Smooth Fano polytopes arising from finite partially ordered sets

TL;DR

This paper introduces Gorenstein Fano polytopes derived from finite partially ordered sets and investigates which of these sets produce smooth Fano polytopes, contributing to the understanding of their geometric properties.

Contribution

It is the first to systematically study the conditions under which finite partially ordered sets generate smooth Gorenstein Fano polytopes.

Findings

Identification of classes of partially ordered sets that produce smooth Fano polytopes

Characterization of Gorenstein Fano polytopes from posets

New insights into the structure of smooth Fano polytopes

Abstract

Gorenstein Fano polytopes arising from finite partially ordered sets will be introduced. Then we study the problem which partially ordered sets yield smooth Fano polytopes.

Equations

$$\hat{P}=P\cup \{\hat{0},\hat{1}\},$$

$$\rho (e)=\{\begin{array}{ll}{\mathbf{e}}_i& \text{if}j=d+1,\\ -{\mathbf{e}}_j& \text{if}i=0,\\ {\mathbf{e}}_i-{\mathbf{e}}_j& \text{if}1\le i,j\le d.\end{array}$$

$$\{\rho (e):\text{e is an edge of}\hat{P}\}.$$

$$\sum _{j=1}^{m-1}\rho (\{y_{i_j},y_{i_{j+1}}\})=(0,\dots 0).$$

$$\sum _e{c}_e\rho (e)=(0,\dots 0),$$

$$\sum _{j=1}^m{r}_{i_j}{q}_{i_j}={\alpha }_1=1,$$

$$\sum _{j=1}^{m^{\prime }}{r}_{i_j}{q}_{i_j}=1,$$

$$♯\{j:y_{i_j}<y_{i_{j+1}},1\le j\le m-1\}=♯\{k:y_{i_k}>y_{i_{k+1}},1\le k\le m-1\}.$$

$${\mu }_{\Gamma }:\{y_{i_1},y_{i_2},\dots ,y_{i_m}\}\to \{0,1,2,\dots \}$$

$$\sum _{i=1}^m{q}_j=\sum _{i=1}^m(a_{i_j}-a_{i_{j+1}})=0.$$

$$-a_{i_2}+\sum _{j=2}^{r-1}(a_{i_j}-a_{i_{j+1}})+a_{i_r}=0.$$

$$-a_{i_2}+\sum _{j=2}^{r-1}(a_{i_j}-a_{i_{j+1}})+a_{i_r}=1+\sum _{j=2}^{r-1}{q}_j+1$$

$$=-{\mu }_C(y_0)+{\mu }_C(y_{i_2})+\sum _{j=2}^{r-1}({\mu }_C(y_{i_{j+1}})-{\mu }_C(y_{i_j}))+{\mu }_C(y_{d+1})-{\mu }_C(y_{i_r})$$

$$={\mu }_C(y_{d+1})-{\mu }_C(y_0).$$

$$y=z_0<z_1<\cdots <z_s=z.$$

$${\mu }_C(y_{i_a})-{\mu }_C(y_{i_b})\le {\mathrm{dist}}_{\hat{P}}(y_{i_b},y_{i_a})$$

$${\mu }_C(y_{i_a})-{\mu }_C(y_{i_b})\le {\mathrm{dist}}_{\hat{P}}(y_0,y_{i_a})+{\mathrm{dist}}_{\hat{P}}(y_{i_b},y_{d+1})$$

$${\mu }_{\Gamma }(y_{i_a})-{\mu }_{\Gamma }(y_{i_b})\le {\mathrm{dist}}_{\hat{P}}(y_{i_b},y_{i_a})$$

$$\sum _{j=1}^m{q}_j\rho (\{y_{i_j},y_{i_{j+1}}\})=(0,\dots ,0),$$

$$\underset{1\le a\le m}{\max }({\mu }_C(y_{i_a})-{\mathrm{dist}}_{\hat{P}}(y_0,y_{i_a}))\le \underset{1\le b\le m}{\min }({\mu }_C(y_{i_b})+{\mathrm{dist}}_{\hat{P}}(y_{i_b},y_{d+1})).$$

$$\underset{1\le a\le m}{\max }({\mu }_C(y_{i_a})-{\mathrm{dist}}_{\hat{P}}(y_0,y_{i_a}))\le a\le \underset{1\le b\le m}{\min }({\mu }_C(y_{i_b})+{\mathrm{dist}}_{\hat{P}}(y_{i_b},y_{d+1})).$$

$$-a_{i_j}\le {\mathrm{dist}}_{\hat{P}}(y_0,y_{i_j}),a_{i_j}\le {\mathrm{dist}}_{\hat{P}}(y_{i_j},y_{d+1}).$$

$$\sum _{i_j\notin \{0,d+1\}}{a}_{i_j}{x}_{i_j}=1.$$

$$a_i=\max (\{a_{i_j}-{\mathrm{dist}}_{\hat{P}}(y_{i_j},y_i):y_{i_j}<y_i\}\cup \{0\}).$$

$$a_i=\min (\{a_{i_k}+{\mathrm{dist}}_{\hat{P}}(y_i,y_{i_k}):y_i<y_{i_k}\}\cup \{0\}).$$

$$b_i=\max (\{a_{i_j}-{\mathrm{dist}}_{\hat{P}}(y_{i_j},y_i):y_{i_j}<y_i\}\cup \{0\})$$

$$c_i=\min (\{a_{i_k}+{\mathrm{dist}}_{\hat{P}}(y_i,y_{i_k}):y_i<y_{i_k}\}\cup \{0\})$$

$${\mu }_C(y_{i_k})-{\mu }_C(y_{i_j})>{\mathrm{dist}}_{\hat{P}}(y_{i_j},y_{i_k}).$$

$$a_j\ge \max \{a_i-1,0\},$$

$$a_i\le \min \{a_j+1,0\},$$

$$\sum _{j=1}^{m-1}{q}_j\rho (\{y_{i_j},y_{i_{j+1}}\})=(0,\dots ,0),$$

Full text

2000 Mathematics Subject Classification: Primary 14J45, 52B20; Secondary 06A11.

      Keywords: smooth Fano polytope, ${\mathbb{Q}}$-factorial Fano polytope, Gorenstein Fano polytope, totally unimodular matrix, finite partially ordered set.

