[labelstyle=]

# The minimal projective bundle dimension
and toric $2$-Fano manifolds

Carolina Araujo Carolina Araujo, IMPA, Estrada Dona Castorina 110, 22460-320
Rio de Janeiro, Brazil<EMAIL_ADDRESS>,  Roya Beheshti Roya Beheshti,
Department of Mathematics & Statistics, Washington University in St. Louis,
St. Louis, MO, 63130<EMAIL_ADDRESS>,  Ana-Maria Castravet Ana-Maria
Castravet, Université Paris-Saclay, UVSQ, Laboratoire de Mathématiques de
Versailles, 45 Avenue des États Unis, 78035 Versailles, France  ana-
<EMAIL_ADDRESS>,  Kelly Jabbusch Kelly Jabbusch, Department of
Mathematics & Statistics, University of Michigan–Dearborn, 4901 Evergreen Rd,
Dearborn, Michigan, 48128, USA<EMAIL_ADDRESS>,  Svetlana Makarova
Svetlana Makarova, Department of Mathematics, University of Pennsylvania, 209
S 33rd St, Philadelphia, PA 19104, USA<EMAIL_ADDRESS>,  Enrica Mazzon
Enrica Mazzon, Fakultät für Mathematik, Universität Regensburg,
Universitätsstrasse 31, 93040 Regensburg, Germany, and Department of
Mathematics, University of Michigan, Ann Arbor, Michigan, 48109, USA
<EMAIL_ADDRESS>and  Nivedita Viswanathan Nivedita
Viswanathan, Department of Mathematical Sciences, Loughborough University,
Loughborough, LE11 3TU, UK and School of Mathematical Sciences, University
Park Campus, The University of Nottingham, Nottingham, NG7 2RD, UK
<EMAIL_ADDRESS>
###### Abstract.

Motivated by the problem of classifying toric $2$-Fano manifolds, we introduce
a new invariant for smooth projective toric varieties, the minimal projective
bundle dimension. This invariant $m(X)\in\\{1,\dots,\dim(X)\\}$ captures the
minimal degree of a dominating family of rational curves on $X$ or,
equivalently, the minimal length of a centrally symmetric primitive relation
for the fan of $X$. We classify smooth projective toric varieties with
$m(X)\geq\dim(X)-2$, and show that projective spaces are the only $2$-Fano
manifolds among smooth projective toric varieties with
$m(X)\in\\{1,\dim(X)-2,\dim(X)-1,\dim(X)\\}$.

###### Contents

  1. 1 Introduction
  2. 2 Primitive collections
    1. 2.1 Notation and background
    2. 2.2 The minimal $\mathbb{P}$-dimension
    3. 2.3 Some properties of primitive collections
    4. 2.4 Primitive collections on toric Fano manifolds
  3. 3 Toric Fano manifolds with $m(X)=1$
  4. 4 Proof of 1.3
    1. 4.1 First case: $\\{x,y,z\\}$ is a primitive collection
      1. 4.1.1 Degree one
      2. 4.1.2 Degree two
    2. 4.2 Second case: none of $\\{x,y,z\\}$ form a primitive collection
  5. 5 Toric Fano manifolds with $m(X)=n-2$
  6. A Code for computing primitive collections

##  1\. Introduction

Fano varieties are projective varieties with positive first Chern class. Over
the complex numbers, this condition is equivalent to the existence of a metric
with positive Ricci curvature. Basic examples of Fano varieties include
projective spaces and Grassmannians. The positivity condition has further
geometric implications, e.g., Fano varieties over the complex numbers are
simply connected. This has an analogue on the algebro-geometric side: any Fano
variety is covered by rational curves [Mor79], and is in fact rationally
connected [KMM92, Cam92], i.e., there are rational curves connecting any two
of its points. In a series of papers, de Jong and Starr introduce and
investigate possible candidates for the notion of _higher rational
connectedness_ [dHS11, dS07, dS06a, dS06b, Sta06], inspired by the natural
analogue in topology. In particular, in [dS06a] they define $2$-Fano
manifolds. A smooth projective variety $X$ is $2$-Fano if it is Fano and its
second Chern character
$\operatorname{ch}_{2}(T_{X})=\frac{1}{2}c_{1}(T_{X})^{2}-c_{2}(T_{X})$ is
positive, i.e., $\operatorname{ch}_{2}(T_{X})\cdot S>0$ for every surface $S$
in $X$. In a similar way, one can define $k$-Fano varieties for any $k\geq 2$,
and aim at their classification. For instance, $\mathbb{P}^{n}$ is $n$-Fano,
and it is conjectured that it is the only $n$-dimensional $n$-Fano manifold.
The geometry of higher Fano manifolds has been fairly investigated, and in
several special cases they are shown to enjoy the expected nice properties.
For instance, $2$-Fano manifolds satisfying some mild assumptions are covered
by rational surfaces [dS07], and similar results hold for higher Fano
manifolds [Suz21], [Nag19]. There is a classification of $2$-Fano manifolds of
high index [AC13] and, more recently, a classification of homogeneous $2$-Fano
manifolds [Ara+22]. On the other hand, very few examples of higher Fano
manifolds are known. Quite strikingly, all known examples of $2$-Fano
manifolds have Picard rank $1$ and relatively large index.

It is natural for algebraic geometers to turn to the pool of toric varieties
when looking for intuition or examples. It is well known that projective
spaces are the only projective toric manifolds with Picard rank $1$. Thus, a
classification of toric $2$-Fano manifolds could either provide the first
examples of $2$-Fano manifolds with higher Picard rank, or it could be an
evidence that every $2$-Fano manifold has Picard rank $1$. Geometric
properties of a toric variety can often be checked in the combinatorics of the
associated fan. This bridge has been exploited in search of new examples of
toric $2$-Fano manifolds [Nob11], [Nob12], [Sat12], [Sat16], [SS20], [SSS21],
[Shr20]. Despite the efforts, a complete (computer aided) classification is
only known up to dimension $8$ [Nob11], [SSS21], and projective spaces remain
the only known examples of toric $2$-Fano manifolds. The sparsity of higher
Fano manifolds leads to the following conjecture.

######  Conjecture 1.1.

([SSS21, Conjecture 4.3]) The only toric 2-Fano manifolds are projective
spaces.

In this paper, we propose a new strategy to approach Conjecture 1.1. We follow
the philosophy introduced in [AC12], namely, to investigate $2$-Fano manifolds
by studying their _minimal dominating families of rational curves_. By
[CFH14], minimal dominating families of rational curves on a smooth projective
toric variety $X$ correspond to primitive relations of the form

(1) $x_{0}+\dots+x_{m}=0,$

satisfied by some of the primitive integral generators $x_{i}$ of the
corresponding fan. These primitive relations are called _centrally symmetric
of order $m+1$_. By [CFH14], a centrally symmetric primitive relation of order
$m+1$ yields a $\mathbb{P}^{m}$-bundle structure $X^{\circ}\to T$ on a dense
open subset $X^{\circ}$ of $X$. If $\dim(T)\geq 1$, and the complement
$X\setminus X^{\circ}$ has codimension at least $2$ in $X$, then one can
construct a complete surface $S\subset X^{\circ}$ such that
$\operatorname{ch}_{2}(T_{X})\cdot S\leq 0$, showing that $X$ is not $2$-Fano.
So our basic strategy consists of trying to describe, in a rather explicit
way, a suitable birational map $\varphi:X\dashrightarrow Y$ transforming $X$
into a projective toric variety $Y$ admitting a $\mathbb{P}^{m}$-bundle
structure on a big open subset. We then hope to be able to compare the second
Chern characters $\operatorname{ch}_{2}(T_{X})$ and
$\operatorname{ch}_{2}(T_{Y})$ to show that $X$ is not $2$-Fano, except if
$X=\mathbb{P}^{m}$ and $\varphi$ is the identity.

To follow this strategy, we introduce a new invariant of a smooth projective
toric variety $X$, the _minimal projective bundle dimension_ of $X$, _minimal
$\mathbb{P}$-dimension_ in short, which is of independent interest (2.10):

$m(X)\ =\ \min\big{\\{}\ m\in\mathbb{Z}_{>0}\ \big{|}\ \text{ there is a
relation as in (\ref{CSR}) }\big{\\}}\ \in\ \\{1,\dots,\dim X\\}.$

By going through the database of toric Fano manifolds of low dimension and
computing their primitive collections, one obtains Table 1, indicating the
number of Fano manifolds for each value of $m$. Appendix A contains the code
used to compute primitive collections.

$\dim(X)$ | # Fanos | #($m$=1) | #($m$=2) | #($m$=3) | #($m$=4) | #($m$=5) | #($m$=6)
---|---|---|---|---|---|---|---
4 | 124 | 107 | 15 | 1 | 1 |  |
5 | 866 | 744 | 112 | 8 | 1 | 1 |
6 | 7622 | 6333 | 1174 | 105 | 8 | 1 | 1
Table 1. The minimal $\mathbb{P}$-dimension of toric Fano manifolds of low
dimension.

When $m(X)=1$, Casagrande constructs in [Cas03a] a sequence of blowdowns and
flips from $X$ to a toric variety admitting a global $\mathbb{P}^{1}$-bundle
structure. This allows us to make the basic strategy work, yielding the
following result.

######  Theorem 1.2.

Let $X$ be a smooth toric Fano variety with $m(X)=1$. Then $X$ is not
$2$-Fano.

Table 1 suggests that this result covers “most” toric varieties, and not just
the fringe cases.

Next we turn our attention to toric Fano manifolds $X$ with large values of
$m(X)$. Projective spaces are the only smooth projective toric varieties
admitting a centrally symmetric primitive relation of order $\dim(X)+1$. In
[CFH14, Proposition 3.8], Chen, Fu and Hwang classify toric Fano manifolds
admitting a centrally symmetric primitive relation of order $\dim(X)$. There
are three such varieties, and two of them also admit a centrally symmetric
primitive relation of order $2$. As a consequence, the only $n$-dimensional
toric Fano manifold $X$ with $m(X)=n-1$ is the blowup of $\mathbb{P}^{n}$
along a linear $\mathbb{P}^{n-2}$. In [BW22], Beheshti and Wormleighton
investigate smooth projective toric varieties admitting a centrally symmetric
primitive relation of order $\dim(X)-1$, showing that they have Picard rank
$\rho(X)\leq 5$. Most of these varieties also admit centrally symmetric
primitive relations of order $2$ or $3$, and we prove the following bound for
the remaining ones. 1.4 shows that this bound is sharp.

######  Theorem 1.3.

Let $X$ be a smooth toric Fano variety with $\dim(X)=n\geq 6$ and $m(X)\geq
3$. If $X$ has a centrally symmetric primitive relation of order $n-1$,

$x_{0}+x_{1}+\cdots+x_{n-2}=0,$

then $\rho(X)\leq 3$. Moreover, $m(X)=n-2$ and the above relation is the only
centrally symmetric primitive relation of $X$.

Using Theorem 1.3 and Batyrev’s description of smooth projective toric
varieties with Picard rank $3$, we are able to classify $n$-dimensional smooth
toric Fano varieties with $m(X)=n-2$. There are eight distinct isomorphism
classes when $n\geq 6$, which can be explicitly described. The following
statement summarizes the classification of toric Fano manifolds with
$m(X)\geq\dim(X)-2$.

######  Theorem 1.4.

We have the following classification of smooth toric Fano varieties with
$m(X)\geq\dim(X)-2$.

  1. (1)

The only $n$-dimensional smooth toric Fano variety $X$ with $m(X)=n$ is
$\mathbb{P}^{n}$.

  2. (2)

For $n\geq 3$, the only $n$-dimensional smooth toric Fano variety $X$ with
$m(X)=n-1$ is the blowup of $\mathbb{P}^{n}$ along a linear
$\mathbb{P}^{n-2}$.

  3. (3)

For $n\geq 6$, there are eight distinct isomorphism classes of $n$-dimensional
smooth toric Fano varieties $X$ with $m(X)=n-2$. Namely:

    1. (a)

$X=\mathbb{P}_{S}(\mathcal{E})$ is a $\mathbb{P}^{n-2}$-bundle over a toric
surface $S$, where $(S,\mathcal{E})$ is one of the following:

       * •

$S=\mathbb{P}^{2}$ and
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-2}$,

       * •

$S=\mathbb{P}^{2}$ and
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-3}$,

       * •

$S=\mathbb{P}^{2}$ and
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}(2)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-2}$,

       * •

$S=\mathbb{P}^{1}\times\mathbb{P}^{1}$ and
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,1)\oplus\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}^{\oplus
n-2}$,

       * •

$S=\mathbb{P}^{1}\times\mathbb{P}^{1}$ and
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,0)\oplus\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(0,1)\oplus\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}^{\oplus
n-3}$,

       * •

$S=\mathbb{F}_{1}$ and
$\mathcal{E}=\mathcal{O}_{\mathbb{F}_{1}}(e+f)\oplus\mathcal{O}_{\mathbb{F}_{1}}^{\oplus
n-2}$, where $e\subset\mathbb{F}_{1}$ is the $-1$-curve, and
$f\subset\mathbb{F}_{1}$ is a fiber of $\mathbb{F}_{1}\to\mathbb{P}^{1}$.

