We discuss the problem on the connectedness of various webs of lattice polytopes by
introducing a geometric point of view from the toric Mori theory.
To this end,
we provide a combinatorial description of toric Sarkisov links
in terms of certain sets of lattice points, which we call primitive generating sets.
In two dimensions, the description is further translated into the language of lattice polygons.
As an application, we prove in two ways (constructive and non-constructive) that reflexive
or terminal polygons form a single connected web via inclusion relations
even without taking modulo unimodular equivalences.

§ INTRODUCTION

As a byproduct of their famous works on the classification of reflexive polytopes,
Kreuzer and Skarke have shown that
the web of $d$-dimensional reflexive polytopes is connected modulo unimodular equivalences if $d\le 4$
This is important in relation to the unsolved mathematical problem, so-called Reid's fantasy [18],
which asks whether the web of smooth Calabi–Yau 3-folds is connected via geometric transitions.
In fact,
their result
implies that the web of
smooth Calabi–Yau 3-folds
described as anticanonical hypersurfaces
in toric varieties (in the sense of Batyrev [2] and
Fredrickson [7])
is connected via geometric transitions and flops.
Since their argument is purely combinatorial and relies on the results of computer-aided classification
of reflexive polytopes,
it is still an interesting problem to analyze
the web of reflexive polytopes from a more geometric point of view.

The present paper provides an argument based on the perspectives from the Mori theory in birational geometry.
Let $Z$ be a smooth or mildly singular projective variety over the field of complex numbers $\bC$.
By running the minimal model program (MMP, for short),
one conjecturally obtains a sequence of elementary birational maps directed
by the canonical divisor $K_Z$,

\begin{equation}
\label{eq:mmp}
\varphi: Z=X_0 \dashrightarrow X_1 \dashrightarrow \cdots \dashrightarrow X_m =X,
\end{equation}

$X$ is either a minimal model or a Mori fiber space $p: X\rightarrow S$.
Although the output $X$ is not unique,
it is known that
different birational minimal models are related by a sequence of flops [13]
and different birational Mori fiber spaces
are related by a sequence
of Sarkisov links [6], [10].
Thus, in either case, one obtains a single web connecting all birational models via
elementary birational maps if the MMP works.
For projective toric varieties,
the Mori theory has been established by the seminal paper of Reid
[17] and developed
through a number of studies; for example,
<cit.>, [8], <cit.>.
By running MMP starting from a $\bQ$-factorial projective toric variety $Z$,
one always ends up with a toric Mori fiber space (we do not assume that it has only terminal singularities).
The resulting web of birational models
can be coarse-grained into the web of Fano polytopes
via inclusion relations.
Here a Fano polytope (also known as a $\bQ$-Fano polytope) is a lattice polytope containing the
origin as an interior point whose vertices are all primitive lattice points.
This coarse-graining is done
by taking the convex hull $\Conv G(\Sigma)$ of
the set of primitive ray generators $G(\Sigma)$ of each projective fan $\Sigma$.
For an elementary birational map $X_{\Sigma} \dashrightarrow X_{\Sigma'}$ in eq:mmp,
we have inclusions  $G(\Sigma)\supset G(\Sigma')$ and $\Conv G(\Sigma)\supset \Conv G(\Sigma')$.

The web of Fano polytopes is connected
since there is a sequence
$\nabla \subset \Conv(\nabla\cup \nabla') \supset \nabla'$ for any pair of Fano polytopes $\nabla$ and $\nabla'$.
Hence, a natural problem is whether
the web of a certain restricted class of Fano polytopes is connected.
If the answer to the following pb:connectedness is yes,
we say that the class of lattice polytopes is globally connected.

For a given class of lattice polytopes,
do they form a
single connected web via inclusion relations?

We introduce five known classes of lattice polytopes including reflexive polytopes,
for which one could naturally ask pb:connectedness.
Let $N\simeq \bZ^d$ and $M=\Hom(N,\bZ)$
be the dual pair of free abelian groups of rank $d$ and
$N_\bR\coloneqq N\otimes_\bZ \bR$.
The polar dual and the Mavlyutov dual of a lattice polytope $\nabla$ in $N_\bR$ is defined as

\begin{equation}
\nabla^* \coloneqq  \lc u\in M_\bR \relmid \la u, v \ra \ge -1 \text{ for all }  v\in \nabla \rc
\quad \text{ and } \quad [\nabla^*] \coloneqq \Conv (\nabla^* \cap M),
\end{equation}

A reflexive (resp. pseudoreflexive) polytope is defined such as
the polar duality (resp. the Mavlyutov duality) holds among the same class of lattice polytopes.
Namely, $\nabla$ is reflexive if $\nabla^*$ is also a lattice polytope,
and pseudoreflexive if $[[\nabla^*]^*] = \nabla$ holds.
While these polytopes are well suited to mirror symmetry as shown by [2] and [16],
the following polytopes are more fitted to the arguments in the toric Mori theory.
A lattice polytope $\nabla$ is called canonical
if the origin is the unique interior lattice point,
if $\nabla\cap N$ consists of
the vertices of $\nabla$ and the origin,
and almost pseudoreflexive
if the Mavlyutov dual $[\nabla^*]$
contains the origin as an interior point.
We shortly remark the geometric meaning of these three classes of lattice polytopes.
A canonical (resp. terminal) polytope $\nabla$ corresponds to
a toric Fano variety with at worst canonical (resp. terminal) singularities [17],
which is the anticanonical model of
the toric variety defined by a projective simplicial fan $\Sigma$ satisfying $\Conv G(\Sigma)=\nabla$.
An almost pseudoreflexive polytope $\nabla$ corresponds to
a toric Fano variety whose general elephant (i.e.,
a member of the anticanonical linear system $\abs{-K}$) is a Calabi–Yau variety.
In fact, the latter is equivalent to that
the Newton polytope $[\nabla^*]$ is
almost pseudoreflexive (and hence, pseudoreflexive in this case) as proved
by <cit.> (and by <cit.> for one direction).

There are the following implications among these five classes of polytopes:

\begin{equation}
\text{terminal} \ \Rightarrow \ \text{canonical} \ \Leftarrow \ \text{almost pseudoreflexive}
\ \Leftarrow \ \text{pseudoreflexive} \ \Leftarrow \ \text{reflexive}.
\end{equation}

Only the second implication is nontrivial, which follows because $[\nabla^*]^*\supset \nabla$ contains only the origin as
an internal lattice point
by <cit.>.
A Fano polytope contained in an almost pseudoreflexive (resp. canonical, terminal) polytope
is again an almost pseudoreflexive (resp. canonical, terminal) polytope.
Hence, by running MMP, pb:connectedness for these three classes of
Fano polytopes comes down to pb:sarkisov below.

For any pair of almost pseudoreflexive (resp. canonical, terminal) polytopes
represented by toric Mori fiber spaces,
is there a pair of representatives
connected by
a sequence of Sarkisov links
consisting only of birational models whose associated polytopes are
almost pseudoreflexive (resp. canonical, terminal)?

Note that the global connectedness of $d$-dimensional almost pseudoreflexive polytopes implies
that of $d$-dimensional pseudoreflexive polytopes.
In fact, this is clear
by taking the pseudoreflexive hull $[[\nabla^*]^*]$ for
each almost pseudoreflexive polytope $\nabla$
in a sequence
connecting a pair of pseudoreflexive polytopes.
Together with the fact that a pseudoreflexive polytope is always reflexive when $d\le 4$ [19],
for almost pseudoreflexive polytopes possibly re-proves the connectedness of
the web of four-dimensional reflexive polytopes due to [14]
without taking modulo unimodular equivalences nor using the classification.

In order to make pb:sarkisov more precise,
we explore the combinatorial description of
Sarkisov links for toric Mori fiber spaces.
However, it turns out that there is an obstacle in the description of Sarkisov links only in terms of
Fano polytopes (eg:obstacle).
On the other hand, we obtain
a satisfactory description of Sarkisov links
in terms of primitive generating sets in arbitrary dimensions,
which are the combinatorial objects (corresponding to $G(\Sigma)$ above)
that appear between projective fans and Fano polytopes.
In sc:pgs, we introduce and develop the notion of primitive generating sets.
In sc:sarkisov, we prove th:sarkisovA, which gives a combinatorial description of Sarkisov links
in terms of primitive generating sets.

In two dimensions, things becomes very simple.

Especially, the description in th:sarkisovA can be further translated into the
language of Fano polygons without any additional conditions (pr:twodim).

Using this simplicity,
we give two proofs (constructive and non-constructive) for the following theorem in
sc:reflexive by solving pb:sarkisov in two dimensions:

The web of reflexive polygons
are connected via inclusion relations
without taking modulo unimodular equivalences.
The same is true for the web of terminal polygons.

The latter proof also implies that

elliptic elephants in smooth projective rational surfaces
can be connected
via smooth transitions associated with blow-ups and blow-downs
decomposing a given birational map between ambient spaces (cr:ellip).
This could be regarded as a toy version of Reid's fantasy in one dimension.

Note that, for a lattice polygons, reflexive is equivalent to canonical,
so that th:polygon yields a complete answer to pb:connectedness
for all the five classes of polytopes in two dimensions.
On the other hand,
the proof is still based on the classification of smooth toric Mori fiber surfaces
and the hand-picked choice of sequences of Sarkisov links.
Hopefully, more detailed study of the Sarkisov program could improve the argument
and provide a way to solve pb:connectedness
in higher dimensions and also a new perspective to Reid's
fantasy for Calabi–Yau 3-folds.

This paper is dedicated to Professor Shinobu Hosono on the occasion of his sixtieth birthday.
The author thanks Atsushi Ito for helpful discussions.
He was supported by Grants-in-Aid for Scientific Research (21K03156).

