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\title{Semistable reduction in characteristic 0}

\author{Karim Adiprasito\thanks{\href{mailto:adiprasito@math.huji.ac.il}{adiprasito@math.huji.ac.il}. K.A. was supported by ERC StG 716424 - CASe and ISF Grant 1050/16.}\addressmark{1},
	Gaku Liu\thanks{\href{mailto:gakuliu@gmail.com}{gakuliu@gmail.com}. G.L. was supported by NSF grant 1440140.}\addressmark{2}, \and
	Michael Temkin\thanks{\href{mailto:michael.temkiin@mail.huji.ac.il}{michael.temkiin@mail.huji.ac.il}. M.T. was supported by ERC Consolidator Grant 770922 - BirNonArchGeom and ISF Grant 1159/15.}\addressmark{3}}

\address{\addressmark{1}Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Giv'at Ram, Jerusalem, 91904, Israel \\
	\addressmark{2}Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany \\
	\addressmark{3}Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Giv'at Ram, Jerusalem, 91904, Israel}
	
\received{\today}

\abstract{We address the semistable reduction conjecture of Abramovich and Karu: we prove that every surjective morphism of complex projective varieties can be modified to a semistable one. The key ingredient is a combinatorial result on triangulating lattice Cayley polytopes.}

\keywords{semistable reduction, Cayley polytopes, mixed subdivisions}
%\thanks{K.A. was supported by ERC StG 716424 - CASe and ISF Grant 1050/16. G.L. was supported by NSF grant 1440140.
%M.T. was supported by ERC Consolidator Grant 770922 - BirNonArchGeom and ISF Grant 1159/15.
%}

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\begin{document}

\maketitle

\section{Introduction}

Motivated by the desire to replace unwieldy morphisms of varieties with sufficiently nice ones, Abramovich and Karu \cite{AK} proposed a best possible conjecture.

\begin{conj}\label{conj:AK}
Let $X\rightarrow B$ denote a surjective morphism of complex projective varieties with geometrically integral generic fiber.  Then there is a projective alteration $B_1 \rightarrow B$, and a projective modification $Y\rightarrow X\times_B B_1$ such that
$Y \rightarrow B_1$ is semistable.
\end{conj}

Abramovich and Karu \cite{AK} themselves proved a weak version of this conjecture, namely the existence to a weak semistable morphism. These are defined as follows:

A morphism $f:X\to B$ is called {\it weakly semistable} if
\begin{compactenum}
\item the varieties $X$ and $B$ admit toroidal structures $U_X\subset
X$ and $U_B\subset B$, with $U_X=f^{-1}U_B$;
\item with this structure, the morphism $f$ is toroidal;
\item the morphism $f$ is equidimensional;
\item all the fibers of the morphism $f$ are reduced;
%\item the generic fiber is nonsingular;
and
\item $B$ is nonsingular.
\end{compactenum}
However, the case when $X$ is additionally nonsingular, and in which the morphism $f:X\to B$ is
called {\it semistable}, was left as an open conjecture. We build on the combinatorial methods of \cite{ALPT} to resolve \cref{conj:AK} positively. The present manuscript is an extended abstract of the full paper, which can be found here \cite{ALT}. 

\subsubsection*{History}
In addition to the work of Abramovich and Karu, a few other cases are known; see \cite{AK} for a more detailed history. In particular, we know the cases $\dim B=1$ and $\dim
X=2$ (see \cite{aw}), $\dim B=1$ (see \cite{KKMS}), and the case of codimension one \cite{AK}.

De Jong proved the case $\dim X = \dim B + 1$ \cite{dJ}. Using work of Alexeev, Koll\'ar and
Shepherd-Barron (see \cite{alex1}, \cite{alex2}), one obtains a version
of the case $\dim X = \dim B + 2$.

Abramovich and Karu were the first to provide substantial progress to the case $\dim X
>\dim B+2$, explaining their dissatisfaction with the quote ``since
we do not have a semistable reduction result over a base of higher
dimension, we will work around it in the following technical
manner...''.

Finally, Karu \cite{Karu} succeeded in another breakthrough: he proved semistable reduction in the case $\dim X- \dim B\le 3$ based on reduction to combinatorics.