In the first three cases, $\rho(X)=2$, while in the latter three cases,
$\rho(X)=3$.

    2. (b)

Let
$Y\simeq\mathbb{P}_{\mathbb{P}^{2}}\big{(}\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-2}\big{)}$ be the blowup of $\mathbb{P}^{n}$ along a linear subspace
$L=\mathbb{P}^{n-3}$, and denote by $E\subset Y$ the exceptional divisor. Then
$X$ is the blowup of $Y$ along a codimension $2$ center $Z\subset Y$, where:

       * •

$Z$ is the intersection of $E$ with the strict transform of a hyperplane of
$\mathbb{P}^{n}$ containing the linear subspace $L$, or

       * •

$Z$ is the intersection of the strict transforms of two hyperplanes of
$\mathbb{P}^{n}$, one containing the linear subspace $L$, and the other one
not containing it.

In both cases, $\rho(X)=3$.

######  Corollary 1.5.

The projective space $\mathbb{P}^{n}$ is the only smooth $n$-dimensional toric
$2$-Fano variety with $m(X)\in\\{1,n-2,n-1,n\\}$.

This paper is organized as follows. In Section 2, we review some results from
toric geometry and fix notation. In particular, we discuss centrally symmetric
primitive relations on smooth projective toric varieties, describing
explicitly their open subsets admitting a projective space bundle structure
(Proposition 2.13). In Section 3, we study smooth toric Fano varieties with
$m(X)=1$, and prove Theorem 1.2. In Section 4, we investigate smooth
projective $n$-dimensional toric varieties admitting a centrally symmetric
primitive relation of order $n-1$, and prove Theorem 1.3. In Section 5, we use
this result, together with Batyrev’s description of smooth projective toric
varieties with Picard rank $3$, to prove Theorem 1.4.

Acknowledgements. This collaboration started as a working group at the ICERM
“Women in Algebraic Geometry Collaborative Research Workshop” in July 2020. A
great part of this work was developed during the follow-up “Collaborate@ICERM”
meeting in May 2022. We thank ICERM for the financial support and the great
working conditions provided to us during our visit. We are very grateful to
Cinzia Casagrande and Milena Hering for many enlightening discussions and
explanations about toric varieties, and to Will Reynolds for running his code
and providing us with precious data about primitive relations of toric Fano
manifolds.

Carolina Araujo was partially supported by CAPES/COFECUB, CNPq and FAPERJ
Research Fellowships. Roya Beheshti was supported by NSF grant DMS-2101935.
Ana-Maria Castravet was partially supported by the ANR 20-CE40-0023 grant
_FanoHK_. Enrica Mazzon was supported by the collaborative research center SFB
1085 _Higher Invariants - Interactions between Arithmetic Geometry and Global
Analysis_ funded by the Deutsche Forschungsgemeinschaft. Nivedita Viswanathan
was supported by the EPSRC New Horizons Grant No.EP/V048619/1.

##  2\. Primitive collections

###  2.1. Notation and background

A toric variety is a normal complex variety $X$ that contains a torus
$T=(\mathbb{C}^{*})^{n}$ as a dense open subset, together with an action of
$T$ on $X$ that extends the natural action of $T$ on itself. There is a one-
to-one correspondence between $n$-dimensional toric varieties and _fans_ in
$\mathbb{Q}^{n}$. Let $N$ be a free abelian group of rank $n$, and consider
the vector space $N_{\mathbb{Q}}=N\otimes_{\mathbb{Z}}\mathbb{Q}$. A _fan_ in
$N_{\mathbb{Q}}$ is a nonempty finite collection $\Sigma$ of strongly convex
polyhedral cones in $N_{\mathbb{Q}}$ such that every face of a cone in
$\Sigma$ is also a cone in $\Sigma$, and the intersection of two cones in
$\Sigma$ is a face of each. We write $\delta\prec\tau$ to express that
$\delta$ is a face of $\tau$. One-dimensional cones in $\Sigma$ are called
rays, and each ray is generated by a primitive vector in $N$. The set of
primitive vectors of $N$ generating rays of $\Sigma$ is denoted by
$G(\Sigma)$. We will write a cone $\tau\in\Sigma$ in terms of its primitive
generators, $\tau=\langle v_{1},\dots,v_{l}\rangle$, saying that the $v_{i}$’s
generate $\tau$, and setting $G(\tau)\coloneqq\\{v_{1},\dots,v_{l}\\}\subseteq
G(\Sigma)$.

We denote by $X_{\Sigma}$ the toric variety corresponding to a fan $\Sigma$.
Conversely, given a toric variety $X$, we denote by $\Sigma_{X}$ the fan
associated to $X$. There is a one-to-one inclusion-reversing correspondence
between cones in $\Sigma$ and $T$-orbit closures in $X_{\Sigma}$. Given a cone
$\tau\in\Sigma$, we write $V(\tau)\subset X_{\Sigma}$ for the corresponding
$T$-orbit closure, or $V(v_{1},\dots,v_{l})$ when
$G(\tau)=\\{v_{1},\dots,v_{l}\\}$. Note that
$\dim(\tau)=\operatorname{codim}_{X_{\Sigma}}V(\tau)$. We refer to [Ful93] and
[CLS11] for the background on toric varieties.

In this paper, we are mostly interested in _smooth_ and _proper_ toric
varieties. The smoothness conditions translates into the fan $\Sigma$ being
_regular_ , i.e., for each cone $\tau\in\Sigma$, the set of generators
$G(\tau)$ is part of a basis of $N$ ([CLS11, Definition 1.2.16]). The
properness condition translates into the fan $\Sigma$ being _complete_ , i.e.,
its support being the whole $N_{\mathbb{Q}}$. In what follows, smooth and
proper toric varieties will be simply called _toric manifolds_. We would like
to classify toric $2$-Fano manifolds. Given a toric manifold $X$, there is an
exact sequence ([CLS11, Theorem 8.1.1]):

$0\to\Omega_{X}^{1}\to
N^{\vee}\otimes_{\mathbb{Z}}\mathcal{O}_{X}\to\bigoplus_{v\in
G(\Sigma_{X})}\mathcal{O}_{V(v)}\to 0\,,$

from which one easily computes:

$c_{1}(X)\,=\sum_{v\in G(\Sigma_{X})}V(v)\quad\text{ and
}\quad\operatorname{ch}_{2}(X)\,=\,\frac{1}{2}\sum_{v\in
G(\Sigma_{X})}V(v)^{2}.$

######  Definition 2.1.

([Bat91, Definition 2.6]) Let $\Sigma$ be a regular complete fan in
$N_{\mathbb{Q}}$. A _primitive collection_ $P\subseteq G(\Sigma)$ is a
nonempty set of primitive vectors of $N$ that does not generate a cone of
$\Sigma$, but such that any proper subset of $P$ does. Equivalently,
$P=\\{v_{1},\dots,v_{r}\\}\subseteq G(\Sigma)$ is a primitive collection if
and only if

$\langle
v_{1},\dots,v_{r}\rangle\notin\Sigma\quad\quad\text{and}\quad\quad\langle
v_{1},\dots,\check{v_{i}},\dots,v_{r}\rangle\in\Sigma$

for any $i=1,\dots,r$. We denote by $\mathrm{PC}(\Sigma)$ the set of primitive
collections of $\Sigma$. For a toric manifold $X$, we will talk about
primitive collections of $X$ and write $\mathrm{PC}(X)$, meaning
$\mathrm{PC}(\Sigma_{X})$.

######  Definition 2.2.

Let $\Sigma$ be a regular complete fan in $N_{\mathbb{Q}}$. Given a primitive
collection $P=\\{v_{1},\dots,v_{r}\\}\in\mathrm{PC}(\Sigma)$, let
$\sigma(P)=\langle w_{1},\dots,w_{s}\rangle$ be the minimal cone of $\Sigma$
such that $v_{1}+\cdots+v_{r}\in\sigma(P)$. Then there is a relation

$r(P)\colon\,v_{1}+\cdots+v_{r}=\mu_{1}w_{1}+\cdots+\mu_{s}w_{s},$

where $\mu_{j}\in\mathbb{Z}_{\geq 0}$ for $j=1,\dots,s$. We call $r(P)$ the
_primitive relation_ associated to $P$. We define the _order_ of $P$ as
$\mathrm{ord}(P)=|P|=r$, while the _degree_ of $P$ as
$\deg(P)=r-\sum_{j=1}^{s}\mu_{j}$.

By [Bat91, Proposition 3.1], for any primitive collection $P$, we have
$P\cap\sigma(P)=\varnothing$. In particular,
$\\{v_{1},\dots,v_{r}\\}\cap\\{w_{1},\dots,w_{s}\\}=\varnothing$.

######  Definition 2.3.

Let $\Sigma$ be a regular complete fan in $N_{\mathbb{Q}}$. A primitive
collection $P=\\{x_{0},\dots,x_{k}\\}$ of $\Sigma$ is called _centrally
symmetric_ if $\sigma(P)=\\{0\\}$, i.e.

$r(P)\colon x_{0}+\dots+x_{k}=0.$

######  Lemma 2.4.

Let $\Sigma$ be a regular complete fan in $N_{\mathbb{Q}}$, and let $P$, $Q$
be two distinct centrally symmetric primitive collections. Then $P\cap
Q=\varnothing$.

###### Proof.

Write $r(P)\colon x_{0}+\cdots+x_{k}=0$ and $r(Q)\colon y_{0}+\cdots+y_{l}=0$.
Assume that $P\cap Q\neq\varnothing$, then without loss of generality we may
assume that $x_{0}=y_{0}$. But then subtracting this vector from both
relations, we get

$x_{1}+\cdots+x_{k}=y_{1}+\cdots+y_{l},$

which shows that interiors of two distinct cones intersect. This is a
contradiction. ∎

######  Lemma 2.5.

Let $\Sigma$ be a regular complete fan in $N_{\mathbb{Q}}$, and let $P$, $Q$
be two distinct centrally symmetric primitive collections. Then
$\operatorname{Span}P\cap\operatorname{Span}Q=\\{0\\}$, in particular
$|P|+|Q|-2\leq\dim N_{\mathbb{Q}}$.

###### Proof.

Write $r(P)\colon x_{0}+\cdots+x_{k}=0$ and $r(Q)\colon y_{0}+\cdots+y_{l}=0$.
Take any vector $v\in\operatorname{Span}P\cap\operatorname{Span}Q$, so we can
write it as

$v=\sum a_{i}x_{i}=\sum b_{j}y_{j}.$

By possibly adding $r(P)$ and $r(Q)$ to the sums, we can get that all
$a_{i},b_{j}\geq 0$, and up to relabelling the $a_{i}$, $b_{j}$, we can assume
$a_{0}=b_{0}=0$. But this shows that $v$ is in the intersection of two cones,
$\langle x_{1},\dots,x_{k}\rangle\cap\langle y_{1},\dots,y_{l}\rangle$, and
the sets of generators are disjoint by 2.4, so $\langle
x_{1},\dots,x_{k}\rangle\cap\langle y_{1},\dots,y_{l}\rangle=\\{0\\}$ and
$v=0$.

The last claim follows from considering the dimensions of
$\operatorname{Span}P$ and $\operatorname{Span}Q$. ∎

Let $A_{1}(X_{\Sigma})$ be the group of algebraic 1-cycles on $X_{\Sigma}$
modulo numerical equivalence, and set
$\mathcal{N}_{1}(X_{\Sigma})=A_{1}(X_{\Sigma})\otimes_{\mathbb{Z}}\mathbb{Q}$.
The Mori cone
$\operatorname{NE}(X_{\Sigma})\subset\mathcal{N}_{1}(X_{\Sigma})$ is the cone
generated by the classes of effective curves. A primitive integral class
generating an extremal ray of $\operatorname{NE}(X_{\Sigma})$ is called an
_extremal class_. There is an exact sequence:  Thus the elements of
$A_{1}(X_{\Sigma})$ are identified with integral relations between the
elements of $G(\Sigma)$. If the class $[C]$ corresponds to the relation
$\sum_{v}\nu_{v}v=0$, then we have $-K_{X_{\Sigma}}\cdot C=\sum_{v}\nu_{v}$.

######  Proposition 2.6.

([Cas03, Lemma 1.4]) Let $\Sigma$ be a regular complete fan in
$N_{\mathbb{Q}}$. A relation

$\alpha_{1}x_{1}+\dots+\alpha_{l}x_{l}-\beta_{1}y_{1}-\dots-\beta_{m}y_{m}=0,$

with $\alpha_{i},\beta_{j}\in\mathbb{Z}_{>0}$, defines an effective class in
$\mathcal{N}_{1}(X_{\Sigma})$ provided that $\langle y_{1},\dots,y_{m}\rangle$
is a cone of $\Sigma$.

