}) & $\mathcal{D}_8$ \\ \hline
(22) & none & $\times$ (See Proposition \ref{faoverp2}) & $\mathcal{D}_9$ \\ \hline
(23) & $\mathcal{B}_5$ & unknown & $\mathcal{D}_{10}$ \\ \hline
(24) & $\mathcal{B}_5,\ \mathcal{C}_2$ & unknown & $\mathcal{D}_{11}$ \\ \hline
(25) & none & $\times$ (See Proposition \ref{faoverp2}) & $\mathcal{D}_{12}$ \\ \hline
(26) & none & $\exists$ (See Proposition \ref{dptimesdp}) & $\mathcal{D}_{13}$ \\ \hline
(27) & $\mathcal{B}_4$ & unknown & $\mathcal{D}_{14}$ \\ \hline
(28) & $\mathcal{C}_4$ & $\exists$ (See Proposition \ref{dptimesdp}) & $\mathcal{D}_{15}$ \\ \hline
(29) & $\mathcal{C}_3$ & $\times$ (See Proposition \ref{faoverp2}) & $\mathcal{D}_{16}$ \\ \hline
(30) & $\mathcal{B}_5$ & unknown & $\mathcal{D}_{17}$ \\ \hline
(31) & $\mathcal{C}_1$ & unknown & $\mathcal{D}_{18}$ \\ \hline
(32) & $\mathcal{C}_2$ & $\times$ (See Kajiwara \cite{kajiwara1}) & $\mathcal{D}_{19}$ \\ \hline
(33) & none & $\times$ (See Example \ref{g1noexist}) & $\mathcal{G}_1$ \\ \hline
(34) & $\mathcal{C}_2$ & $\times$ (See Kajiwara \cite{kajiwara1}) & $\mathcal{G}_2$ \\ \hline
(35) & none & unknown & $\mathcal{G}_3$ \\ \hline
(36) & $\mathcal{C}_2$ & unknown & $\mathcal{G}_4$ \\ \hline
(37) & $\mathcal{C}_3$ & unknown & $\mathcal{G}_5$ \\ \hline
(38) & $\mathcal{C}_4$ & $\times$ (See Kajiwara \cite{kajiwara1}) & $\mathcal{G}_6$ \\ \hline
(39) & $\mathcal{D}_2$ & $\times$ & $\mathcal{H}_1$ \\ \hline
(40) & $\mathcal{D}_3$ & $\times$ & $\mathcal{H}_2$ \\ \hline
(41) & $\mathcal{D}_1,\ \mathcal{D}_5$ & $\times$ & $\mathcal{H}_3$ \\ \hline
(42) & $\mathcal{D}_8,\ \mathcal{D}_9$ & $\times$ & $\mathcal{H}_4$ \\ \hline
(43) & $\mathcal{D}_6,\ \mathcal{D}_{12},\ \mathcal{D}_{16}$ & $\times$ & $\mathcal{H}_5$ \\ \hline
(44) & $\mathcal{D}_3,\ \mathcal{D}_9$ & $\times$ & $\mathcal{H}_6$ \\ \hline
(45) & $\mathcal{D}_2,\ \mathcal{D}_5,\ \mathcal{D}_{18}$ & $\times$ & $\mathcal{H}_7$ \\ \hline
(46) & $\mathcal{D}_{13},\ \mathcal{D}_{15}$ & $\exists$ (See Proposition \ref{dptimesdp}) & $\mathcal{H}_8$ \\ \hline
(47) & $\mathcal{D}_8,\ \mathcal{D}_{12},\ \mathcal{D}_{19}$ & $\times$ & $\mathcal{H}_9$ \\ \hline
(48) & $\mathcal{D}_9,\ \mathcal{D}_{16}$ & $\times$ & $\mathcal{H}_{10}$ \\ \hline
(49) & none & $\times$ (See $(c)$ in Section \ref{secex}) & $\mathcal{L}_1$ \\ \hline