A breakthrough was obtained in the recent work of the three authors and Pak, who proved a local version in \cite{ALPT} here using recent constructions of Haase, Paffenholz, Piechnik and Santos \cite{Santos}. We expand further on this here, and obtain a complete solution to the conjecture.

\section{Proof}

\subsection{Statement of the conjecture}

Following the usual approach, Abramovich and Karu translated \cref{conj:AK} to a combinatorial one. We shall not repeat that translation here, and instead refer to \cite{AK} and review the combinatorial statement of the conjecture. We follow the notation and terminology on rational cones, polyhedral complexes, etc., from this paper. Recall that a \emph{rational conical polyhedral complex}, or \emph{conical complex}, is an (abstract) polyhedral complex formed by gluing together finitely many polyhedral cells $\sigma$, each of which is equipped with a lattice $N_\sigma \cong \bfZ^{d_\sigma}$ such that $\sigma$ is a full-dimensional, rational, strictly convex polyhedral cone in $N_\sigma \otimes \bfR$, and such that if $\tau$ is a face of $\sigma$, then $N_\tau = N_\sigma|_{\Span(\tau)}$. We will denote such a conical complex by $\{(\sigma,N_\sigma)\}$. A \emph{map} $f : X \to Y$ of conical complexes $X = \{(\sigma,N_\sigma)\}$ and $Y = \{(\rho,N_\rho)\}$ is a collection of group homomorphisms $f_\sigma : N_\sigma \to N_\rho$, one for each $\sigma \in X$ (and with $\rho$ chosen for each $\sigma$), such that the extension $f_\sigma : N_\sigma \otimes \bfR \to N_\rho \otimes \bfR$ maps $\sigma$ into $\rho$, and such that if $\tau$ is a face of $\sigma$, then $f_\tau = f_\sigma|_\tau$. A conical complex is \emph{nonsingular} if each of its cells $\sigma$ is generated by a lattice basis of $N_\sigma$.

\begin{defin} \label{def:semistable}
Let $X$, $B$ be conical complexes and $f : X \to B$ a map of conical complexes. We say that $f$ is \emph{semistable} if the following conditions hold.
\begin{enumerate}
\item $f^{-1}(0) = \{0\}$.
\item For every $\sigma \in X$, we have $f(\sigma) \in B$.
\item For every $\sigma \in X$, we have $f(N_\sigma) = N_{f(\sigma)}$.
\item $X$ and $B$ are nonsingular.
\end{enumerate}
\end{defin}

\begin{defin}
Let $X$ be a conical complex. A \emph{lattice alteration} is a map $X_1 \to X$ where $X_1$ is a conical complex of the form $\{ (\sigma,N_\sigma^1) : \sigma \in X \}$ and each $N_\sigma^1$ is a sublattice of $N_\sigma$. An \emph{alteration} is a composition $X_1 \to X' \to X$ of a lattice alteration $X_1 \to X'$ with a subdivision $X' \to X$. The alteration is \emph{projective} if the subdivision $X' \to X$ is projective.
\end{defin}

Fix a map $f : X \to Y$ of conical complexes. For any alteration $g : Y_1 \to Y$, there is an alteration $X_1 \to X$ with
\[
X_1 = \{ (\sigma \cap f^{-1}(g(\tau)), N_\sigma \cap f^{-1}(g(N_\tau)) : \sigma \in X, \tau \in Y_1 \}
\]
which is the unique minimal alteration admitting a map $X_1 \to Y_1$. We call $X_1 \to X$ the alteration \emph{induced} by $g$ (with respect to $f$). If $g$ is projective, then $X_1 \to X$ is as well.

We can now state the conjecture of Abramovich and Karu.

\begin{conj} \label{conj:cones}
Let $f : X \to B$ be a map of conical complexes with $f^{-1}(0) = \{0\}$. Then there exists a projective alteration $B_1 \to B$, with induced alteration $X_1 \to X$, and a projective subdivision $Y \to X_1$ such that $Y \to B_1$ is semistable.
\end{conj}


\subsection{Polytopal complexes}

Our next step is to translate the problem from conical complexes to polytopal complexes, which will be more natural for our constructions. A \emph{lattice polytope} is a pair $(P,N_P)$ (usually denoted as just $P$) where $N_P \cong \bfZ^{d_P}$ is an affine lattice and $P$ is a polytope in $N_P \otimes \bfR$ with vertices in $N_P$. A \emph{face} of a lattice polytope $(P,N_P)$ is a lattice polytope $(F,N_F)$ where $F$ is a face of $P$ and $N_F = N_P |_{\AffSpan(F)}$. A \emph{lattice polytopal complex}, or \emph{polytopal complex}, is a polyhedral complex formed by gluing together finitely many lattice polytopes (with the usual gluing conditions of polyhedral complexes) such that lattices agree on common faces and each polytope $P$ is full-dimensional in $N_P$. Maps between polytopal complexes are defined analogously to the conical case.