We will usually write the above relation as

$\alpha_{1}x_{1}+\dots+\alpha_{l}x_{l}=\beta_{1}y_{1}+\dots+\beta_{m}y_{m}.$

It follows that primitive relations correspond to effective curve classes. By
abuse of notation, we will identify a primitive relation $r(P)$ with the
corresponding curve class. Note that $\deg(P)=-K_{X_{\Sigma}}\cdot r(P)$. In
the projective case we have the following description of
$\operatorname{NE}(X_{\Sigma})$.

######  Proposition 2.7.

([Bat91, Theorem 2.15]) Let $\Sigma$ be a regular complete fan in
$N_{\mathbb{Q}}$, and assume that $X_{\Sigma}$ is projective. Then the Mori
cone is generated by primitive relations:

$\operatorname{NE}(X_{\Sigma})=\sum_{P\in\mathrm{PC}(X_{\Sigma})}\mathbb{Q}_{\geq
0}\,r(P).$

######  Proposition 2.8.

([Rei83, Theorem 2.4]) Let $\Sigma$ be a regular complete fan in
$N_{\mathbb{Q}}$, and assume that $X_{\Sigma}$ is projective. Let $\gamma$ be
an extremal class in $\operatorname{NE}(X_{\Sigma})$ whose corresponding
primitive relation is

$r(P)\colon\,v_{1}+\dots+v_{r}=\mu_{1}w_{1}+\dots+\mu_{s}w_{s}.$

Let $\tau=\langle z_{1},\dots,z_{l}\rangle$ be a cone of $\Sigma$ such that
$G(\tau)\cap P=G(\tau)\cap G(\sigma(P))=\varnothing$, and such that
$\langle\sigma(P),\tau\rangle=\langle
w_{1},\dots,w_{s},z_{1},\dots,z_{l}\rangle$ is a cone of $\Sigma$. Then, for
each $i=1,\dots,r$,

$\langle P\setminus\\{v_{i}\\},\sigma(P),\tau\rangle=\langle
v_{1},\dots,\check{v_{i}},\dots,v_{r},w_{1},\dots,w_{s},z_{1},\dots,z_{l}\rangle$

is also a cone of $\Sigma$.

###  2.2. The minimal $\mathbb{P}$-dimension

Let $X$ be a toric manifold with regular complete fan $\Sigma_{X}$ in
$N_{\mathbb{Q}}$. In this section, we discuss centrally symmetric primitive
collections, introduced in 2.3.

######  Proposition 2.9.

([Bat91, Proposition 3.2]) If $X$ is projective, then $\Sigma_{X}$ has a
centrally symmetric primitive collection of order $k+1$

(2) $r(P)\colon\,x_{0}+\dots+x_{k}=0$

for some $k\in\\{1,\dots,\dim(X)\\}$.

######  Definition 2.10.

For a projective toric manifold $X$, we define the _minimal
$\mathbb{P}$-dimension_ as

$m(X)\coloneqq\min\Big{\\{}m\in\mathbb{Z}_{>0}\,\Big{|}\
\begin{gathered}\text{$\Sigma_{X}$ has a centrally symmetric }\\\ \text{
primitive collection of order $m+1$ }\end{gathered}\ \Big{\\}}.$

The next remark explains the terminology of 2.10 and highlights the
significance of studying centrally symmetric primitive collections.

######  Remark 2.11.

In [CFH14], Chen, Fu and Hwang provide a new geometric proof of 2.9 by
relating centrally symmetric primitive collections to _minimal dominating
families of rational curves_. We review some aspects of the theory of rational
curves on varieties and refer to [Kol96] for details. Given a smooth and
proper uniruled variety $X$, there is a scheme
$\operatorname{RatCurves}^{n}(X)$ parametrizing rational curves on $X$. A
_dominating family of rational curves_ on $X$ is an irreducible component of
$\operatorname{RatCurves}^{n}(X)$ parametrizing rational curves that sweep out
a dense open subset of $X$. A dominating family of rational curves $H$ is said
to be _minimal_ if, for a general point $x\in X$, the subvariety of $H$
parametrizing curves through $x$ is proper. When $X$ is projective, there
always exists a minimal dominating family of rational curves on $X$. For
instance, one can take $H$ to be a dominating family of rational curves on $X$
having minimal degree with respect to some fixed ample line bundle on $X$.

When $X=X_{\Sigma}$ is a toric variety, there is a one-to-one correspondence
between minimal dominating families of rational curves $H$ on $X$ and
centrally symmetric primitive collections of $\Sigma$ ([CFH14, Proposition
3.2]). Moreover, if the centrally symmetric primitive collection has order
$k+1$ as in Equation 2 above, then there is a dense $T$-invariant open subset
$U$ of $X$ and a $\mathbb{P}^{k}$-bundle $\pi\colon U\to W$ such that the
general curve parametrized by $H$ is a line on a general fiber $\pi$ ([CFH14,
Corollary 2.6]).

It follows from this discussion that the minimal $\mathbb{P}$-dimension $m(X)$
is the smallest integer $k$ such that $X$ admits a generic
$\mathbb{P}^{k}$-bundle structure. We have

$m(X)\in\ \\{1,\dots,n=\dim(X)\\},$

and $m(X)=n$ if and only if $X\simeq\mathbb{P}^{n}$. By [CFH14, Proposition
3.8], there are three toric Fano manifolds admitting a centrally symmetric
primitive relation of order $n=\dim(X)$, namely:
$\mathbb{P}^{n-1}\times\mathbb{P}^{1}$, the blowup of
$\mathbb{P}^{n-1}\times\mathbb{P}^{1}$ along a linear $\mathbb{P}^{n-2}$, and
the blowup of $\mathbb{P}^{n}$ along a linear $\mathbb{P}^{n-2}$. The first
two varieties also admit a generic $\mathbb{P}^{1}$-bundle structure. As a
consequence, the only $n$-dimensional toric Fano manifold $X$ with $m(X)=n-1$
is the blowup of $\mathbb{P}^{n}$ along a linear $\mathbb{P}^{n-2}$. In
Section 5, we shall classify $n$-dimensional toric Fano manifolds $X$ with
$m(X)=n-2$.

Let $P\in\mathrm{PC}(X)$ be a centrally symmetric primitive collection of
order $k+1$. As explained in 2.11, $P$ induces a $\mathbb{P}^{k}$-bundle
structure on a dense $T$-invariant open subset $U$ of $X$. In [CFH14,
Corollary 2.6], the $T$-invariant open subset $U$ was taken as small as
possible, namely, $U\cong\mathbb{P}^{k}\times(\mathbb{C}^{*})^{n-k}$. For our
purposes, we want to take $U$ as big as possible. So our next goal is to
describe explicitly the biggest $T$-invariant open subset of $X$ on which $P$
induces a $\mathbb{P}^{k}$-bundle structure.

######  Notation 2.12.

Let $P=\\{x_{0},\dots,x_{k}\\}\in\mathrm{PC}(X)$ be a centrally symmetric
primitive collection. Denote by $\mathcal{E}_{P}$ the set of cones
$\sigma=\langle v_{1},\dots,v_{r}\rangle\in\Sigma_{X}$ such that $P\cap
G(\sigma)=\varnothing$, and
$\\{v_{1},\dots,v_{r},x_{j_{1}},\dots,x_{j_{s}}\\}\in\mathrm{PC}(X)$ for some
$s\geq 1$, i.e.,

$\mathcal{E}_{P}\coloneqq\left\\{\sigma\in\Sigma_{X}\mid P\cap
G(\sigma)=\varnothing\text{ and }\exists P^{\prime}\subsetneq P\text{ such
that }P^{\prime}\cup G(\sigma)\in\mathrm{PC}(X)\right\\}.$

We write

$V(\mathcal{E}_{P})\coloneqq\bigcup_{\sigma\in\mathcal{E}_{P}}V(\sigma)\subset
X.$

######  Proposition 2.13.

Let $P=\\{x_{0},\dots,x_{k}\\}\in\mathrm{PC}(X)$ be a centrally symmetric
primitive collection, and let $V(\mathcal{E}_{P})$ be as in Notation 2.12.
Then the open subset $U=X\setminus V(\mathcal{E}_{P})$ admits a
$\mathbb{P}^{k}$-bundle structure over a smooth toric variety.

In order to prove 2.13, we first prove two auxiliary lemmas.

######  Lemma 2.14.

Let $P=\\{x_{0},\dots,x_{k}\\}\in\mathrm{PC}(X)$ be a centrally symmetric
primitive collection, let $V(\mathcal{E}_{P})$ be as in Notation 2.12, and set
$U=X\setminus V(\mathcal{E}_{P})$. Then the fan $\Sigma_{U}$ of $U$ consists
of all cones of $\Sigma_{X}$ of the form

(3) $\tau^{\prime}=\langle\tau,x_{j_{1}},\dots,x_{j_{m}}\rangle,$

where $0\leq m\leq k$, and $\tau\in\Sigma_{X}$ is such that
$\langle\tau,P\setminus\\{x_{i}\\}\rangle\in\Sigma$ for every
$i\in\\{0,\dots,k\\}$. (When $m=0$, Equation 3 means that
$\tau^{\prime}=\tau$.)

###### Proof.

Recall that a cone $\sigma\in\Sigma_{X}$ corresponds to a $T$-orbit, which is
dense and open in $V(\sigma)$. Hence, a cone $\sigma\in\Sigma_{X}$ is in
$\Sigma_{U}$ if and only if the corresponding orbit does not intersect
$V(\mathcal{E}_{P})$, which is equivalent to saying that
$V(\sigma)\not\subseteq V(\mathcal{E}_{P})$. It is immediate that the cones of
the form (3) define a fan $\Sigma^{\prime}\subset\Sigma_{X}$ in
$N_{\mathbb{Q}}$, and $X_{\Sigma^{\prime}}$ is a dense open subset of $X$. We
now prove that the toric variety $X_{\Sigma^{\prime}}$ coincides with $U$ by
showing that a cone $\sigma\in\Sigma_{X}$ is of the form (3) if and only if
$V(\sigma)\not\subseteq V(\mathcal{E}_{P})$.

Consider $\sigma\in\Sigma_{X}\setminus\Sigma^{\prime}$, which means that
$\langle G(\sigma)\cup P\setminus\\{x_{i}\\}\rangle\notin\Sigma_{X}$ for some
$i$. Then the set $G(\sigma)\cup P\setminus\\{x_{i}\\}$ contains a primitive
collection $S$, so the cone $\tau\coloneqq\langle S\setminus P\rangle$ is in
$\mathcal{E}_{P}$. But notice that $\tau\prec\sigma$, so $V(\sigma)\subseteq
V(\tau)\subseteq V(\mathcal{E}_{P})$ and hence $\sigma$ is not in
$\Sigma_{U}$. Conversely, if $\sigma\in\Sigma_{X}\setminus\Sigma_{U}$, then
$V(\sigma)\subseteq V(\mathcal{E}_{P})$, hence there exists
$\tau\in\mathcal{E}_{P}$ such that $V(\sigma)\subseteq V(\tau)$ and
$G(\tau)\cup P^{\prime}\in\mathrm{PC}(X)$ for some $P^{\prime}\subset P$.
Since $G(\tau)\subseteq G(\sigma)$, we conclude that $\langle G(\sigma)\cup
P^{\prime}\rangle\notin\Sigma_{X}$, i.e., $\sigma\not\in\Sigma^{\prime}$. ∎

Consider the sequences

${0}$${N_{P}\coloneqq\ker(\phi)}$${N}$${\overline{N}=N/\mathbb{Z}\langle
x_{0},\dots,x_{k}\rangle}$${0,}$${0}$${(N_{P})_{\mathbb{Q}}}$${N_{\mathbb{Q}}}$${\overline{N}_{\mathbb{Q}}}$${0,}$${\Sigma_{0}}$${\Sigma_{U}}$${\overline{\Sigma}_{U},}$$\scriptstyle{\phi}$$\scriptstyle{\phi_{\mathbb{Q}}}$

where $\phi$ is the quotient map, the fan $\Sigma_{0}$ of
$(N_{P})_{\mathbb{Q}}\simeq\mathbb{Q}^{k+1}$ is the subfan of $\Sigma_{U}$ of
cones of the form (3) with $\tau=\\{0\\}$ (in particular note that
$X_{\Sigma_{0}}\simeq\mathbb{P}^{k}$), and
$\overline{\Sigma}_{U}=\\{\phi_{\mathbb{Q}}(\sigma)\,|\,\sigma\in\Sigma_{U}\\}$.

######  Lemma 2.15.

Let the notation be as above. Then $\overline{\Sigma}_{U}$ is a toric fan, and
the linear map $\phi_{\mathbb{Q}}$ is compatible with the fans $\Sigma_{U}$
and $\overline{\Sigma}_{U}$.

###### Proof.

The cones of $\overline{\Sigma}_{U}$ are exactly $\phi_{\mathbb{Q}}(\tau)$ for
$\tau\in\Sigma_{U}$ such that $G(\tau)\cap P=\varnothing$, so for simplicity
we only consider these $\tau$.