Introduction

An integral (or lattice) polytope is a convex polytope all of whose vertices have integer coordinates. Let ${\mathcal{P}}\subset{\mathbb{R}}^{d}$ be an integral convex polytope of dimension $d$.

• •

We say that ${\mathcal{P}}$ is a Fano polytope if the origin of ${\mathbb{R}}^{d}$ is the unique integer point belonging to the interior of ${\mathcal{P}}$.

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A Fano polytope ${\mathcal{P}}$ is called terminal if each integer point belonging to the boundary of ${\mathcal{P}}$ is a vertex of ${\mathcal{P}}$.

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A Fano polytope is called Gorenstein if its dual polytope is integral. (Recall that the dual polytope ${\mathcal{P}}^{\vee}$ of a Fano polytope ${\mathcal{P}}$ is the convex polytope which consists of those $x\in{\mathbb{R}}^{d}$ such that $\langle x,y\rangle\leq 1$ for all $y\in{\mathcal{P}}$, where $\langle x,y\rangle$ is the usual inner product of ${\mathbb{R}}^{d}$.)

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A ${\mathbb{Q}}$-factorial Fano polytope is a simplicial Fano polytope, i.e., a Fano polytope each of whose faces is a simplex.

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A smooth Fano polytope is a Fano polytope such that the vertices of each facet form a ${\mathbb{Z}}$-basis of ${\mathbb{Z}}^{d}$. (Sometimes, smooth polytopes denote simple polytopes, which are dual polytopes of simplicial polytopes.)

Thus in particular a smooth Fano polytope is ${\mathbb{Q}}$-factorial, Gorenstein and terminal.

Øbro [10] succeeded in finding an algorithm which yields the classification list of the smooth Fano polytopes for given $d$. It is proved in Casagrande [2] that the number of vertices of a Gorenstein ${\mathbb{Q}}$-factorial Fano polytope is at most $3d$ if $d$ is even, and at most $3d-1$ if $d$ is odd. B. Nill and M. Øbro [9] classified the Gorenstein ${\mathbb{Q}}$-factorial Fano polytopes of dimension $d$ with $3d-1$ vertices. Gorenstein Fano polytopes are classified when $d\leq 4$ by Kreuzer and Skarke [6], [7] and mirro symmetry is studied as the relevance of Gorenstein Fano polytopes by Batyrev [1]. The study on the classification of terminal or canonical Fano polytopes was done by Kasprzyk [4], [5]. The combinatorial conditions for what it implies to be terminal and canonical are explained in Reid [11].

In the present paper, given a finite partially ordered set $P$ we associate a terminal Fano polytope ${\mathcal{Q}}_{P}$. By using the theory of totally unimodular matrices, it turns out that these Fano polytopes are Gorenstein. Then we study the problem of which partially ordered sets yield ${\mathbb{Q}}$-factorial Fano polytopes. Finally, it turns out that the Fano polytope ${\mathcal{Q}}_{P}$ is smooth if and only if ${\mathcal{Q}}_{P}$ is ${\mathbb{Q}}$-factorial.

1. Fano polytopes arising from finite partially ordered sets

Let $P=\{y_{1},\ldots,y_{d}\}$ be a finite partially ordered set and

[TABLE]

where $\hat{0}$ (resp. $\hat{1}$) is a unique minimal (resp. maximal) element of $\hat{P}$ with $\hat{0}\not\in P$ (resp. $\hat{1}\not\in P$). Let $y_{0}=\hat{0}$ and $y_{d+1}=\hat{1}$. We say that $e=\{y_{i},y_{j}\}$, where $0\leq i,\,j\leq d+1$ with $i\neq j$, is an edge of $\hat{P}$ if $e$ is an edge of the Hasse diagram of $\hat{P}$. (The Hasse diagram of a finite partially ordered set can be regarded as a finite nondirected graph.) In other words, $e=\{y_{i},y_{j}\}$ is an edge of $\hat{P}$ if $y_{i}$ and $y_{j}$ are comparable in $\hat{P}$, say, $y_{i}<y_{j}$, and there is no $z\in P$ with $y_{i}<z<y_{j}$.

Definition 1.1**.**

Let $\hat{P}=\{y_{0},y_{1},\ldots,y_{d},y_{d+1}\}$ be a finite partially ordered set with $y_{0}=\hat{0}$ and $y_{d+1}=\hat{1}$. Let ${\bf e}_{i}$ denote the $i$th canonical unit coordinate vector of ${\mathbb{R}}^{d}$. Given an edge $e=\{y_{i},y_{j}\}$ of $\hat{P}$ with $y_{i}<y_{j}$, we define $\rho(e)\in{\mathbb{R}}^{d}$ by setting

[TABLE]

Moreover, we write ${\mathcal{Q}}_{P}\subset{\mathbb{R}}^{d}$ for the convex hull of the finite set

[TABLE]

Example 1.2**.**

Let $P=\{y_{1},y_{2},y_{3}\}$ be the finite partially ordered set with the partial order $y_{1}<y_{2}$. Then ${\hat{P}}$ together with $\rho(e)$’s and ${\mathcal{Q}}_{P}$ are drawn below: **

P=$$y_{2}$$y_{1}$$y_{3}$$\hat{P}=$$y_{2}$$y_{1}$$y_{3}$$\hat{1}=y_{4}$$\hat{0}=y_{0}$$(0,1,0)$$(0,0,1)$$(0,0,-1)$$(-1,0,0)$$(1,-1,0)$${\bf e}_{3}$${\bf e}_{2}$${\bf e}_{1}-111-1-11${\mathcal{Q}}_{P}=$

Let $P$ be a finite partially ordered set. A subset $Q$ of $P$ is called a chain of $P$ if $Q$ is a totally ordered subset of $P$. The length of a chain $Q$ is $\ell(Q)=\sharp(Q)-1$. A chain $Q$ of $P$ is saturated if $x,y\in Q$ with $x<y$, then there is no $z\in P$ with $x<z<y$. A maximal chain of ${\hat{P}}$ is a saturated chain $Q$ of ${\hat{P}}$ with $\{{\hat{0}},{\hat{1}}\}\subset Q$.

