Nef-partitions arising from unimodular configurations

TL;DR

This paper introduces a new method for constructing nef-partitions of reflexive polytopes using Gröbner basis techniques, expanding the class of known nef-partitions from unimodular configurations.

Contribution

It provides a large family of nef-partitions derived from unimodular configurations through Gröbner basis methods, enhancing combinatorial and algebraic tools for mirror symmetry.

Findings

Constructed numerous nef-partitions from unimodular configurations.

Applied Gröbner basis techniques to combinatorial geometry.

Extended the class of nef-partitions available for mirror symmetry studies.

Abstract

Reflexive polytopes have been studied from viewpoints of combinatorics, commutative algebra and algebraic geometry. A nef-partition of a reflexive polytope $\mathcal{P}$ is a decomposition $\mathcal{P}=\mathcal{P}_1+\cdots+\mathcal{P}_r$ such that each $\mathcal{P}_i$ is a lattice polytope containing the origin. Batyrev and van Straten gave a combinatorial method for explicit constructions of mirror pairs of Calabi-Yau complete intersections obtained from nef-partitions. In the present paper, by means of Gr\"{o}bner basis techniques, we give a large family of nef-partitions arising from unimodular configurations.

Equations

$${\mathcal{P}}^{\vee }=\{\mathbf{x}\in {\mathbb{R}}^d:⟨\mathbf{x},\mathbf{y}⟩\le 1,\forall \mathbf{y}\in \mathcal{P}\}$$

$${\mathcal{P}}_1\ast \cdots \ast {\mathcal{P}}_r:=\mathrm{conv}(\{\{{\mathbf{e}}_1\}\times {\mathcal{P}}_1,\dots ,\{{\mathbf{e}}_{r-1}\}\times {\mathcal{P}}_{r-1},\{0\}\times {\mathcal{P}}_r\})\subset {\mathbb{R}}^{r-1}\times {\mathbb{R}}^d,$$

$$({\mathcal{P}}_1\cap {\mathbb{Z}}^d)+\cdots +({\mathcal{P}}_r\cap {\mathbb{Z}}^d)=({\mathcal{P}}_1+\cdots +{\mathcal{P}}_r)\cap {\mathbb{Z}}^d.$$

$$A_{{\mathcal{P}}}:=\left(\begin{array}[]{ccc}{\mathbf{a}}_{1}&\cdots&{\mathbf{a}}_{n}\\ 1&\cdots&1\end{array}\right)\in{\mathbb{Z}}^{(d+1)\times n}.$$

$$A^{\pm}:=\left(\begin{array}[]{ccccccc}{\mathbf{a}}_{1}&\cdots&{\mathbf{a}}_{n}&-{\mathbf{a}}_{1}&\cdots&-{\mathbf{a}}_{n}&{\bf 0}\\ 1&\cdots&1&1&\cdots&1&1\end{array}\right)\in{\mathbb{Z}}^{(d+1)\times(2n+1)}.$$

$${\phi }_{A^{\pm }}:K[x_1,\dots ,x_n,y_1,\dots ,y_n,z]\to K[t_1^{\pm 1},\dots ,t_d^{\pm 1},s]$$

$$\{x_i{y}_i-z^2:i\in [n]\}\cup \{g_1,\dots ,g_s\},$$

$$g=p-q=x_1^{u_1}\dots x_n^{u_n}{y}_1^{v_1}\dots y_n^{v_n}-z^{\ell }{x}_1^{u_1^{\prime }}\dots x_n^{u_n^{\prime }}{y}_1^{v_1^{\prime }}\dots y_n^{v_n^{\prime }}$$

$$\sum _{k=1}^n{u}_k{\mathbf{a}}_k+\sum _{k=1}^n{v}_k(-{\mathbf{a}}_k)=\sum _{k=1}^n{u}_k^{\prime }{\mathbf{a}}_k+\sum _{k=1}^n{v}_k^{\prime }(-{\mathbf{a}}_k)$$

$$\sum _{k=1}^n{u}_k-\sum _{k=1}^n{v}_k=\sum _{k=1}^n{u}_k^{\prime }-\sum _{k=1}^n{v}_k^{\prime }.$$

$$\mathrm{supp}({\mathbf{x}}^{{\mathbf{w}}_1})\subset \mathrm{supp}({\mathbf{x}}^{{\mu }_1}),\mathrm{supp}({\mathbf{x}}^{{\mathbf{w}}_1^{\prime }})\subset \mathrm{supp}({\mathbf{x}}^{{\eta }_2}),\mathrm{supp}({\mathbf{x}}^{{\mathbf{w}}_2})\subset \mathrm{supp}({\mathbf{x}}^{{\mu }_2}),\mathrm{supp}({\mathbf{x}}^{{\mathbf{w}}_2^{\prime }})\subset \mathrm{supp}({\mathbf{x}}^{{\eta }_1}).$$

$${\phi }_{\ast }:K[x_1,\dots ,x_n,y_1,\dots ,y_n]\to K[t_0^{\pm 1},t_1^{\pm 1},\dots ,t_d^{\pm 1},s]$$

$${\mathcal{G}}^{\prime }=\{x_i{y}_i-x_1{y}_1:i=2,\dots ,n\}\cup \{g_1,\dots ,g_s\},$$

$${\phi }_{\ast }:K[x_0,x_1,\dots ,x_n,y_0,y_1,\dots ,y_n]\to K[t_0^{\pm 1},t_1^{\pm 1},\dots ,t_d^{\pm 1},s]$$

$${\mathcal{G}}^{\prime }=\{x_i{y}_i-x_0{y}_0:i\in [n]\}\cup \{g_1,\dots ,g_s\},$$

$$f=x_0^{u_0}{x}_1^{u_1}\dots x_n^{u_n}{y}_0^{v_0}{y}_1^{v_1}\dots y_n^{v_n}-x_0^{u_0^{\prime }}{x}_1^{u_1^{\prime }}\dots x_n^{u_n^{\prime }}{y}_0^{v_0^{\prime }}{y}_1^{v_1^{\prime }}\dots y_n^{v_n^{\prime }}.$$

$$\sum _{k=1}^n{u}_k{\mathbf{a}}_k+\sum _{k=1}^n{v}_k(-{\mathbf{a}}_k)=\sum _{k=1}^n{u}_k^{\prime }{\mathbf{a}}_k+\sum _{k=1}^n{v}_k^{\prime }(-{\mathbf{a}}_k),$$

$$\sum _{k=1}^n{u}_k-\sum _{k=1}^n{v}_k=\sum _{k=1}^n{u}_k^{\prime }-\sum _{k=1}^n{v}_k^{\prime }.$$

$$f=(x_0{y}_0{)}^{u_0}{x}_1^{u_1}\dots x_n^{u_n}{y}_1^{v_1}\dots y_n^{v_n}-x_1^{u_1^{\prime }}\dots x_n^{u_n^{\prime }}{y}_1^{v_1^{\prime }}\dots y_n^{v_n^{\prime }}.$$