  * -

It is immediate that the cones of $\overline{\Sigma}_{U}$ are rational
polyhedral, and that the faces of $\phi_{\mathbb{Q}}(\tau)$ are
$\phi_{\mathbb{Q}}(\delta)$, for all subcones $\delta\prec\tau$.

  * -

We need to show that the cone $\phi_{\mathbb{Q}}(\tau)$ is strongly convex,
i.e., if $y\in\phi_{\mathbb{Q}}(\tau)$ and $-y\in\phi_{\mathbb{Q}}(\tau)$,
then $y=0$. This follows automatically from the fact that the images of the
generators
$\phi_{\mathbb{Q}}(G(\tau))=\\{\overline{v}_{1},\dots,\overline{v}_{r}\\}$ are
linearly independent, which we prove by contradiction. If they are linearly
dependent, then there exist $a_{1},\dots,a_{r}\in\mathbb{Q}$, not all $0$,
such that $\Sigma_{i=1}^{r}a_{i}\overline{v}_{i}=0$ in
$\overline{N}_{\mathbb{Q}}$. This implies that there exist
$b_{j}\in\mathbb{Q}$ for $j=1,\dots,k$ such that
$\Sigma_{i=1}^{r}a_{i}v_{i}=\sum_{j=1}^{k}b_{j}x_{j}$, which is a
contradiction since $\langle G(\tau)\cup
P\setminus\\{x_{0}\\}\rangle\in\Sigma$ and hence its primitive generators are
linearly independent in $N_{\mathbb{Q}}$.

  * -

$\Sigma_{U}$ and $\overline{\Sigma}_{U}$ are compatible with
$\phi_{\mathbb{Q}}$ as we have
$\phi_{\mathbb{Q}}(\tau^{\prime})\in\overline{\Sigma}_{U}$ for any
$\tau^{\prime}\in\Sigma_{U}$.

∎

###### Proof of 2.13.

Let the notation be as above. Let $\hat{\Sigma}_{U}$ be the collection of fans
of the form (3) above with $m=0$. It follows from the description of the cones
of $\Sigma_{U}$ in 2.14 that

  1. (1)

$\phi_{\mathbb{Q}}$ maps each cone $\hat{\tau}\in\hat{\Sigma}_{U}$ bijectively
to a cone $\overline{\tau}\in\overline{\Sigma}_{U}$ such that
$\phi(\hat{\tau}\cap N)=\overline{\tau}\cap\overline{N}$. Furthermore, the map
$\hat{\tau}\mapsto\overline{\tau}$ defines a bijection
$\hat{\Sigma}_{U}\to\overline{\Sigma}_{U}$;

  2. (2)

given cones $\hat{\tau}\in\hat{\Sigma}_{U}$ and $\tau_{0}\in\Sigma_{0}$, the
sum $\hat{\tau}+\tau_{0}$ lies in $\Sigma_{U}$ and every cone of $\Sigma_{U}$
arises in this way.

In the notation of [CLS11, Definition 3.3.18], we say that _$\Sigma_{U}$ is
split by $\overline{\Sigma}_{U}$ and $\Sigma_{0}$_. We conclude by [CLS11,
Theorem 3.3.19] that $U=X\setminus V(\mathcal{E}_{P})$ is a locally trivial
fiber bundle over $X_{\overline{\Sigma}_{U}}$ with fiber
$X_{\Sigma_{0}}\simeq\mathbb{P}^{k}$. It follows automatically that
$X_{\overline{\Sigma}_{U}}$ is smooth, since it is the base of a locally
trivial fibration with a smooth total space. ∎

###  2.3. Some properties of primitive collections

Before focusing on toric Fano manifolds, we collect here two useful properties
of primitive collections of arbitrary toric manifolds. The first one, by Sato,
describes the behaviour of primitive collections under a smooth toric
blowdown. The second one, by Batyrev, describes primitive collections on toric
manifolds of Picard rank $3$.

######  Proposition 2.16.

([Sat00, Corollary 4.9]) Let $X$ be a toric manifold, and let $f\colon
X\rightarrow Y$ be the contraction associated to an extremal class in
$\mathrm{NE}(X)$, corresponding to a primitive relation of the form

$r(Q)\colon t_{1}+\dots+t_{s}=z.$

Then the fan $\Sigma_{Y}$ is obtained from $\Sigma_{X}$ by removing the ray
generated by $z$, and $X$ is the blowup of $Y$ along $V(t_{1},\dots,t_{s})$.
Furthermore, the primitive collections of $Y$ are precisely the following
$P_{Y}\in\mathrm{PC}(Y)$:

  * •

$P_{Y}=P_{X}$ for some $P_{X}\in\mathrm{PC}(X)$ such that $z\notin P_{X}$ and
$P_{X}\neq Q=\\{t_{1},\dots,t_{s}\\}$;

  * •

$P_{Y}=(P_{X}\setminus\\{z\\})\cup\\{t_{1},\dots,t_{r}\\}$ for some
$P_{X}\in\mathrm{PC}(X)$ such that $z\in P_{X}$ and
$(P_{X}\setminus\\{z\\})\cup S\notin\mathrm{PC}(X)$ for any subset
$S\subsetneq\\{t_{1},\dots,t_{r}\\}$.

######  Proposition 2.17.

([Bat91, Theorem 5.7, Theorem 6.6]) Let $X$ be a projective toric manifold
with $\rho(X)=3$. Then the number of primitive collections of $\Sigma_{X}$ is
either $l=3$ or $l=5$. Moreover, the set of generators $G(\Sigma_{X})$ can be
written as a disjoint union of $l$ nonempty subsets

$G(\Sigma_{X})=X_{0}\sqcup\cdots\sqcup X_{l-1}$

that define primitive collections and relations as follows:

  * •

Case $l=3$. Each $X_{0}$, $X_{1}$, $X_{2}$ is a primitive collection, and the
corresponding primitive relations are extremal.

  * •

Case $l=5$. There are five primitive collections of the form $X_{i}\sqcup
X_{i+1}$, $0\leq i\leq 4$, where $X_{5}\coloneqq X_{0}$. To describe the
primitive relations of $X$, we use the following notation. We fix a labelling
$(v_{1},\dots,v_{k})$ for the elements of $X_{i}$. If
$\overline{c}=(c_{1},\dots,c_{k})\in\mathbb{Z}^{k}$, then $\overline{c}\cdot
X_{i}$ stands for $c_{1}v_{1}+\dots+c_{k}v_{k}$. Moreover, we set
$\overline{1}=(1,\dots,1)$. Then there are vectors $\overline{c}$ and
$\overline{b}$ of nonnegative integers such that at least one entry in
$\overline{c}$ is zero (up to relabelling, we may assume that $c_{1}=0$), and
the primitive relations of $X$ are the following:

$\displaystyle r_{0}$ $\displaystyle\colon\quad\overline{1}\cdot
X_{0}+\overline{1}\cdot X_{1}=\overline{c}\cdot
X_{2}+(\overline{b}+\overline{1})\cdot X_{3}$ $\displaystyle r_{1}$
$\displaystyle\colon\quad\overline{1}\cdot X_{1}+\overline{1}\cdot
X_{2}=\overline{1}\cdot X_{4},$ $\displaystyle r_{2}$
$\displaystyle\colon\quad\overline{1}\cdot X_{2}+\overline{1}\cdot X_{3}=0,$
$\displaystyle r_{3}$ $\displaystyle\colon\quad\overline{1}\cdot
X_{3}+\overline{1}\cdot X_{4}=\overline{1}\cdot X_{1},$ $\displaystyle r_{4}$
$\displaystyle\colon\quad\overline{1}\cdot X_{4}+\overline{1}\cdot
X_{0}=\overline{c}\cdot X_{2}+\overline{b}\cdot X_{3}.$

The relations $r_{0}$, $r_{1}$ and $r_{3}$ are extremal, while
$r_{2}=r_{1}+r_{3}$ and $r_{4}=r_{0}+r_{3}$.

######  Remark 2.18.

In [SS20, Corollary 1.2], Sato and Suyama use Proposition 2.17 to show that
projective spaces are the only toric $2$-Fano manifolds with Picard rank
$\rho\leq 3$.

###  2.4. Primitive collections on toric Fano manifolds

Let $X$ be a projective toric manifold with regular complete fan $\Sigma_{X}$
in $N_{\mathbb{Q}}$.

######  Proposition 2.19.

([Bat99, Proposition 2.3.6]) The toric variety $X$ is Fano if and only if all
primitive collections of $\Sigma_{X}$ have strictly positive degree.

######  Proposition 2.20.

([Cas03, Corollary 4.4]) Assume that $X$ is Fano, and let
$P\in\mathrm{PC}(X)$. If $\deg(P)=1$, then the corresponding curve class is
extremal.

######  Proposition 2.21.

Assume that $X$ is Fano, and let $x\in G(\Sigma_{X})$.

  1. (1)

There is at most one primitive collection of order $2$ and degree $2$
containing $x$. If it exists, then it is of the form $x+(-x)=0$, and $m(X)=1$.

  2. (2)

([Cas03a, Lemma 3.3]) There are at most two primitive collections of order $2$
and degree $1$ containing $x$. If there are exactly two of them, then they are
of the form $x+y=(-w)$ and $x+w=(-y)$, and $m(X)=1$.

######  Corollary 2.22.

Assume that $X$ is Fano and $m(X)>1$. Then any $x\in G(\Sigma_{X})$ is
contained in at most one primitive collection of order 2. If there is such a
primitive collection, then it is of the form $x+y=z$.

######  Definition 2.23.

Let $x\in G(\Sigma_{X})$. We say that $y\in G(\Sigma_{X})$ is an _opponent_ of
$x$ if $\langle x,y\rangle\notin\Sigma_{X}$.

######  Notation 2.24.

Assume that $X$ is Fano and $m(X)>1$. By 2.22, each vector $x\in
G(\Sigma_{X})$ has at most one opponent. If such an opponent exists, we denote
it by $x^{\prime}$.

######  Lemma 2.25.

Assume that $X$ is Fano and $m(X)>1$. Consider a pair of opponents
$x,x^{\prime}\in G(\Sigma_{X})$. If there exist $y,z\in G(\Sigma_{X})$ such
that $x+x^{\prime}=y+z$, then $z=y^{\prime}$.

###### Proof.

If $y=x$ or $y=x^{\prime}$, the claim follows automatically. So we assume
otherwise. Note that $\\{x,x^{\prime}\\}\in\mathrm{PC}(X)$, and $y$ and $z$ do
not form a cone, as otherwise $x+x^{\prime}=y+z$ would give us a primitive
relation of degree $0$, which is impossible for a toric Fano manifold. ∎

######  Lemma 2.26.

Assume that $X$ is Fano and $m(X)>1$. Assume that there exist $x,y,z,u,v$ in
$G(\Sigma_{X})$ such that $(*)$ $x+y+z=u+v$ and such that $\langle
u,v\rangle\in\Sigma_{X}$. Then exactly one of the following must happen:

  * a.

The vectors $x,y,z$ are pairwise distinct, and $\\{x,y,z\\}$ is a primitive
collection with primitive relation $(*)$. In particular, the corresponding
curve class is extremal.

  * b.

Up to relabeling, $v=z$, $y=x^{\prime}$ and $x+x^{\prime}=u$.

###### Proof.

Assume two of $\\{x,y,z\\}$ do not form a cone. For example, assume $x$, $y$
do not form a cone. Then $y=x^{\prime}$, the opponent of $x$. Let
$x+x^{\prime}=\alpha$, for some $\alpha\in G(\Sigma_{X})$. We have that
$\alpha+z=u+v$. As $\langle u,v\rangle\in\Sigma_{X}$, 2.6 implies that
$\alpha+z=u+v$ corresponds to an effective class of degree $0$, which
therefore implies that $\\{\alpha,z\\}=\\{u,v\\}$. Up to relabeling, we may
assume $v=z$, and hence, $x+x^{\prime}=u$ and we are in the situation b.

Assume now that any two vectors in $\\{x,y,z\\}$ form a cone. Then $x,y,z$ are
mutually disjoint and $\langle x,y,z\rangle\notin\Sigma_{X}$, as otherwise by
2.6 we obtain an effective curve class of degree $-1$, contradicting the fact
that $X$ is Fano. It follows that $\\{x,y,z\\}$ is a primitive collection.
Since $\langle u,v\rangle\in\Sigma_{X}$, it follows that $(*)$ is the
associated primitive relation. 2.20 now implies that the corresponding curve
class is extremal. ∎

##  3\. Toric Fano manifolds with $m(X)=1$

In this section, we study toric Fano manifolds with $m(X)=1$, and follow the
strategy outlined in the introduction to show that they cannot be $2$-Fano.

For any $x\in G(\Sigma_{X})$, the set of primitive collections containing $x$
is denoted by

$\mathrm{PC}_{x}(X)=\\{P\in\mathrm{PC}(X)\ |\ x\in P\\}.$

######  Proposition 3.1.

([Cas03a, Lemma 3.1]) Assume that $X$ is a toric Fano manifold and that
$P=\\{x,-x\\}\in\mathrm{PC}(X)$.