Let $(P,N_P)$ be a lattice polytope. We let $L_P$ denote the affine sublattice of $N_P$ spanned by the vertices of $P$. We say that the \emph{index} of $P$ is the index 
\[
[N_P \cap \AffSpan(P) : L_P].
\] 
A \emph{unimodular simplex} is a lattice simplex of index 1. A \emph{unimodular triangulation} is a polytopal complex all of whose elements are unimodular simplices. If the vertices of $P$ are contained in some sublattice $N'$ of $N_P$, then we say that the \emph{index of $P$ with respect to $N'$} is the index of $(P,N')$. We define unimodularity with respect to $N'$ accordingly.

Usually, we will specify an origin in $N_P$, which allows us to define dilations of lattice polytopes and Minkowski sums of polytopes with the same lattice. Since all of our results will be invariant under translation, the choice of origin will not matter. If $X = \{(P,N_P)\}$ is a polytopal complex, then we can define a polytopal complex $cX = \{(cP,N_{P})\}$. Given a map $f : X \to Y$ of polytopal complexes, there is the obvious induced map $cX \to cY$; we denote this map by $cf$.

We define lattice alterations and alterations of polytopal complexes analogously to the conical case. For induced alterations, we need to proceed more carefully. Let $f : X \to Y$ be a map of polytopal complexes and $g : Y_1 \to Y$ an alteration. Consider the rational subdivision
\begin{equation} \label{eq:induced}
X_1 := \{ (P \cap f^{-1}(g(Q)), N_P \cap f^{-1}(g(N_Q))) : P \in X, Q \in Y_1 \}.
\end{equation}
The vertices of $X_1$ are not necessarily lattice points. However, if they are, then $X_1$ is a lattice polytopal complex, and $X_1 \to X$ is the unique minimal alteration admitting a map $X_1 \to Y_1$. In this case we say that $X_1 \to X$ is \emph{induced} by $g$ (with respect to $f$).

We say a map $f : X \to Y$ of polytopal complexes is \emph{good} if for every $P \in X$, we have $f(P) \in Y$. We have the following.

\begin{prop} \label{prop:good}
Let $f : X \to Y$ be a map of polytopal complexes. Then there exists a positive integer $c$ and a projective subdivision $Y' \to cY$ which induces with respect to $cf$ an alteration $X' \to cX$ such that $X' \to Y'$ is good.
\end{prop}

\begin{proof}
It is easy to see that there is a rational projective subdivision $\tilde{Y}$ of $Y$ such that for every $P \in X$, we have that $f(X)$ is a union of cells of $Y$. This induces as in \eqref{eq:induced} a rational subdivision $\tilde{X}$ of $X$. For some $c$, we have that $c\tilde{X}$ and $c\tilde{Y}$ are lattice subdivisions of $cX$ and $cY$, and the map $c\tilde{X} \to c\tilde{Y}$ gives the result.
\end{proof}

\begin{prop} \label{prop:induced}
Let $f : X \to Y$ be a good map of polytopal complexes and $Y_1 \to Y$ an alteration. Then there exists a positive integer $c$ such that if $Y_1 \to cY_1$ is the lattice alteration given by $(P,N_P) \cong (cP,cN_P) \mapsto (cP,N_P)$ and $g$ is the alteration $Y_1 \to cY_1 \to cY$, then $g$ induces with respect to $cf$ an alteration $X_1 \to cX$.
\end{prop}

\begin{proof}
As in the previous proof, there is $c$ such that $cY_1 \to cY$ induces a lattice subdivision $X' \to cX$. Then the lattice alteration $Y_1 \to cY_1$ induces an alteration $X_1 \to X'$, and $X_1 \to X' \to cX$ is the desired alteration.
\end{proof}