$$f^{\prime }=z^{2u_0}{x}_1^{u_1}\dots x_n^{u_n}{y}_1^{v_1}\dots y_n^{v_n}-x_1^{u_1^{\prime }}\dots x_n^{u_n^{\prime }}{y}_1^{v_1^{\prime }}\dots y_n^{v_n^{\prime }}$$

$$L_{\mathcal{P}}(m)=|m\mathcal{P}\cap {\mathbb{Z}}^d|,$$

$${\text{Ehr}}_{\mathcal{P}}(t)=1+\sum _{k=1}^{\infty }{L}_{\mathcal{P}}(k)t^k$$

$${\text{Ehr}}_{\mathcal{P}}(t)=\frac{h^{\ast }(\mathcal{P},t)}{(1-t{)}^{d+1}}.$$

$$K[x_1,\dots ,x_n,y_1,\dots ,y_n,z]/{\mathrm{in}}_{<}(I_{A^{\pm }})$$

$$K[x_0,x_1,\dots ,x_n,y_0,y_1,\dots ,y_n]/{\mathrm{in}}_{{<}^{\prime }}(I_{{\mathcal{P}}_{A_0}\ast (-{\mathcal{P}}_{A_0})})$$

$$h(K[A^{\pm }],t)$$

$$=$$

$$K[x_1,\dots ,x_n,y_1,\dots ,y_n,z]/{\mathrm{in}}_{<}(I_{A^{\pm }})$$

$$K[x_1,\dots ,x_n,y_1,\dots ,y_n]/{\mathrm{in}}_{{<}^{\prime }}(I_{{\mathcal{P}}_A\ast (-{\mathcal{P}}_A)})$$

$$h(K[x_1,\dots ,x_n,y_1,\dots ,y_n,z]/{\mathrm{in}}_{<}(I_{A^{\pm }}),t)$$

Full text

Introduction

A lattice polytope is a convex polytope all of whose vertices have integer coordinates. A lattice polytope ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ of dimension $d$ is called reflexive if the origin ${\bf 0}$ of ${\mathbb{R}}^{d}$ belongs to the interior of ${\mathcal{P}}$ and if the dual polytope

[TABLE]

is again a lattice polytope. Here $\langle{\bf x},{\bf y}\rangle$ is the canonical inner product of ${\mathbb{R}}^{d}$. It is known that reflexive polytopes correspond to Gorenstein toric Fano varieties, and they are related to mirror symmetry (see, e.g., [1, 7]). Combinatorial aspects of mirror symmetry motivate the following definition: a nef-partition of length $r$ is a decomposition ${\mathcal{P}}={\mathcal{P}}_{1}+\cdots+{\mathcal{P}}_{r}$ of a $d$-dimensional reflexive polytope ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ into a Minkowski sum of $r$ lattice polytopes ${\mathcal{P}}_{1},\ldots,{\mathcal{P}}_{r}$ such that each ${\bf 0}\in{\mathcal{P}}_{i}$. Nef-partitions give many explicit constructions of mirrors of Calabi-Yau complete intersections ([3, 4, 5]).

In the present paper, making use of algebraic techniques involving Gröbner bases, we give a large family of nef-partitions arising from unimodular configurations. Given positive integers $d$ and $n$, let ${\mathbb{Z}}^{d\times n}$ denote the set of all $d\times n$ integer matrices. A configuration of ${\mathbb{R}}^{d}$ is a matrix $A\in{\mathbb{Z}}^{d\times n}$, for which there exists an affine hyperplane ${\mathcal{H}}\subset{\mathbb{R}}^{d}$ not passing ${\bf 0}$ such that each column vector of $A$ lies ${\mathcal{H}}$. An integer matrix $A\in{\mathbb{Z}}^{d\times n}$ of rank $d$ is called unimodular if all nonzero maximal minors of $A$ have the same absolute value. Unimodular matrices are important in polyhedral combinatorics and combinatorial optimization [18]. Given an integer matrix $A=({\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n})\in{\mathbb{Z}}^{d\times n}$, let ${\mathcal{P}}_{A}:=\operatorname{conv}(\{{\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n}\})$. A famous example of a unimodular matrix is the incidence matrix $A_{G}$ of a bipartite graph $G$ (by deleting redundant rows) and ${\mathcal{P}}_{A_{G}}$ is called the edge polytope of $G$ (see Section 3).

In order to give a family of nef-partitions, we consider whether the Cayley sum of given lattice polytopes is Gorenstein. A lattice polytope ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ is called Gorenstein of index $r$ if $r{\mathcal{P}}$ is unimodularly equivalent to a reflexive polytope. In particular, a reflexive polytope is Gorenstein of index $1$. Gorenstein polytopes are of interest in combinatorial commutative algebra, mirror symmetry, and tropical geometry (we refer to [2, 3, 14]). Given $r$ lattice polytopes ${\mathcal{P}}_{1},\ldots,{\mathcal{P}}_{r}\subset{\mathbb{R}}^{d}$, the Cayley sum of ${\mathcal{P}}_{1},\ldots,{\mathcal{P}}_{r}$ is the lattice polytope

[TABLE]

where ${\mathbf{e}}_{1},\ldots,{\mathbf{e}}_{r-1}$ are the standard basis of ${\mathbb{R}}^{r-1}$. Then it is known that ${\mathcal{P}}_{1}+\cdots+{\mathcal{P}}_{r}$ is Gorenstein of index $1$ if (and only if) ${\mathcal{P}}_{1}*\cdots*{\mathcal{P}}_{r}$ is Gorenstein of index $r$ ([3, Theorem 2.6]). On the other hand, our interest is in whether ${\mathcal{P}}_{1}+\cdots+{\mathcal{P}}_{r}$ possesses a regular unimodular triangulation. A unimodular simplex is a lattice simplex which is unimodularly equivalent to the standard simplex $\Delta^{d}\subset{\mathbb{R}}^{d}$, the convex hull of ${\bf 0}$ together with ${\mathbf{e}}_{1},\ldots,{\mathbf{e}}_{d}$. Equivalently, a full-dimensional lattice simplex in ${\mathbb{R}}^{d}$ is unimodular if and only if it has the minimal possible Euclidean volume, $1/d!$. A triangulation of a lattice polytope is called unimodular if every maximal simplex is unimodular. A full-dimensional lattice polytope ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ is called spanning if it holds $\sum_{{\mathbf{a}}\in{\mathcal{P}}\cap{\mathbb{Z}}^{d}}{\mathbb{Z}}({\mathbf{a}},1)={\mathbb{Z}}^{d+1}$. In general, we say that a lattice polytope is spanning if it is unimodularly equivalent to a full-dimensional spanning polytope. Sturmfels gave a one-to-one correspondence between a regular triangulation of a lattice polytope and the radical of an initial ideal of its toric ideal ([20, Section 8]). In particular, for a spanning polytope, its regular triangulation is unimodular if and only if the associated initial ideal is squarefree. Moreover, it follows that if ${\mathcal{P}}_{1}*\cdots*{\mathcal{P}}_{r}$ possesses a regular unimodular triangulation for spanning polytopes ${\mathcal{P}}_{1},\ldots,{\mathcal{P}}_{r}\subset{\mathbb{R}}^{d}$ of dimension $d$, then so does ${\mathcal{P}}_{1}+\cdots+{\mathcal{P}}_{r}$ ([13, Theorem 2.2]). In particular, from [21, Theorem 0.4] one has

[TABLE]

In [15], Oda asked when the equation (1) holds. Recently, this question and the following conjecture are among the current trends of the research on lattice polytopes.