(50) & $\mathcal{D}_7$ & $\times$ (See $(c)$ in Section \ref{secex}) & $\mathcal{L}_2$ \\ \hline
(51) & $\mathcal{D}_6$ & $\times$ & $\mathcal{L}_3$ \\ \hline
(52) & $\mathcal{D}_8,\ \mathcal{D}_{10},\ \mathcal{D}_{11}$ & $\times$ & $\mathcal{L}_4$ \\ \hline
(53) & none & $\times$ (See Example \ref{l5noexist}) & $\mathcal{L}_5$ \\ \hline
(54) & $\mathcal{D}_{12},\ \mathcal{D}_{14}$ & $\times$ & $\mathcal{L}_6$ \\ \hline
(55) & $\mathcal{D}_{15}$ & $\exists$ (See Proposition \ref{dptimesdp}) & $\mathcal{L}_7$ \\ \hline
(56) & none & $\exists$ (See Proposition \ref{dptimesdp}) & $\mathcal{L}_8$ \\ \hline
(57) & $\mathcal{D}_{13}$ & $\exists$ (See Proposition \ref{dptimesdp}) & $\mathcal{L}_9$ \\ \hline
(58) & $\mathcal{D}_{10},\ \mathcal{D}_{17}$ & $\times$ (See $(c)$ in Section \ref{secex}) & $\mathcal{L}_{10}$ \\ \hline
(59) & $\mathcal{D}_{14}$ & unknown & $\mathcal{L}_{11}$ \\ \hline
(60) & $\mathcal{D}_{11},\ \mathcal{D}_{17},\ \mathcal{D}_{19}$ & $\times$ (See $(d)$ in Section \ref{secex}) & $\mathcal{L}_{12}$ \\ \hline
(61) & $\mathcal{D}_7$ & unknown & $\mathcal{L}_{13}$ \\ \hline
(62) & $\mathcal{D}_4$ & $\times$ & $\mathcal{I}_1$ \\ \hline
(63) & $\mathcal{D}_1,\ \mathcal{D}_6$ & $\times$ & $\mathcal{I}_2$ \\ \hline
(64) & $\mathcal{D}_3,\ \mathcal{D}_8$ & $\times$ & $\mathcal{I}_3$ \\ \hline
(65) & $\mathcal{D}_{10}$ & $\times$ (See $(a)$ in Section \ref{secex}) & $\mathcal{I}_4$ \\ \hline
(66) & $\mathcal{E}_2,\ \mathcal{D}_4,\ \mathcal{D}_{10}$ & $\times$ & $\mathcal{I}_5$ \\ \hline
(67) & $\mathcal{D}_{10}$ & $\times$ (See $(a)$ in Section \ref{secex}) & $\mathcal{I}_6$ \\ \hline
(68) & $\mathcal{D}_5,\ \mathcal{D}_{12}$ & $\times$ & $\mathcal{I}_7$ \\ \hline
(69) & $\mathcal{D}_8,\ \mathcal{D}_{16}$,\ $\mathcal{G}_4$ & $\times$ & $\mathcal{I}_8$ \\ \hline
(70) & $\mathcal{D}_{14}$ & unknown & $\mathcal{I}_9$ \\ \hline
(71) & $\mathcal{D}_6,\ \mathcal{D}_{15},\ \mathcal{G}_5$ & $\times$ & $\mathcal{I}_{10}$ \\ \hline
(72) & $\mathcal{D}_9,\ \mathcal{D}_{12}$ & $\times$ & $\mathcal{I}_{11}$ \\ \hline
(73) & $\mathcal{D}_{15},\ \mathcal{D}_{19},\ \mathcal{G}_6$ & $\times$ (See $(a)$ in Section \ref{secex}) & $\mathcal{I}_{12}$ \\ \hline
(74) & $\mathcal{D}_{12},\ \mathcal{D}_{13}$ & $\times$ & $\mathcal{I}_{13}$ \\ \hline