§ PRIMITIVE GENERATING SETS

We call a finite set $A\subset  N$ a primitive generating set of $N_\bR$ if $A$
consists of primitive lattice points and generates $N_\bR$ as a cone, i.e.,
$N_\bR = \lc \sum_{v\in A}\lambda_v v \relmid \lambda_v \ge 0 \text{ for all } v\in A \rc$.

An inclusion $A\supset A'$ of primitive generating sets
is said to be a reduction of $A$
if $\abs{A}=\abs{A'}+1$.
We denote by $A\supsetdot A'$ a reduction of $A$.

For a primitive generating set $A$,
an inclusion $\Af \subset A$ is said to be
a fiber structure on $A$ if $\Af\ne \emptyset$
is a primitive generating set of the linear span $L$ of $\Af$ and $\Af = L\cap A$.
A fiber structure $\Af \subset A$ defines
the associated exact sequence

\begin{equation}
\label{eq:exact}
\begin{tikzcd}
0\arrow[r] & \Nf \arrow[r] & N \arrow[r,"\pi"] & \Nb \arrow[r] & 0,
\end{tikzcd}
\end{equation}

where $\Nf \coloneqq N\cap L$ and $\Nb \coloneqq N/\Nf$.
For any $v\in A \setminus \Af$, the ray $\bR_+\pi(v)$ in $(\Nb)_\bR$ has the unique
primitive generator $\pibar(v) \in \Nb$.
Thus there is a map

\begin{equation}
\label{eq:pibar}
\pibar: A \setminus \Af\rightarrow
\Ab \coloneqq \lc \pibar(v) \relmid v \in A\setminus \Af  \rc \subset \Nb.
\end{equation}

We call $\Ab$ the base of the fiber structure $\Af \subset A$,
which turns out to be a primitive generating set of $(\Nb)_\bR$.
The map eq:pibar is also denoted by $\pibar: A \dashrightarrow \Ab$.

A fiber structure $\Af\subset A$ is called irreducible if $\abs{A}=\abs{\Af}+\abs{\Ab}$ holds.
A Mori fiber structure is an irreducible fiber structure
satisfying $\abs{\Af} = \dim L+1$.
We call a primitive generating set $A$ that admits a Mori fiber structure
a Mori fiber primitive generating set.
A Mori fiber structre $\Af \subset A$ on a Mori fiber primitive generating set $A$
is denoted by $\Af \subsetm A$.

By abuse of notation,
we use all the notions in df:reduction and df:fiberstr
also for Fano polytopes
by replacing
a primitive generating set $\nabla \cap \Nprim$
a Fano polytope $\nabla$,
where $\Nprim$ is the set of primitive lattice points in $N$.
Thus, for a reduction $\nabla\supsetdot \nabla'$ of Fano polytopes,
$\nabla'$ is a Fano polytope defined as
the convex hull of all primitive lattice points in $\nabla\setminus \lc v\rc$
for a vertex $v\in \nabla$.
Similarly, for a Mori fiber structure $\nablaf \subsetm \nabla$ of Mori fiber Fano polytopes
(or Mori fiber polytopes, for short),
$\nablaf$ is a terminal simplex and $\abs{\nabla\cap \Nprim}=\abs{\Ab} + \dim L +1$ holds,
where $\Ab$ is the base of $\nablaf\subsetm\nabla$.
Notice that, even for a fiber structure $\nablaf\subset \nabla$ of Fano polytopes,
the base $\Ab$ needs not to be a Fano polytope.

Let $X_\Sigma$ be a $\bQ$-factorial projective toric variety defined by a simplicial projective fan $\Sigma$.
Denote by $\overline{\NE}(X_\Sigma)$
the closed convex cone generated by
numerical equivalence classes of curves.
The following
is evident from the description of extremal contractions,
e.g., <cit.>.

Let $\varphi_R: X_\Sigma \rightarrow X_{\Sigma'}$ be an extremal contraction of
a $\bQ$-factorial projective toric variety $X_\Sigma$ with respect to
an extremal ray $R \subset \overline{\NE}(X_\Sigma)$.
If $\varphi_R$ is a small contraction, it preserves $G(\Sigma) = G(\Sigma')$.
If $\varphi_R$ is a divisorial contraction, it gives a reduction $G(\Sigma) \supsetdot G(\Sigma')$.
If $\varphi_R$ is a fibering contraction,
it gives a Mori fiber structure $G(\Sigmaf) \subsetm G(\Sigma)$ with the base $G(\Sigma')$
for a naturally defined subfan $\Sigmaf$ of $\Sigma$.

Note that,
if $\varphi_R$ is a fibering contraction (i.e., the last case in lm:contractions),
$X_\Sigma$ is automatically a toric Mori fiber space
with the fibers isomorphic to $X_{\Sigmaf}$, that is, $K_{X_{\Sigma}}.R<0$.
In fact, $R$ is the outer normal of a facet of the pseudoeffective cone
and $-K_{X_\Sigma}$ is big so that its numerical class is contained in the interior of $\overline{\Eff}(X_{\Sigma})$.
Since $\overline{\Eff}(X_\Sigma)$ is strongly convex, $K_{X_{\Sigma}}.R<0$ always holds.

For any primitive generating set $A$,
there exits a simplicial projective fan $\Sigma$ such that $G(\Sigma) =A$,
which we call a projective $A$-maximal fan,
following the case
where $A$ is (the set of primitive lattice points in) a reflexive polytope
The following is a key lemma which shows the converse of lm:contractions also holds in a sense:

For any primitive generating set $A$ and
any pair of
projective $A$-maximal fans $\Sigma$ and $\Sigma'$,
the natural birational map
$\varphi: X_\Sigma \dashrightarrow X_{\Sigma'}$ (induced by the identity map of $N$) is decomposed into
a sequence of flips, flops and inverse flips.
For any reduction $A\supsetdot A'$ and
any projective $A'$-maximal fan $\Sigma'$,
there exists an extremal divisorial contraction
$X_{\Sigma} \rightarrow X_{\Sigma'}$
such that $\Sigma$ is a projective $A$-maximal fan.
For any Mori fiber structure $\Af \subsetm A$ with the base $\Ab$
and any projective $\Ab$-maximal fan $\Sigmab$,
there exists a toric Mori fiber space $X_\Sigma \rightarrow X_{\Sigmab}$ with fibers
isomorphic to $X_{\Sigmaf}$
such that $\Sigma$ is a projective $A$-maximal fan and
$\Sigmaf\subset \Sigma$ is a projective $\Af$-maximal fan.

All the statements in lm:lift follow from the description of the movable cone
$\overline{\Mov}(X_\Sigma)$ of a $\bQ$-factorial projective toric variety $X_\Sigma$
as the corresponding cone
$\Mov_{\mathrm{GKZ}}$ in the secondary fan $\Sigma_{\mathrm{GKZ}}$
defined by $A=G(\Sigma)$
(see (15.1.5), Proposition 15.1.4 and Theorem 15.1.10(c) in [5]).
The first statement follows because
$\Mov_{\mathrm{GKZ}}$ is a full-dimensional convex rational polyhedral cone
decomposed into finite chambers corresponding to projective $A$-maximal fans.
In fact, along a general path between the two chambers corresponding to $\Sigma$ and $\Sigma'$,
one obtains a sequence of flips, flops and inverse flips, which decomposes
$\varphi: X_\Sigma \dashrightarrow X_{\Sigma'}$ (see <cit.>).
The second statement is a simple
extension of <cit.> and the same proof works.
That is, since $\Mov_{\mathrm{GKZ}}$ is full-dimensional and convex, there exists a chamber
containing the cone corresponding to $\Sigma'$ as a face, which gives
an extremal divisorial contraction
(refer to <cit.> for the face structure of the secondary fan).
Exactly the same proof works for the third statement
if one starts from a generalized fan
$\pi^{-1} \Sigmab \coloneqq \lc \pi_\bR^{-1}(\sigma) \relmid \sigma \in \Sigmab \rc$ instead of $\Sigma'$,
where $\pi_\bR:N_\bR \rightarrow (\Nb)_\bR$ is the natural projection defined by eq:exact.

The shed of a fan $\Sigma$
is defined as
$\shed(\Sigma) = \bigcup_{\sigma\in \Sigma} %\shed(\sigma)$,
%where $\shed(\sigma) =
\Conv \lb \lc 0\rc \cup  G(\sigma) \rb \subset \Conv G(\Sigma)$.

As observed originally by <cit.>,
for a birational contraction $\varphi_R: X_{\Sigma} \rightarrow X_{\Sigma'}$ of an extremal ray
$R\subset \overline{\NE}(X_\Sigma)$,
$\shed(\Sigma') \subsetneq \shed(\Sigma)$
(resp. $\shed(\Sigma') = \shed(\Sigma)$, $\shed(\Sigma') \supsetneq \shed(\Sigma)$)
if and only if
$K_{X_\Sigma}.R <0$ (resp. $K_{X_\Sigma}.R =0$, $K_{X_\Sigma}.R >0$).
Be aware that the sign of $K_\Sigma.R$
for a divisorial contraction $\varphi_R: X_{\Sigma} \rightarrow X_{\Sigma'}$
actually depends not only on the primitive generating sets
$G(\Sigma)$ and $G(\Sigma')$ but
also on $\shed(\Sigma')$ in general.
However, for a reduction $\nabla \supsetdot \nabla'$ of Fano polytopes,
any divisorial contraction $\varphi_R: X_{\Sigma} \rightarrow X_{\Sigma'}$
with $\Conv G(\Sigma) = \nabla$ and $\Conv G(\Sigma') = \nabla'$
satisfies $K_\Sigma.R < 0$ since $\shed(\Sigma') \subset \nabla'$ does not contain
$v\in \nabla\setminus \nabla'$.