Our goal now is to prove the following theorem:

\begin{theor} \label{thm:polytopes}
Let $f : X \to B$ be a map of polytopal complexes. Then there exists a positive integer $c$, a projective alteration $B_1 \to cB$ which induces with respect to $cf$ an alteration $X_1 \to cX$, and a projective subdivision $Y \to X_1$ such that $Y$ and $B_1$ are both unimodular triangulations.
\end{theor}

\begin{prop}
\cref{thm:polytopes} implies \cref{conj:cones}.
\end{prop}

\begin{proof}
Let $\hat f : \hat X \to \hat B$ be a map of conical complexes with $\hat f^{-1}(0) = \{0\}$. Then we can construct a map $f : X \to B$ of polytopal complexes such that there are bijections $\hat X \to X$ and $\hat B \to B$ which preserve the combinatorial structure and such that each cone $\sigma$ is mapped to the intersection of $\sigma$ and some hyperplane $H_\sigma$. Applying \cref{thm:polytopes}, we have a positive integer $c$, a projective alteration $B_1 \to cB$ with induced alteration $X_1 \to cX$, and a projective subdivision $Y \to X_1$ such that $Y$ and $B_1$ are unimodular triangulations. Note that $Y$ and $B_1$ are alterations of the polytopal complexes formed by intersecting each $\sigma$ in $\hat X$ and $\hat B$ with the hyperplane $cH_\sigma$. Hence, by coning $Y$ and $B_1$, we have alterations $\hat Y \to \hat X$ and $\hat B_1 \to \hat B$, respectively. Since $B_1$ is unimodular, the map $\hat Y \to \hat B_1$ satisfies (2) and (3) of \cref{def:semistable}. Since $Y$ is also unimodular, $\hat Y$ and $\hat B_1$ are nonsingular.
\end{proof}



\subsection{Canonical subdivisions}

As we will see, much of our proof relies on being able to construct ``canonical'' subdivisions for polytopes. We now formalize his notion.

An \emph{ordered polytope} is a polytope along with a total order on its vertices. A \emph{face} of an ordered polytope is a face of the underlying polytope along with the induced ordering. Let $\calP$ to be the category whose objects are ordered lattice polytopes and whose morphisms are $F \to P$ where $F$ is a face of $P$. Let $\calS$ be the category whose objects are subdivisions of ordered lattice polytopes and whose morphisms are $F' \to P'$ where $F'$ is the subdivision induced on a face of the underlying ordered polytope of $P'$.

Let $\Gamma : \calF \to \calP$ be a full and faithful functor for some category $\calF$. A \emph{canonical subdivision} of $\Gamma$ is a functor $\Sigma: \calF \to \calS$ such that $\Sigma(P)$ is a subdivision of $\Gamma(P)$ for all $P \in \Ob(\calF)$. If $\Sigma(P)$ is projective for all $P$, then we call $\Sigma$ \emph{projective}. If $\Sigma(P)$ is a triangulation for all $P$, then we call $\Sigma$ a \emph{canonical triangulation}.




\subsubsection{Canonical triangulations of dilated simplices}

Let $\Delta$ be the category defined as follows. The objects are ordered pairs $(P,c)$ where $P$ is an ordered lattice simplex and $c$ is a positive integer. The morphisms are $(P',c) \to (P,c)$ where $P'$ is a face of $P$. We have a full and faithful functor $\mu : \Delta \to \calP$ defined by $\mu(P,c) = cP$. We will assume the origin is in $L_P$, so that $cP$ has vertices in $L_P$.

The following is a key result from Haase et al.\ \cite{Santos}.

\begin{theor} \label{thm:cansimp}
There is a projective canonical triangulation $\Sigma$ of $\mu$ such that for all $(P,c) \in \Ob(\Delta)$, we have that $\Sigma(P,c)$ is unimodular with respect to $L_P$.
\end{theor}

Later we will prove a generalization of this to polysimplices, \cref{lem:polysubdv}. For now, we will state a modified version of this Theorem. Let $\Delta'$ be the full subcategory of $\Delta$ whose objects are $(P,c) \in \Ob(\Delta)$ with $c \ge \dim(P) + 1$. Let $\mu'$ be the restriction of $\mu$ to $\Delta'$. Then we have the following.