Lemma 1.3**.**

The convex polytope ${\mathcal{Q}}_{P}$ is a Fano polytope.

Proof.

Let $e=\{y_{i},y_{j}\}$ be an edge of ${\hat{P}}$ with $y_{i}<y_{j}$. Let $c_{e}$ denote the number of maximal chains $Q$ of ${\hat{P}}$ with $\{y_{i},y_{j}\}\subset Q$. If $\{y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}}\}$ is a maximal chain of ${\hat{P}}$ with $y_{0}=y_{i_{1}}<y_{i_{2}}<\ldots<y_{i_{m}}=y_{d+1}$, then

[TABLE]

Hence

[TABLE]

where $e$ ranges all edges of ${\hat{P}}$. Thus the origin of ${\mathbb{R}}^{d}$ belongs to the interior of ${\mathcal{Q}}_{P}$. Since ${\mathcal{Q}}_{P}$ is a convex polytope which is contained in the convex hull of the finite set $\{\sum_{i=1}^{d}\varepsilon_{i}{\bf e}_{i}:\varepsilon_{i}\in\{0,1,-1\}\}$ in ${\mathbb{R}}^{d}$, it follows that the origin of ${\mathbb{R}}^{d}$ is the unique integer point belonging to the interior of ${\mathcal{Q}}_{P}$. Thus ${\mathcal{Q}}_{P}$ is a Fano polytope, as desired.

Lemma 1.4**.**

The Fano polytope ${\mathcal{Q}}_{P}$ is terminal.

Proof.

Suppose that ${\mathcal{Q}}_{P}$ contains an integer point $\alpha=(\alpha_{1},\ldots,\alpha_{d})\in{\mathbb{Z}}^{d}$ with $\alpha\not=(0,\ldots,0).$ Then, obviously, $\alpha_{1},\ldots,\alpha_{d}\in\{-1,0,1\}$. Let, say, $\alpha_{1}=1$. Let $e_{1},\ldots,e_{n}$ be all edges of $\hat{P}$ and $e_{i_{1}},\ldots,e_{i_{m}}$ the edges with $y_{1}\in e_{i_{j}}$ for $j=1,\ldots,m$. If we set $e_{i_{j}}=\{y_{i_{j}},y_{i_{j^{\prime}}}\}$ with $y_{i_{j}}<y_{i_{j^{\prime}}}$, since $\alpha$ belongs to the convex hull of $\{\rho(e_{1}),\ldots,\rho(e_{n})\}$, then one has

[TABLE]

where $0\leq r_{i_{1}},\ldots,r_{i_{m}}\leq 1$ and $q_{i_{j}}=1$ (resp. $q_{i_{j}}=-1$) if $y_{1}<y_{i_{j^{\prime}}}$ (resp. $y_{i_{j}}<y_{1}$). By removing all $r_{i_{j}}$ with $r_{i_{j}}=0$, we may assume that

[TABLE]

where $0<r_{i_{1}},\ldots,r_{i_{m^{\prime}}}\leq 1$. Since $\sum_{j=1}^{m^{\prime}}r_{i_{j}}\leq 1$, there is no $j$ with $q_{i_{j}}=-1$. Hence $\sum_{j=1}^{m^{\prime}}r_{i_{j}}=1$. If $m^{\prime}>1$, then $0<r_{i_{1}},\ldots,r_{i_{m^{\prime}}}<1$. Thus $\sum_{j=1}^{m^{\prime}}r_{i_{j}}\rho(e_{i_{j}})=\alpha\not\in{\mathbb{Z}}^{d}$. Thus $m^{\prime}=1$. In other words, if ${\mathcal{Q}}_{P}$ contains an integer point $\alpha\not=(0,\ldots,0)$, then $\alpha$ must be one of $\rho(e_{1}),\ldots,\rho(e_{n})$ and $\rho(e_{1}),\ldots,\rho(e_{n})$ are precisely the vertices of ${\mathcal{Q}}_{P}$.

Lemma 1.5**.**

The Fano polytope ${\mathcal{Q}}_{P}$ is Gorenstein.

Proof.

Via the theory of totally unimodular matrices ([12, Chapter 9]), it follows that the equation of each supporting hyperplane of ${\mathcal{Q}}_{P}$ is of the form $a_{1}x_{1}+\cdots+a_{d}x_{d}=1$ with each $a_{i}\in{\mathbb{Z}}$. In other words, the dual polytope of ${\mathcal{Q}}_{P}$ is integral. Hence ${\mathcal{Q}}_{P}$ is Gorenstein, as required.

Remark 1.6**.**

There is a well-known integral convex polytope arising from a finite partially ordered set $P$, which is called an order polytope ${\mathcal{O}}_{P}$. (See [13, Chapter 4] and [14].) The authors propose to consult [3], [8], [15] and [16] on the related work on order polytopes. One can verify immediately that the primitive outer normals of each facet of ${\mathcal{O}}_{P}$ one-to-one corresponds to each vertex of ${\mathcal{Q}}_{P}$. Now ${\mathcal{O}}_{P}$ is Gorenstein if and only if $P$ is pure, i.e., all maximal chains of $\hat{P}$ have the same length. When $P$ is pure, let $l$ denote the length of each maximal chain of $\hat{P}$. Then the dilated polytope $l{\mathcal{O}}_{P}$ contains a unique integer point $\alpha\in{\mathbb{Z}}^{d}$, where $d$ is the cardinality of $P$, belonging to the interior of $l{\mathcal{O}}_{P}$. Then the dual polytope of the Gorenstein Fano polytope $l{\mathcal{O}}_{P}-\alpha$ coincides with ${\mathcal{Q}}_{P}$. Thus, when $P$ is pure, we can associate ${\mathcal{Q}}_{P}$ with the dual polytope of an order polytope ${\mathcal{O}}_{P}$.