§ DESCRIPTION OF TORIC SARKISOV LINKS

A Sarkisov link is one of the following four types of diagrams of normal projective varieties
that connects two Mori fiber spaces $p: X \rightarrow S$ and $p': X' \rightarrow S'$:

\begin{equation}
\label{eq:links}
\arraycolsep=8pt
\begin{array}{cccc}
\text{type I} & \text{type II} & \text{type III} & \text{type IV} \\
\begin{tikzcd}[column sep=tiny]
X \arrow[rr, dashrightarrow] \arrow[d, "p"'] && X'' \arrow[d]\\
S \arrow[dr] && X' \arrow[ld, "p'"]\\
& S'&
\end{tikzcd} &
\begin{tikzcd}[column sep=tiny]
X'' \arrow[rr, dashrightarrow] \arrow[d] && X''' \arrow[d]\\
X \arrow[d, "p"'] && X' \arrow[d, "p'"]\\
S \arrow[rr, equal] & \phantom{.} &S'
\end{tikzcd} &
\begin{tikzcd}[column sep=tiny]
X'' \arrow[rr, dashrightarrow] \arrow[d] && X' \arrow[d, "p'"]\\
X \arrow[dr, "p"'] && S' \arrow[ld]\\
& S&
\end{tikzcd} &
\begin{tikzcd}[column sep=tiny]
X \arrow[rr, dashrightarrow] \arrow[d, "p"'] && X' \arrow[d, "p'"]\\
S \arrow[dr] && S' \arrow[ld]\\
& R&
\end{tikzcd}
\end{array}
\end{equation}

all the varieties except $R$ are $\bQ$-factorial.
Every dashed arrow denotes a sequence of flips, flops and inverse flips,
every solid arrow denotes an extremal contraction.
In the latter case, each solid arrow between $X$'s is a divisorial contraction of a $K$-negative extremal ray,
each arrow between $S$'s is either divisorial or fibering contraction, and
each arrow toward $R$ is either small or fibering contraction.

Let $\Af \subsetm A$ and $\Af' \subsetm A'$ be Mori fiber primitive generating sets
associated with toric Mori fiber spaces $p: X\rightarrow S$ and $p': X'\rightarrow S'$, respectively.
Suppose that

these are connected
by a single Sarkisov link
in eq:links.
Then $\Af \subsetm A$ and $\Af' \subsetm A'$
coincide or
are related by one of the following diagrams and their inverses
(i.e., the diagrams switching $\Af\subsetm A$ and $\Af'\subsetm A'$):

\begin{equation}
\label{eq:sarkisovA}
\arraycolsep=8pt\def\arraystretch{1.3}
\begin{array}{ccccc}
\text{type I}{}_\mathrm{d} & \text{type I}{}_\mathrm{m} & \text{type II}{}_{\mathrm{irr}}
& \text{type II}{}_{\mathrm{ni}}  &  \text{type IV}{}_ {\mathrm{m}}\\
\begin{tikzcd}[row sep = small, column sep=-7pt]
A & \supsetdot &  A'\\
\Af \arrow[u, phantom, sloped, "\subsetm"]  & = & \Af'\arrow[u, phantom, sloped, "\subsetm"]
\end{tikzcd} &
\begin{tikzcd}[row sep = small, column sep=-7pt]
A & = &  A'' & \supsetdot & A'\\
\Af \arrow[u, phantom, sloped, "\subsetm"]
&\subsetm& \Af''\arrow[u, phantom, sloped, "\subset"] & \supsetdot & \Af'\arrow[u, phantom, sloped, "\subsetm"]
\end{tikzcd} &
\begin{tikzcd}[row sep = small, column sep=-7pt]
A & \subsetdot &  A'' & \supsetdot & A'\\
\Af \arrow[u, phantom, sloped, "\subsetm"]
&\subsetdot & \Af''\arrow[u, phantom, sloped, "\subset"] & \supsetdot & \Af'\arrow[u, phantom, sloped, "\subsetm"]
\end{tikzcd} &
\begin{tikzcd}[row sep = small, column sep=-7pt]
A & \subsetdot &  A'' & \supsetdot & A'\\
\Af \arrow[u, phantom, sloped, "\subsetm"]
&=& \Af''\arrow[u, phantom, sloped, "\subset"] & = & \Af'\arrow[u, phantom, sloped, "\subsetm"]
\end{tikzcd} &
\begin{tikzcd}[row sep = small, column sep=-7pt]
A & = &  A'' & = & A'\\
\Af \arrow[u, phantom, sloped, "\subsetm"]
&\subsetm& \Af''\arrow[u, phantom, sloped, "\subset"] &\supsetm&
\Af'\arrow[u, phantom, sloped, "\subsetm"]
\end{tikzcd}
\end{array}
\end{equation}

We call one of the diagrams in eq:sarkisovA and their inverses an elementary link for
Mori fiber primitive generating sets.
The inverse of an elementary link of type I${}_d$ (resp. type I${}_m$) is referred to as
that of type III${}_d$ (resp. type III${}_m$).
It is natural to add the trivial link, $A=A$ with $\Af=\Af'$, as an elementary link of type IV${}_\mathrm{s}$.
The scripts “$\mathrm{s}$”, “$\mathrm{d}$”, “$\mathrm{m}$”
indicate that the contraction of $S$'s in the corresponding Sarkisov link is
a small contraction, a divisorial contraction, a Mori fiber space, respectively.
Also, “$\mathrm{irr}$” and “$\mathrm{ni}$” for type II links indicate
that all fibers of the composite contractions $X''\rightarrow S$ and $X'''\rightarrow S'$ are irreducible, and
that the fibers are not irreducible over a common invariant prime divisor of the base $S=S'$, respectively.

We begin with describing the composition of two extremal contractions $q$ and $q'$,

\begin{equation}
\begin{tikzcd}
Y \arrow[r, "q"] & Y' \arrow[r, "q'"] & Y''
\end{tikzcd}
\end{equation}

appeared in one of the Sarkisov links in eq:links for
$\bQ$-factorial projective toric varieties.
There are four types of such compositions:

*  $q$ is a fibering contraction and $q'$ is a small contraction,

*  $q$ is a fibering contraction and $q'$ is a divisorial contraction,

*  $q$ is a divisorial contraction and $q'$ is a fibering contraction,

*  both $q$ and $q'$ are fibering contractions.

By lm:contractions,
the contractions $q$ and $q'$ are described by primitive generating sets as

it:1 $\Bf\subsetm B$ and $\Bb=\Bb'$,

it:2 $\Bf\subsetm B$ and $\Bb\supsetdot \Bb'$,

it:3 $B\supsetdot B'$ and $\Bf' \subsetm B'$,

it:4 $\Bf \subsetm B$ and $B_{\mathrm{b, f}}\subsetm \Bb$,

for each type of composition, respectively.
We shall combine two of such descriptions into
an elementary link for Mori fiber primitive generating sets.
Hereafter, we denote by
$\Af \subsetm A$ and $\Af' \subsetm A'$
the two Mori fiber structures
corresponding to $p$ and $p'$
in each Sarkisov link,

A composition of type it:1 only appears in the Sarkisov link of type IV${}_\mathrm{s}$.
In this Sarkisov link,
both compositions
are of type it:1
so that $A = A'=B$ and $\Ab = \Ab'=\Bb$ by lm:contractions, and hence, $\Af = \Af'$ as well.
This yields the trivial elementary link of type IV${}_\mathrm{s}$.

A composition of type it:2 only appears in the Sarkisov link of type I${}_\mathrm{d}$
(or type III${}_\mathrm{d}$ as its inverse).
In this case, we have a reduction $B\supsetdot B'\coloneqq \pi^{-1}(\Bb'\cup \lc0 \rc)\cap B$
the fiber structure $\Bf\subsetm B$ is irreducible so that
$B\setminus B' = \pi^{-1}(\Bb\setminus \Bb')\cap B$ consists of exactly one point.
Thus we obtain the elementary link of type I${}_\mathrm{d}$,

\begin{equation}
\label{eq:diagram1}
\begin{tikzcd}[row sep = small, column sep=-5pt]
B & \supsetdot & B'\\
\Bf \arrow[u, phantom, sloped, "\subsetm"]  & = & \Bf'\arrow[u, phantom, sloped, "\subsetm"]
\end{tikzcd},
\end{equation}

with $\Bb \supsetdot \Bb'$ and
$\Bf'\coloneqq B'\cap L$, where $L$ is the linear span of $\Bf$.
Clearly, the fiber structure $\Bf' \subset B'$ is irreducible and
its base coincides with $\Bb'$ given in advance.
We may put $A=B$ and $A' = B'$ (resp. $A=B'$ and $A' =B$)
for the Sarkisov link of type I${}_\mathrm{d}$ (resp. type III${}_\mathrm{d}$).
In either case, the opponent composition is of type it:3 below.

For a type it:3 composition, we have a sequence
$B \supsetdot B' \supsetm \Bf'$.  Let $v$ be the unique element in
$B\setminus B'$ and $L'$ be the linear span of $\Bf'$.
depending on whether $v \not \in L'$ or $v\in L'$,
we can make the following diagrams, respectively:

\begin{equation}
\label{eq:diagram2}
\begin{tikzcd}[row sep = small, column sep=-5pt]
B & \supsetdot & B'\\
\Bf \arrow[u, phantom, sloped, "\subset"]  & = & \Bf'\arrow[u, phantom, sloped, "\subsetm"]
\end{tikzcd}, \quad \text{or} \quad
\begin{tikzcd}[row sep = small, column sep=-5pt]
B & \supsetdot & B'\\
\Bf \arrow[u, phantom, sloped, "\subset"]  & \supsetdot & \Bf'\arrow[u, phantom, sloped, "\subsetm"]
\end{tikzcd}.
\end{equation}

In fact,
in the case of
$v \not\in L'$,
it is clear that $\Bf\coloneqq \Bf' \subset B$ defines a fiber structure.
Furthermore, if the fiber structure is irreducible,
the left diagram in eq:diagram2
coincides with the elementary link of type I${}_\mathrm{d}$ eq:diagram1,
which is already discussed.