Oda Conjecture. Every smooth polytope has the integer decomposition property.

For a lattice polytope ${\mathcal{P}}$, we let $-{\mathcal{P}}:=\{-{\mathbf{a}}:{\mathbf{a}}\in{\mathcal{P}}\}$. The main results of the present paper are the following two theorems.

Theorem 0.1**.**

Let $A=({\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n})\in{\mathbb{Z}}^{d\times n}$ be a unimodular configuration. Assume that ${\mathcal{P}}_{A}\cap{\mathbb{Z}}^{d}=\{{\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n}\}$ and ${\mathcal{P}}_{A}$ is spanning. Then

1. (1)

${\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})$* is Gorenstein of index $2$ with a regular unimodular triangulation*;** 2. (2)

${\mathcal{P}}_{A}+(-{\mathcal{P}}_{A})$* is reflexive with a regular unimodular triangulation. In particular, $({\mathcal{P}}_{A}-{\mathbf{a}})+(-{\mathcal{P}}_{A}+{\mathbf{a}})$ is a nef-partition, where ${\mathbf{a}}$ is an arbitrary lattice point in ${\mathcal{P}}_{A}$;* 3. (3)

One has $({\mathcal{P}}_{A}\cap{\mathbb{Z}}^{d})+(-{\mathcal{P}}_{A}\cap{\mathbb{Z}}^{d})=({\mathcal{P}}_{A}+(-{\mathcal{P}}_{A}))\cap{\mathbb{Z}}^{d}$.

Theorem 0.2**.**

Let $A=({\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n})\in{\mathbb{Z}}^{d\times n}$ be a unimodular configuration and set $A_{0}=(A,{\bf 0})\in{\mathbb{Z}}^{d\times(n+1)}$. Assume that ${\mathcal{P}}_{A_{0}}\cap{\mathbb{Z}}^{d}=\{{\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n},{\bf 0}\}$ and ${\mathcal{P}}_{A_{0}}$ is spanning. Then

1. (1)

${\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})$* is Gorenstein of index $2$ with a regular unimodular triangulation*;** 2. (2)

${\mathcal{P}}_{A_{0}}+(-{\mathcal{P}}_{A_{0}})$* is reflexive with a regular unimodular triangulation. In particular, ${\mathcal{P}}_{A_{0}}+(-{\mathcal{P}}_{A_{0}})$ is a nef-partition*;** 3. (3)

One has $({\mathcal{P}}_{A_{0}}\cap{\mathbb{Z}}^{d})+(-{\mathcal{P}}_{A_{0}}\cap{\mathbb{Z}}^{d})=({\mathcal{P}}_{A_{0}}+(-{\mathcal{P}}_{A_{0}}))\cap{\mathbb{Z}}^{d}$.

Remark 0.3**.**

In general, for a lattice polytope ${\mathcal{P}}\subset{\mathbb{R}}^{d}$, if ${\mathcal{P}}+(-{\mathcal{P}})$ is reflexive, then $({\mathcal{P}}-{\mathbf{a}})+(-{\mathcal{P}}+{\mathbf{a}})$ is a nef-partition, where ${\mathbf{a}}$ is an arbitrary lattice point in ${\mathcal{P}}$.

In Section 3, we will apply these theorems to unimodular configurations arising from finite simple graphs. In fact, we will show that for any finite simple graph $G$ all pairs of whose odd cycles have a common vertex, $({\mathcal{P}}_{A_{G}}-{\mathbf{a}})+(-{\mathcal{P}}_{A_{G}}+{\mathbf{a}})$ is a nef-partition, where ${\mathbf{a}}$ is an arbitrary lattice point in ${\mathcal{P}}_{A_{G}}$ (Theorem 3.3). Note that for such a graph, the edge polytope is unimodular, i.e., all triangulations are unimodular. Moreover, we will show that for any finite bipartite graphs $G$, ${\mathcal{P}}_{(A_{G})_{0}}+(-{\mathcal{P}}_{(A_{G})_{0}})$ is a nef-partition (Theorem 3.6).

The present paper is organized as follows: In Section 1, we will investigate three types of toric ideals arising from unimodular configurations. In particular, they possess squarefree initial ideals with respect to some reverse lexicographic orders (Theorems 1.2, 1.3 and 1.4). In Section 2, we compare $h^{*}$-polynomials with $h$-polynomials for the ideals and initial ideals considered in Theorems 1.2, 1.3 and 1.4, thus proving Theorems 0.1 and 0.2. Finally, in Section 3, we apply these theorems to unimodular configurations arising from finite simple graphs.

1. Reverse lexicographic Gröbner bases of unimodular configurations

Let $A=({\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n})\in{\mathbb{Z}}^{d\times n}$ be a configuration. From now on, we always assume that $A$ has no repeated columns. Let $K[{\mathbf{t}}^{\pm}]=K[t_{1}^{\pm 1},\dots,t_{d}^{\pm 1}]$ be a Laurent polynomial ring over a field $K$. Given an integer vector $\alpha=(\alpha_{1},\dots,\alpha_{d})\in{\mathbb{Z}}^{d}$, let ${\mathbf{t}}^{\alpha}=t_{1}^{\alpha_{1}}\dots t_{d}^{\alpha_{d}}$ be a Laurent monomial in $K[{\mathbf{t}}^{\pm}]$. The toric ring $K[A]\subset K[{\mathbf{t}}^{\pm}]$ of $A$ is a semigroup ring generated by ${\mathbf{t}}^{{\mathbf{a}}_{1}},\dots,{\mathbf{t}}^{{\mathbf{a}}_{n}}$ over $K$. Let $K[{\mathbf{x}}]=K[x_{1},\dots,x_{n}]$ be a polynomial ring over $K$ with each ${\rm deg}(x_{i})=1$. Then the toric ideal $I_{A}$ of $A$ is the kernel of a surjective ring homomorphism $\varphi_{A}:K[{\mathbf{x}}]\rightarrow K[A]$ defined by $\varphi_{A}(x_{i})={\mathbf{t}}^{{\mathbf{a}}_{i}}$ for each $1\leq i\leq n$. It is known that $I_{A}$ is generated by homogeneous binomials (of degree $\geq 2$) if $I_{A}\neq\{0\}$. In addition, any reduced Gröbner basis of $I_{A}$ consists of homogeneous binomials. See [11, Section 3] for the introduction to toric rings and ideals. For a lattice polytope ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ with ${\mathcal{P}}\cap{\mathbb{Z}}^{d}=\{{\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n}\}$, we define a configuration

[TABLE]

Then the toric ring $K[{\mathcal{P}}]$ of ${\mathcal{P}}$ is $K[A_{{\mathcal{P}}}]$ and the toric ideal $I_{{\mathcal{P}}}$ of ${\mathcal{P}}$ is $I_{A_{{\mathcal{P}}}}$.