2. When is ${\mathcal{Q}}_{P}$ ${\mathbb{Q}}$-factorial?

Let $P=\{y_{1},\ldots,y_{d}\}$ be a finite partially ordered set and ${\hat{P}}=P\cup\{y_{0},y_{d+1}\}$, where $y_{0}={\hat{0}}$ and $y_{d+1}={\hat{1}}$. A sequence $\Gamma=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}})$ is called a path in ${\hat{P}}$ if $\Gamma$ is a path in the Hasse diagram of ${\hat{P}}$. In other words, $\Gamma=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}})$ is a path in ${\hat{P}}$ if $y_{i_{j}}\neq y_{i_{k}}$ for all $1\leq j<k\leq m$ and if $\{y_{i_{j}},y_{i_{j+1}}\}$ is an edge of ${\hat{P}}$ for all $1\leq j\leq m-1$. In particular, if $\{y_{i_{1}},y_{i_{m}}\}$ is also an edge of $\hat{P}$, then $\Gamma$ is called a cycle. The length of a path $\Gamma=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}})$ is $\ell(\Gamma)=m-1$ or $\ell(\Gamma)=m$ if $\Gamma$ is a cycle.

A path $\Gamma=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m+1}})$ is called special if

[TABLE]

Given a special path $\Gamma=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}})$, there exists a unique function

[TABLE]

such that

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$\mu_{\Gamma}(y_{i_{j+1}})=\mu_{\Gamma}(y_{i_{j}})+1$ (resp. $\mu_{\Gamma}(y_{i_{j}})=\mu_{\Gamma}(y_{i_{j+1}})+1$) if $y_{i_{j}}<y_{i_{j+1}}$ (resp. $y_{i_{j}}>y_{i_{j+1}}$);

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$\min\{\mu_{\Gamma}(y_{i_{1}}),\mu_{\Gamma}(y_{i_{2}}),\ldots,\mu_{\Gamma}(y_{i_{m}})\}=0$.

In particular, $\Gamma$ is special if and only if $\mu_{\Gamma}(y_{i_{1}})=\mu_{\Gamma}(y_{i_{m}})$.

Similary, a special cycle is defined and given a special cycle $C$, there exists a unique function $\mu_{C}$ which is defined the same way as above.

Example 2.1**.**

Among the two paths and three cycles drawn below, each of one path and two cycles on the left-hand side is special; none of one path and one cycle on the right-hand side is special.

We say that a path $\Gamma=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m+1}})$ or a cycle $C=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}})$ of ${\hat{P}}$ belongs to a facet of ${\mathcal{Q}}_{P}$ if there is a facet ${\mathcal{F}}$ of ${\mathcal{Q}}_{P}$ with $\rho(\{y_{i_{j}},y_{i_{j+1}}\})\in{\mathcal{F}}$ for all $1\leq j\leq m$, where $y_{i_{m+1}}=y_{i_{1}}$.

We say that a cycle $C=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}})$ is very special if $C$ is special and if $\{y_{0},y_{d+1}\}\not\subset\{y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}}\}$.

Lemma 2.2**.**

(a)* Let $C=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}})$ be a cycle in ${\hat{P}}$. If $C$ belongs to a facet of ${\mathcal{Q}}_{P}$, then $C$ is a special cycle. In particular, $C$ is a very special cycle or $C$ contains a special path $(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{r+1}})$ with $y_{i_{1}}=y_{0}$ and $y_{i_{r+1}}=y_{d+1}$.*

(b)* Let $\Gamma=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}})$ with $y_{i_{1}}=y_{0}$ and $y_{i_{m}}=y_{d+1}$ be a path in ${\hat{P}}$. If $\Gamma$ belongs to a facet of ${\mathcal{Q}}_{P}$, then $\Gamma$ is a special path.*

Proof.

(a) Let $a_{1}x_{1}+\cdots+a_{d}x_{d}=1$ with each $a_{i}\in{\mathbb{Q}}$ denote the equation of the supporting hyperplane of ${\mathcal{Q}}_{P}$ which defines the facet. Since $\{y_{i_{j}},y_{i_{j+1}}\}$ are edges of ${\hat{P}}$ for $1\leq j\leq m$, where $y_{i_{m+1}}=y_{i_{1}}$, it follows that $a_{i_{j}}-a_{i_{j+1}}=q_{j}$, where $q_{j}=1$ if $y_{i_{j}}<y_{i_{j+1}}$ and $q_{j}=-1$ if $y_{i_{j}}>y_{i_{j+1}}$. Now,

[TABLE]

Hence $C$ must be special.

Suppose that $\{y_{0},y_{d+1}\}\subset\{y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}}\}$. Let $y_{i_{1}}=y_{0}$ and $y_{i_{r+1}}=y_{d+1}$. Since $\{y_{i_{j}},y_{i_{j+1}}\}$, $1\leq j\leq r$, are edges of $\hat{P}$, one has $-a_{i_{2}}=1$, $a_{i_{r}}=1$ and $a_{i_{j}}-a_{i_{j+1}}=q_{j}$ for $j=2,3,\ldots,r-1.$ On the one hand, one has

[TABLE]

On the other hand, one has

[TABLE]

It then follows that one must be $\mu_{C}(y_{0})=\mu_{C}(y_{d+1})$. Let $\Gamma=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{r+1}}).$ Then it is clear that $\mu_{\Gamma}(y_{0})=\mu_{\Gamma}(y_{d+1}).$ Thus $\Gamma$ is a special path. Hence $C$ contains a special path $\Gamma$.

(b) A proof can be given by the similar way of a proof of (a).

Let $P$ be a finite partially ordered set and $y,z\in\hat{P}$ with $y<z$. The distance of $y$ and $z$ in $\hat{P}$ is the smallest integer $s$ for which there is a saturated chain $Q=\{z_{0},z_{1},\ldots,z_{s}\}$ with

[TABLE]

Let $\operatorname{dist}_{\hat{P}}(y,z)$ denote the distance of $y$ and $z$ in $\hat{P}$.

Theorem 2.3**.**

Let $P=\{y_{1},\ldots,y_{d}\}$ be a finite partially ordered set and $\hat{P}=P\cup\{y_{0},y_{d+1}\}$, where $y_{0}={\hat{0}}$ and $y_{d+1}={\hat{1}}$. Then the following conditions are equivalent:

1. (i)

${\mathcal{Q}}_{P}$* is ${\mathbb{Q}}$-factorial;* 2. (ii)

${\mathcal{Q}}_{P}$* is smooth;* 3. (iii)

${\hat{P}}$* possesses no very special cycle $C=(y_{i_{1}},\ldots,y_{i_{m}})$ such that *

[TABLE]

for all $1\leq a,\,b\leq m$ with $y_{i_{b}}<y_{i_{a}}$, and

[TABLE]

for all $1\leq a,\,b\leq m$, and no special path $\Gamma=(y_{i_{1}},\ldots,y_{i_{m}})$ with $y_{i_{1}}=y_{0}$ and $y_{i_{m}}=y_{d+1}$ such that

[TABLE]

for all $1\leq a,\,b\leq m$ with $y_{i_{b}}<y_{i_{a}}$.