If the fiber structure is not irreducible,
we call the diagram a non-irreducible half-link,
which should be coupled with another half-link.
In the case of $v\in L'$,
we have a reduction
$\Bf\coloneqq B\cap L' \supsetdot \Bf'$ as in
the right diagram in eq:diagram2, and
$\Bf \subset B$ becomes an irreducible fiber structure.
We call the diagram an irreducible half-link I.

Finally, in the case of a type it:4,
we denote by $\Bb'$ the base of the fiber structure $B_\mathrm{b, f}\subsetm \Bb$.
Accordingly, we set a diagram of lattices and fit all relevant sets of lattice points together:

\begin{equation}
\begin{tikzcd}[row sep = small]
N \arrow[r, twoheadrightarrow, "\pi"] & \Nb \arrow[r, twoheadrightarrow, "\pi_\mathrm{b}"] & \Nb' \\
\Nf' \arrow[u, phantom, sloped, "\subset"]  \arrow[r, twoheadrightarrow] &  N_{\mathrm{b,f}}
\arrow[u, phantom, sloped, "\subset"]  & \\
%0 \arrow[u, phantom, sloped, "\subset"] \\
\Nf\arrow[u, phantom, sloped, "\subset"] & &
%0 \arrow[u, phantom, sloped, "\subset"] &
\end{tikzcd}
, \qquad
\begin{tikzcd}[row sep = small]
B \arrow[r, dashrightarrow, "\pibar"]& \Bb \arrow[r, dashrightarrow, "\overline{\pi}_\mathrm{b}"] & \Bb' \\
\Bf' \arrow[r, dashrightarrow] \arrow[u, phantom, sloped, "\subset"]  &  B_{\mathrm{b,f}}
\arrow[u, phantom, sloped, "\subsetm"] & \\
\Bf\arrow[u, phantom, sloped, "\subset"] &&
\end{tikzcd},
\end{equation}

where $\Nf' = \ker(\pi_\mathrm{b}\circ \pi)$ and $\Bf' = B \cap \Nf'$.
$B_{\mathrm{b,f}}\subset N_{\mathrm{b,f}}$ is a primitive generating set,
so is $\Bf'\subset \Nf'$, and hence,
$\Bf'\subset B$ and $\Bf \subset \Bf'$ are fiber structures.
By definition, Mori fiber structures $\Bf \subsetm B$ and $B_{\mathrm{b,f}}\subsetm \Bb$
are irreducible, and hence, the equalities
$\abs{B}=\abs{\Bf}+\abs{\Bb} = \abs{\Bf} + \abs{B_{\mathrm{b,f}}} + \abs{\Bb'}$ hold.
Then the inequalities
$\abs{B} \ge \abs{\Bf'} + \abs{\Bb'}$ and $\abs{\Bf'} \ge \abs{\Bf} + \abs{B_{\mathrm{b,f}}}$
for fiber structures
$\Bf'\subset B$ and $\Bf \subset \Bf'$
must be equalities, that is,
both $\Bf'\subset B$ and $\Bf \subset \Bf'$ are also irreducible.
In particular, $\Bf \subset \Bf'$ is a Mori fiber structure.
Thus we obtain another diagram:

\begin{equation}
\label{eq:diagram3}
\begin{tikzcd}[row sep = small, column sep=-5pt]
B & = & B'\\
\Bf \arrow[u, phantom, sloped, "\subsetm"]  & \subsetm & \Bf'\arrow[u, phantom, sloped, "\subset"]
\end{tikzcd}.
\end{equation}

We call the diagram an irreducible half-link II.

Now we combine two of half-links to compose remaining elementary links
between $\Af \subsetm A$ and $\Af'\subsetm A'$, which is very easy.
If the fiber structure in the middle is not irreducible,
there is only one way to combine them,
resulting the elementary link of type II${}_\mathrm{ni}$ in eq:sarkisovA.
If the fiber structure in the middle is irreducible,
one has the remaining three ways,
I-I, I-II, and II-II,
except switching how to set $\Af \subsetm A$ and $\Af'\subsetm A'$,
which results the remaining elementary links
of type II${}_\mathrm{irr}$, type I${}_\mathrm{m}$, and type IV${}_\mathrm{m}$
in eq:sarkisovA, respectively.
This completes the proof.

For any pair of Mori fiber primitive generating sets
$\Af \subsetm A$ and $\Af' \subsetm A'$ in $N_\bR$,
there exists a sequence of elementary links connecting them.

By lm:lift, there exists a pair of toric Mori fiber spaces
$p: X\rightarrow S$ and $p': X' \rightarrow S'$ which represent
$\Af \subsetm A$ and $\Af' \subsetm A'$, respectively.
The $p$ and $p'$ are connected by a sequence of toric Sarkisov links by the Sarkisov program for toric varieties
<cit.>, [10].
By coarse-graining the resulting sequence of Sarkisov links, one obtains a sequence of elementary links connecting
$\Af \subsetm A$ and $\Af' \subsetm A'$ by th:sarkisovA.

We define a log Sarkisov link as one of the four types of diagrams in eq:links
but each divisorial contraction of $X$'s is not imposed to be $K$-negative.

For any sequence of elementary links connecting two Mori fiber primitive generating sets
$\Af\subsetm A$ and $\Af' \subsetm A'$,
there exists a sequence of log Sarkisov links connecting two toric Mori fiber spaces
$p: X\rightarrow S$ and
$p': X'\rightarrow S'$
which represent
$\Af\subsetm A$ and $\Af' \subsetm A'$, respectively.

Assume $\Af =\Af'$ and $A=A'$ first.
There is a natural birational map
$\varphi_\mathrm{b}: S\dashrightarrow S'$ induced by the identity map of $\Nb$.
By the first statement of lm:lift,
$\varphi_\mathrm{b}$ is decomposed into flips, flops, and inverse flips.
Since each intermediate birational model $S''$ also represents the base $\Ab$ of $\Af\subsetm A$,
there exists a toric Mori fiber space $p'': X''\rightarrow S''$ which represent
$\Af\subsetm A$ by the third statement of lm:lift.
Hence, by replacing $p'$ with $p''$,
one may assume $\varphi_\mathrm{b}$ is either a single flip, flop, or inverse flip.
In either case, it fits a log Sarkisov link of type IV${}_\mathrm{s}$ because
the natural map $\varphi: X\dashrightarrow X'$ is decomposed into flips, flops, and inverse flips
and $\varphi_\mathrm{b}\circ p = p' \circ \varphi$ holds.

Next, assume that
$\Af\subsetm A$ and $\Af' \subsetm A'$ are related by a single elementary link.
As in the first case,
one can make a log Sarkisov link from the bottom of its diagram
for each elementary link by applying lm:lift repeatedly. For this purpose,
it is helpful to rewrite the elementary links in eq:sarkisovA into
alternative descriptions by using the bases of fiber structures:

\begin{equation}
\label{eq:sarkisovA2}
\arraycolsep=9pt\def\arraystretch{1.3}
\begin{array}{cccc}
\text{type I}{}_\mathrm{d} & \text{type I}{}_\mathrm{m} & \text{type II}{}_{\mathrm{irr}}
\text{/II}{}_{\mathrm{ni}}  &  \text{type IV}{}_ {\mathrm{m}}\\
\begin{tikzcd}[row sep = 15pt, column sep=-3pt]
A \arrow[d,dashrightarrow] & \supsetdot &  A'\arrow[d,dashrightarrow] \\
\Ab  & \supsetdot & \Ab'
\end{tikzcd} &
\begin{tikzcd}[row sep = 15pt, column sep=-7pt]
A \arrow[d,dashrightarrow]& = &  A'' \arrow[d,dashrightarrow]& \supsetdot & A'\arrow[d,dashrightarrow]\\
\Ab
&\dashrightarrow & \Ab'' & \begin{minipage}{20pt}\begin{center}=\end{center}\end{minipage} & \Ab'
\end{tikzcd} &
\begin{tikzcd}[row sep = 15pt, column sep=-3pt]
A \arrow[d,dashrightarrow]& \subsetdot &  A'' \arrow[d,dashrightarrow]& \supsetdot & A'\arrow[d,dashrightarrow]\\
\Ab
&=& \Ab''
& = & \Ab'
\end{tikzcd} &
\begin{tikzcd}[row sep = 15pt, column sep=-7pt]
A \arrow[d,dashrightarrow]& = &  A'' \arrow[d,dashrightarrow]& = & A'\arrow[d,dashrightarrow]\\
\Ab &\dashrightarrow & \Ab''
&\dashleftarrow & \Ab'
\end{tikzcd}
\end{array}
\end{equation}

Finally, in general case, the log Sarkisov links obtained from each elementary link
are glued together by the sequence of log Sarkisov links of type IV${}_\mathrm{s}$ in the first case,
so that one obtains a sequence of log Sarkisov links connecting $p: X \rightarrow S$ and $p':X'\rightarrow S'$.

For any sequence of elementary links connecting two Mori fiber polytopes
$\nablaf\subsetm \nabla$ and $\nablaf' \subsetm \nabla'$
consisting of only Fano polytopes,
there exists a sequence of Sarkisov links connecting two toric Mori fiber spaces
$p: X\rightarrow S$ and
$p': X'\rightarrow S'$
which represent
$\nablaf\subsetm \nabla$ and $\nablaf' \subsetm \nabla'$, respectively.

This is an immediate consequence of cr:sarkisovlift
and the last sentence of rm:shed.