Given an integer matrix $A=({\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n})\in{\mathbb{Z}}^{d\times n}$, let $A^{\pm}$ be a configuration

[TABLE]

The configuration $A^{\pm}$ is called the centrally symmetric configuration of $A$. The toric ideal $I_{A^{\pm}}$ of a configuration $A^{\pm}$ is the kernel of a ring homomorphism

[TABLE]

defined by $\varphi_{A^{\pm}}(x_{i})={\mathbf{t}}^{{\mathbf{a}}_{i}}s$, $\varphi_{A^{\pm}}(y_{i})={\mathbf{t}}^{-{\mathbf{a}}_{i}}s$ for each $1\leq i\leq n$, and $\varphi_{A^{\pm}}(z)=s$. The following proposition is given in [17, Theorems 2.7, 2.15 and Corollary 2.8].

Proposition 1.1**.**

Let $A\in{\mathbb{Z}}^{d\times n}$ be a unimodular matrix. Then we have the following:**

• (a)

The initial ideal of $I_{A^{\pm}}$ is squarefree with respect to a reverse lexicographic order such that the smallest variable is $z$;

• (b)

The toric ring $K[A^{\pm}]$ is normal and Gorenstein. In particular the $h$-polynomial $h(K[A^{\pm}],t)$ of $K[A^{\pm}]$ is a palindromic polynomial of degree $d$, i.e., $h(K[A^{\pm}],t)=t^{d}h(K[A^{\pm}],t^{-1})$.

In [17], Proposition 1.1 (a) is shown by studying the corresponding triangulation of ${\mathcal{P}}_{A^{\pm}}$, and Proposition 1.1 (b) is shown by using Proposition 1.1 (a). We now study the reverse lexicographic Gröbner bases of $I_{A^{\pm}}$ in detail when $A$ is a unimodular configuration.

Theorem 1.2**.**

Let $A\in{\mathbb{Z}}^{d\times n}$ be a unimodular configuration. Let $<$ be the reverse lexicographic order induced by the ordering of variables $z<x_{1}<y_{1}<\cdots<x_{n}<y_{n}$. Then the reduced Gröbner basis ${\mathcal{G}}$ of $I_{A^{\pm}}$ with respect to $<$ is of the form

[TABLE]

where $[n]:=\{1,\ldots,n\}$, $g_{i}\in K[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]$, ${\rm in}_{<}(g_{i})\in K[x_{2},\dots,x_{n},y_{2},\dots,y_{n}]$, and any monomial of $g_{i}$ is squarefree.

Proof.

Let ${\mathcal{G}}$ be the reduced Gröbner basis and let ${\mathcal{G}}_{1}=\{x_{i}y_{i}-z^{2}:i\in[n]\}$. It is easy to see that ${\mathcal{G}}_{1}$ is a subset of ${\mathcal{G}}$. Let

[TABLE]

be a binomial in ${\mathcal{G}}\setminus{\mathcal{G}}_{1}$ with ${\rm in}_{<}(g)=p$. Since $g$ belongs to the reduced Gröbner basis, (i) $g$ must be irreducible, (ii) $p$ belongs to the minimal set of monomial generators of ${\rm in}_{<}(I_{A^{\pm}})$, and (iii) $q\notin{\rm in}_{<}(I_{A^{\pm}})$. By Proposition 1.1 (a), $p$ is squarefree, that is, $u_{i},v_{i}\in\{0,1\}$. Moreover, if $u_{i}=v_{i}=1$, then $p$ is divisible by the initial monomial $x_{i}y_{i}$ of $x_{i}y_{i}-z^{2}$, a contradiction to the assumption that ${\mathcal{G}}$ is the reduced Gröbner basis. Hence $(u_{i},v_{i})\in\{(0,0),(0,1),(1,0)\}$ for each $i$. Since $g$ belongs to $I_{A^{\pm}}$, we have

[TABLE]

and $\sum_{k=1}^{n}u_{k}+\sum_{k=1}^{n}v_{k}=\ell+\sum_{k=1}^{n}u_{k}^{\prime}+\sum_{k=1}^{n}v_{k}^{\prime}$. Moreover since $A$ is a configuration, there exists a vector ${\mathbf{c}}\in{\mathbb{R}}^{d}$ such that the inner product $\left<{\mathbf{a}}_{i},{\mathbf{c}}\right>=1$ for all $1\leq i\leq n$. Taking the inner product of equation (3) with ${\mathbf{c}}$, we have

[TABLE]

Thus $\ell=2(\sum_{k=1}^{n}u_{k}-\sum_{k=1}^{n}u_{k}^{\prime})$, and hence $\ell$ is even.

Suppose that $\ell>0$. Since $\ell$ is even, we have $\ell\geq 2$. Let $k=\min\{i\in[n]:u_{i}\mbox{ or }v_{i}\mbox{ is }1\}$. If $u_{k}=1$, then $h=p/x_{k}-qy_{k}/z^{2}$ belongs to $I_{A^{\pm}}$. Since $v_{k}=0$, we have $h\neq 0$. Thus ${\rm in}_{<}(h)=p/x_{k}$ divides $p$, a contradiction. If $v_{k}=1$, then $h^{\prime}=p/y_{k}-qx_{k}/z^{2}$ belongs to $I_{A^{\pm}}$. Since $u_{k}=0$, we have $h^{\prime}\neq 0$. Thus ${\rm in}_{<}(h^{\prime})=p/y_{k}$ divides $p$, a contradiction. Therefore $\ell=0$, and hence $g\in K[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]$.

Since $g$ is irreducible and $<$ is a reverse lexicographic order, $x_{1}$ does not appear in $p$. Suppose that $y_{1}$ appears in $p$. Since $<$ is a reverse lexicographic order, $q$ is divisible by $x_{1}$. Then $h=pz^{2}/y_{1}-qx_{1}$ belongs to $I_{A^{\pm}}$. The initial monomial of $h$ is $qx_{1}$ that is not squarefree. Since the initial ideal is squarefree, $q$ belongs to the initial ideal. This contradicts that $g$ belongs to the reduced Gröbner basis. Thus $p\in K[x_{2},\dots,x_{n},y_{2},\dots,y_{n}]$.