(75) & $\mathcal{E}_3,\ \mathcal{D}_{10},\ \mathcal{D}_{14}$ & $\times$ & $\mathcal{I}_{14}$ \\ \hline
(76) & $\mathcal{D}_{18},\ \mathcal{D}_{19},\ \mathcal{G}_2$ & $\times$ (See $(a)$ in Section \ref{secex}) & $\mathcal{I}_{15}$ \\ \hline
(77) & none & $\times$ (See $(e)$ in Section \ref{secex}) & $\mathcal{M}_1$ \\ \hline
(78) & none & $\times$ (See $(e)$ in Section \ref{secex}) & $\mathcal{M}_2$ \\ \hline
(79) & $\mathcal{G}_3,\ \mathcal{G}_5$ & $\times$ (See $(e)$ in Section \ref{secex}) & $\mathcal{M}_3$ \\ \hline
(80) & $\mathcal{G}_3$ & $\times$ (See $(e)$ in Section \ref{secex}) & $\mathcal{M}_4$ \\ \hline
(81) & $\mathcal{G}_4,\ \mathcal{G}_6$ & $\times$ (See $(f)$ in Section \ref{secex}) & $\mathcal{M}_5$ \\ \hline
(82) & $\mathcal{G}_1,\ \mathcal{G}_3$ & $\times$ & $\mathcal{J}_1$ \\ \hline
(83) & $\mathcal{G}_3$ & $\times$ (See $(b)$ in Section \ref{secex}) & $\mathcal{J}_2$ \\ \hline
(84) & $\mathcal{L}_2$ & $\times$ & $\mathcal{Q}_1$ \\ \hline
(85) & $\mathcal{H}_4,\ \mathcal{L}_4$ & $\times$ & $\mathcal{Q}_2$ \\ \hline
(86) & $\mathcal{L}_1,\ \mathcal{L}_5$ & $\times$ & $\mathcal{Q}_3$ \\ \hline
(87) & $\mathcal{L}_3$ & $\times$ & $\mathcal{Q}_4$ \\ \hline
(88) & $\mathcal{H}_5,\ \mathcal{L}_3,\ \mathcal{L}_6$ & $\times$ & $\mathcal{Q}_5$ \\ \hline
(89) & $\mathcal{L}_6$ & $\times$ & $\mathcal{Q}_6$ \\ \hline
(90) & $\mathcal{L}_7$ & $\times$ & $\mathcal{Q}_7$ \\ \hline
(91) & $\mathcal{L}_5,\ \mathcal{L}_9$ & $\times$ & $\mathcal{Q}_8$ \\ \hline
(92) & $\mathcal{L}_3,\ \mathcal{L}_7,\ \mathcal{I}_{10}$ & $\times$ & $\mathcal{Q}_9$ \\ \hline
(93) & $\mathcal{H}_8,\ \mathcal{L}_7,\ \mathcal{L}_9$ & $\exists$ (See Proposition \ref{dptimesdp}) & $\mathcal{Q}_{10}$ \\ \hline
(94) & $\mathcal{L}_8,\ \mathcal{L}_9$ & $\exists$ (See Proposition \ref{dptimesdp}) & $\mathcal{Q}_{11}$ \\ \hline
(95) & $\mathcal{L}_{10},\ \mathcal{L}_{12},\ \mathcal{I}_6$ & $\times$ & $\mathcal{Q}_{12}$ \\ \hline
(96) & $\mathcal{L}_2,\ \mathcal{L}_5,\ \mathcal{L}_{13}$ & $\times$ & $\mathcal{Q}_{13}$ \\ \hline
(97) & $\mathcal{H}_9,\ \mathcal{L}_4,\ \mathcal{L}_6,\ \mathcal{L}_{12},\ \mathcal{I}_{14}$ & $\times$ & $\mathcal{Q}_{14}$ \\ \hline
(98) & $\mathcal{L}_6,\ \mathcal{L}_9,\ \mathcal{L}_{11},\ \mathcal{I}_{13}$ & $\times$ & $\mathcal{Q}_{15}$ \\ \hline
