Cayley decompositions of lattice polytopes and upper bounds for $h^{*}$-polynomials

This follows from [11, Theorem 1.7]. ∎

The relative interior of the cone spanned by the vertices of $S$ that are not in $Z$ lies in the interior of $\sigma$. Therefore, the sum of these vertices is in the interior of $(n+1-|Z|)P$. Since $(n-d)P$ contains no interior lattice points, it follows that $|Z|$ is less than or equal to $d$. ∎

The cone spanned by $F_{0}\cup Z^{+}(y)\cup(V\smallsetminus V^{-}(y))$ meets the relative interior of the cone spanned by $y$ together with $F_{0}\cup Z^{-}(y)\cup V^{-}(y)$, and thus meets the interior of $\sigma$, since the cone spanned by $F_{0}\cup V$ contains $x$ in its relative interior. Therefore,

$$p=\{x\}+\sum_{v_{i}\in Z^{+}(y)}v_{i}+\sum_{v_{j}\in(V\smallsetminus V^{-}(y))}v_{j}$$

is a lattice point in the interior of $\sigma$. The last coordinate of $p$ is

$$n+1-d+|Z^{+}(y)|-|V^{-}(y)|.$$

Since $(n-d)P$ contains no interior lattice points, it follows that $|V^{-}(y)|$ is less than or equal to $|Z^{+}(y)|$, as required. ∎

Suppose $y$ is in the cone spanned by vertices $w_{1},\ldots,w_{k}$ of $P$. Now, the cone spanned by $w_{1},\ldots,w_{k}$ together with $F_{0}\cup Z^{-}(y)\cup(V\smallsetminus V^{+}(y))$ meets the relative interior of the cone spanned by $F_{0}\cup Z^{+}(y)\cup V^{+}(y)$, and hence meets the interior of $\sigma$. Therefore,

$$w_{1}+\cdots+w_{k}+\{x\}+\sum_{v_{i}\in Z^{-}(y)}v_{i}+\sum_{v_{j}\in(V\smallsetminus V^{+}(y))}v_{j}$$

is a lattice point in the interior of $\sigma$, whose last coordinate is

$$n+1-d+|Z^{-}(y)|-|V^{+}(y)|+k.$$

Since $(n-d)P$ contains no interior lattice points, it follows that $|V^{+}(y)|$ is less than or equal to $|Z^{-}(y)|+k$. ∎

Let $\pi$ be the projection from $\mathbb{R}^{n+1}$ to the subspace spanned by $V^{-}(y)$, taking $a_{0}v_{0}+\cdots+a_{n}v_{n}$ to $\displaystyle\sum_{v_{i}\in V^{-}(y)}a_{i}v_{i}.$ By Carathéodory’s Theorem, for some $k\leq|V^{-}(y)|$ there are vertices $w_{1},\ldots,w_{k}$ of $P$ such that

$$\pi(y)=c_{1}\pi(w_{1})+\cdots+c_{k}\pi(w_{k}),$$

for some positive real numbers $c_{1},\ldots,c_{k}$. Let $y^{\prime}=c_{1}w_{1}+\cdots+c_{k}w_{k}$. By construction, $V^{-}(y^{\prime})$ contains $V^{-}(y)$. Furthermore

$$|V^{+}(y^{\prime})|\leq|Z^{-}(y^{\prime})|+k,$$

by Lemma 2.5. Now $k$ is less than or equal to $|V^{-}(y^{\prime})|$, which is at most $|Z^{+}(y^{\prime})|$, by Lemma 2.4. Therefore, $|V^{+}(y^{\prime})|$ is bounded above by $|Z|$, as required. ∎

For $y\in\sigma$, let $V(y)=V^{+}(y)\cup V^{-}(y)$ be the set of vertices of $V$ appearing with nonzero coefficients in the unique expression $y=b_{0}v_{0}+\cdots+b_{n}v_{n}$. We prove the proposition by choosing a “greedy” sequence of elements $y_{0},\ldots,y_{k}$ in $\sigma$ such that

$$R(j)=V^{-}(y_{j})\smallsetminus\bigcup_{i=0}^{j-1}V(y_{i})$$

is as large as possible at each step. By Lemma 2.4 and Lemma 2.6, we may choose these $y_{i}$ such that $|V^{+}(y_{i})|\leq|Z|$ and $|V^{-}(y_{i})|\leq|Z|$ for each $i$.