\begin{lem} \label{lem:stcan}
There is a projective canonical triangulation $\Sigma'$ of $\mu'$ such that for all $(P,c) \in \Ob(\Delta')$ and all full-dimensional simplices $Q$ of $\Sigma'(P,c)$, we have the following:
\begin{enumerate}
\item $L_P = L_Q$. (That is, $\Sigma'(P,c)$ is unimodular with respect to $L_P$.)
\item If the vertices of $Q$ are ordered $v_1$, $v_2$, ~\dots, $v_{\dim(P)+1}$, then for all $i = 1$, ~\dots, $\dim(P)$, every face of $P$ which contains $v_i$ also contains $v_{i+1}$.
\end{enumerate}
\end{lem}

\begin{proof}
Note: This proof uses ideas and notation from the next section. We have put the proof in this section for the sake of organization.

We construct the triangulation $\Sigma'(P,c)$ of $cP$ as follows. For each face $F$ of $P$, let $O_F$ be the barycenter of $F$. If $F$ has dimension $k$ and $F'$ is a face of $F$, then we note that
\[
\phi(F',F) := (c-k-1)F' + (k+1)O_F
\]
is a lattice polytope contained in $cF$.

Let $d = \dim(P)$. Let $F_{r}$, $F_{r+1}$, ~\dots, $F_d$ be a sequence of nonempty faces of $P$ with $F_r < ~\dotsb < F_d$ and $\dim(F_i) = i$ for all $r \le i \le d$. We define
\[
(cP)_{F_r,\dots,F_d} := \conv \bigcup_{i=r}^d \phi(F_{r},F_i).
\]
Then the collection of all such $(cP)_{F_{r},\dots,F_d}$ are the full-dimensional cells of a subdivision $\Sigma$ of $cP$. 

The final step is to refine $\Sigma$ to a triangulation. When viewed as lattice polytopes in $L_P$, each $(cP)_{F_r,\dots,F_d}$ is lattice equivalent to the Cayley polytope
\[
\Cay( (c-r-1)F_r, (c-r-2)F_r, \dots, (c-d-1)F_r ).
\]
Thus, by \cref{lem:polysubdv}, there are canonical triangulations of each $(cP)_{F_r,\dots,F_d}$ which are unimodular with respect to $L_P$. These extend to a triangulation of $\Sigma$ which is unimodular in $L_P$. The fact that this triangulation satisfies property (2) is easy to check, as is canonicity.
\end{proof}



\subsection{Cayley polytopes}

Let $(P_1,\bfZ^d)$, $(P_2,\bfZ^d)$ ~\dots, $(P_n,\bfZ^d)$ be lattice polytopes. To simplify some statements later, we will assume that $L_{P_j}$ contains the origin for all $j$. (Note that we can always translate the $P_j$ so that this holds.) We define the \emph{Cayley polytope} $\Cay(P_1,\dots,P_n)$ to be the polytope
\[
\Cay(P_1,\dots,P_n) := \conv \left( \bigcup_{i=1}^n P_i \times e_i \right) \subset \bfR^d \times \bfR^n
\]
where $\conv$ denotes convex hull and $e_i$ is the $i$-th standard basis vector of $\bfR^n$. We make $\Cay(P_1,\dots,P_n)$ a lattice polytope by equipping it with the affine lattice $\bfZ^d \times \Lambda^{n-1}$, where $\Lambda^{n-1} := \{ x \in \bfZ^n : x_1 + \dots + x_n = 1 \} \cong \bfZ^{n-1}$.

If $P_1$, ~\dots, $P_n$ are ordered polytopes, then we make $\Cay(P_1.\dots,P_n)$ and ordered polytope with the following ordering: First the vertices of $P_1 \times e_1$ in the order given by $P_1$, then the vertices of $P_2 \times e_2$ in the order given by $P_2$, and so on. Note that these are precisely the vertices of $\Cay(P)$.

A \emph{polysimplex}, or \emph{product of simplices}, is a polytope of the form $\sum_i P_i$, where $\{P_i\}$ is an affinely independent set of simplices and $\sum$ denotes Minkowski sum. If $\{P_i\}$ is an ordered set of such simplices and each $P_i$ is an ordered polytope, then we make $\sum_i P_i$ an ordered polytope by lexicographic ordering on its vertices.