Finally, suppose that $q$ is not squarefree. Let $g={\mathbf{x}}^{{\mu}_{1}}{\mathbf{y}}^{\eta_{1}}-{\mathbf{x}}^{\mu_{2}}{\mathbf{y}}^{\eta_{2}}$. By equation (3), ${\mathbf{x}}^{\mu_{1}+\eta_{2}}-{\mathbf{x}}^{\mu_{2}+\eta_{1}}$ belongs to the toric ideal $I_{A}$ and ${\mathbf{x}}^{\mu_{1}+\eta_{2}}-{\mathbf{x}}^{\mu_{2}+\eta_{1}}$ has a monomial that is not squarefree. An irreducible binomial $h$ of $I_{A}$ is called a circuit of $I_{A}$ if there is no binomial $0\neq h^{\prime}\in I_{A}$ such that ${\rm supp}(h^{\prime})\subsetneq{\rm supp}(h)$. Here ${\rm supp}(h)$ is the set of all variables appearing in $h$. By [11, Lemma 4.32] there exists a circuit $f={\mathbf{x}}^{{\mathbf{w}}_{1}+{\mathbf{w}}_{1}^{\prime}}-{\mathbf{x}}^{{\mathbf{w}}_{2}+{\mathbf{w}}_{2}^{\prime}}\in I_{A}$ such that

[TABLE]

Since $A$ is unimodular, by [11, Theorem 4.35], both ${\mathbf{x}}^{{\mathbf{w}}_{1}+{\mathbf{w}}_{1}^{\prime}}$ and ${\mathbf{x}}^{{\mathbf{w}}_{2}+{\mathbf{w}}_{2}^{\prime}}$ are squarefree. Hence we have $\deg(f)<\deg(g)$. Note that $f^{\prime}={\mathbf{x}}^{{\mathbf{w}}_{1}}{\mathbf{y}}^{{\mathbf{w}}_{2}^{\prime}}z^{\alpha}-{\mathbf{x}}^{{\mathbf{w}}_{2}}{\mathbf{y}}^{{\mathbf{w}}_{1}^{\prime}}z^{\beta}$ belongs to $I_{A^{\pm}}$ for some $\alpha,\beta\in{\mathbb{Z}}_{\geq 0}$ with $\alpha\beta=0$. If $\alpha>0$, then ${\mathbf{x}}^{{\mathbf{w}}_{2}}{\mathbf{y}}^{{\mathbf{w}}_{1}^{\prime}}$ belongs to the initial ideal, and hence so does $q$. This contradicts that $g$ belongs to the reduced Gröbner basis. If $\beta>0$, then ${\mathbf{x}}^{\mu_{1}}{\mathbf{y}}^{\eta_{1}}$ is divisible by ${\rm in}_{<}(f^{\prime})={\mathbf{x}}^{{\mathbf{w}}_{1}}{\mathbf{y}}^{{\mathbf{w}}_{2}^{\prime}}$ and hence ${\mathbf{x}}^{\mu_{1}}{\mathbf{y}}^{\eta_{1}}={\rm in}_{<}(f^{\prime})$. Then $q$ is the initial monomial of a binomial $f^{\prime}-g\in I_{A^{\pm}}$, a contradiction. If $\alpha=\beta=0$, then $\deg(f^{\prime})<\deg(g)$ and either $p$ or $q$ is divisible by ${\rm in}_{<}(f^{\prime})$, a contradiction. Thus it follows that $q$ is squarefree. ∎

Let $A=({\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n})\in{\mathbb{Z}}^{d\times n}$ be a unimodular configuration. Assume that ${\mathcal{P}}_{A}\cap{\mathbb{Z}}^{d}=\{{\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n}\}$. The toric ideal $I_{{\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})}$ of ${\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})$ is the kernel of a ring homomorphism

[TABLE]

defined by $\varphi_{*}(x_{i})=t_{0}{\mathbf{t}}^{{\mathbf{a}}_{i}}s$, $\varphi_{*}(y_{i})={\mathbf{t}}^{-{\mathbf{a}}_{i}}s$ for each $1\leq i\leq n$.

Theorem 1.3**.**

Work with the same notation as above. Let $<^{\prime}$ be the reverse lexicographic order induced by the ordering of variables $x_{1}<^{\prime}y_{1}<^{\prime}\cdots<^{\prime}x_{n}<^{\prime}y_{n}$. Then the reduced Gröbner basis of $I_{{\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})}$ with respect to $<^{\prime}$ is

[TABLE]

where $\{g_{1},\dots,g_{s}\}$ is such that $\{x_{i}y_{i}-z^{2}:i\in[n]\}\cup\{g_{1},\dots,g_{s}\}$ is the reduced Gröbner basis of $I_{A^{\pm}}$ from Theorem 1.2.

Proof.

It is easy to see that $x_{i}y_{i}-x_{1}y_{1}$ belongs to $I_{{\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})}$. In the proof of Theorem 1.2, it is proved that each $g_{i}$ is of the form (2) with $0=\ell=2(\sum_{k=1}^{n}u_{k}-\sum_{k=1}^{n}u_{k}^{\prime})$. Thus $\sum_{k=1}^{n}u_{k}=\sum_{k=1}^{n}u_{k}^{\prime}$. Hence each $g_{i}$ belongs to $I_{{\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})}$ and ${\mathcal{G}}^{\prime}$ is a subset of $I_{{\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})}$. The initial monomial of each binomial in ${\mathcal{G}}^{\prime}$ is ${\rm in}_{<^{\prime}}(x_{i}y_{i}-x_{1}y_{1})=x_{i}y_{i}$ and ${\rm in}_{<^{\prime}}(g_{i})={\rm in}_{<}(g_{i})$.

Suppose that ${\mathcal{G}}^{\prime}$ is not a Gröbner basis of $I_{{\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})}$. By [11, Theorem 3.11], there exists an irreducible binomial $0\neq f=u-v\in I_{{\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})}$ such that $u,v\notin\left<{\rm in}_{<^{\prime}}({\mathcal{G}}^{\prime})\right>$. Then $f\in I_{{\mathcal{P}}_{A^{\pm}}}\cap K[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]$. Since ${\mathcal{G}}$ in Theorem 1.2 is a Gröbner basis of $I_{{\mathcal{P}}_{A^{\pm}}}$, there exists $g\in{\mathcal{G}}$ such that ${\rm in}_{<}(g)$ divides ${\rm in}_{<}(f)={\rm in}_{<^{\prime}}(f)$. If $g\neq x_{1}y_{1}-z^{2}$, then there exists $g^{\prime}\in{\mathcal{G}}^{\prime}$ such that ${\rm in}_{<}(g)={\rm in}_{<^{\prime}}(g^{\prime})$, a contradiction. Hence $g=x_{1}y_{1}-z^{2}$ and ${\rm in}_{<^{\prime}}(f)$ is divisible by $x_{1}y_{1}$. Since $f$ is irreducible, the monomial $f-{\rm in}_{<^{\prime}}(f)$ does not contain $x_{1}$. This contradicts the definition of $<^{\prime}$. Thus ${\mathcal{G}}^{\prime}$ is a Gröbner basis of $I_{{\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})}$. Moreover, since $\{x_{i}y_{i}-z^{2}:i\in[n]\}\cup\{g_{1},\dots,g_{s}\}$ is the reduced Gröbner basis of $I_{A^{\pm}}$ and $x_{1}$ and $y_{1}$ do not appear in any ${\rm in}_{<^{\prime}}(g_{i})$, it follows that ${\mathcal{G}}^{\prime}$ is reduced . ∎

Let $A=({\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n})\in{\mathbb{Z}}^{d\times n}$ be a unimodular configuration and set $A_{0}=(A,{\bf 0})\in{\mathbb{Z}}^{d\times(n+1)}$. Assume that ${\mathcal{P}}_{A_{0}}\cap{\mathbb{Z}}^{d}=\{{\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n},{\bf 0}\}$. The toric ideal $I_{{\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})}$ of ${\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})$ is the kernel of a ring homomorphism

[TABLE]

defined by $\varphi_{*}(x_{i})=t_{0}{\mathbf{t}}^{{\mathbf{a}}_{i}}s$, $\varphi_{*}(y_{i})={\mathbf{t}}^{-{\mathbf{a}}_{i}}s$ for each $1\leq i\leq n$, and $\varphi_{*}(x_{0})=t_{0}s$, $\varphi_{*}(y_{0})=s$.