For $i<j$, $V^{-}(y_{i}+y_{j})$ contains both $R(j)$ and also $R(i)\smallsetminus V^{+}(y_{j})$. Since $y_{i}$ was chosen over $y_{i}+y_{j}$ at the $i$-th step, it then follows from the greedy hypothesis that

$$|R(j)|\,\leq\,|R(i)\cap V^{+}(y_{j})|.$$

Now the sets $R(i)\cap V^{+}(y_{j})$ are disjoint for fixed $j$, so $|V^{+}(y_{j})|\geq j\cdot|R(j)|$. In particular, since $|V^{+}(y_{j})|$ is at most $|Z|$, it follows that $R(j)$ is empty for $j>|Z|$. We set $z:=\max(j\,:\,R(j)\not=\emptyset)\leq|Z|$. Let

$$R^{\prime}(j)=V^{+}(y_{j})\smallsetminus\bigcup_{i=0}^{j-1}V(y_{i}).$$

Then, for any point $y\in\sigma$, $V^{-}(y)$ is contained in the union of the sets $R(j)\cup R^{\prime}(j)$, for $j=0,\ldots,z$. It remains to show

$$\sum_{j=0}^{z}(|R(j)|+|R^{\prime}(j)|)\,\leq\,(|Z|^{2}+5|Z|)/2,$$

since then we can take $F_{2}$ to be the face spanned by $F_{1}$, $Z$, and $\bigcup_{j=0}^{z}R(j)\cup R^{\prime}(j)$.

To see this inequality, note that $|R(0)|$ and $|R^{\prime}(0)|$ are each at most $|Z|$. For $0<j\leq z$, we have $|R^{\prime}(j)|\leq|V^{+}(y_{j})|-\sum_{i<j}|R(i)\cap V^{+}(y_{j})|$, which is bounded above by $|Z|-j|R(j)|$. In particular, $|R(j)|+|R^{\prime}(j)|$ is at most $|Z|-j+1$. Therefore,

$$\sum_{j=0}^{z}(|R(j)|+|R^{\prime}(j)|)\ \leq\ 2|Z|+\big{(}|Z|+(|Z|-1)+\cdots+1\big{)},$$

and the proposition follows. ∎

We choose a greedy sequence of vertices $w_{1},\ldots,w_{r}$ of $S$ that are not in $F_{2}$ such that

$$\tilde{R}(j)=V^{+}(w_{j})\smallsetminus\big{(}F_{2}\cup\bigcup_{i=1}^{j-1}V^{+}(w_{i})\big{)}$$

is as large as possible at each step. By Lemma 2.5, $|V^{+}(w_{1}+\cdots+w_{j})|\leq|Z|+j$. Since $b_{i}(w_{1}+\cdots+w_{j})$ is positive for $v_{i}\in\tilde{R}(1)\sqcup\cdots\sqcup\tilde{R}(j)$, it follows that $|\tilde{R}(j)|$ is at most one for $j$ greater than $|Z|$. In particular, we may take $F_{3}$ to be the face spanned by $F_{2}$ together with the union $\tilde{R}(1)\sqcup\cdots\sqcup\tilde{R}(|Z|)$, which has size at most $2|Z|$. ∎

Let $G$ be the set of vertices $v_{i}$ of $S$ that are not in $F_{3}$ such that $b_{i}(w)\geq 2$ for some vertex $w$ of $P$. It remains to show that $|G|\leq 2d-2|Z|$, since then we may take $F$ to be the face of $S$ spanned by $F_{3}$ and $G$, and projection along $F$ will map $P$ onto a unimodular simplex, as required.

Number the vertices of $S$ so that $G=\{v_{1},\ldots,v_{r}\}$, $|G|=r$, and choose vertices $w_{1},\ldots,w_{r}$ of $P$ such that $b_{i}(w_{i})\geq 2$. Then let

$$p=w_{1}/b_{1}(w_{1})+\cdots+w_{r}/b_{r}(w_{r}),$$

so $p$ is a rational point in $\sigma$ with $b_{i}(p)=1$ for $1\leq i\leq r$.