In order to establish the whole picture
only in terms of Fano polytopes,
the following two problems could be keys,
the latter of which is regarded as the Fano polytope version of cr:seqlinks.
We say that a Fano polytope $\nabla$ is minimal if there is no reduction $\nabla \supsetdot \nabla'$.

For any minimal Fano polytope $\nabla$,  is there a Mori fiber structure $\nablaf\subsetm \nabla$?

For any pair of Mori fiber polytopes
$\nablaf \subsetm \nabla$ and $\nablaf' \subsetm \nabla'$ in $N_\bR$,
is there a sequence of elementary links consisting only of Fano polytopes
which connects $\nablaf\subsetm \nabla$ and $\nablaf' \subsetm \nabla'$?

pb:polytopeMMP is required to guarantee that the MMP
works only inside the category of Fano polytopes.
Namely, it assures that
one ends up with a Mori fiber polytope
after finite times of reductions of a given Fano polytope.
Note that, by triangulating a minimal Fano polytope $\nabla$ using only its vertices,
one can always find a fiber structure $\nablaf \subset \nabla$ with
a minimal Fano simplex $\nablaf$ (see <cit.>, and also <cit.>
for the structure of minimal primitive generating sets).
Hence,  one part of the problem is whether
a minimal Fano simplex $\nablaf$ allows a fiber structure
$\nablaf'\subset \nablaf$ such that $\nablaf'$ is a terminal simplex.

The following example shows the subtleties of pb:mfpconnectedness in three or higher dimensions.
The obstacle stems from the fact that, for a relevant Fano polytope $\nabla$,
a projective simplicial terminal fan $\Sigma$ satisfying $\Conv G(\Sigma)=\nabla$
needs not to be a projective $\nabla$-maximal fan.

$v_1 = (1,0,0)$,
$v_2 = (0,1,0)$,
$v_3 = (-1,-1,0)$,
$v_4 = (-1,0,0)$,
$v_5 = (0,0,1)$,
$v_6 = (-1,0,-1)$, and
$v_7= (-2,0,-1)$
in $N\simeq \bZ^3$.
As an abbreviation, we write
Fano polytopes
$\nabla_{i_1\cdots i_r} = \Conv(v_{i_1}, \dots, v_{i_r})$
primitive generating sets
$A_{i_1\cdots i_r} = \lc v_{i_1}, \dots, v_{i_r} \rc$

with concatenated subscripts for any indices $i_1, \dots, i_r$.
In the following,
the latter symbol is used if and only
if $A_{i_1\cdots i_r} \ne \nabla_{i_1\cdots i_r}\cap \Nprim$.
Then we have a sequence of elementary links
connecting a pair of Mori fiber polytopes $\nabla_{14}\subsetm \nabla_{123457}$
and $\nabla_{123}\subsetm \nabla_{12356}$,

\begin{equation}
\label{eq:fano}
\begin{tikzcd}[row sep = small, column sep=-5pt]
\nabla_{123457} & = & \nabla_{123457} & \supsetdot & A_{12357} & \subsetdot
&A_{123567} & \supsetdot & \nabla_{12356}\\
\nabla_{14} \arrow[u, phantom, sloped, "\subsetm"]
& \subsetm & \nabla_{1234} \arrow[u, phantom, sloped, "\subset"]
& \supsetdot & \nabla_{123} \arrow[u, phantom, sloped, "\subsetm"]
& = & \nabla_{123} \arrow[u, phantom, sloped, "\subset"]
& = & \nabla_{123} \arrow[u, phantom, sloped, "\subsetm"]
\end{tikzcd}.
\end{equation}

The sequence  eq:fano corresponds to that of Sarkisov links between toric Mori fiber spaces,
which are all projective bundles (cf. <cit.>),

\begin{equation}
\begin{tikzcd}
\bP_{\bP^1\times \bP^1}\lb \cO \oplus \cO(-1,-2) \rb \arrow[r,dashrightarrow,"\text{I}_{\mathrm{d}}"] &
\bP_{\bP^1}\lb \cO \oplus \cO \oplus  \cO(-2) \rb \arrow[r,dashrightarrow,"\text{II}_{\mathrm{ni}}"] &
\bP_{\bP^1}\lb \cO \oplus \cO \oplus  \cO(-1) \rb,
\end{tikzcd}
\end{equation}

where we put the types of the Sarkisov links on dashed arrows.
Note that the coarse-graining of the sequence eq:fano into Fano polytopes does not attain
a sequence of elementary links.
On the other hand, one can also take another sequence of elementary links
connecting the same pair of Mori fiber polytopes
which consists only of Fano polytopes:

\begin{equation}
\label{eq:fano2}
\begin{tikzcd}[row sep = small, column sep=-5pt]
\nabla_{123457} & \subsetdot & \nabla_{1234567} & \supsetdot & \nabla_{123456} & =
&\nabla_{123456} & \supsetdot & \nabla_{12356}\\
\nabla_{14} \arrow[u, phantom, sloped, "\subsetm"]
& = & \nabla_{14} \arrow[u, phantom, sloped, "\subset"]
& = & \nabla_{14} \arrow[u, phantom, sloped, "\subsetm"]
& \subsetm & \nabla_{1234} \arrow[u, phantom, sloped, "\subset"]
& \supsetdot & \nabla_{123} \arrow[u, phantom, sloped, "\subsetm"]
\end{tikzcd}.
\end{equation}

The sequence eq:fano2 corresponds to the following sequence of Sarkisov links:

\begin{equation}
\begin{tikzcd}
\bP_{\bP^1\times \bP^1}\lb \cO \oplus \cO(-1,-2) \rb \arrow[r,dashrightarrow,"\text{II}_{\mathrm{ni}}"] &
\bP_{\bP^1\times \bP^1}\lb \cO \oplus  \cO(-1, -1) \rb \arrow[r,dashrightarrow,"\text{I}_{\mathrm{d}}"] &
\bP_{\bP^1}\lb \cO \oplus \cO \oplus  \cO(-1) \rb.
\end{tikzcd}
\end{equation}

Now, let us see that the situation becomes better
in two dimensions
once we use the classification of
the Sarkisov links for smooth rational Mori fiber surfaces:

\begin{equation}
\label{eq:2dimlinks}
\arraycolsep=8pt \def\arraystretch{1.2}
\begin{array}{cccc}
\text{type I}_\mathrm{m} & \text{type II}_\mathrm{ni} & \text{type III}_\mathrm{m} &
\text{type IV}_\mathrm{m} \vspace{5pt}\\
\begin{tikzcd}[column sep= 20pt, row sep=20pt]
\bF_1 \arrow[d] \arrow[dr]\\
\bP^1 \arrow[dr] & \bP^2 \arrow[d]\\
& pt
\end{tikzcd} &
\begin{tikzcd}[column sep=10pt, row sep=20pt]
& S_x \arrow[ld, "\mathrm{bl}_x"'] \arrow[dr]&\\
\bF_m \arrow[dr] && \bF_{m\pm 1} \arrow[dl]\\
& \bP_1&
\end{tikzcd} &
\begin{tikzcd}[column sep=20pt, row sep=20pt]
& \bF_1 \arrow[d] \arrow[dl]\\
\bP^2 \arrow[d] & \bP^1 \arrow[dl]\\
\end{tikzcd} &
\begin{tikzcd}[column sep=0pt, row sep=20pt]
&\bP^1 \times \bP^1 \arrow[dl, "p_1"'] \arrow[dr, "p_2"]&\\
\bP^1 \arrow[dr] && \bP^1 \arrow[ld]\\
& pt&
\end{tikzcd}
\end{array}
\end{equation}

Here $\bF_m = \bP_{\bP^1}\lb \cO \oplus \cO(-m) \rb$ is the Hirzebruch surface for $m\ge 0$,
$p_i: \bF_0 = \bP^1\times \bP^1\rightarrow \bP^1$ is the $i$-th projection ($i=1,2$),
and $\mathrm{bl}_x: S_x\rightarrow \bF_m$
is a blow-up at a point $x\in \bF_m$.
The Sarkisov link of type II${}_\mathrm{ni}$ in eq:2dimlinks is classically known as an
elementary transform, that is,
a blow-up at a point $x\in \bF_m$ followed by the contraction of the strict transform of the fiber
passing through $x$
with respect to a ruling $\bF_m\rightarrow \bP^1$ <cit.>.

We denote it by

\begin{equation}
\label{eq:elem}
\el_x: \bF_m \dashrightarrow \bF_{m\pm 1}
\end{equation}

in the following.
Note that, for $m\ge 1$,
the elementary transform at $x\in \bF_m$ results $\bF_{m+1}$ or $\bF_{m-1}$
depending on whether $x$ lies on the section $D\subset \bF_m$ with negative self-intersection, $D^2=-m$.

pb:polytopeMMP and
pb:mfpconnectedness are affirmative in two dimensions.

It is easy to verify that
all smooth toric Mori fiber surfaces $\bP^2$ and $\bF_m$ ($m\ge 0$) are described by Mori fiber polygons,
all toric Sarkisov links in eq:2dimlinks

are described by elementary links consisting only of Fano polygons (see sc:reflexive for concrete forms).