Proof.

((i) $\Rightarrow$ (iii)) If $C=(y_{i_{1}},\ldots,y_{i_{m}})$ is a cycle in ${\hat{P}}$ with $y_{i_{m+1}}=y_{1}$, then

[TABLE]

where $q_{j}=1$ if $y_{i_{j}}<y_{i_{j+1}}$ and $q_{j}=-1$ if $y_{i_{j}}>y_{i_{j+1}}$. Thus in particular $\rho(\{y_{i_{j}},y_{i_{j+1}}\})$, $1\leq j\leq m$, cannot be affinely independent if $C$ is special.

Now, suppose that ${\hat{P}}$ possesses a very special cycle $C=(y_{i_{1}},\ldots,y_{i_{m}})$ which satisfies the inequalities (1) and (2). Our work is to show that ${\mathcal{Q}}_{P}$ is not simplicial. Let $v_{j}=\rho(\{y_{i_{j}},y_{i_{j+1}}\})$, $1\leq j\leq m$, where $y_{i_{m+1}}=y_{i_{1}}$. Since $v_{1},\ldots,v_{m}$ cannot be affinely independent, to show that ${\mathcal{Q}}_{P}$ is not simplicial, what we must prove is the existence of a face of ${\mathcal{Q}}_{P}$ which contains the vertices $v_{1},\ldots,v_{m}$.

Let $a_{1},\ldots,a_{d}$ be integers. Write ${\mathcal{H}}\subset{\mathbb{R}}^{d}$ for the hyperplane defined the equation $a_{1}x_{1}+\cdots+a_{d}x_{d}=1$ and ${\mathcal{H}}^{(+)}\subset{\mathbb{R}}^{d}$ for the closed half-space defined by the inequality $a_{1}x_{1}+\cdots+a_{d}x_{d}\leq 1$. We will determine $a_{1},\ldots,a_{d}$ such that ${\mathcal{H}}$ is a supporting hyperplane of a face ${\mathcal{F}}$ of ${\mathcal{Q}}_{P}$ with $\{v_{1},\ldots,v_{m}\}\subset{\mathcal{F}}$ and with ${\mathcal{Q}}_{P}\subset{\mathcal{H}}^{(+)}$.

First Step. It follows from $(\ref{bbbbb})$ that

[TABLE]

By using $(\ref{aaaaa})$, if $y_{0}\in\{y_{i_{1}},\ldots,y_{i_{m}}\}$, then the left-hand side of $(\ref{ccccc})$ is equal to $\mu_{C}(y_{0})$. Similarly, if $y_{d+1}\in\{y_{i_{1}},\ldots,y_{i_{m}}\}$, then the right-hand side of $(\ref{ccccc})$ is equal to $\mu_{C}(y_{d+1})$.

Now, fix an arbitrary integer $a$ with

[TABLE]

However, exceptionally, if $y_{0}\in\{y_{i_{1}},\ldots,y_{i_{m}}\}$, then $a=\mu_{C}(y_{0})$. If $y_{d+1}\in\{y_{i_{1}},\ldots,y_{i_{m}}\}$, then $a=\mu_{C}(y_{d+1})$. Let $a_{i_{j}}=a-\mu_{C}({y_{i_{j}}})$ for $1\leq j\leq m$. Then one has

[TABLE]

Moreover, it follows easily that each $v_{j}$ lies on the hyperplane of ${\mathbb{R}}^{d}$ defined by the equation

[TABLE]

Second Step. Let $A={\hat{P}}\setminus(\{y_{0},y_{d+1}\}\cup\{y_{i_{1}},\ldots,y_{i_{m}}\})$ and $y_{i}\in A$.

• •

Suppose that there is $y_{i_{j}}$ with $y_{i_{j}}<y_{i}$ and that there is no $y_{i_{k}}$ with $y_{i_{k}}>y_{i}$. Then we define $a_{i}$ by setting

[TABLE]

• •

Suppose that there is no $y_{i_{j}}$ with $y_{i_{j}}<y_{i}$ and that there is $y_{i_{k}}$ with $y_{i_{k}}>y_{i}$. Then we define $a_{i}$ by setting

[TABLE]

• •

Suppose that there is $y_{i_{j}}$ with $y_{i_{j}}<y_{i}$ and that there is $y_{i_{k}}$ with $y_{i_{k}}>y_{i}$. Then either

[TABLE]

or

[TABLE]

must be zero. In fact, if $b_{i}\neq 0$ and $c_{i}\neq 0$, then there are $j$ and $k$ with $a_{i_{j}}>\operatorname{dist}_{\hat{P}}(y_{i_{j}},y_{i})$ and $-a_{i_{k}}>\operatorname{dist}_{\hat{P}}(y_{i},y_{i_{k}})$. Since $\mu_{C}(y_{i_{k}})-\mu_{C}(y_{i_{j}})=a_{i_{j}}-a_{i_{k}}$ and since $\operatorname{dist}_{\hat{P}}(y_{i_{j}},y_{i})+\operatorname{dist}_{\hat{P}}(y_{i},y_{i_{k}})\geq\operatorname{dist}_{\hat{P}}(y_{i_{j}},y_{i_{k}})$, it follows that

[TABLE]

This contradicts $(\ref{aaaaa})$. Hence either $b_{i}=0$ or $c_{i}=0$. If $b_{i}\neq 0$, then we set $a_{i}=b_{i}$. If $c_{i}\neq 0$, then we set $a_{i}=c_{i}$. If $b_{i}=c_{i}=0$, then we set $a_{i}=0$.