Theorem 1.4**.**

Work with the same notation as above. Let $<^{\prime}$ be the reverse lexicographic order induced by the ordering of variables $x_{0}<^{\prime}y_{0}<^{\prime}\cdots<^{\prime}x_{n}<^{\prime}y_{n}$. Then the reduced Gröbner basis of $I_{{\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})}$ with respect to $<^{\prime}$ is

[TABLE]

where $\{g_{1},\dots,g_{s}\}$ is such that $\{x_{i}y_{i}-z^{2}:i\in[n]\}\cup\{g_{1},\dots,g_{s}\}$ is the reduced Gröbner basis of $I_{A^{\pm}}$ from Theorem 1.2.

Proof.

Each $g_{i}$ belongs to $I_{{\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})}$ since $I_{{\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})}\subset I_{{\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})}$. Since each $x_{i}y_{i}-x_{0}y_{0}$ belongs to $I_{{\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})}$, the set ${\mathcal{G}}^{\prime}$ is a subset of $I_{{\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})}$. The initial monomial of each binomial in ${\mathcal{G}}^{\prime}$ is ${\rm in}_{<^{\prime}}(x_{i}y_{i}-x_{0}y_{0})=x_{i}y_{i}$ and ${\rm in}_{<^{\prime}}(g_{i})={\rm in}_{<}(g_{i})$. Note that ${\rm in}_{<}({\mathcal{G}})={\rm in}_{<^{\prime}}({\mathcal{G}}^{\prime})$.

Suppose that ${\mathcal{G}}^{\prime}$ is not a Gröbner basis of $I_{{\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})}$. By [11, Theorem 3.11], there exists an irreducible binomial $0\neq f=u-v\in I_{{\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})}$ such that $u,v\notin\left<{\rm in}_{<^{\prime}}({\mathcal{G}}^{\prime})\right>$. Let

[TABLE]

Then we have

[TABLE]

$\sum_{k=0}^{n}u_{k}=\sum_{k=0}^{n}u_{k}^{\prime}$ and $\sum_{k=0}^{n}v_{k}=\sum_{k=0}^{n}v_{k}^{\prime}$. Moreover since $A$ is a configuration, there exists a vector ${\mathbf{c}}\in{\mathbb{R}}^{d}$ such that the inner product $\left<{\mathbf{a}}_{i},{\mathbf{c}}\right>=1$ for all $1\leq i\leq n$. Taking the inner product of equation (4) with ${\mathbf{c}}$, we have

[TABLE]

Thus $v_{0}-u_{0}=v_{0}^{\prime}-u_{0}^{\prime}$.

If $u_{0}=u_{0}^{\prime}=v_{0}=v_{0}^{\prime}=0$, then $f\in K[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]$. Then $f$ belongs to $I_{A^{\pm}}$. This contradicts that ${\mathcal{G}}$ is a Gröbner basis of $I_{A^{\pm}}$. Hence at least one of $u_{0}$, $u_{0}^{\prime}$, $v_{0}$, $v_{0}^{\prime}$ is positive. Since $f$ is irreducible, we may assume that $u_{0}>0$ and $u_{0}^{\prime}=0$. Then $v_{0}=u_{0}+v_{0}^{\prime}>0$ and hence $v_{0}^{\prime}=0$. Thus $u_{0}=v_{0}>0$ and $u_{0}^{\prime}=v_{0}^{\prime}=0$, that is,

[TABLE]

It then follows that

[TABLE]

belongs to $I_{A^{\pm}}$. Since ${\mathcal{G}}$ is a Gröbner basis of $I_{A^{\pm}}$, there exists $g\in{\mathcal{G}}$ such that ${\rm in}_{<}(g)$ divides ${\rm in}_{<}(f^{\prime})=x_{1}^{u_{1}^{\prime}}\dots x_{n}^{u_{n}^{\prime}}y_{1}^{v_{1}^{\prime}}\dots y_{n}^{v_{n}^{\prime}}$. Since there exists $g^{\prime}\in{\mathcal{G}}^{\prime}$ such that ${\rm in}_{<}(g)={\rm in}_{<^{\prime}}(g^{\prime})$, this is a contradiction. ∎

2. Proofs of Theorems

In the present section, we give proofs of Theorems 0.1 and 0.2 by using the Gröbner bases constructed in Section 1. First, we recall the theory of the $h^{*}$-polynomials of lattice polytopes. Two lattice polytopes ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ and ${\mathcal{P}}^{\prime}\subset{\mathbb{R}}^{d^{\prime}}$ are said to be unimodularly equivalent if there exists an affine map from the affine span ${\rm aff}({\mathcal{P}})$ of ${\mathcal{P}}$ to the affine span ${\rm aff}({\mathcal{P}}^{\prime})$ of ${\mathcal{P}}^{\prime}$ that maps ${\mathbb{Z}}^{d}\cap{\rm aff}({\mathcal{P}})$ bijectively onto ${\mathbb{Z}}^{d^{\prime}}\cap{\rm aff}({\mathcal{P}}^{\prime})$ and that maps ${\mathcal{P}}$ to ${\mathcal{P}}^{\prime}$. Note that every lattice polytope is unimodularly equivalent to a full-dimensional one. Let ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ be a lattice polytope of dimension $d$. Given a positive integer $m$, we define

[TABLE]

where $m{\mathcal{P}}:=\{m{\mathbf{a}}:{\mathbf{a}}\in{\mathcal{P}}\}$. Ehrhart [9] proved that $L_{{\mathcal{P}}}(m)$ is a polynomial in $m$ of degree $d$ with the constant term $1$. We say that $L_{{\mathcal{P}}}(m)$ is the Ehrhart polynomial of ${\mathcal{P}}$. Clearly, if ${\mathcal{P}}$ and ${\mathcal{Q}}$ are unimodularly equivalent, then one has $L_{{\mathcal{P}}}(m)=L_{{\mathcal{Q}}}(m)$. The generating function of the lattice point enumerator, i.e., the formal power series

[TABLE]

is called the Ehrhart series of ${\mathcal{P}}$. It is well known that it can be expressed as a rational function of the form

[TABLE]

Then $h^{*}({\mathcal{P}},t)$ is a polynomial in $t$ of degree at most $d$ with nonnegative integer coefficients ([19]) and it is called the $h^{*}$-polynomial (or the $\delta$-polynomial) of ${\mathcal{P}}$. A characterization of Gorenstein polytopes in terms of their $h^{*}$-polynomials is known. In fact, a lattice polytope ${\mathcal{P}}$ of dimension $d$ is Gorenstein of index $r$ if and only if its $h^{*}$-polynomial is of degree $d+1-r$ and palindromic, i.e., $h^{*}({\mathcal{P}},t)=t^{d+1-r}h^{*}({\mathcal{P}},t^{-1})$ ([8]).