§ TWO PROOFS OF TH:POLYGON

In this section,
we give a constructive proof of th:polygon first, and then a non-constructive proof of it.
The latter proof uses a stronger claim (lm:smallseq), which also implies
the connectedness of the web of elliptic elephants in smooth projective rational surfaces

Recall that reflexive is equivalent to canonical for lattice polygons.
By repeating reductions for a canonical polygon,
one ends up with a canonical Mori fiber polygon by pr:twodim.
Then it suffices to show that any pair of canonical Mori fiber polygons
is connected via a sequence of elementary links whose intermediate polygons are all canonical.
The same is true for terminal polygons.
Up to unimodular equivalences, there are only
three terminal Mori fiber polygons $\nabla_{-\infty}$, $\nabla_0$ and $\nabla_1$ and
four canonical Mori fiber polygons $\nabla_{-\infty}$, $\nabla_0$, $\nabla_1$ and $\nabla_2$.
Here, for any integer $m\ge 0$
and a fixed lattice basis $\lc e_1, e_2 \rc$ of $N\simeq \bZ^2$, we set

\begin{equation}
\label{eq:polygons}
\nabla_{-\infty} \coloneqq \Conv (e_1, e_2, -e_1-e_2) \quad
\text{and} \quad \nabla_m\coloneqq \Conv(e_1, e_2, -e_1, -m e_1-e_2)
\end{equation}

corresponding to $\bP^2$ and $\bF_m$, respectively.
There is an elementary link of type II${}_\mathrm{ni}$ connecting $\nabla_{m}$ and $\nabla_{m+1}$,

\begin{equation}
\label{eq:linkm}
\begin{tikzcd}[column sep = -7pt, row sep = small]
\nabla_{m} & \subsetdot \  & \Conv(\nabla_{m}\cup \nabla_{m+1}) & \  \supsetdot & \nabla_{m+1}\\
\Conv(\pm e_1)  \arrow[u, phantom, sloped, "\subsetm"] & = & \Conv(\pm e_1)  \arrow[u, phantom, sloped, "\subset"]
& = &\Conv(\pm e_1) \arrow[u, phantom, sloped, "\subsetm"].
\end{tikzcd}
\end{equation}

The elementary link eq:linkm
and its inverse are denoted by $\ell_m^{+}$ and $\ell_m^{-}$, respectively.
It is easy to observe that all polygons in $\ell_0^{\pm}$ are terminal and
all polygons in $\ell_1^{\pm}$ are canonical.
By lm:unimodular below,
we may assume each Mori fiber polygon is moved to standard form eq:polygons in advance.
By using $\ell_m^\pm$ and another elementary link $\ell_{-\infty}^+$ of type III${}_\mathrm{m}$,

\begin{equation}
\label{eq:link0}
\begin{tikzcd}[column sep = -7pt, row sep = small]
\nabla_{-\infty} & \subsetdot \  & \nabla_{1} & \  = & \nabla_{1}\\
\nabla_{-\infty}  \arrow[u, phantom, sloped, "\subsetm"] & \subsetdot & \nabla_{1}  \arrow[u, phantom, sloped, "\subset"]
& \supsetm &\Conv(\pm e_1) \arrow[u, phantom, sloped, "\subsetm"],
\end{tikzcd}
\end{equation}

with its inverse $\ell_{-\infty}^-$,
each pair of terminal Mori fiber polygons from $\lc \nabla_{-\infty}, \nabla_0, \nabla_1 \rc$
(resp. canonical Mori fiber polygons from $\lc \nabla_{-\infty}, \nabla_0, \nabla_1, \nabla_2 \rc$)
can be connected
via a sequence of elementary links whose intermediate polygons are all terminal (resp. canonical).
This completes the proof.

With the same notation as above,
any unimodular transformation of $\nabla_{-\infty}$ (resp. $\nabla_m$ for $m\ge 0$) is decomposed into
a sequence of elementary links unimodular equivalent to $\ell^\pm$, $\ell_{-\infty}^\pm$ and $\ell_0^\pm$ (resp.
$\ell^{\pm}$, $\ell_j^\pm$ with $0\le j\le \max\lc 1,m-1 \rc$), where $\ell^+$ is the following
elementary link of type IV${}_\mathrm{m}$,

\begin{equation}
\begin{tikzcd}[column sep = -7pt, row sep = small]
\nabla_0 & = \  & \nabla_0 & \  = & \nabla_0\\
\Conv(\pm e_1)  \arrow[u, phantom, sloped, "\subsetm"] & \subsetm & \nabla_0
\arrow[u, phantom, sloped, "\subset"]
& \supsetm &\Conv(\pm e_2) \arrow[u, phantom, sloped, "\subsetm"],
\end{tikzcd}
\end{equation}

and $\ell^-$ is its inverse.
In particular, any terminal (resp. canonical) Mori fiber polygons is connected to
that in standard form eq:polygons
through a sequence of elementary links whose intermediate polygons are all terminal (resp. canonical).

It is enough to show the claim only for
the pairs of polygons, either
$\nabla_{-\infty}$ or $\nabla_m$ in standard form eq:polygons
and one of those
moved by the action of a generator of $GL(N) \simeq GL(2,\bZ)$, say,

\begin{equation}
S = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}, \quad
T = \begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix} \quad \text{or} \quad
U = \begin{pmatrix}
-1 & 0 \\
0 & 1
\end{pmatrix}.
\end{equation}

Indeed, any other pair of transformed polygons
is related by a finite composition of some conjugates of the above moves,
and, for such a conjugate move, it can be decomposed into a sequence of
the corresponding conjugates of the elementary links.
Thus the claim can be verified by direct computation.
For example, the pair $\nabla_{-\infty}$ and $S\nabla_{-\infty}$ is connected by a sequence,

\begin{equation}
\label{eq:cremona}
\begin{tikzpicture}[scale=0.5, baseline={(0,0)}]
\fill[gray] (1,0) -- (0,1) -- (-1,-1) --  cycle;
\draw[black] (1,0) -- (0,1) -- (-1,-1) --  cycle;
%  \draw[lightgray] (0,0) -- (1,0);
%  \draw[lightgray] (0,0) -- (0,1);
%  \draw[lightgray] (0,0) -- (-1,-1);
\fill (0,0) circle (2.0pt);
\fill (1,0) circle (2.0pt);
\fill (0,1) circle (2.0pt);
\fill (-1,-1) circle (2.0pt);
\node at (0,-1.5) {$\scriptscriptstyle{\mathrm{Mfp}}$};
\end{tikzpicture}
\,\subsetdot \,
\begin{tikzpicture}[scale=0.5, baseline={(0,0)}]
\fill[gray] (1,0) -- (0,1) -- (-1,0) -- (-1,-1) --  cycle;
\draw[black] (1,0) -- (0,1) -- (-1, 0) --(-1,-1) --  cycle;
%  \draw[lightgray] (0,0) -- (1,0);
%  \draw[lightgray] (0,0) -- (0,1);
%  \draw[lightgray] (0,0) -- (-1,0);
%  \draw[lightgray] (0,0) -- (-1,-1);
\fill (0,0) circle (2.0pt);
\fill (1,0) circle (2.0pt);
\fill (0,1) circle (2.0pt);
\fill (-1,0) circle (2.0pt);
\fill (-1,-1) circle (2.0pt);
\end{tikzpicture}
\,= \,
\begin{tikzpicture}[scale=0.5, baseline={(0,0)}]
\draw[black] (1,0) -- (0,1) -- (-1, 0) --(-1,-1) --  cycle;
\draw[thick, gray] (0,0) -- (1,0);
%  \draw[lightgray] (0,0) -- (0,1);
\draw[thick, gray] (0,0) -- (-1,0);
%  \draw[lightgray] (0,0) -- (-1,-1);
\fill (0,0) circle (2.0pt);
\fill (1,0) circle (2.0pt);
\fill (0,1) circle (2.0pt);
\fill (-1,0) circle (2.0pt);
\fill (-1,-1) circle (2.0pt);
\node at (0,-1.5) {$\scriptscriptstyle{\mathrm{Mfp}}$};
\end{tikzpicture}
\,\subsetdot \,
\begin{tikzpicture}[scale=0.5, baseline={(0,0)}]
\draw[black] (1,0) -- (0,1) -- (-1, 0) --(-1,-1) -- (0,-1) -- cycle;
\draw[thick, gray] (0,0) -- (1,0);
%  \draw[lightgray] (0,0) -- (0,1);
\draw[thick, gray] (0,0) -- (-1,0);
%  \draw[lightgray] (0,0) -- (-1,-1);
%  \draw[lightgray] (0,0) -- (0,-1);
\fill (0,0) circle (2.0pt);
\fill (1,0) circle (2.0pt);
\fill (0,1) circle (2.0pt);
\fill (0,-1) circle (2.0pt);
\fill (-1,0) circle (2.0pt);
\fill (-1,-1) circle (2.0pt);
\end{tikzpicture}
\,\supsetdot \,
\begin{tikzpicture}[scale=0.5, baseline={(0,0)}]
\draw[black] (1,0) -- (0,1) -- (-1, 0) -- (0,-1) -- cycle;
\draw[thick, gray] (0,0) -- (1,0);
%  \draw[lightgray] (0,0) -- (0,1);
\draw[thick, gray] (0,0) -- (-1,0);
%  \draw[lightgray] (0,0) -- (0,-1);
\fill (0,0) circle (2.0pt);
\fill (1,0) circle (2.0pt);
\fill (0,1) circle (2.0pt);
\fill (0,-1) circle (2.0pt);
\fill (-1,0) circle (2.0pt);
\node at (0,-1.5) {$\scriptscriptstyle{\mathrm{Mfp}}$};
\end{tikzpicture}
\,\subsetdot \,
\begin{tikzpicture}[scale=0.5, baseline={(0,0)}]
\draw[black] (1,0) -- (0,1) -- (-1, 0) -- (0,-1) -- (1,-1) -- cycle;
\draw[thick, gray] (0,0) -- (1,0);
%  \draw[lightgray] (0,0) -- (0,1);
\draw[thick, gray] (0,0) -- (-1,0);
%  \draw[lightgray] (0,0) -- (0,-1);
%  \draw[lightgray] (0,0) -- (1,-1);
\fill (0,0) circle (2.0pt);
\fill (1,0) circle (2.0pt);
\fill (0,1) circle (2.0pt);
\fill (0,-1) circle (2.0pt);
\fill (-1,0) circle (2.0pt);
\fill (1,-1) circle (2.0pt);
\end{tikzpicture}
\,\supsetdot \,
\begin{tikzpicture}[scale=0.5, baseline={(0,0)}]
\draw[black] (1,0) -- (0,1) -- (-1, 0) -- (1,-1) -- cycle;
\draw[thick, gray] (0,0) -- (1,0);
%  \draw[lightgray] (0,0) -- (0,1);
\draw[thick, gray] (0,0) -- (-1,0);
%  \draw[lightgray] (0,0) -- (1,-1);
\fill (0,0) circle (2.0pt);
\fill (1,0) circle (2.0pt);
\fill (0,1) circle (2.0pt);
\fill (-1,0) circle (2.0pt);
\fill (1,-1) circle (2.0pt);
\node at (0,-1.5) {$\scriptscriptstyle{\mathrm{Mfp}}$};
\end{tikzpicture}
\,= \,
\begin{tikzpicture}[scale=0.5, baseline={(0,0)}]
\fill[gray] (1,0) -- (0,1) -- (-1,0) -- (1,-1) --  cycle;
\draw[black] (1,0) -- (0,1) -- (-1, 0) -- (1,-1) -- cycle;
%  \draw[lightgray] (0,0) -- (1,0);
%  \draw[lightgray] (0,0) -- (0,1);
%  \draw[lightgray] (0,0) -- (-1,0);
%  \draw[lightgray] (0,0) -- (1,-1);
\fill (0,0) circle (2.0pt);
\fill (1,0) circle (2.0pt);
\fill (0,1) circle (2.0pt);
\fill (-1,0) circle (2.0pt);
\fill (1,-1) circle (2.0pt);
\end{tikzpicture}
\,\supsetdot \,
\begin{tikzpicture}[scale=0.5, baseline={(0,0)}]
\fill[gray]  (0,1) -- (-1,0) -- (1,-1) --  cycle;
\draw[black] (0,1) -- (-1, 0) -- (1,-1) -- cycle;
%  \draw[lightgray] (0,0) -- (0,1);
%  \draw[lightgray] (0,0) -- (-1,0);
%  \draw[lightgray] (0,0) -- (1,-1);
\fill (0,0) circle (2.0pt);
\fill (0,1) circle (2.0pt);
\fill (-1,0) circle (2.0pt);
\fill (1,-1) circle (2.0pt);
\node at (0,-1.5) {$\scriptscriptstyle{\mathrm{Mfp}}$};
\end{tikzpicture},
\end{equation}