(99) & $\mathcal{L}_{11},\ \mathcal{L}_{13},\ \mathcal{I}_9$ & unknown & $\mathcal{Q}_{16}$ \\ \hline
(100) & $\mathcal{L}_7,\ \mathcal{L}_{12},\ \mathcal{I}_{12}$ & $\times$ & $\mathcal{Q}_{17}$ \\ \hline
(101) & $\mathcal{H}_1,\ \mathcal{H}_3,\ \mathcal{H}_7$ & $\times$ & $\mathcal{K}_1$ \\ \hline
(102) & $\mathcal{H}_2,\ \mathcal{H}_6,\ \mathcal{H}_{10}$ & $\times$ & $\mathcal{K}_2$ \\ \hline
(103) & $\mathcal{H}_4,\ \mathcal{H}_5,\ \mathcal{H}_9$ & $\times$ & $\mathcal{K}_3$ \\ \hline
(104) & $\mathcal{H}_8$ & $\exists$ (See Proposition \ref{dptimesdp}) & $\mathcal{K}_4$ \\ \hline
(105) & $\mathcal{M}_3$ & $\times$ & $\mathcal{R}_1$ \\ \hline
(106) & $\mathcal{M}_2,\ \mathcal{M}_4$ & $\times$ & $\mathcal{R}_2$ \\ \hline
(107) & $\mathcal{M}_1,\ \mathcal{M}_4$ & $\times$ & $\mathcal{R}_3$ \\ \hline
(108) & $\mathcal{I}_{11},\ \mathcal{I}_{13}$ & $\times$ & $\mathcal{P}$ \\ \hline
(109) & $\mathcal{Q}_1,\ \mathcal{Q}_3,\ \mathcal{Q}_{13}$ & $\times$ & $\mathcal{U}_1$ \\ \hline
(110) & $\mathcal{Q}_2,\ \mathcal{Q}_5,\ \mathcal{Q}_{14},\ \mathcal{K}_3$ & $\times$ & $\mathcal{U}_2$ \\ \hline
(111) & $\mathcal{Q}_4,\ \mathcal{Q}_9$ & $\times$ & $\mathcal{U}_3$ \\ \hline
(112) & $\mathcal{Q}_{10},\ \mathcal{K}_4$ & $\times$ & $\mathcal{U}_4$ \\ \hline
(113) & $\mathcal{Q}_{11}$ & $\exists$ (See Proposition \ref{dptimesdp}) & $\mathcal{U}_5$ \\ \hline
(114) & $\mathcal{Q}_6,\ \mathcal{Q}_8,\ \mathcal{Q}_{15}$ & $\times$ & $\mathcal{U}_6$ \\ \hline
(115) & $\mathcal{Q}_7,\ \mathcal{Q}_{12},\ \mathcal{Q}_{17}$ & $\times$ & $\mathcal{U}_7$ \\ \hline
(116) & $\mathcal{Q}_{16}$ & unknown & $\mathcal{U}_8$ \\ \hline
(117) & none & $\times$ (See Kajiwara \cite{kajiwara2} and \cite{kajiwara1}) & ${\widetilde{V}}^{4}$ \\ \hline
(118) & none & unknown & $V^{4}$ \\ \hline
(119) & $\mathcal{Q}_{10},\ \mathcal{Q}_{11}$ & $\exists$ (See Proposition \ref{dptimesdp}) & $S_2\times S_2$ \\ \hline
(120) & $\mathcal{U}_4,\ \mathcal{U}_5,\ S_2\times S_2$ & $\exists$ (See Proposition \ref{dptimesdp}) & $S_2\times S_3$ \\ \hline
(121) & $S_2\times S_3$ & $\exists$ (See Proposition \ref{dptimesdp}) & $S_3\times S_3$ \\ \hline
(122) & $\mathcal{G}_6$ & unknown & $\mathcal{Z}_1$ \\ \hline
(123) & $\mathcal{G}_4$ & unknown & $\mathcal{Z}_2$ \\ \hline
(124) & $\mathcal{Z}_1$ & unknown & $\mathcal{W}$ \\ \hline