We say that a lattice polytope ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ possesses the integer decomposition property if for any $k\in{\mathbb{Z}}_{>0}$ and for any ${\mathbf{x}}\in k{\mathcal{P}}\cap{\mathbb{Z}}^{d}$, there exist $k$ lattice points ${\mathbf{x}}_{1},\ldots,{\mathbf{x}}_{k}$ in ${\mathcal{P}}$ with ${\mathbf{x}}={\mathbf{x}}_{1}+\cdots+{\mathbf{x}}_{k}$, and we call a lattice polytope with the integer decomposition property an IDP polytope. IDP polytopes turn up in many fields of mathematics such as algebraic geometry, where they correspond to projectively normal embeddings of toric varieties, and commutative algebra, where they correspond to standard graded Cohen-Macaulay domains (see [6]). Moreover, the integer decomposition property is particularly important in the theory and application of integer programing [18, §22.10]. Note that a lattice polytope with a unimodular triangulation is IDP, and an IDP polytope is spanning. Moreover, a lattice polytope is IDP if and only if its $h^{*}$-polynomial coincides with the $h$-polynomial of its toric ring.

We turn to the review of indispensable lemmata for our proofs of Theorems 0.1 and 0.2.

Lemma 2.1** ([10, Corollary 6.1.5]).**

Let $S$ be a polynomial ring and $I\subset S$ a graded ideal of $S$. Fix a monomial order $<$ on $S$. Then $S/I$ and $S/{\rm in}_{<}(I)$ have the same Hilbert function, in particular, they have the same $h$-polynomial.

Since a lattice polytope ${\mathcal{P}}$ is unimodularly equivalent to a lattice polytope ${\mathcal{P}}^{\prime}$ if and only if ${\mathcal{P}}*(-{\mathcal{P}})$ is unimodularly equivalent to ${\mathcal{P}}^{\prime}*(-{\mathcal{P}}^{\prime})$ and since any initial ideal given in Section 1 is squarefree, we can use the following lemma.

Lemma 2.2** ([13, Theorem 2.2]).**

Let ${\mathcal{P}}_{1},\dots,{\mathcal{P}}_{r}\subset{\mathbb{R}}^{d}$ be spanning lattice polytopes of dimension $d$. Suppose that the toric ideal of ${\mathcal{P}}_{1}*\cdots*{\mathcal{P}}_{r}$ has a squarefree initial ideal. Then the toric ideal of ${\mathcal{P}}_{1}+\cdots+{\mathcal{P}}_{r}$ has a squarefree initial ideal, and both ${\mathcal{P}}_{1}*\cdots*{\mathcal{P}}_{r}$ and ${\mathcal{P}}_{1}+\cdots+{\mathcal{P}}_{r}$ have a regular unimodular triangulation and are IDP.

Now, we prove Theorems 0.1 and 0.2.

Proof of Theorem 0.2.

Let $A=({\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n})\in{\mathbb{Z}}^{d\times n}$ be a unimodular configuration and set $A_{0}=(A,{\bf 0})\in{\mathbb{Z}}^{d\times(n+1)}$. Assume that ${\mathcal{P}}_{A_{0}}\cap{\mathbb{Z}}^{d}=\{{\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n},{\bf 0}\}$ and ${\mathcal{P}}_{A_{0}}$ is spanning. From Theorem 1.4, the toric ideal $I_{{\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})}$ has a squarefree initial ideal. Hence by Lemma 2.2, the toric ideal of ${\mathcal{P}}_{A_{0}}+(-{\mathcal{P}}_{A_{0}})$ has a squarefree initial ideal, and both ${\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})$ and ${\mathcal{P}}_{A_{0}}+(-{\mathcal{P}}_{A_{0}})$ have a regular unimodular triangulation and are IDP. Then [21, Theorem 0.4] guarantees that $({\mathcal{P}}_{A_{0}}\cap{\mathbb{Z}}^{d})+(-{\mathcal{P}}_{A_{0}}\cap{\mathbb{Z}}^{d})=({\mathcal{P}}_{A_{0}}+(-{\mathcal{P}}_{A_{0}}))\cap{\mathbb{Z}}^{d}$.

On the other hand, the minimal set of monomial generators of the initial ideals ${\rm in}_{<}(I_{A^{\pm}})$ and ${\rm in}_{<^{\prime}}(I_{{\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})})$ given in Theorems 1.2 and 1.4 are the same. Thus

[TABLE]

and

[TABLE]

Since ${\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})$ is IDP, it follows from Lemma 2.1 that

[TABLE]

Here (5) and (7) follow from Lemma 2.1, (6) follows from the above, and (8) holds since ${\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})$ is IDP. From Proposition 1.1, $h(K[A^{\pm}],t)$ is palindromic and of degree $d$, and so is $h^{*}({\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}}),t)$. Since the dimension of ${\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})$ is $d+1$, ${\mathcal{P}}_{A_{0}}*(-{\mathcal{P}}_{A_{0}})$ is Gorenstein of index $2$. By [3, Theorem 2.6], ${\mathcal{P}}_{A_{0}}+(-{\mathcal{P}}_{A_{0}})$ is reflexive. ∎

Proof of Theorem 0.1.

The proof is the same as the proof of Theorem 0.2 except for the following discussion. Comparing the initial ideals ${\rm in}_{<}(I_{A^{\pm}})$ and ${\rm in}_{<^{\prime}}(I_{{\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})})$ given in Theorems 1.2 and 1.3, it follows that

[TABLE]

and

[TABLE]

Then $h(K[x_{1},y_{1}]/\left<x_{1}y_{1}\right>,t)=t+1$, and hence

[TABLE]

Since ${\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})$ is IDP, it follows from Lemma 2.1 that

[TABLE]

From Proposition 1.1, $h(K[A^{\pm}],t)$ is palindromic and of degree $d$, and hence $h^{*}({\mathcal{P}}_{A}*(-{\mathcal{P}}_{A}),t)$ is palindromic and of degree $d-1$. Since the dimension of ${\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})$ is $d$, ${\mathcal{P}}_{A}*(-{\mathcal{P}}_{A})$ is Gorenstein of index $2$. ∎

From the proof of Theorems 0.1 and 0.2 we obtain the following Corollary.