Let $p^{\prime}=p+\sum_{v_{i}\not\in Z}c_{i}v_{i}$, where

$$c_{i}=\left\{\begin{array}[]{ll}1&\mbox{ if }b_{i}(p)\in\mathbb{Z}_{\leq 0}\\ 1-\{b_{i}(p)\}&\mbox{ if }b_{i}(p)\in\mathbb{Q}\smallsetminus\mathbb{Z}\\ 0&\mbox{ otherwise.}\end{array}\right.$$

Then $p^{\prime}$ is a lattice point in the interior of $\sigma$, with last coordinate

$$1/b_{1}(w_{1})+\cdots+1/b_{r}(w_{r})+\sum c_{i},$$

which is at most $r/2+n+1-r-|Z|=n+1-(r/2)-|Z|$. Therefore, $(r/2)+|Z|\leq d$, and hence $r\leq 2d-2|Z|$.

Therefore, $P$ decomposes as a Cayley sum of lattice polytopes in $\mathbb{R}^{q}$ for some $q\leq 6d-2+(|Z|^{2}+7|Z|)/2$. Since $|Z|$ is at most $d$, the theorem follows. ∎

Let $P$ be a $n$-dimensional Gorenstein polytope of degree $d$ in $\mathbb{R}^{n+1}$ that is contained in the affine hyperplane $x_{n+1}=1$. Then the polar dual of the cone over $P$ is the cone over a dual Gorenstein polytope $P^{*}$, also of dimension $n$ and degree $d$. See [2] for this and other basic facts about duality for Gorenstein polytopes.

Let $x$ be the unique lattice point in the relative interior of $(n+1-d)P^{*}$. Choose a $n$-dimensional lattice simplex $S\subset P^{*}$ such that $x$ is contained in the cone over $S$, and order the vertices $v_{0},\ldots,v_{n}$ of $S$ so that

$$x=v_{0}+\cdots+v_{s}+\{x\},$$

where the fractional part can be written $\{x\}=\sum b_{i}v_{i}$ with $0\leq b_{i}<1$. By [5, Corollary 3.11], the sum of the coefficients $b_{i}$ appearing in this expression for $\{x\}$ is at most $d$. Therefore $s$ is at least $n-2d$, and hence $x$ can be written as a sum of $n-2d+2$ nonzero lattice points in the cone over $P^{*}$. Therefore, by [4, Proposition 2.3], $P$ must decompose as a Cayley sum of $n+2-2d$ lattice polytopes in $\mathbb{R}^{2d-1}$, and the theorem follows. ∎

Similar to the proof of Theorem 4.1, below, except that if $P$ is Gorenstein then $q$ is less than or equal to $N=2d-1$. ∎

Let $P$ be a $n$-dimensional lattice polytope in $\mathbb{R}^{n}$ such that $h^{*}_{P}$ has degree $d$ and leading coefficient $k$. By Theorem 1.2, $P$ decomposes as a Cayley sum $P\cong P_{0}*\cdots*P_{s}$ of lattice polytopes in $\mathbb{R}^{q}$, for some $q\leq N$. We may choose $q$ as small as possible, so $n=q+s$.

Let $\pi$ be the projection from $\mathbb{R}^{n}$ to $\mathbb{R}^{s}$ induced by the Cayley decomposition, and let $S$ be the standard unimodular simplex in $\mathbb{R}^{s}$ that is the image of $P$. Set $r=n+1-d$, so $rP$ contains exactly $k$ interior lattice points. The interior lattice points in $rP$ are exactly the interior lattice points in $\pi^{-1}(\lambda)\cap rP$, for interior lattice points $\lambda\in rS$. Say $\lambda=(\lambda_{1},\ldots,\lambda_{s})$ is an interior lattice point of $rS$ such that $\pi^{-1}(\lambda)$ contains an interior lattice point of $P$. Then $\lambda_{1},\ldots,\lambda_{s}$ and

$$\lambda_{0}=r-\lambda_{1}-\cdots-\lambda_{s}$$

are positive integers, and $\pi^{-1}(\lambda)\cap P$ is naturally identified with the Minkowski sum $\lambda_{0}P_{0}+\cdots+\lambda_{s}P_{s}$.