where we encode the fiber structure $\nablaf \subset \nabla$ as
the thick interval overlaid in gray if $\dim \nablaf =1$ and as
the polytope filled in gray if $\nablaf = \nabla$.
Also, the marker “Mfp” indicates that $\nablaf \subsetm \nabla$ is a Mori fiber polygon.
If we write the conjugate and the composition of elementary links as in ordinary maps,
the sequence eq:cremona is expressed as
$S\nabla = (U l_{-\infty}^- U^{-1})\circ (U l_0^+ U^{-1})\circ l_0^-\circ l_{-\infty}^+(\nabla)$.
Similarly, as another example, the pair $\nabla_m$ and $S\nabla_m$ is connected by a sequence
$S\nabla_m=(Sl_{m-1}^+S^{-1})\circ \cdots \circ (S l_1^+ S^{-1})\circ l^+ \circ l_1^- \circ \cdots \circ l_{m-1}^-(\nabla_m)$.
Remaining four pairs are left to the reader.

Let us set $\bF_{-\infty} = \bP^2$ for convenience.
For any sequence $s$ of Sarkisov links in eq:2dimlinks,
we write $M(s)$ as the maximal $m$ such that $\bF_m$
appears in the sequence $s$, and $M(\bF_m)=m$ for all $m\in \lc -\infty, 0, 1, \dots \rc$.

Let $p: X\rightarrow S$ and $p':X'\rightarrow S'$ be smooth rational Mori fiber surfaces and
$\varphi: X \dashrightarrow X'$ be any birational map.
Then $\varphi$ is decomposed into a sequence $s$ of Sarkisov links
such that $M(s) \le \max\lc 1, M(X), M(X')\rc$.

By the Sarkisov program in two dimensions,
any birational map $\varphi: X\dashrightarrow X'$ can be decomposed into
a sequence of Sarkisov links.
Suppose $M(s) > \max\lc 1,M(X), M(X')\rc$ for a sequence $s$ decomposing $\varphi$.
Then $s$ should be locally in the following form:

\begin{equation}
\label{eq:maxM}
\begin{tikzcd}
\cdots \arrow[r,dashrightarrow] &
\bF_{M(s)-1} \arrow[r,dashrightarrow, "\el_x"] &
\bF_{M(s)} \arrow[r,dashrightarrow, "\el_y"] & \bF_{M(s)-1} \arrow[r,dashrightarrow] &\cdots
\end{tikzcd},
\end{equation}

where $x\in \bF_{M(s)-1}$ is on the curve of negative self-intersection,
$y\in \bF_{M(s)}$ is away from the curve of negative self-intersection,
and $\el_x$ and $\el_y$ are elementary transforms eq:elem.
If $y$ is not an infinitely near point to $x$ (i.e,
if $y$ does not lie on the strict transform of the exceptional curve of the blow-up of $\bF_{M(s)-1}$ at $x$),
one can replace $s$ with the following sequence $s'$ by switching the order of taking elementary transforms:

\begin{equation}
\label{eq:maxMfinite}
\begin{tikzcd}
\cdots \arrow[r,dashrightarrow] &
\bF_{M(s)-1} \arrow[r,dashrightarrow, "\el_y"] &
\bF_{M(s)-2} \arrow[r,dashrightarrow, "\el_x"] & \bF_{M(s)-1} \arrow[r,dashrightarrow] &\cdots
\end{tikzcd},
\end{equation}

where we denote the corresponding points $x \in \bF_{M(s)-2}, y\in \bF_{M(s)-1}$
as the same symbols.
Similarly, if $y$ is an infinitely near point to $x$ (which is only redundant in toric case),
by taking another elementary transform at $z\in \bF_{M(s)-1}$ in advance,
one can replace $s$ with the following sequence $s'$
(by using elementary relations in the sense of [11] twice):

\begin{equation}
\label{eq:maxMfinite}
\begin{tikzcd}[column sep = 20pt]
\cdots \arrow[r,dashrightarrow] &
\bF_{M(s)-1} \arrow[r,dashrightarrow, "\el_z"] &
\bF_{M(s)-2} \arrow[r,dashrightarrow, "\el_x"] &
\bF_{M(s)-1} \arrow[r,dashrightarrow, "\el_y"] &
\bF_{M(s)-2} \arrow[r,dashrightarrow, "\el_z^{-1}"] & \bF_{M(s)-1} \arrow[r,dashrightarrow] &\cdots
\end{tikzcd},
\end{equation}

where $z$ is chosen away from neither the curve of negative
self-intersection nor the fiber passing through $x$,
and $\el_z^{-1}$ denotes the inverse of $\el_z$

away from
the fiber passing through $x$.
In either case, the number of times that $\bF_{M(s)}$ appears is strictly decreased in $s'$.
Thus, after a finite number of replacing, one may have $M(s')<M(s)$.
Repeating the same process,
one obtains a sequence $s$ that satisfies $M(s) \le \max\lc 1, M(X), M(X')\rc$ in the end.

Let $\nabla$ and $\nabla'$ be any pair of reflexive, and hence, canonical polygons.
By pr:twodim,
one may assume that
those are canonical Mori fiber polygons by
repeating reductions in advance.
By lm:lift,
for any pair of canonical Mori fiber polygons $\nablaf\subsetm \nabla$ and $\nablaf'\subsetm \nabla'$,
there is a pair of smooth toric Mori fiber surfaces $p: X\rightarrow S$ and $p': X'\rightarrow S'$
which respectively represent $\nablaf\subsetm \nabla$ and $\nablaf'\subsetm \nabla'$
and the natural birational map $\varphi: X\dashrightarrow X'$
(all of which are uniquely determined in two dimensions).
Note that $X$ and $X'$ are either $\bF_m$ ($m\le 2$) in this case.
Clearly from the proof, the sequence in lm:smallseq
can be chosen as a sequence of toric Sarkisov links for
the equivariant birational map $\varphi: X \dashrightarrow X'$.
the sequence of Sarkisov links can be coarse-grained into a sequence of elementary links
consisting only of Fano polygons, which
connects $\nablaf\subsetm \nabla$ and $\nablaf'\subsetm \nabla'$ again by pr:twodim.
From the concrete description of elementary links,
all the intermediate Fano polygons turn ont to be canonical.
The same argument also works for terminal Mori fiber polygons.

This completes the proof.

lm:smallseq is a variant of the classical Noether–Castelnuovo theorem, which states that
the Cremona group $\Bir(\bP^2)$ is generated by automorphisms in $\Aut(\bP^2)\simeq PGL(3,\bC)$
and the standard quadratic transformation $\sigma: \bP^2 \dashrightarrow \bP^2$ written as
$[a:b:c]\mapsto [\frac{1}{a}:\frac{1}{b}:\frac{1}{c}]$.
In fact,
for a given birational automorphism of $\bP^2$,
one can easily rewrite the sequence obtained in lm:smallseq
into the composition of
the transformations described as

\begin{equation}
\label{eq:quadratic}
\begin{tikzcd}
h^{-1} \circ \sigma \circ  g: \bP^2 \arrow[r,dashrightarrow, "\mathrm{bl}_x^{-1}"] &
\bF_1 \arrow[r,dashrightarrow, "\el_y"] &
\bF_0 \arrow[r,dashrightarrow, "\el_z"] & \bF_1 \arrow[r, rightarrow] & \bP^2
\end{tikzcd}
\end{equation}

by inserting redundant Sarkisov links.
Here in eq:quadratic, $x,y,z \in \bP^2$ are any three points not on a line,
$g\in \Aut(\bP^2)$ is an automorphism which moves $x, y, z$ to the torus fixed points, and
$h\in \Aut(\bP^2)$ is defined to produce $\sigma$
by fixing torus action depending on the choice of $g$.