Let $A$ be an $m \times n$ matrix with nonnegative integer entries. We define
\begin{align*}
\Cay(P_1,\dots,P_n,A) &:= \Cay \left( (P_1, \dotsb, P_n)  A^T \right) \\
&:= \Cay \left( \sum_{j=1}^n A_{1j}P_j, \sum_{j=1}^n A_{2j}P_j, \dots, \sum_{j=1}^n A_{mj}P_j \right).
\end{align*}
If $P_1$, ~\dots, $P_n$ are affinely independent ordered simplices, then $\Cay(P_1,\dots,P_n,A)$ is an ordered polytope as described in the previous two paragraphs. In addition, with the assumption that $L_{P_j}$ contains the origin for all $j$, the vertices of $\Cay(P_1,\dots,P_n,A)$ are contained in $L_{\tilde{P}} \times \Lambda^{m-1}$, where $\tilde P := P_1 + \dots + P_n$.

Let $A$ be as above. For each row $A_i$ of $A$, let $\supp A_i $ denote the set of column indices at which $A_i$ is nonzero. We say that $A$ is \emph{full} if
\[
[n] = \supp A_1 \supseteq \supp A_2 \supseteq \dotsb \supseteq \supp A_m.
\]


\subsubsection{Canonical triangulations of polysimplices}

We now turn to subdividing Cayley polytopes of polysimplices. Let $\calF$ be the category whose objects are tuples $(P_1,\dots,P_n,A)$ satisfying the following.

\begin{enumerate}
\item $P_1$, ~\dots, $P_n$ are ordered lattice simplices with the same lattice.
\item $P_1$, ~\dots, $P_n$ are affinely independent.
\item $A$ is an $m \times n$ matrix with nonnegative integer entries.
\end{enumerate}
The morphisms in $\calF$ are $(F_1,\dots,F_n,A') \to (P_1,\dots,P_n,A)$, where $F_i$ is a face of $P_i$ for all $i$, and $A'$ is obtained from $A$ by taking a subset of the rows of $A$. We have a full and faithful functor $\Cay : \calF \to \calP$ given by $(P_1,\dots,P_n,A) \mapsto \Cay(P_1,\dots,P_n,A)$.

We generalize \cref{thm:cansimp} as follows.

\begin{lem} \label{lem:polysubdv}
There is a projective canonical subdivision $\Sigma$ of $\Cay$ such that for all $(P_1,\dots,P_n,A) \in \Ob(\calF)$, we have that $\Sigma(P_1,\dots,P_n,A)$ is unimodular with respect to $L_{\tilde{P}} \times \Lambda^{m-1}$, where $\tilde P := P_1 + \dots + P_n$.
\end{lem}

\begin{proof}
See \cite{ALT}.
\end{proof}

A \emph{box point} of a lattice polytope $P$ is a nonzero element of $N_P / L_P$. If $F$ is a face of $P$, then there is a natural inclusion $N_F / L_F \into N_P / L_P$, and so any box point of $F$ can be regarded as a box point of $P$. Moreover, if we have two polytopes $P$, $Q$ with $L_P = L_Q$, then we identify the box points of $P$ with the box points of $Q$.

Let $\calF^\ast$ denote the full subcategory of $\calF$ whose objects are $(P_1,\dots,P_n,A) \in \Ob(\calF)$ such that
\begin{enumerate}
\item For each entry $a$ of $A$, either $a = 0$ or $a \ge \max_j \dim P_j$.
\item $A$ is full.
\end{enumerate}
Let $\Cay^\ast$ be the restriction of $\Cay$ to $\calF^\ast$.


We now state the main result of this section.

\begin{lem} \label{lem:box}
Let $\mathbf{m}$ be a box point for some polytope. Then there is a projective canonical triangulation $\Sigma^{\mathbf{m}}$ of $\Cay^\ast$ such that for all $(P_1,\dots,P_n,A) \in \Ob(\calF^\ast)$ and all full-dimensional simplices $Q$ in $\Sigma^{\mathbf{m}}(P_1,\dots,P_n,A)$, we have the following.
\begin{itemize}
\item If $\mathbf{m}$ is a box point of $\tilde P := P_1 + \dotsb + P_n$, then
\[
{\rm{index}}(Q) < {\rm{index}}(\tilde P).
\]
\item If $\mathbf{m}$ is not a box point of $\tilde P$, then $L_Q = L_{\tilde P} \times \Lambda^{m-1}$.
\end{itemize}
\end{lem}

\begin{proof}
See \cite{ALT}.
\end{proof}


\subsection{Proof of \cref{thm:polytopes}}

We are now ready to prove \cref{thm:polytopes}. By \cref{prop:good}, we may assume $f : X \to Y$ is a good map.