\end{longtable}



\begin{thebibliography}{99}



\bibitem{batyrev3} V. V. Batyrev, On the classification of smooth projective toric varieties, Tohoku Math. J. 43 (1991), 569--585.
\bibitem{batyrev4} V. V. Batyrev, On the classification of toric Fano 4-folds, Algebraic Geometry, 9, J. Math. Sci. (New York) 94 (1999), 1021--1050.


\bibitem{horrocks1} G. Horrocks and D. Mumford, A rank $2$ vector bundle on ${\bf P}^{4}$ with 15,000 symmetries, Topology 12 (1973), 63--81.

\bibitem{hulek1} K. Hulek, Abelian surfaces in products of projective spaces, in Algebraic Geometry, L'Aquila (A. J. Sommese, A. Biancofiore, E. L. Livorni, eds.) Lecture Notes in Math. 1417, 129--137, Springer, Berlin 1988.

\bibitem{kajiwara2} T. Kajiwara, Abelian surfaces in toric 4-folds (Japanese), Su-rikaisekikenkyu-sho Ko-kyu-roku 1175 (2000), 123-131.


\bibitem{kajiwara1} T. Kajiwara, Notes on embedding of abelian surfaces to projective smooth toric $4$-folds, preprint.



\bibitem{lange1} H. Lange, Abelian surfaces in ${\bf P}^{1}\times{\bf P}^{3}$, Arch. Math. 63 (1994), 80--84.

\bibitem{mumford1} D. Mumford, Abelian Varieties (Second edition), Oxford Univ. Press, 1974.


\bibitem{oda2} T. Oda, Convex Bodies and Algebraic Geometry---An introduction to the theory of toric varieties, Ergebnisse Math. Grenzgeb. (3), Vol. 15, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988.

\bibitem{reid1} M. Reid, Decomposition of toric morphisms, in Arithmetic and Geometry, papers dedicated to I. R. Shafarevich on the occasion of his 60th birthday (M. Artin and J. Tate, eds.), vol. II, Geometry, Progress in Math. 36, Birkh\"{a}user, Boston, Basel, Stuttgart, 1983, 395--418.

\bibitem{sankaran1} G. K. Sankaran, Abelian surfaces in toric $4$-folds, Math. Ann. 313 (1999), 409--427.

\bibitem{sato1} H. Sato, Toward the classification of higher-dimensional toric Fano varieties, Tohoku Math. J. 52 (2000), 383--413.

\bibitem{vandeven1} A. Van de Ven, On the embedding of abelian varieties in projective space, Annali di Matematica Pura ed Applicata 103 (1971), 127--129.

\end{thebibliography}

\bigskip

\begin{flushleft}
\begin{sc}
Department of Mathematics,\\
Tokyo Institute of Technology,\\
2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan.
\end{sc}

\medskip
{\it E-mail address}: hirosato@math.titech.ac.jp
\end{flushleft}

\end{document}


math0201257.extracted_bib
\begin{thebibliography}{99}



\bibitem{batyrev3} V. V. Batyrev, On the classification of smooth projective toric varieties, Tohoku Math. J. 43 (1991), 569--585.
\bibitem{batyrev4} V. V. Batyrev, On the classification of toric Fano 4-folds, Algebraic Geometry, 9, J. Math. Sci. (New York) 94 (1999), 1021--1050.


\bibitem{horrocks1} G. Horrocks and D. Mumford, A rank $2$ vector bundle on ${\bf P}^{4}$ with 15,000 symmetries, Topology 12 (1973), 63--81.

\bibitem{hulek1} K. Hulek, Abelian surfaces in products of projective spaces, in Algebraic Geometry, L'Aquila (A. J. Sommese, A. Biancofiore, E. L. Livorni, eds.) Lecture Notes in Math. 1417, 129--137, Springer, Berlin 1988.

\bibitem{kajiwara2} T. Kajiwara, Abelian surfaces in toric 4-folds (Japanese), Su-rikaisekikenkyu-sho Ko-kyu-roku 1175 (2000), 123-131.


\bibitem{kajiwara1} T. Kajiwara, Notes on embedding of abelian surfaces to projective smooth toric $4$-folds, preprint.



\bibitem{lange1} H. Lange, Abelian surfaces in ${\bf P}^{1}\times{\bf P}^{3}$, Arch. Math. 63 (1994), 80--84.

\bibitem{mumford1} D. Mumford, Abelian Varieties (Second edition), Oxford Univ. Press, 1974.


\bibitem{oda2} T. Oda, Convex Bodies and Algebraic Geometry---An introduction to the theory of toric varieties, Ergebnisse Math. Grenzgeb. (3), Vol. 15, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988.

\bibitem{reid1} M. Reid, Decomposition of toric morphisms, in Arithmetic and Geometry, papers dedicated to I. R. Shafarevich on the occasion of his 60th birthday (M. Artin and J. Tate, eds.), vol. II, Geometry, Progress in Math. 36, Birkh\"{a}user, Boston, Basel, Stuttgart, 1983, 395--418.

\bibitem{sankaran1} G. K. Sankaran, Abelian surfaces in toric $4$-folds, Math. Ann. 313 (1999), 409--427.

\bibitem{sato1} H. Sato, Toward the classification of higher-dimensional toric Fano varieties, Tohoku Math. J. 52 (2000), 383--413.

\bibitem{vandeven1} A. Van de Ven, On the embedding of abelian varieties in projective space, Annali di Matematica Pura ed Applicata 103 (1971), 127--129.

\end{thebibliography}