• •

Suppose that there is no $y_{i_{j}}$ with $y_{i_{j}}<y_{i}$ and that there is no $y_{i_{k}}$ with $y_{i_{k}}>y_{i}$. Then we set $a_{i}=0$.

Third Step. Finally, we finish determining the integers $a_{1},\ldots,a_{d}$. Let ${\mathcal{H}}\subset{\mathbb{R}}^{d}$ denote the hyperplane defined by the equation $a_{1}x_{1}+\ldots+a_{d}x_{d}=1$ and ${\mathcal{H}}^{(+)}\subset{\mathbb{R}}^{d}$ the closed half-space defined by the inequality $a_{1}x_{1}+\ldots+a_{d}x_{d}\leq 1$. Since each $v_{j}$ lies on the hyperplane ${\mathcal{H}}$, in order for ${\mathcal{F}}={\mathcal{H}}\cap{\mathcal{Q}}_{P}$ to be a face of ${\mathcal{Q}}_{P}$, it is required to show ${\mathcal{Q}}_{P}\subset{\mathcal{H}}^{(+)}$. Let $\{y_{i},y_{j}\}$ with $y_{i}<y_{j}$ be an edge of ${\hat{P}}$.

• •

Let $y_{i}\in\{y_{i_{1}},\ldots,y_{i_{m}}\}$ with $y_{j}\not\in\{y_{i_{1}},\ldots,y_{i_{m}}\}$. If $y_{j}\neq y_{d+1}$, then

[TABLE]

where $a_{0}=0$. Thus $a_{i}-a_{j}\leq 1$. If $y_{j}=y_{d+1}$, then by using $(\ref{qqqqq})$ one has $a_{i}\leq 1$, as desired.

• •

Let $y_{j}\in\{y_{i_{1}},\ldots,y_{i_{m}}\}$ with $y_{i}\not\in\{y_{i_{1}},\ldots,y_{i_{m}}\}$. If $y_{i}\neq y_{0}$, then

[TABLE]

where $a_{d+1}=0$. Thus $a_{i}-a_{j}\leq 1$. If $y_{i}=y_{0}$, then by using $(\ref{qqqqq})$ one has $-a_{j}\leq 1$, as desired.

Let $A^{\prime}={\hat{P}}\setminus\{y_{i_{1}},\ldots,y_{i_{m}}\}$. Write $B$ for the subset of $A^{\prime}$ consisting of those $y_{i}\in A^{\prime}$ such that there is $j$ with $y_{i_{j}}<y_{i}$. Write $C$ for the subset of $A^{\prime}$ consisting of those $y_{i}\in A^{\prime}$ such that there is $k$ with $y_{i}<y_{i_{k}}$. Again, let $e=\{y_{i},y_{j}\}$ with $y_{i}<y_{j}$ be an edge of ${\hat{P}}$. In each of the nine cases below, a routine computation easily yields that $\rho(e)\in{\mathcal{H}}^{(+)}$.

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$y_{i}\in B\setminus C$ and $y_{j}\in B\setminus C$;

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$y_{i}\in C\setminus B$ and $y_{j}\in C\setminus B$;

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$y_{i}\in C\setminus B$ and $y_{j}\in B\setminus C$;

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$y_{i}\in C\setminus B$ and $y_{j}\in B\cap C$;

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$y_{i}\in C\setminus B$ and $y_{j}\not\in B\cup C$;

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$y_{i}\in B\cap C$ and $y_{j}\in B\cap C$;

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$y_{i}\in B\cap C$ and $y_{j}\in B\setminus C$;

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$y_{i}\not\in B\cup C$ and $y_{j}\in B\setminus C$;

• •

$y_{i}\not\in B\cup C$ and $y_{j}\not\in B\cup C$.

For example, in the first case, a routine computation is as follows. Let $y_{j}\neq y_{d+1}$. Let $a_{i}=0$. Then, since $a_{j}\geq 0$, one has $a_{i}-a_{j}\leq 1$. Let $a_{i}>0$. Then, since $a_{j}\geq a_{i}-1$, one has $a_{i}-a_{j}\leq 1$. Let $y_{j}=y_{d+1}$ and $a_{i}>0$. Then there is $j$ with $a_{i}=a_{i_{j}}-\operatorname{dist}_{\hat{P}}(y_{i_{j}},y_{i})$. By using $(\ref{qqqqq})$ one has $a_{i_{j}}\leq\operatorname{dist}_{\hat{P}}(y_{i_{j}},y_{d+1})$. Thus $a_{i}\leq\operatorname{dist}_{\hat{P}}(y_{i_{j}},y_{d+1})-\operatorname{dist}_{\hat{P}}(y_{i_{j}},y_{i})$. Hence $a_{i}\leq 1$, as required.

Fourth step. Suppose that ${\hat{P}}$ possesses a special path $\Gamma=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}})$ with $y_{i_{1}}=y_{0}$ and $y_{i_{m}}=y_{d+1}$ which satisfies the inequalities (3). Then one has

[TABLE]

where $q_{j}=1$ if $y_{i_{j}}<y_{i_{j+1}}$ and $q_{j}=-1$ if $y_{i_{j}}>y_{i_{j+1}}$. Thus $\rho(\{y_{i_{j}},y_{i_{j+1}}\})$, $1\leq j\leq m-1$, cannot be affinely independent. Our work is to show that ${\mathcal{Q}}_{P}$ is not simplicial. In this case, however, the same discussion can be given as the case which ${\hat{P}}$ possesses a very special cycle. (We should set $a=\mu_{\Gamma}(y_{0})$ ($=\mu_{\Gamma}(y_{d+1})$).)