  1. (1)

Any $Q\in\mathrm{PC}_{x}(X)\setminus\\{P\\}$ has degree 1 (hence is extremal
by 2.20), and $r(Q)$ is of the form

$r(Q)\colon x+{\underbrace{y_{1}+\dots+y_{h}}_{\scriptstyle\in\,\langle
Q\setminus\\{x\\}\rangle}}={\underbrace{z_{1}+\dots+z_{h}}_{\scriptstyle\in\sigma(Q)}},$

where we denote $\langle Q\setminus\\{x\\}\rangle\coloneqq\langle
y_{1},\dots,y_{h}\rangle$ and $\sigma(Q)\coloneqq\langle
z_{1},\dots,z_{h}\rangle$.

  2. (2)

For any $R\in\mathrm{PC}_{x}(X)\setminus\\{P,Q\\}$, we have

$V(R\setminus\\{x\\})\cap V(Q\setminus\\{x\\})=\varnothing\quad\text{ and
}\quad V(R\setminus\\{x\\})\cap V(\sigma(Q))=\varnothing.$

  3. (3)

For any $Q\in\mathrm{PC}_{x}(X)\setminus\\{P\\}$ with $r(Q)\colon
x+y_{1}+\dots+y_{h}=z_{1}+\dots+z_{h}$ we have
$Q^{\prime}=\\{-x,z_{1},\dots,z_{h}\\}\in\mathrm{PC}_{-x}(X)$, $Q^{\prime}$
has degree 1 (hence is extremal) and

$r(Q^{\prime})\colon-x+{\underbrace{z_{1}+\dots+z_{h}}_{\scriptstyle\in\langle
Q^{\prime}\setminus\\{-x\\}\rangle=\sigma(Q)}}={\underbrace{y_{1}+\dots+y_{h}}_{\scriptstyle\in\sigma(Q^{\prime})=\langle
Q\setminus\\{x\\}\rangle}}.$

######  Corollary 3.2.

Let $X$ be a toric manifold, and $P=\\{x,-x\\}\in\mathrm{PC}(X)$. With 2.12,

(4) $\mathcal{E}_{P}=\left\\{\langle Q\setminus\\{v\\}\rangle\mid
Q\in\mathrm{PC}_{v}(X)\setminus\\{P\\},\quad v=\pm x\right\\}.$

If moreover $X$ is Fano, then $V(\mathcal{E}_{P})$ has $0$, $2$ or $4$
components of codimension $1$ in $X$.

###### Proof.

Let $X$ be a toric manifold, and $P=\\{x,-x\\}\in\mathrm{PC}(X)$. The
description of $V(\mathcal{E}_{P})$ in Equation 4 follows from 2.12. In the
Fano case, the number of components of codimension $1$ of $V(\mathcal{E}_{P})$
equals the number of primitive collections of order $2$ and degree $1$
containing $x$ or $-x$, which is $0$, $2$ or $4$ by 2.21 and 3.1. ∎

######  Proposition 3.3.

Let $X$ be a toric Fano manifold, and $P=\\{x,-x\\}\in\mathrm{PC}(X)$. Then
there exists a birational morphism $f\colon X\rightarrow Y$ such that

  * •

$P_{Y}\coloneqq\\{x,-x\\}\in\mathrm{PC}(Y)$,

  * •

$V(\mathcal{E}_{P_{Y}})$ has codimension $\geq 2$ in $Y$,

  * •

$f$ is a composition of at most two blow-downs with disjoint centers and
smooth target:

$\displaystyle\mathrm{Exc}(f)$
$\displaystyle=\bigcup_{\begin{subarray}{c}Q\in\mathrm{PC}_{x}(X)\setminus\\{P\\}\,:\,\\\
\mathrm{ord}(Q)=2\end{subarray}}V(\sigma(Q))\,\subset X,$ (5) $\displaystyle
f(\mathrm{Exc}(f))$
$\displaystyle=\bigcup_{\begin{subarray}{c}Q\in\mathrm{PC}_{x}(X)\setminus\\{P\\}\,:\,\\\
\mathrm{ord}(Q)=2\end{subarray}}V(Q)\,\subset Y.$

###### Proof.

If $V(\mathcal{E}_{P})$ has codimension $\geq 2$ in $X$, i.e., if $P$ is the
unique primitive collection of order $2$ containing $x$, then the statement
holds with $f=\mathrm{Id}$.

Assume now that $V(\mathcal{E}_{P})$ has $2$ components of codimension $1$,
i.e. we have primitive relations

$r(P)\colon x+(-x)=0,\quad\quad r(Q_{1})\colon x+y=z,\quad\quad
r(Q_{1}^{\prime})\colon-x+z=y,$

and, for any other $R\in\mathrm{PC}_{x}(X)\cup\mathrm{PC}_{-x}(X)$, one has
$\mathrm{ord}(R\setminus\\{\pm x\\})\geq 2$. Let $f_{1}\colon X\rightarrow Y$
be the smooth blow-down induced by the extremal ray of $\mathrm{NE}(X)$
corresponding to $r(Q_{1})$. By 2.16 and 3.1,
$\mathrm{PC}_{x}(Y)\cup\mathrm{PC}_{-x}(Y)$ consists of

  * -

$P_{Y}=\\{x,-x\\}$;

  * -

$R_{Y}=R_{X}$ for some
$R_{X}\in\mathrm{PC}_{x}(X)\cup\mathrm{PC}_{-x}(X)\setminus\\{Q_{1}\\}$ such
that $z\notin R_{X}$. In particular we have $\mathrm{ord}(R_{Y}\setminus\\{\pm
x\\})\geq 2$;

  * -

$R_{Y}=(R_{X}\setminus\\{z\\})\cup\\{x,y\\}$ for some
$R_{X}\in\mathrm{PC}_{z}(X)$ such that
$(R_{X}\setminus\\{z\\})\cup\\{x\\}\notin\mathrm{PC}(X)$ and
$(R_{X}\setminus\\{z\\})\cup\\{y\\}\notin\mathrm{PC}(X)$. In particular, we
have $\langle R_{Y}\setminus\\{x\\}\rangle=\langle
R_{X}\setminus\\{z\\},y\rangle$, so $\mathrm{ord}(R_{Y}\setminus\\{x\\})\geq
2$.

It follows that $V(\mathcal{E}_{P_{Y}})$ has codimension $\geq 2$ in $Y$, so
the proposition holds with $f=f_{1}$.

Assume now that $V(\mathcal{E}_{P})$ has $4$ components of codimension $1$,
i.e., we have

$\displaystyle r(P)\colon x+(-x)$ $\displaystyle=0$ $\displaystyle
r(Q_{1})\colon x+y$ $\displaystyle=-w$ $\displaystyle
r(Q_{1}^{\prime})\colon-x+(-w)$ $\displaystyle=y$ $\displaystyle y+(-y)$
$\displaystyle=0$ $\displaystyle r(Q_{2})\colon x+w$ $\displaystyle=-y$
$\displaystyle r(Q_{2}^{\prime})\colon-x+(-y)$ $\displaystyle=w$
$\displaystyle w+(-w)$ $\displaystyle=0$ $\displaystyle w+y$
$\displaystyle=-x$ $\displaystyle-y+(-w)$ $\displaystyle=x$

by 2.21. By [Cas03a, p.1487, Case (3)], any other primitive collection
$R\in\mathrm{PC}(X)$ is disjoint from $\\{x,-x,y,-y,w,-w\\}$. Let $f_{1}\colon
X\rightarrow X_{1}$ be the smooth blow-down induced by the extremal ray of
$\mathrm{NE}(X)$ corresponding to $r(Q_{1})$. By 2.16 the primitive
collections of $X_{1}$ containing $x$ or $-x$ are only

$P_{X_{1}}=P\quad\quad(Q_{2})_{X_{1}}=Q_{2}\quad\quad(Q^{\prime}_{2})_{X_{1}}=Q^{\prime}_{2}.$

It follows that $r(Q_{2})$ corresponds to an extremal curve class in
$\mathrm{NE}(X_{1})$. Let $f_{2}\colon X_{1}\rightarrow X_{2}$ be the smooth
blow-down induced by the extremal ray of $\mathrm{NE}(X)$ corresponding to
$r(Q_{2})$. By 2.16 $P_{X_{2}}=\\{x,-x\\}$ is the only primitive collection in
$\mathrm{PC}_{x}(X_{2})\cup\mathrm{PC}_{-x}(X_{2})$. This implies that
$V(\mathcal{E}_{P_{X_{2}}})=\varnothing$, so the proposition holds with
$Y=X_{2}$ and $f=f_{2}\circ f_{1}$. ∎

######  Construction 3.4.

Let $Y$ be a projective toric manifold of dimension $\geq 3$, and
$P=\\{x,-x\\}\in\mathrm{PC}(Y)$ a centrally symmetric primitive collection of
order $2$. Let $V(\mathcal{E}_{P})\subset Y$ be the closed subset defined in
2.12, set $U:=Y\setminus V(\mathcal{E}_{P})$, and let $\pi:U\rightarrow W$ be
the $\mathbb{P}^{1}$-bundle given by 2.13.

Assume that $V(\mathcal{E}_{P})$ has codimension $\geq 2$ in $Y$, and let
$Z\subset Y$ be any given closed subset of codimension $\geq 2$ in $Y$. Note
that a toric manifold is rational, hence rationally connected, so by [Kol96,
Proposition II.3.7], there is a smooth (very free) rational curve $C\subset
U\setminus Z$. Consider the surface $S\coloneqq\pi^{-1}(\pi(C))\subset U$, and
let $n\colon\tilde{S}\to S$ be its normalization. Then $\tilde{S}$ is a
Hirzebruch surface with $\mathbb{P}^{1}$-bundle structure
$\tilde{\pi}:\tilde{S}\to\mathbb{P}^{1}$ induced by $\pi$. The curve class of
the image of a fiber of $\tilde{\pi}$ on $Y$ corresponds to the centrally
symmetric relation $x+(-x)=0$. By taking $C$ general, we may assume that
$\pi(C)$ and $\pi(Z)$ meet transversely in at most finitely many general
points. Hence $S$ and $Z$ meet transversely in at most finitely many points.

######  Proposition 3.5.

Let $Y$ be a projective toric manifold of dimension $\geq 3$, and
$P_{Y}=\\{x,-x\\}\in\mathrm{PC}(Y)$ a centrally symmetric primitive collection
of order $2$. Assume that $V(\mathcal{E}_{P})$ has codimension $\geq 2$ in
$Y$, and let $S\subset Y$ be as in 3.4. Then
$S\cdot\operatorname{ch}_{2}(Y)=0$.

###### Proof.

Let $S=\pi^{-1}(\pi(C))$ be the surface from 3.4, $n\colon\tilde{S}\to S$ its
normalization, $\tilde{\pi}\colon\tilde{S}\to\mathbb{P}^{1}$ the
$\mathbb{P}^{1}$-bundle structure induced by $\pi$, and $F$ a fiber of
$\tilde{\pi}$. Our goal is to compute

$S\cdot\operatorname{ch}_{2}(Y)=\frac{1}{2}\sum_{v\in G(\Sigma_{Y})}S\cdot
V(v)^{2}=\frac{1}{2}\sum_{v\in G(\Sigma_{Y})}\big{(}n^{*}V(v)\big{)}^{2}.$

Recall that the curve class of the image of $F$ in $Y$ is associated to the
relation $x+(-x)=0$. By restricting the divisors $n^{*}V(v)$ to $F$, we have:

$\begin{cases}n^{*}V(v)\cdot F=0&\text{ if }v\neq x,-x,\\\ n^{*}V(x)\cdot
F=1,&\\\ n^{*}V(-x)\cdot F=1.&\end{cases}$

Hence there are sections $\sigma,\sigma^{\prime}$ of
$\tilde{\pi}\colon\tilde{S}\to\mathbb{P}^{1}$, and
$\alpha,\beta,\gamma\in\mathbb{Z}$ such that, on $\tilde{S}$,

$\begin{cases}n^{*}V(v)=\alpha F&\text{ if }v\neq x,-x,\\\
n^{*}V(x)=\sigma+\beta F,&\\\ n^{*}V(-x)=\sigma^{\prime}+\gamma
F.&\end{cases}$

Therefore

$\displaystyle\sum_{v\in G(\Sigma_{Y})}\big{(}n^{*}V(v)\big{)}^{2}$
$\displaystyle=\sum_{v\neq
x,-x}\cancel{\big{(}n^{*}V(v)\big{)}^{2}}+\big{(}n^{*}V(x)\big{)}^{2}+\big{(}n^{*}V(-x)\big{)}^{2}$
$\displaystyle=\sigma^{2}+2\beta+\sigma^{\prime 2}+2\gamma$
$\displaystyle=\sigma^{2}-2(\sigma\cdot\sigma^{\prime})+\sigma^{\prime
2}\quad\text{ as }\sigma\cdot\sigma^{\prime}+\beta+\gamma=n^{*}V(x)\cdot
n^{*}V(-x)=0$ $\displaystyle=(\sigma-\sigma^{\prime})^{2}=0\qquad\quad\hskip
5.0pt\text{ as $\sigma-\sigma^{\prime}$ is a multiple of $F$},$

and this concludes the proof. ∎

######  Lemma 3.6.

([dS06, Lemma 5.1]) Consider the blowup diagram  where both $Y$ and $Z$ are
smooth projective varieties and $\operatorname{codim}_{Y}Z=c\geq 2$. Then we
have the following relation between the 2nd Chern characters of $X$ and $Y$:

$\operatorname{ch}_{2}(X)=f^{*}\operatorname{ch}_{2}(Y)+\frac{c+1}{2}E^{2}-j_{*}\pi^{*}\operatorname{c}_{1}(\mathcal{N}_{Z/Y}).$

######  Corollary 3.7.

In the setting of Lemma 3.6, assume that $c=2$. Let $S\subset Y$ be a surface
that intersects $Z$ at most transversely at $k\geq 0$ points, and let
$S_{X}\subset X$ be its strict transform. Then

$\operatorname{ch}_{2}(X)\cdot S_{X}=\operatorname{ch}_{2}(Y)\cdot
S-\frac{3}{2}\cdot k.$

###### Proof.

Write $S\cap Z=\\{p_{1},\dots,p_{k}\\}$. Then $S_{X}$ is isomorphic to the
blowup of $S$ at $p_{1},\dots,p_{k}$, $S_{X}\cap E=\cup_{i=1}^{k}e_{i}$, where
$e_{i}\simeq\mathbb{P}^{1}$ is the exceptional curve over $p_{i}$, and
$(e_{i}^{2})_{S_{X}}=-1$. By 3.6,

$\displaystyle\operatorname{ch}_{2}(X)\cdot S_{X}$
$\displaystyle=f^{*}\operatorname{ch}_{2}(Y)\cdot
S_{X}+\frac{c+1}{2}E^{2}\cdot
S_{X}-j_{*}\pi^{*}\operatorname{c}_{1}(\mathcal{N}_{Z/Y})\cdot S_{X}$
$\displaystyle=\operatorname{ch}_{2}(Y)\cdot
S+\frac{3}{2}(E_{|S_{X}})^{2}-\pi^{*}\operatorname{c}_{1}(\mathcal{N}_{Z/Y})\cdot{S_{X}}_{|E}$
$\displaystyle=\operatorname{ch}_{2}(Y)\cdot
S+\frac{3}{2}\sum_{i=1}^{k}(e_{i})^{2}-\pi^{*}\operatorname{c}_{1}(\mathcal{N}_{Z/Y})\cdot\sum_{i=1}^{k}e_{i}$
$\displaystyle=\operatorname{ch}_{2}(Y)\cdot
S-\frac{3}{2}k-\sum_{i=1}^{k}\cancel{\operatorname{c}_{1}(\mathcal{N}_{Z/Y})\cdot
p_{i}}.$

∎

We are ready to prove 1.2: a toric Fano manifold $X$ with $m(X)=1$ is not
$2$-Fano.

###### Proof of 1.2.

Let $X$ be a toric Fano manifold with $m(X)=1$, and fix a centrally symmetric
primitive relation $r(P)\colon x+(-x)=0$. Let $f\colon X\rightarrow Y$ be as
in 3.3, and $\pi\colon U=Y\setminus V(\mathcal{E}_{P_{Y}})\rightarrow W$ the
$\mathbb{P}^{1}$-bundle structure induced by $r(P_{Y})\colon x+(-x)=0$ (see
2.13). By • ‣ 3.3, $Z\coloneqq f(\mathrm{Exc}(f))$ has codimension $\geq 2$ in
$Y$. Let $S\subset Y$ be the surface given by 3.4. Then $S$ and $Z$ meet
transversely in at most finitely many points. It follows from 3.5 that
$\operatorname{ch}_{2}(Y)\cdot S=0$. Let $S_{X}$ be the strict transform of
$S$ in $X$. By 3.7, $\operatorname{ch}_{2}(X)\cdot
S_{X}\leq\operatorname{ch}_{2}(Y)\cdot S=0$, and so $X$ is not $2$-Fano. ∎

##  4\. Proof of 1.3

In this section, we work in the setting of 1.3:

  1. (I)

$X$ is a Fano manifold of dimension $n\geq 6$ with fan $\Sigma=\Sigma_{X}$ in
$N_{\mathbb{Q}}$;

  2. (II)

$m(X)\geq 3$;

  3. (III)

$r(P)\colon x_{0}+\cdots+x_{n-2}=0$ is a centrally symmetric primitive
relation in $\Sigma$.

The equality $m(X)=n-2$, as well as uniqueness of the centrally symmetric
primitive collection, follows immediately from 2.5. Now the goal is to prove
that $\rho(X)\leq 3$, and this will follow from 4.11, 4.16 and 4.17. By 2.18,
we conclude that the projective space $\mathbb{P}^{n}$ is the only smooth
$n$-dimensional toric $2$-Fano variety with $m(X)\geq n-2$ (1.5).

Let $P\coloneqq\\{x_{0},\dots,x_{n-2}\\}$ and set
$\Gamma=\operatorname{Span}P\subset N_{\mathbb{Q}}$. Consider the quotient

$\pi\colon
N_{\mathbb{Q}}\simeq\mathbb{Q}^{n}\to\mathbb{Q}^{n}/\Gamma\simeq\mathbb{Q}^{2}.$

Since $X$ is complete, the support of $\Sigma$ is equal to $N_{\mathbb{Q}}$,
and hence we can find generators

  1. (IV)

$x,y,z\in G(\Sigma)\setminus P$, for which

  2. (V)

$0\in\operatorname{Conv}(\pi(x),\pi(y),\pi(z))$.

######  Lemma 4.1.

The nonnegative span $\langle x,y,z\rangle$ is not a cone in $\Sigma$.

###### Proof.

We will argue by contradiction and assume that $\langle
x,y,z\rangle\in\Sigma$. By (V), we can find a nonnegative triple of constants
$(c_{1},c_{2},c_{3})$ such that $c_{1}\pi(x)+c_{2}\pi(y)+c_{3}\pi(z)=0$, or in
other words

$v\coloneqq c_{1}x+c_{2}y+c_{3}z=a_{0}x_{0}+\cdots+a_{n-2}x_{n-2}$

for some constants $a_{i}$. Note that by adding some multiple of $r(P)$ (III)
to the right hand side, we can assume the $a_{i}$’s are nonnegative and such
that at least one of them, say $a_{j}$, is $0$. It follows that $v$ lies in
two cones of $\Sigma$, namely

$v\in\langle x,y,z\rangle\cap\langle
x_{0},\dots,\check{x_{j}},\dots,x_{n-2}\rangle,$

which is impossible since $\\{x,y,z\\}\cap S=\varnothing$. ∎

Since $\\{x,y,z\\}$ does not span a cone, we conclude that

  1. (1)

either it is a primitive collection,

  2. (2)

or two of these vectors do not form a cone.

The former case is the more technical one, and we start with it in Section
4.1. After we are done analyzing it, we can assume that none of the triples
$\\{x,y,z\\}$ as in (IV) and (V) form a primitive collection, and this case
will be treated in Section 4.2.

###  4.1. First case: $\\{x,y,z\\}$ is a primitive collection

We recall that if $X$ is a projective toric Fano manifold and $m(X)>1$, then
any $x\in G(\Sigma)$ has at most one opponent by 2.22, where the opponent of
$x$ is an element $x^{\prime}\in G(\Sigma)$ such that
$\\{x,x^{\prime}\\}\in\mathrm{PC}(X)$. When we write $\\{x,x^{\prime}\\}$, we
mean either the set of two elements if $x^{\prime}$ exists, or the singleton
$\\{x\\}$ if $x^{\prime}$ does not exist.

######  Remark 4.2.

In the setting (I)—(V), pick a generator $u\in G(\Sigma)$. Then (V) and plane
geometry imply that the convex hull of $\pi(u)$ together with two of the
vectors $\pi(x)$, $\pi(y)$, $\pi(z)$ contains $0$.

######  Lemma 4.3.

In the setting (I)—(V), assume in addition that $Q\coloneqq\\{x,y,z\\}$ is a
primitive collection. Then the corresponding primitive relation is

  * •

either $r(Q)\colon\,x+y+z=x_{i}+x_{j}$ for possibly equal $x_{i},x_{j}\in P$,

  * •

or $r(Q)\colon\,x+y+z=v$ for some $v\in G(\Sigma)$.

###### Proof.

Since $X$ is Fano (I), the degree of the primitive relation $x+y+z=A$ is
positive, so we can have only three possibilities for $A$. The first one with
$A=0$ is actually not possible by our assumption (II). The second is $A=v$,
and we cannot say much about $v$ at the moment. The last possibility is

$r(Q)\colon\,x+y+z=u+v$

for some, possibly equal, $u,v\in G(\Sigma)$. By 2.20, this is an extremal
primitive relation, so by 2.8 applied to $r(Q)$ and $\tau=\\{0\\}$, we get
that $\langle u,x,y\rangle$, $\langle u,y,z\rangle$, $\langle
u,x,z\rangle\in\Sigma$. By 4.2, we may assume without loss of generality that
$0\in\operatorname{Conv}(\pi(u),\pi(x),\pi(y))$, hence by 4.1, we get $u\in
P$. The same argument applies to conclude $v\in P$. ∎

Hence we have two cases to consider: when $\deg(Q)$ is $1$ and when it is $2$.

####  4.1.1. Degree one

In this case, by 4.3, we have

  1. (VI)

a primitive relation $r(Q)\colon\,x+y+z=x_{i}+x_{j}$ for possibly equal
$x_{i},x_{j}\in P$.

######  Lemma 4.4.

In the setting (I)—(VI), assume in addition that $G(\Sigma)$ is contained in
$P\cup\\{x,y,z,x^{\prime},y^{\prime},z^{\prime}\\}$. Then $x^{\prime}$,
$y^{\prime}$, $z^{\prime}$ do not exist. Consequently, $\rho(X)\leq 2$.

###### Proof.

By contradiction, assume that $x^{\prime}\in G(\Sigma)$ exists, so
$\\{x,x^{\prime}\\}$ is a primitive collection, and let $x+x^{\prime}=\alpha$
be the corresponding primitive relation. Clearly $\alpha\neq x$, $x^{\prime}$.
The relation $r(Q)$ (VI) gives

$\alpha+y+z=x_{i}+x_{j}+x^{\prime},$

which shows $\alpha\not\in P\cup\\{y,z\\}$ since otherwise the left hand side
(LHS) forms a cone and we get an effective class of degree zero. Indeed, if
$\alpha\in\\{y,z\\}$ it is clear that the LHS would form a cone; if $\alpha\in
P$ applying 2.8 to $r(Q)$ and $\tau=\langle\alpha\rangle$ would follow that
the LHS is a cone.

So without loss of generality, we can assume $x+x^{\prime}=y^{\prime}$.
Applying the same argument to $y^{\prime}$, we get that
$y+y^{\prime}=x^{\prime}$ or $y+y^{\prime}=z^{\prime}$. The former would imply
$x+y=0$, which is not possible by (II), so $y+y^{\prime}=z^{\prime}$. Again,
applying the same argument to $z^{\prime}$, we get $z+z^{\prime}=x^{\prime}$.
Summing the three primitive relations, we obtain $x+y+z=0$, which contradicts
(VI). ∎

This leaves us with the case when

  1. (VII)

there exists $u\in
G(\Sigma)\setminus(P\cup\\{x,y,z,x^{\prime},y^{\prime},z^{\prime}\\})$.

By 4.2, we can assume without loss of generality that

  1. (VIII)

$0\in\operatorname{Conv}(\pi(x),\pi(y),\pi(u))$.

######  Lemma 4.5.

Assume (I)—(VIII), then $R\coloneqq\\{x,y,u\\}$ is a primitive collection with
primitive relation $r(R)\colon\,x+y+u=v$ for some $v\in G(\Sigma)$.

###### Proof.

By 4.1, we have $\langle x,y,u\rangle\notin\Sigma$, and since $u\neq
x^{\prime},y^{\prime}$, we conclude that $\\{x,y,u\\}$ is a primitive
collection. By 4.3, we can have either $r(R)\colon\,x+y+u=x_{k}+x_{l}$ or
$r(R)\colon\,x+y+u=v$. In the former case, combining $r(R)$ with $r(Q)$ (VI)
provides us with a relation

$u+x_{i}+x_{j}=z+x_{k}+x_{l}.$

By applying 2.8 to $r(Q)$ (VI) and $\tau=\langle x_{k},x_{l}\rangle$, we get
$\langle z,x_{k},x_{l}\rangle\in\Sigma$ (here we are using the assumption
$\dim(X)=n\geq 6$ (I)). Hence, by 2.6, we get an effective curve class of
degree $0$, contradicting the Fano assumption (I). ∎

We will write down the result of 4.5 as an additional assumption, remembering
that it is implied by the previous assumptions:

  1. (IX)

We have a primitive relation $r(R)\colon\,x+y+u=v$ for some $v\in G(\Sigma)$.