Let $\omega_{ij}$ be the width of $P_{j}$ with respect to the $i$-th coordinate on $\mathbb{R}^{q}$, the difference between the maximum and the minimum of the $i$-th coordinates of points in $P_{j}$. By [9, Theorem 2] there is a choice of coordinates on $\mathbb{R}^{q}$ such that $\lambda_{0}P_{0}+\cdots+\lambda_{s}P_{s}$ is contained in the standard cube $[0,C]^{q}$ with the side length $C$ being equal to $q$ times the normalized volume of $\lambda_{0}P_{0}+\cdots+\lambda_{s}P_{s}$. By the bounds given in [9, Theorem 1], respectively in [12], we may choose

$$C=q\,q!\,\min\left(k\,\big{(}7(k+1)\big{)}^{q\cdot 2^{\scriptstyle q+1}},\;k\,\big{(}8q\big{)}^{q}\,15^{q\cdot 2^{\scriptstyle 2q+1}}\right).$$

Since widths are additive and each $\lambda_{j}$ is a positive integer, it follows that $\omega_{i0}+\cdots+\omega_{is}\leq C$, for $1\leq i\leq q$.

Now $P$ projects onto $S$, so we can express the normalized volume of $P$ as an integral

$$\operatorname{Vol}(P)=n!\cdot\int_{S}\operatorname{vol}\big{(}\pi^{-1}(\lambda)\cap P\big{)}\,d\lambda,$$

where $\operatorname{vol}$ is the ordinary euclidean volume. The volume of $\pi^{-1}(\lambda)\cap P$ is bounded by the product of its coordinate widths, which is $\prod_{i=1}^{s}(\lambda_{0}\omega_{i0}+\cdots+\lambda_{s}\omega_{is})$. Expanding the product and substituting into the integral above gives

(1)

$$\operatorname{Vol}(P)\ \leq n!\cdot\sum_{j_{1},\ldots,j_{q}}\bigg{(}\omega_{1j_{1}}\cdots\omega_{qj_{q}}\int_{S}\lambda_{j_{1}}\cdots\lambda_{j_{q}}\,d\lambda\bigg{)},$$

where the sum is over $(j_{1},\ldots,j_{q})\in\{0,\ldots,s\}^{q}$. Now it follows from Hölder’s inequality that the integral over $S$ of the monomial $\lambda_{j_{1}}\cdots\lambda_{j_{q}}$ is bounded above by the integral of $\lambda_{1}^{q}$, and a straightforward induction shows that

$$\int_{S}\lambda_{1}^{q}\,d\lambda=q!/n!.$$

Substituting into (1) then gives

$$\operatorname{Vol}(P)\ \leq\ q!\cdot\sum_{j_{1},\ldots,j_{q}}(\omega_{1j_{1}}\cdots\omega_{qj_{q}}).$$

The sum on the right hand side may be written as $\prod_{i=1}^{q}(\omega_{i0}+\cdots+\omega_{is})$, which is bounded above by $C^{q}$ since, for each $i$, $\omega_{i0}+\cdots+\omega_{is}$ is less than or equal to $C$. We conclude that $\operatorname{Vol}(P)$ is bounded above by $q!\cdot C^{q}$. Now the theorem follows, since $q\leq N$. ∎

Suppose $(X,L)$ is a polarized toric variety and $K_{X}+(n-d)L$ has no nonzero sections. Then $L$ corresponds to an $n$-dimensional lattice polytope $P$ such that $h^{*}_{P}$ has degree at most $d$. By Theorem 1.2, $P$ has a Cayley decomposition

$$P\cong P_{0}*\cdots*P_{s},$$

for some lattice polytopes $P_{0},\ldots,P_{s}$ in $\mathbb{R}^{q}$, with $q\leq(d^{2}+19d-4)/2$. Let $Y$ be the toric variety associated to the Minkowski sum $P_{0}+\cdots+P_{s}$, and let $L_{i}$ be the line bundle on $Y$ corresponding to $P_{i}$. Then $P$ is the polytope associated to $\mathcal{O}(1)$ on the toric variety

$$X^{\prime}\cong\mathbb{P}_{Y}(L_{0}\oplus\cdots\oplus L_{s}).$$

It follows that there is a proper birational toric morphism $\pi:X^{\prime}\rightarrow X$ with $\pi^{*}L\cong\mathcal{O}(1)$, as required. ∎