To clarify the relationship between lm:smallseq and Reid's fantasy,
we introduce the notion of geometric transitions for Calabi–Yau elephants in arbitrary dimensions.

Let $X$ and $X'$ be normal projective varieties
whose general elephants $D$ and $D'$ are Calabi–Yau varieties,
and $\varphi: X \rightarrow X'$ be a birational contraction.
We say that the Calabi–Yau varieties $D$ and $D'$ are related by a
geometric transition associated with $\varphi$
if $\varphi|_D: D \rightarrow \varphi_* D$ is a birational contraction
to a normal variety,
and $\varphi_*D$ and $D'$ are related by a flat deformation inside $X'$.
The opposite operation is also called a geometric transition associated with $\varphi^{-1}$.
if $\varphi|_D$ is an isomorphism, and $\varphi_*D$ and $D'$ are related by a smooth deformation inside $X'$,
we say that $D$ and $D'$ are related by a
smooth transition.

Let $X$ and $X'$ be smooth projective rational surfaces
whose general elephants $E$ and $E'$ are smooth elliptic curves.
Then any birational map $\varphi: X\dashrightarrow X'$
is decomposed into a sequence of blow-ups and blow-downs
such that  $E$ and $E'$ are connected
via a sequence of smooth transitions associated with them.

Let $X$ be a smooth projective rational surface whose general elephant $E$ is a smooth elliptic curve.
Then any $(-1)$-curve $C$ on $X$ intersects with $E$ at one point by the adjunction formula $C.(C+K)=-2$.
Hence, a contraction of $(-1)$-curves sends $E$ to its isomorphic image
without changing the property of being an elephant.
By contracting all $(-1)$-curves in advance,
one may assume that $X$ and $X'$ are smooth rational Mori fiber surfaces
whose elephants are elliptic curves, i.e., $\bF_m$ ($m\le 2$).
Now, the birational map $\varphi: X\dashrightarrow X'$ is decomposed into a sequence of
Sarkisov links obtained by lm:smallseq.
In each Sarkisov link in the sequence,
all birational models allow elliptic elephants since they are described by reflexive polygons in
its toric description.
Thus, the elliptic elephants are related via a smooth transition at each step of the Sarkisov links.
More precisely, for a blow-up at a point,
one can smoothly specialize $E$ to another elephant passing through the center of the blow-up
and send it to its isomorphic strict transform that is again an elephant in a new ambient space.

cr:ellip is compared with the Sarkisov program for Mori fibered
Calabi–Yau pairs established in arbitrary dimensions by [4],
which shows,
in the setting of cr:ellip,
$E$ and $E'$ are connected only by a sequence of strict transforms if
$\varphi: (X, E) \dashrightarrow (X',E)$ is a volume-preserving birational map of Calabi–Yau pairs, i.e.,
$p^*(K_X + E)= {q}^*(K_{X'}+E')$ for a common log resolution $(p,q):W\rightarrow X\times X'$.

[1]
Michela Artebani, Paola Comparin, and Robin Guilbot, Families of
Calabi-Yau hypersurfaces in $\Bbb{Q}$-Fano toric varieties, J. Math.
Pures Appl. (9) 106 (2016), no. 2, 319–341. 3515305

[2]
Victor V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau
hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994),
no. 3, 493–535. MR1269718 (95c:14046)

[3]
Victor Batyrev, The stringy Euler number of Calabi-Yau
hypersurfaces in toric varieties and the Mavlyutov duality, Pure Appl.
Math. Q. 13 (2017), no. 1, 1–47. 3858013

[4]
Alessio Corti and Anne-Sophie Kaloghiros, The Sarkisov program for
Mori fibred Calabi-Yau pairs, Algebr. Geom. 3 (2016), no. 3,
370–384. 3504536

[5]
David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties,
Graduate Studies in Mathematics, vol. 124, American Mathematical Society,
Providence, RI, 2011. 2810322

[6]
Alessio Corti, Factoring birational maps of threefolds after Sarkisov,
J. Algebraic Geom. 4 (1995), no. 2, 223–254. 1311348

[7]
Karl Fredrickson, Generalized compactifications of Batyrev hypersurface
families, 2015, arXiv/1410.8287.

[8]
Osamu Fujino and Hiroshi Sato, Introduction to the toric Mori theory,
Michigan Math. J. 52 (2004), no. 3, 649–665. 2097403

[9]
Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977,
Graduate Texts in Mathematics, No. 52. 0463157 (57 #3116)

[10]
Christopher D. Hacon and James McKernan, The Sarkisov program, J.
Algebraic Geom. 22 (2013), no. 2, 389–405. 3019454

[11]
Anne-Sophie Kaloghiros, Relations in the Sarkisov program, Compos.
Math. 149 (2013), no. 10, 1685–1709. 3123306

[12]
Alexander M. Kasprzyk, Canonical toric Fano threefolds, Canad. J.
Math. 62 (2010), no. 6, 1293–1309. 2760660

[13]
Yujiro Kawamata, Flops connect minimal models, Publ. Res. Inst. Math.
Sci. 44 (2008), no. 2, 419–423. 2426353

[14]
Maximilian Kreuzer and Harald Skarke, Reflexive polyhedra, weights and
toric Calabi-Yau fibrations, Rev. Math. Phys. 14 (2002), no. 4,
343–374. 1901220 (2003b:14062)

[15]
Kenji Matsuki, Introduction to the Mori program, Universitext,
Springer-Verlag, New York, 2002. 1875410

[16]
Anvar R. Mavlyutov, Mirror Symmetry for Calabi–Yau complete
intersections in Fano toric varieties, 2011, arXiv/1103.2093.

[17]
Miles Reid, Decomposition of toric morphisms, Arithmetic and geometry,
Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983,
pp. 395–418. 717617

[18]
, The moduli space of $3$-folds with $K=0$ may nevertheless be
irreducible, Math. Ann. 278 (1987), no. 1-4, 329–334. 909231

[19]
Harald Skarke, Weight systems for toric Calabi-Yau varieties and
reflexivity of Newton polyhedra, Modern Phys. Lett. A 11 (1996),
no. 20, 1637–1652. 1397227

[1]
Michela Artebani, Paola Comparin, and Robin Guilbot, Families of
Calabi-Yau hypersurfaces in $\Bbb{Q}$-Fano toric varieties, J. Math.
Pures Appl. (9) 106 (2016), no. 2, 319–341. 3515305

[2]
Victor V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau
hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994),
no. 3, 493–535. MR1269718 (95c:14046)

[3]
Victor Batyrev, The stringy Euler number of Calabi-Yau
hypersurfaces in toric varieties and the Mavlyutov duality, Pure Appl.
Math. Q. 13 (2017), no. 1, 1–47. 3858013

[4]
Alessio Corti and Anne-Sophie Kaloghiros, The Sarkisov program for
Mori fibred Calabi-Yau pairs, Algebr. Geom. 3 (2016), no. 3,
370–384. 3504536

[5]
David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties,
Graduate Studies in Mathematics, vol. 124, American Mathematical Society,
Providence, RI, 2011. 2810322

[6]
Alessio Corti, Factoring birational maps of threefolds after Sarkisov,
J. Algebraic Geom. 4 (1995), no. 2, 223–254. 1311348

[7]
Karl Fredrickson, Generalized compactifications of Batyrev hypersurface
families, 2015, arXiv/1410.8287.

[8]
Osamu Fujino and Hiroshi Sato, Introduction to the toric Mori theory,
Michigan Math. J. 52 (2004), no. 3, 649–665. 2097403

[9]
Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977,
Graduate Texts in Mathematics, No. 52. 0463157 (57 #3116)

[10]
Christopher D. Hacon and James McKernan, The Sarkisov program, J.
Algebraic Geom. 22 (2013), no. 2, 389–405. 3019454

[11]
Anne-Sophie Kaloghiros, Relations in the Sarkisov program, Compos.
Math. 149 (2013), no. 10, 1685–1709. 3123306

[12]
Alexander M. Kasprzyk, Canonical toric Fano threefolds, Canad. J.
Math. 62 (2010), no. 6, 1293–1309. 2760660

[13]
Yujiro Kawamata, Flops connect minimal models, Publ. Res. Inst. Math.
Sci. 44 (2008), no. 2, 419–423. 2426353

[14]
Maximilian Kreuzer and Harald Skarke, Reflexive polyhedra, weights and
toric Calabi-Yau fibrations, Rev. Math. Phys. 14 (2002), no. 4,
343–374. 1901220 (2003b:14062)

[15]
Kenji Matsuki, Introduction to the Mori program, Universitext,
Springer-Verlag, New York, 2002. 1875410

[16]
Anvar R. Mavlyutov, Mirror Symmetry for Calabi–Yau complete
intersections in Fano toric varieties, 2011, arXiv/1103.2093.

[17]
Miles Reid, Decomposition of toric morphisms, Arithmetic and geometry,
Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983,
pp. 395–418. 717617

[18]
, The moduli space of $3$-folds with $K=0$ may nevertheless be
irreducible, Math. Ann. 278 (1987), no. 1-4, 329–334. 909231

[19]
Harald Skarke, Weight systems for toric Calabi-Yau varieties and
reflexivity of Newton polyhedra, Modern Phys. Lett. A 11 (1996),
no. 20, 1637–1652. 1397227