((iii) $\Rightarrow$ (i)) Now, suppose that ${\mathcal{Q}}_{P}$ is not ${\mathbb{Q}}$-factorial. Thus ${\mathcal{Q}}_{P}$ possesses a facet ${\mathcal{F}}$ which is not a simplex. Let $v_{1},\ldots,v_{n}$ denote the vertices of ${\mathcal{F}}$, where $n>d$, and $e_{j}$ the edge of ${\hat{P}}$ with $v_{j}=\rho(e_{j})$ for $1\leq j\leq n$. Let $a_{1}x_{1}+\cdots+a_{d}x_{d}=1$ denote the equation of the supporting hyperplane ${\mathcal{H}}\subset{\mathbb{R}}^{d}$ of ${\mathcal{Q}}_{P}$ with ${\mathcal{F}}={\mathcal{Q}}_{P}\cap{\mathcal{H}}$ and with ${\mathcal{Q}}_{P}\subset{\mathcal{H}}^{(+)}$, where ${\mathcal{H}}^{(+)}\subset{\mathbb{R}}^{d}$ is the closed-half space defined by the inequality $a_{1}x_{1}+\cdots+a_{d}x_{d}\leq 1$. Since $v_{1},\ldots,v_{n}$ are not affinely independent, there is $(r_{1},\ldots,r_{n})\in{\mathbb{Z}}^{n}$ with $(r_{1},\ldots,r_{n})\neq(0,\ldots,0)$ such that $r_{1}v_{1}+\cdots+r_{n}v_{n}=(0,\ldots,0)$. By removing $r_{j}$ with $r_{j}=0$, we may assume that $r_{1}v_{1}+\cdots+r_{n^{\prime}}v_{n^{\prime}}=(0,\ldots,0)$, where $r_{j}\neq 0$ for $1\leq j\leq n^{\prime}$ with $r_{1}+\cdots+r_{n^{\prime}}=0$. Let $e_{j}=\{y_{i_{j}},y_{i_{j^{\prime}}}\}$ with $1\leq i_{j},i_{j^{\prime}}\leq d$. If either $y_{i_{j}}$ or $y_{i_{j^{\prime}}}$ appears only in $e_{j}$ among the edges $e_{1},\ldots,e_{n^{\prime}}$, then $r_{j}=0$. Hence both $y_{i_{j}}$ and $y_{i_{j^{\prime}}}$ must appear in at least two edges among $e_{1},\ldots,e_{n^{\prime}}$. Let $G$ denote the subgraph of the Hasse diagram of ${\hat{P}}$ with the edges $e_{1},\ldots,e_{n^{\prime}}$. Then there is no end point of $G$ in $P$. Thus $G$ possesses a cycle of $\hat{P}$ or $G$ is a path of $\hat{P}$ from $y_{0}$ to $y_{d+1}$. Since $v_{1},\ldots,v_{n^{\prime}}$ are contained in the facet ${\mathcal{F}}$, Lemma 2.2 says that every cycle in $G$ is very special or else $G$ contains a special path.

Suppose that $G$ possesses a very special cycle $C=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}})$. Our goal is to show that $C$ satisfies the inequalities $(\ref{aaaaa})$ and $(\ref{bbbbb})$.

Let $y_{k_{0}}<y_{k_{1}}<\cdots<y_{k_{\ell}}$ be a saturated chain of ${\hat{P}}$ with $\ell=\operatorname{dist}_{\hat{P}}(y_{k_{0}},y_{k_{\ell}})$ such that each of $y_{k_{0}}$ and $y_{k_{\ell}}$ belongs to $\{y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}}\}$. We claim

[TABLE]

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Let $y_{0}\neq y_{k_{0}}$ and $y_{d+1}\neq y_{k_{\ell}}$. Since ${\bf e}_{k_{j}}-{\bf e}_{k_{j+1}}\in{\mathcal{Q}}_{P}$, one has $a_{k_{j}}-a_{k_{j+1}}\leq 1$ for each $0\leq j\leq\ell-1$. Hence $a_{k_{0}}-a_{k_{\ell}}\leq\ell$. On the other hand, $a_{k_{0}}-a_{k_{\ell}}=\mu_{C}(y_{k_{\ell}})-\mu_{C}(y_{k_{0}})$. Thus $\mu_{C}(y_{k_{\ell}})-\mu_{C}(y_{k_{0}})\leq\operatorname{dist}_{\hat{P}}(y_{k_{0}},y_{k_{\ell}})$.

• •

Let $y_{0}=y_{k_{0}}$ and $y_{d+1}\neq y_{k_{\ell}}$. Since $-{\bf e}_{k_{1}}\in{\mathcal{Q}}_{P}$, one has $-a_{k_{1}}\leq 1$. Since ${\bf e}_{k_{j}}-{\bf e}_{k_{j+1}}\in{\mathcal{Q}}_{P}$, one has $a_{k_{j}}-a_{k_{j+1}}\leq 1$ for each $1\leq j\leq\ell-1$. Hence $a_{k_{1}}-a_{k_{\ell}}\leq\ell-1$. Thus $-a_{k_{\ell}}\leq\ell$. On the other hand, $-a_{k_{\ell}}=\mu_{C}(y_{k_{\ell}})-\mu_{C}(y_{k_{0}})$. Thus $\mu_{C}(y_{k_{\ell}})-\mu_{C}(y_{k_{0}})\leq\operatorname{dist}_{\hat{P}}(y_{k_{0}},y_{k_{\ell}})$.

• •

Let $y_{0}\neq y_{k_{0}}$ and $y_{d+1}=y_{k_{\ell}}$. Since ${\bf e}_{k_{j}}-{\bf e}_{k_{j+1}}\in{\mathcal{Q}}_{P}$, one has $a_{k_{j}}-a_{k_{j+1}}\leq 1$ for each $0\leq j\leq\ell-2$. Hence $a_{k_{0}}-a_{k_{\ell-1}}\leq\ell-1$. Since ${\bf e}_{k_{\ell-1}}\in{\mathcal{Q}}_{P}$, one has $a_{k_{\ell-1}}\leq 1$. Hence $a_{k_{0}}\leq\ell$. On the other hand, $a_{k_{0}}=\mu_{C}(y_{k_{\ell}})-\mu_{C}(y_{k_{0}})$. Thus $\mu_{C}(y_{k_{\ell}})-\mu_{C}(y_{k_{0}})\leq\operatorname{dist}_{\hat{P}}(y_{k_{0}},y_{k_{\ell}})$.