######  Lemma 4.6.

Assume (I)—(IX), then

$G(\Sigma)\subset
P\cup\\{x,y,z,u,x^{\prime},y^{\prime},u^{\prime},v^{\prime}\\}.$

In particular, $z^{\prime}\in
P\cup\\{x,y,z,u,x^{\prime},y^{\prime},u^{\prime},v^{\prime}\\}$, and since
$v\neq x,y,u,v^{\prime}$, we have that $v\in P$ or
$v\in\\{z,x^{\prime},y^{\prime},u^{\prime}\\}$.

###### Proof.

Take any $w\in
G(\Sigma)\setminus(P\cup\\{x,y,z,u,x^{\prime},y^{\prime},u^{\prime}\\})$. By
4.2, the convex hull of $\pi(w)$ together with two of $\pi(x)$, $\pi(y)$,
$\pi(u)$ contains $0$, yielding an analog of (VIII). By 4.1 and from $w\neq
x^{\prime},y^{\prime},u^{\prime}$, it follows that one of $\\{w,x,u\\}$,
$\\{w,y,u\\}$, $\\{w,x,y\\}$ is a primitive collection.

We will prove that the corresponding primitive relation has the form $w+x+u=b$
or $w+y+u=b$ or $w+x+y=b$ for some $b\in G(\Sigma)$. Assume for a
contradiction that this is not the case. By 4.3, we have that one of $w+x+u$,
$w+y+u$, or $w+x+y$ equals $x_{k}+x_{l}$, for possibly equal $x_{k},x_{l}\in
P$. Hence there exist $a,b\in G(\Sigma)$ (one of which is $w\neq z$) such that
either $x+a+b$ or $y+a+b$ equals $x_{k}+x_{l}$. Assume $x+a+b=x_{k}+x_{l}$.
Combining this with $r(Q)$ (VI), it follows that
$y+z+x_{k}+x_{l}=a+b+x_{i}+x_{j}$. By applying 2.8 to $r(Q)$ (VI) and
$\tau=\langle x_{k},x_{l}\rangle$, we get $\langle
y,z,x_{k},x_{l}\rangle\in\Sigma$ (here we are using the assumption
$\dim(X)=n\geq 6$ (I)). Hence, by 2.6, we get an effective curve class of
degree $0$. The class is non-trivial since $w\neq y,z,x_{k},x_{l}$. This
contradicts the assumption that $X$ is Fano. The case $y+a+b=x_{k}+x_{l}$ is
similar.

So we have a primitive relation of the form $w+x+u=b$ or $w+y+u=b$ or
$w+x+y=b$ for some $b\in G(\Sigma)$. Combining this with $r(R)$ (IX), we get
$b+y=w+v$ or $b+x=w+v$ or $b+u=w+v$. All possibilities imply that
$w=v^{\prime}$, as otherwise, by 2.6, we obtain an effective class of degree
$0$ (non-trivial since $w,v\neq y,u$), which contradicts the Fano assumption
(I). ∎

######  Lemma 4.7.

Assume (I)—(IX). Then $x$, $y$ don’t have opponents,

$G(\Sigma)\subset P\cup\\{x,y,z,u,u^{\prime},v^{\prime}\\},$

and we have a trichotomy: $v\in P$ or $v=z$ or $v=u^{\prime}$.

###### Proof.

We will only show that $x^{\prime}$ does not exist, and the argument for
$y^{\prime}$ is symmetric. Suppose to the contrary that $x^{\prime}\in
G(\Sigma)$, and let $x+x^{\prime}=\alpha$ be the corresponding primitive
relation, so $\alpha\neq x,x^{\prime}$. Then we can substitute
$x=\alpha-x^{\prime}$ into $r(Q)$ (VI) to get

$\alpha+y+z=x^{\prime}+x_{i}+x_{j},$

which shows $\alpha\notin P\cup\\{y,z\\}$, as otherwise we would get an
effective curve class of degree $0$. Substituting $x=\alpha-x^{\prime}$ into
$r(R)$ (IX) gives the relation

(6) $\alpha+y+u=v+x^{\prime}.$

Note that $\langle v,x^{\prime}\rangle\in\Sigma$ by 2.22. We claim that
Equation 6 is an extremal primitive relation by 2.26. Assume not, then 2.26
implies that one of $\\{\alpha,y,u\\}$ is in $\\{v,x^{\prime}\\}$ and the
remaining two vectors are opponents. Since $u\neq y^{\prime}$, then either
$\alpha,y$ are opponents and $u\in\\{v,x^{\prime}\\}$, or $\alpha,u$ are
opponents and $y\in\\{v,x^{\prime}\\}$. From (IX), we have $u\neq v$ and
$y\neq v$; (VII) means $u\neq x^{\prime}$; and (VI) implies $y\neq
x^{\prime}$. So both cases are impossible.

So Equation 6 is an extremal primitive relation. In particular, $\alpha\neq
y^{\prime},u,u^{\prime}$. By 2.8, $v$ forms a cone with $\alpha$, hence
$\alpha\neq v^{\prime}$, which contradicts 4.6. ∎

######  Lemma 4.8.

Assume (I)—(IX). Then we have

$G(\Sigma)\subset P\cup\\{x,y,z,u,v^{\prime}\\}$

and a dichotomy: $v\in P$ or $v=z$. If moreover $u^{\prime}$ exists, we have
$u+u^{\prime}=z$, $v=x_{i}$ and $u=x_{j}^{\prime}$.

###### Proof.

Assume that $u^{\prime}$ exists and let $u+u^{\prime}=\alpha$ be the
corresponding primitive relation. Then substituting it into $r(R)$ (IX) gives
a relation

(7) $x+y+\alpha=u^{\prime}+v.$

By $r(R)$ (IX), $v\neq u$, so $\langle u^{\prime},v\rangle\in\Sigma$. Since
there are no $x^{\prime}$ and $y^{\prime}$, it follows from 2.26 that Equation
7 is an extremal primitive relation. It follows that
$\alpha\notin\\{u,u^{\prime},x,y,v^{\prime}\\}$. Moreover, $\alpha\notin P$,
as otherwise applying 2.8 to $r(Q)$ and $\tau=\langle\alpha\rangle$ would
imply $\langle x,y,\alpha\rangle\in\Sigma$. So by 4.7, the only possibility is
that $\alpha=z$. Therefore $u^{\prime}+v=x_{i}+x_{j}$ by $r(Q)$ (VI), so we
have, after possibly relabeling, that $v=x_{i}$, $u^{\prime}=x_{j}$, which by
4.7 proves the statement.

If instead $u^{\prime}$ does not exist, the statement follows directly from
4.7. ∎

######  Lemma 4.9.

Assume (I)—(IX). Then $z$ doesn’t have an opponent.

###### Proof.

We argue by contradiction and assume that $z^{\prime}$ exists. Clearly
$z^{\prime}\neq x,y,z$ by $r(Q)$ (VI) and $z^{\prime}\neq u$ by (VII).
Furthermore, $z^{\prime}\neq u^{\prime}$, otherwise we have $z=u$ by 2.22,
which contradicts (VII). Finally, $z^{\prime}\notin P$, otherwise
$z^{\prime}=x_{k}$ implies $z=x_{k}^{\prime}$, and $r(Q)$ (VI) becomes

$x+y+x_{k}^{\prime}=x_{i}+x_{j},$

but applying 2.8 to $r(Q)$ and $\tau=\langle x_{k}\rangle$, we get $\langle
x_{k},x_{k}^{\prime}\rangle\in\Sigma$, a contradiction. Thus by 4.8, the only
possibility is $z^{\prime}=v^{\prime}$, so $z=v$ by 2.22. By 4.8, this implies
$G(\Sigma)\subset P\cup\\{x,y,z,u,z^{\prime}\\}$.

Consider the primitive relation $z+z^{\prime}=\beta\in G(\Sigma)$. We will
show that $\beta=u$. Indeed, it is clear that $\beta\neq z,z^{\prime}$.
Combining $z+z^{\prime}=\beta$ with $r(Q)$ (VI), we have

$x+y+\beta=x_{i}+x_{j}+z^{\prime}.$

If $\beta\in\\{x,y\\}$, the left hand side is a cone, and we get a non-trivial
effective relation of degree 0, which is impossible by (I). If $\beta\in P$,
applying 2.8 to $r(Q)$ and $\tau=\langle\beta\rangle$ implies that $\langle
x,y,\beta\rangle\in\Sigma$, and we obtain a contradiction as before.

But now, substituting $z+z^{\prime}=u$ into $r(R)$ (IX) yields
$x+y+z^{\prime}=0$, hence contradicting $m(X)\geq 3$ (II). ∎

######  Lemma 4.10.

Assume (I)—(VII). Then $\rho(X)\leq 3$.

###### Proof.

Let us summarize the consequences of (I)—(VII):

  * (VIII)

$0\in\operatorname{Conv}(\pi(x),\pi(y),\pi(u))$ after possibly relabeling $x$,
$y$, $z$;

  * (IX)

we have a primitive relation $r(R)\colon x+y+u=v$ (4.5);

  * •

$x,y,z$ don’t have opponents (4.7, 4.9);

  * •

$G(\Sigma)\subset P\cup\\{x,y,z,u,v^{\prime}\\}$ (4.8), so $\rho(X)\leq 4$;

  * •

we have $v\in P$ or $v=z$ (4.8), so we can consider two cases.

##### Case $v=z$.

By 4.9, $v^{\prime}=z^{\prime}$ does not exist, hence it follows from 4.8 that
$G(\Sigma)\subset P\cup\\{x,y,z,u\\}$, so $\rho(X)\leq 3$.

##### Case $v\in P$, say $v=x_{l}$.

If $v=x_{l}$ doesn’t have an opponent, then $G(\Sigma)\subset
P\cup\\{x,y,z,u\\}$ and $\rho(X)\leq 3$. So the tricky case is when $v=x_{l}$
has an opponent $x_{l}^{\prime}\notin P$.

Let $v+v^{\prime}=\beta\in G(\Sigma)$. By $r(R)$, we have that
$x+y+u+v^{\prime}=v+v^{\prime}=\beta$. Since $m(X)\geq 3$, $\beta\neq
x,y,u,v^{\prime}$. Hence $\beta\in P\cup\\{z\\}$. If $\beta\in P$, let
$\beta=x_{k}$. Then $v^{\prime}=x_{k}-x_{l}$ and $k\neq l$. Using $r(P)$, we
obtain that $v^{\prime}=2x_{k}+\sum_{t\neq k,l}x_{t}$. As the vectors on the
right hand side form a cone, we get an effective curve class of degree $2-n$,
which is impossible by (I).

So $\beta=z$. By $r(Q)$, we obtain that $x+y+v^{\prime}=x_{i}+x_{j}-x_{l}$. If
$l\neq i,j$, then again, by using $r(P)$ (III), we may write
$x_{i}+x_{j}-x_{l}$ as $x_{i}+x_{j}+\sum_{t\neq l}x_{t}$. As the vectors on
the right hand side form a cone, we obtain an effective curve class of degree
$3-n$, which is impossible by (I).

So $l=i$ or $l=j$. Up to symmetry, we may assume $l=i$, so $v=x_{i}$,
$x_{i}+x^{\prime}_{i}=z$. By $r(Q)$, we have that $x+y+x^{\prime}_{i}=x_{j}$.
Combining this with $r(R)$, we obtain that $x^{\prime}_{i}+x_{i}=u+x_{j}$.
Furthermore, $u\neq v^{\prime}=x^{\prime}_{i}$ and $u\neq x_{i}$. If $u,x_{j}$
form a cone, we obtain an effective non-trivial curve class of degree $0$,
which is impossible by (I). So $u=x^{\prime}_{j}$, $v=x_{i}$ and we have two
primitive relations

$x_{i}+x_{i}^{\prime}=z\quad\mbox{and}\quad x_{j}+x_{j}^{\prime}=z.$

Then either $i=j$, in which case $\rho(X)\leq 3$, or $i\neq j$, and we notice
that they are both degree $1$ and hence extremal, so we can perform the
contraction associated to one of them, say we contract the curve class
$x_{i}+x_{i}^{\prime}=z$:

${X}$${Y,}$${V(z)}$${V(x_{i},x_{i}^{\prime}).}$

By 2.16,

$\displaystyle r(P_{Y})\colon$ $\displaystyle x_{0}+\cdots+x_{n-2}=0,$
$\displaystyle r(R_{Y})\colon$ $\displaystyle x+y+x^{\prime}_{j}=x_{i},$
$\displaystyle r(Q^{\prime})\colon$ $\displaystyle x+y+x^{\prime}_{i}=x_{j},$
$\displaystyle r(Q^{\prime\prime})\colon$ $\displaystyle
x_{j}+x^{\prime}_{j}=x_{i}+x^{\prime}_{i}$

are primitive relations in $Y$. Since $\rho(Y)=3$, we apply 2.17. We adopt the
same notation as in 2.17 and observe that we are in the case $l=5$, hence