Corollary 2.3**.**

Let $A=({\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n})\in{\mathbb{Z}}^{d\times n}$ be a unimodular configuration and set $A_{0}=(A,{\bf 0})\in{\mathbb{Z}}^{d\times(n+1)}$. Assume that ${\mathcal{P}}_{A_{0}}\cap{\mathbb{Z}}^{d}=\{{\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n},{\bf 0}\}$, and both ${\mathcal{P}}_{A}$ and ${\mathcal{P}}_{A_{0}}$ are spanning. Then one has

[TABLE]

3. Unimodular configurations arising from finite simple graphs

Let $G$ be a finite connected simple graph on the vertex set $[d]$ with the edge set $E(G)=\{e_{1},\dots,e_{n}\}$. Here a graph is called simple if $G$ has no loops and no multiple edges. Given an edge $e=\{i,j\}$ of $G$, let $\rho(e)={\mathbf{e}}_{i}+{\mathbf{e}}_{j}\in{\mathbb{R}}^{d}$. The edge polytope of $G$ is ${\mathcal{P}}_{A_{G}}$, where $A_{G}$ is the configuration defined by

[TABLE]

Since ${\mathcal{P}}_{A_{G}}$ is a $(0,1)$-polytope, we have ${\mathcal{P}}_{A_{G}}\cap{\mathbb{Z}}^{d}=\{\rho(e_{1}),\dots,\rho(e_{n})\}$. It is known by [16, Proposition 1.3] that

[TABLE]

A classification of the graphs $G$ such that $A_{G}$ is unimodular is as follows:

Proposition 3.1** ([17, Theorem 3.3]).**

Let $G$ be a finite connected simple graph. Then the following conditions are equivalent:**

• (i)

The toric ring $K[A_{G}^{\pm}]$ is normal;**

• (ii)

The toric ideal $I_{A_{G}^{\pm}}$ has a squarefree initial ideal;**

• (iii)

The configuration $A_{G}$ is unimodular (by deleting a redundant row if $G$ is bipartite);**

• (iv)

All pairs of odd cycles in $G$ have a common vertex.

It follows from [12] that ${\mathcal{P}}_{A_{G}}$ is always spanning.

Lemma 3.2** ([12, Proof of Corollary 3.4]).**

Let $G$ be a finite connected graph. Then ${\mathcal{P}}_{A_{G}}$ is spanning.

Hence we obtain the following.

Theorem 3.3**.**

Let $G$ be a finite connected simple graph on $[d]$. Assume that all pairs of odd cycles in $G$ have a common vertex. Then

1. (1)

${\mathcal{P}}_{A_{G}}*(-{\mathcal{P}}_{A_{G}})$* is Gorenstein of index $2$ with a regular unimodular triangulation*;** 2. (2)

${\mathcal{P}}_{A_{G}}+(-{\mathcal{P}}_{A_{G}})$* is reflexive with a regular unimodular triangulation. In particular, $({\mathcal{P}}_{A_{G}}-{\mathbf{a}})+(-{\mathcal{P}}_{A_{G}}+{\mathbf{a}})$ is a nef-partition, where ${\mathbf{a}}$ is an arbitrary lattice point in ${\mathcal{P}}_{A_{G}}$;* 3. (3)

One has $({\mathcal{P}}_{A_{G}}\cap{\mathbb{Z}}^{d})+(-{\mathcal{P}}_{A_{G}}\cap{\mathbb{Z}}^{d})=({\mathcal{P}}_{A_{G}}+(-{\mathcal{P}}_{A_{G}}))\cap{\mathbb{Z}}^{d}$.

On the other hand, in general, ${\mathcal{P}}_{(A_{G})_{0}}$ is not always spanning even if $A_{G}$ is unimodular.

Example 3.4**.**

Let $G$ be a cycle of length $3$, i.e., $E(G)=\{\{1,2\},\{2,3\},\{1,3\}\}$. Then

[TABLE]

is unimodular. On the other hand, ${\mathcal{P}}_{(A_{G})_{0}}$ is not spanning.

We determine when ${\mathcal{P}}_{(A_{G})_{0}}$ is spanning.

Lemma 3.5**.**

Let $G$ be a finite connected simple graph. Then ${\mathcal{P}}_{(A_{G})_{0}}$ is spanning if and only if $G$ is bipartite.

Proof.

Note that a full-dimensional lattice polytope ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ with ${\bf 0}\in{\mathcal{P}}$ is spanning if and only if it holds $\sum_{{\mathbf{a}}\in{\mathcal{P}}\cap{\mathbb{Z}}^{d}}{\mathbb{Z}}{\mathbf{a}}={\mathbb{Z}}^{d}$.

We assume that $G$ is not bipartite. Then ${\mathcal{P}}_{(A_{G})_{0}}$ is full-dimensional. Since for any lattice point $(a_{1},\ldots,a_{d})\in{\mathcal{P}}_{(A_{G})_{0}}\cap{\mathbb{Z}}^{d}$, $a_{1}+\cdots+a_{d}$ is even, it follows that for any lattice point $(x_{1},\ldots,x_{d})\in\sum_{{\mathbf{a}}\in{\mathcal{P}}_{(A_{G})_{0}}\cap{\mathbb{Z}}^{d}}{\mathbb{Z}}{\mathbf{a}}$, $x_{1}+\cdots+x_{d}$ is even. Hence ${\mathcal{P}}_{(A_{G})_{0}}$ is not spanning.

Conversely, we assume that $G$ is bipartite. Let $A$ be a configuration obtained by deleting a row of $A_{G}$. Then ${\mathcal{P}}_{(A_{G})_{0}}$ is unimodularly equivalent to a full-dimensional lattice polytope ${\mathcal{P}}_{A_{0}}\subset{\mathbb{R}}^{d-1}$. It is known by [18, Example 1 (p.273)] that any nonzero maximal minor of $A_{0}$ is $\pm 1$. Thus $\sum_{{\mathbf{a}}\in{\mathcal{P}}_{A_{0}}\cap{\mathbb{Z}}^{d-1}}{\mathbb{Z}}{\mathbf{a}}={\mathbb{Z}}^{d-1}$, and hence ${\mathcal{P}}_{A_{0}}$ is spanning. ∎

Therefore, we have obtain the following.

Theorem 3.6**.**

Let $G$ be a finite connected simple bipartite graph on $[d]$. Then

1. (1)

${\mathcal{P}}_{(A_{G})_{0}}*(-{\mathcal{P}}_{(A_{G})_{0}})$* is Gorenstein of index $2$ with a regular unimodular triangulation*;** 2. (2)

${\mathcal{P}}_{(A_{G})_{0}}+(-{\mathcal{P}}_{(A_{G})_{0}})$* is reflexive with a regular unimodular triangulation. In particular, ${\mathcal{P}}_{(A_{G})_{0}}+(-{\mathcal{P}}_{(A_{G})_{0}})$ is a nef-partition*;** 3. (3)

One has $({\mathcal{P}}_{(A_{G})_{0}}\cap{\mathbb{Z}}^{d})+(-{\mathcal{P}}_{(A_{G})_{0}}\cap{\mathbb{Z}}^{d})=({\mathcal{P}}_{(A_{G})_{0}}+(-{\mathcal{P}}_{(A_{G})_{0}}))\cap{\mathbb{Z}}^{d}$; 4. (4)

One has $h^{*}({\mathcal{P}}_{(A_{G})_{0}}*(-{\mathcal{P}}_{(A_{G})_{0}}),t)=(1+t)h^{*}({\mathcal{P}}_{A_{G}}*(-{\mathcal{P}}_{A_{G}}),t)$.

Acknowledgment**.**

The authors are grateful to an anonymous referee for his careful reading of the manuscript and for his useful comments. The authors were partially supported by JSPS KAKENHI 18H01134, 19K14505 and 19J00312.