\subsubsection{Reducing the base}

The first step is to alter $B$ so that it is a unimodular triangulation. By the KKMS theorem \cite{KKMS}, there is a positive integer $c$ so that we have a projective unimodular triangulation $B' \to cB$. Now, by \cref{prop:induced}, there is a positive integer $c'$ and an alteration $B' \to c'cB$ such that this alteration induces with respect to $c'cf$ an alteration $X_1 \to c'cX$ and map $X_1 \to B'$. Hence, we may assume $B$ is a unimodular triangulation. By arbitrarily triangulating $X$, we may assume $X$ is a triangulation.

\subsubsection{Lowering the index} \label{sec:lower}

Fix a linear order on the vertices of $B$. By \cref{lem:stcan}, there exists $c$ such that for each $Q \in B$ we have a projective triangulation
\[
\Sigma'(Q,c) \to cQ.
\]
Since these triangulations are canonical, this gives a triangulation $B_1$ of $cB$. By \cref{prop:induced}, for some $c'$, the alteration $B_1 \to c'B_1 \to c'cB$ induces an alteration $X_1 \to c'cX$ and a map $f_1 : X_1 \to B_1$. We may assume $c' > \dim X$.

Suppose $Q \in B$ and $P$ is a full-dimensional simplex of the complex $f^{-1}(Q)$. Let $v_1$, ~\dots, $v_n$ be the vertices of $Q$, and let
\[
P_i := (f^{-1}(v_i) \cap P, N_P)
\]
for all $i$. Note that the $P_i$ are affinely independent simplices and $P = \Cay(P_1,\dots,P_n)$.

Let $v$ be a vertex of $B_1$ contained in $cQ$. Define
\[
P(v) := f^{-1}_1(v) \cap c'cP.
\]
Let $(a_1,\dots,a_n)$ be the barycentric coordinates of $v$ with respect to the vertices $cv_1$, ~\dots, $cv_n$ of $cQ$. Since $Q$ is unimodular, $ca_1$, ~\dots, $ca_n$ are nonnegative integers. From the definition of $X_1$, we have
\begin{align*}
P(v) &= a_1 P(cv_1) + a_2 P(cv_2) + \dotsb + a_kP(cv_n) \\
&= c'c (a_1 P_1 + a_2P_2 + \dotsb + a_nP_n).
\end{align*}
Thus, if $Q_1 \in B_1$ has vertices $u_1$, ~\dots, $u_m$, we have
\begin{align*}
f_1^{-1}(Q_1) \cap c'cP &= \Cay( P(u_1), \dots, P(u_m) ) \\
&= \Cay( P_1, \dots, P_n, A)
\end{align*}
where $A$ is an $m \times n$ matrix of nonnegative integers divisible by $c'$. By \cref{lem:stcan}, $A$ is full. Hence $( P_1, \dots, P_n, A) \in \Ob(\calF^\ast)$.

Let $\mathbf{m}$ be a box point of some $P^0 \in X$. By \cref{lem:box}, we have a projective triangulation $Y \to X_1$ where each $P' \in X_1$ is triangulated into $\Sigma^{\mathbf{m}}(P')$. Hence, every simplex $Q$ of $Y$ has either $L_Q = L_P$ for some simplex $P \neq P^0$ of $X$ or ${\rm{index}}(Q) < {\rm{index}}(P)$ where $P$ is a simplex of $X$ and $\mathbf{m}$ is a box point of $P$.

Now repeat the process of \cref{sec:lower} with $Y$ instead of $X$. Each time we do this procedure, we lower the indices of some of the lattices spanned by elements of $X$ while keeping the other lattices the same. Eventually all lattices will be unimodular, completing the proof.


\sloppy
\printbibliography
\fussy


\end{document}