Finally, fix arbitrary $y_{i_{j}}$ and $y_{i_{k}}$ with $\mu_{C}(y_{i_{j}})<\mu_{C}(y_{i_{k}})$. Then $-a_{i_{k}}\leq\operatorname{dist}_{\hat{P}}(y_{0},y_{i_{k}})$ and $a_{i_{j}}\leq\operatorname{dist}_{\hat{P}}(y_{i_{j}},y_{d+1})$. We claim

[TABLE]

If $y_{i_{j}}\neq y_{0}$ and $y_{i_{k}}\neq y_{d+1}$, then $a_{i_{j}}-a_{i_{k}}=\mu_{C}(y_{i_{k}})-\mu_{C}(y_{i_{j}})$. If $y_{i_{j}}=y_{0}$ and $y_{i_{k}}\neq y_{d+1}$, then $-a_{i_{k}}=\mu_{C}(y_{i_{k}})-\mu_{C}(y_{i_{j}})$. If $y_{i_{j}}\neq y_{0}$ and $y_{i_{k}}=y_{d+1}$, then $a_{i_{j}}=\mu_{C}(y_{i_{k}})-\mu_{C}(y_{i_{j}})$. Hence the required inequality follows immediately.

Suppose that $G$ contains a special path $\Gamma=(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{m}})$ with $y_{i_{1}}=y_{0}$ and $y_{i_{m}}=y_{d+1}$. Our goal is to show that $C$ satisfies the inequalities $(\ref{eeeee})$. Now the same discussion can be given as above.

((i) $\Rightarrow$ (ii)) If $P$ is a totally ordered set, then ${\mathcal{Q}}_{P}$ is a $d$-simplex with the vertices, say, $-{\bf e}_{1},{\bf e}_{1}-{\bf e}_{2},\ldots,{\bf e}_{d-1}-{\bf e}_{d},{\bf e}_{d}$. Thus in particular ${\mathcal{Q}}_{P}$ is smooth.

Now, suppose that $P$ is not a totally ordered set. Then ${\hat{P}}$ possesses a cycle. Let $C=(y_{i_{1}},\ldots,y_{i_{m}})$ be a cycle in ${\hat{P}}$. If $C$ is not special, then Lemma 2.2 (a) says that $C$ cannot belong to a facet of ${\mathcal{Q}}_{P}$. If $C$ is special, then as was shown in the proof of (i) $\Rightarrow$ (iii) it follows that $\rho(\{y_{i_{j}},y_{i_{j+1}}\})$, $1\leq j\leq m$, where $y_{i_{m+1}}=y_{i_{1}}$, are not affinely independent. Hence there is no facet ${\mathcal{F}}$ of ${\mathcal{Q}}_{P}$ with $\rho(\{y_{i_{j}},y_{i_{j+1}}\})\in{\mathcal{F}}$ for all $1\leq j\leq m$.

Let ${\mathcal{F}}$ be an arbitrary facet of ${\mathcal{Q}}_{P}$ with $d$ vertices $v_{j}=\rho(e_{j})$, $1\leq j\leq d$. Let $G$ denote the subgraph of the Hasse diagram of ${\hat{P}}$ with the edges $e_{1},\ldots,e_{d}$ and $V(G)$ the vertex set of $G$. Since ${\mathcal{F}}$ is of dimension $d-1$, it follows that, for each $1\leq i\leq d$, there is a vertex of ${\mathcal{F}}$ whose $i$th coordinate is nonzero. Hence $P\subset V(G)$. Suppose that $P=V(G)$. Since $G$ has $d$ edges, it follows that $G$ possesses a cycle, a contradiction. Hence either $y_{0}\in V(G)$ or $y_{d+1}\in V(G)$.

What we must prove is that the determinant

[TABLE]

is equal to $\pm 1$. Let, say, $e_{1}=\{y_{1},y_{d+1}\}$. Thus $v_{1}=(1,0,\ldots,0)$. Now, since $G$ is a forest, by arranging the numbering of the elements of $P$ if necessary, one has

[TABLE]

with each $a_{ij}\in\{1,0,-1\}$. Since the determinant $(\ref{determinant})$ is nonzero, it follows that the determinant $(\ref{determinant})$ is equal to $\pm 1$, as desired.

((ii) $\Rightarrow$ (i)) In general, every smooth Fano polytope is ${\mathbb{Q}}$-factorial.

Corollary 2.4**.**

Suppose that a finite partially ordered set $P$ is pure. Then the following conditions are equivalent:

1. (i)

${\mathcal{Q}}_{P}$* is ${\mathbb{Q}}$-factorial;* 2. (ii)

${\mathcal{Q}}_{P}$* is smooth;* 3. (iii)

$P$* is a disjoint union of chains.*

Proof.

If $P$ is pure, then every cycle of ${\hat{P}}$ is special and, in addition, satisfies the inequalities $(\ref{aaaaa})$ and $(\ref{bbbbb})$. Moreover, every path from $y_{0}$ to $y_{d+1}$ cannot be special. Hence ${\mathcal{Q}}_{P}$ is ${\mathbb{Q}}$-factorial if and only if there is no very special cycle, i.e., every cycle of ${\hat{P}}$ possesses both ${\hat{0}}$ and ${\hat{1}}$. Now if there is a connected component of $P$ which is not a chain, then $P$ possesses a very special cycle. Thus ${\mathcal{Q}}_{P}$ is ${\mathbb{Q}}$-factorial if and only if $P$ does not possess a connected component which is not a chain. In other words, ${\mathcal{Q}}_{P}$ is ${\mathbb{Q}}$-factorial if and only if $P$ is a disjoint union of chains, as desired.

Example 2.5**.**

Among the five finite partially ordered sets drawn below, each of the three finite partially ordered sets on the left-hand side yields a ${\mathbb{Q}}$-factorial Fano polytope; none of the two finite partially ordered sets on the right-hand side yields a ${\mathbb{Q}}$-factorial Fano polytope.

Let $P$ and $P^{\prime}$ be finite partially ordered sets. Then one can verify easily that ${\mathcal{Q}}_{P}$ is isomorphic with ${\mathcal{Q}}_{P^{\prime}}$ as a convex polytope if and only if $P$ is isomorphic with $P^{\prime}$ or with the dual finite partially ordered set of $P^{\prime}$ as a finite partially ordered set.

On the following table drawn below, the number of finite partially ordered sets with $d(\leq 8)$ elements, up to isomorphic and up to isomorphic with dual finite partially ordered sets, is written in the second row. Moreover, among those, the number of finite partially ordered sets constructing smooth Fano polytopes is written in the third row.

[TABLE]