$G(\Sigma)=\sqcup_{h=0}^{4}X_{h}=\\{x_{0},\dots,\check{x_{j}},\dots,x_{n-2}\\}\sqcup\\{x_{j}\\}\sqcup\\{x,y\\}\sqcup\\{x^{\prime}_{j}\\}\sqcup\\{x^{\prime}_{i}\\},$

and either $X_{2}\sqcup X_{3}=\\{x_{0},\dots,x_{n-2}\\}$, or
$\overline{c}=\overline{b}=\overline{0}$ and $X_{4}\sqcup
X_{0}=\\{x_{0},\dots,x_{n-2}\\}$. However, all possibilities for $\\{x_{j}\\}$
lead to a contradiction. Indeed:

  * -

if $X_{3}=\\{x_{j}\\}$, then we have $r_{3}\colon\,\overline{1}\cdot
X_{3}+\overline{1}\cdot
X_{4}=x_{j}+x^{\prime}_{j}=x_{i}+x^{\prime}_{i}\neq\overline{1}\cdot X_{h}$
for all $h=0,\dots,4$;

  * -

if $X_{2}=\\{x_{j}\\}$, then we have $r_{1}\colon\,\overline{1}\cdot
X_{1}+\overline{1}\cdot
X_{2}=x^{\prime}_{j}+x_{j}=x_{i}+x^{\prime}_{i}\neq\overline{1}\cdot X_{h}$
for all $h=0,\dots,4$;

  * -

if $X_{4}=\\{x_{j}\\}$, then we have $r_{3}\colon\,\overline{1}\cdot
X_{3}+\overline{1}\cdot
X_{4}=x^{\prime}_{j}+x_{j}=x_{i}+x^{\prime}_{i}\neq\overline{1}\cdot X_{h}$
for all $h=0,\dots,4$;

  * -

if $X_{0}=\\{x_{j}\\}$, then we have $r_{0}\colon\,\overline{1}\cdot
X_{0}+\overline{1}\cdot
X_{1}=x_{j}+x^{\prime}_{j}=x_{i}+x^{\prime}_{i}\neq\cancel{\overline{c}\cdot
X_{2}}+(\cancel{\overline{b}}+\overline{1})X_{3}=\overline{1}\cdot X_{3}$.

This concludes the last case, and we get that $\rho(X)\leq 3$. ∎

######  Proposition 4.11.

Let $X$ be a projective toric Fano manifold of $\dim(X)=n\geq 5$, $m(X)\geq
3$, which admits a primitive relation $r(P)\colon x_{0}+x_{1}+\dots+x_{n-2}=0$
(I)—(III). Let $x,y,z\in G(\Sigma)\setminus P$ be such that
$0\in\operatorname{Conv}(\pi(x),\pi(y),\pi(z))$ (IV)—(V). Assume in addition
that $\\{x,y,z\\}$ is a primitive collection of degree $1$ (VI). Then
$\rho(X)\leq 3$.

###### Proof.

We recall that assumptions (I)—(III) imply (IV)—(V). Then either
$G(\Sigma)\subset P\cup\\{x,y,z,x^{\prime},y^{\prime},z^{\prime}\\}$, in which
case $\rho(X)\leq 2$ by 4.4, or there exists a vector $u\in
G(\Sigma)\setminus(P\cup\\{x,y,z,x^{\prime},y^{\prime},z^{\prime}\\})$ (VII),
in which case 4.10 implies $\rho(X)\leq 3$, and we are done. ∎

####  4.1.2. Degree two

Still working in the setting (I)—(V), we assume that $\\{x,y,z\\}$ is a
primitive collection whose primitive relation has degree $2$, and by 4.11, we
can exclude the case when degree one primitive collections as in (IV)—(VI)
exist. In other words, these are the additional assumptions for this part of
the proof:

  1. (X)

we have a primitive relation $r(Q)\colon\,x+y+z=v$ for some $v\in G(\Sigma)$;

  2. (XI)

there is no primitive collection $\\{a,b,c\\}\subset G(\Sigma)\setminus P$
with $0\in\operatorname{Conv}(\pi(a),\pi(b),\pi(c))$ whose primitive relation
has degree $1$.

######  Lemma 4.12.

In the setting (I)—(V) and (X)—(XI), we have

$G(\Sigma)\subset
P\cup\\{x,y,z,x^{\prime},y^{\prime},z^{\prime},v^{\prime}\\}\quad\text{and}\quad
v\in P\cup\\{x^{\prime},y^{\prime},z^{\prime}\\}.$

###### Proof.

If there is a generator $u\in
G(\Sigma)\setminus(P\cup\\{x,y,z,x^{\prime},y^{\prime},z^{\prime}\\})$, then
by 4.2 and (XI) we can assume that we have a primitive relation of the form

$x+y+u=w.$

Combining it with (X), we get $u+v=w+z$, so $u=v^{\prime}$, otherwise $\langle
u,v\rangle\in\Sigma$ and we get an effective non-trivial curve class of degree
zero.

In particular, since $v\neq v^{\prime}$, we have $v\in
P\cup\\{x^{\prime},y^{\prime},z^{\prime}\\}$. ∎

######  Lemma 4.13.

In the setting (I)—(V) and (X)—(XI), assume that $v=x_{m}\in P$. Then
$x_{m}^{\prime}$ does not exist.

###### Proof.

The primitive relation $r(Q)$ (X) becomes $x+y+z=x_{m}$, and by 4.12, we have
$G(\Sigma)\subset
P\cup\\{x,y,z,x^{\prime},y^{\prime},z^{\prime},x_{m}^{\prime}\\}$.

Assume to the contrary that we have $x_{m}^{\prime}\in G(\Sigma)$. Clearly,
$x_{m}^{\prime}\notin P$ because $x_{m}$ forms a $2$-dimensional cone in
$\Sigma$ with any other generator $x_{i}\in P$. By 4.2, $x_{m}^{\prime}$ makes
a primitive collection with two of $x,y,z$. Without loss of generality, assume
we have a primitive relation of the form

$x+y+x_{m}^{\prime}=w.$

Combining it with $r(Q)$ (X), we get $x_{m}^{\prime}+x_{m}=z+w$, so
$w=z^{\prime}$ by 2.25. Let $\alpha=x_{m}+x_{m}^{\prime}=z+z^{\prime}$. Then
$\alpha\not\in P$ because $x_{m}^{\prime}=\alpha-x_{m}$ is not in
$\operatorname{Span}P$. Now substituting $z=\alpha-z^{\prime}$ into $r(Q)$ (X)
yields

(8) $x+y+\alpha=x_{m}+z^{\prime}.$

Since $x_{m}\neq z$ by $r(Q)$ (X), we have $\langle
x_{m},z^{\prime}\rangle\in\Sigma$, so Equation 8 is an extremal primitive
relation. This implies that $\alpha\neq
x,y,z,x^{\prime},y^{\prime},z^{\prime},x^{\prime}_{m}$, a contradiction with
4.12. ∎

######  Lemma 4.14.

In the setting (I)—(V) and (X)—(XI), assume that $v=x_{m}\in P$. Then
$\rho(X)\leq 3$.

###### Proof.

By 4.12 and 4.13, we have $G(\Sigma)\subset
P\cup\\{x,y,z,x^{\prime},y^{\prime},z^{\prime}\\}$, and, as before, the
primitive relation (X) is $r(Q)\colon\,x+y+z=x_{m}$.

We show that at most one of $x^{\prime},y^{\prime},z^{\prime}$ exists. Suppose
to the contrary that for example $x^{\prime},y^{\prime}\in G(\Sigma)$, and let
$\alpha=x+x^{\prime}$, $\beta=y+y^{\prime}\in G(\Sigma)$. Since

$\alpha+y+z=x_{m}+x^{\prime}\quad\text{and}\quad\langle
x_{m},x^{\prime}\rangle\in\Sigma,$

applying 2.26 shows that this is an extremal primitive relation: indeed,
either $\alpha$ and $y$ are opponents and $z\in\\{x_{m},x^{\prime}\\}$, or
$\alpha$ and $z$ are opponents and $y\in\\{x_{m},x^{\prime}\\}$. But
$y,z\notin P\cup\\{x^{\prime}\\}$, so this is a contradiction. It follows that
$\alpha\not\in\\{x,x^{\prime},y,y^{\prime},z,z^{\prime}\\}$, so
$\alpha=x_{l}\in P$; similarly, $\beta=x_{k}\in P$. But we have

$\alpha+\beta+z=x^{\prime}+y^{\prime}+x_{m},$

which is not possible since we show that the right hand side forms a cone.
Indeed, applying 2.8 to $x+x^{\prime}=x_{l}$ and $\tau=\langle
x_{m},x_{k}\rangle$ (we use here that $n\geq 5$), we obtain $\langle
x_{m},x_{k},x^{\prime}\rangle\in\Sigma$, and applying then 2.8 to
$y+y^{\prime}=x_{k}$ and $\tau=\langle x_{m},x^{\prime}\rangle$ we have that
$\langle x_{m},x^{\prime},y^{\prime}\rangle\in\Sigma$. So, after possibly
relabelling $x$, $y$, $z$, we have $G(\Sigma)\subset
P\cup\\{x,y,z,x^{\prime}\\}$, hence $\rho(X)\leq 3$. ∎

######  Lemma 4.15.

In the setting (I)—(V) and (X)—(XI), assume that $v=x^{\prime}$. Then
$\rho(X)\leq 3$.

###### Proof.

We have $r(Q)\colon\,x+y+z=x^{\prime}$. We know $G(\Sigma)\subset
P\cup\\{x,y,z,x^{\prime},y^{\prime},z^{\prime}\\}$ by 4.12. So it is enough to
show $y^{\prime}$ and $z^{\prime}$ do not exist. Suppose to the contrary that,
for example, $y^{\prime}\in G(\Sigma)$, and let $\beta=y+y^{\prime}$. Then

$\beta+x+z=x^{\prime}+y^{\prime}\quad\text{and}\quad\langle
x^{\prime},y^{\prime}\rangle\in\Sigma,$

so again, by 2.26, this is an extremal primitive relation. Indeed, assume the
relation is not primitive. By 2.26, either $\beta$ and $x$ are opponents and
$z\in\\{x^{\prime},y^{\prime}\\}$, or $\beta$ and $z$ are opponents and
$x\in\\{x^{\prime},y^{\prime}\\}$. But $z,x\notin\\{x^{\prime},y^{\prime}\\}$,
since $\\{x,y,z\\}$ is a primitive collection. Hence
$\beta+x+z=y^{\prime}+x^{\prime}$ is a primitive relation. It follows from 2.8
that $\langle x,x^{\prime}\rangle\in\Sigma$, which is a contradiction. ∎

######  Proposition 4.16.

Let $X$ be a projective toric Fano manifold of $\dim(X)=n\geq 5$, $m(X)\geq
3$, which admits a primitive relation $r(P)\colon
x_{0}+x_{1}+\cdots+x_{n-2}=0$ (I)—(III). Assume that any primitive collection
$\\{x,y,z\\}$ such that $x,y,z\in G(\Sigma)\setminus P$ and
$0\in\operatorname{Conv}(\pi(x),\pi(y),\pi(z))$ (IV) —(V) has degree $2$ (XI),
and that there exists such a triple $x,y,z$ (X). Then $\rho(X)\leq 3$.

###### Proof.

The statement follows from 4.12, 4.14 and 4.15. ∎

###  4.2. Second case: none of $\\{x,y,z\\}$ form a primitive collection

######  Proposition 4.17.

Let $X$ be a toric Fano manifold of $\dim(X)=n\geq 5$, $m(X)\geq 3$, which
admits a primitive relation $r(P)\colon x_{0}+x_{1}+\cdots+x_{n-2}=0$
(I)—(III). Assume in addition that none of the triples $\\{x,y,z\\}\subseteq
G(\Sigma)\setminus P$ such that
$0\in\operatorname{Conv}(\pi(x),\pi(y),\pi(z))$ (IV)—(V) form a primitive
collection. Then $\rho(X)\leq 3$.

Before proving 4.17, we will formulate the following useful lemma.

######  Lemma 4.18.

In the setting of 4.17, take any triple $\\{x,y,z\\}\subseteq
G(\Sigma)\setminus P$ with $\langle x,y\rangle\in\Sigma$. Assume that
$0\in\operatorname{Conv}(\pi(x),\pi(y),\pi(z))$, or equivalently,
$\pi(z)\in\langle-\pi(x),-\pi(y)\rangle$. Then $z=x^{\prime}$ or
$z=y^{\prime}$.

###### Proof.

By 4.1, $x$, $y$, $z$ do not span a cone, and by assumption, they do not form
a primitive collection. So two of the vectors must not form a cone. Since we
assumed that $\langle x,y\rangle\in\Sigma$, we must have $z=x^{\prime}$ or
$z=y^{\prime}$. ∎

###### Proof of 4.17.

We select $x,y\in G(\Sigma)$ with $\langle x,y\rangle\in\Sigma$ and such that
the cone generated by $\pi(x)$ and $\pi(y)$ is maximal among cones in
$\mathbb{Q}^{2}\simeq\mathbb{Q}^{n}/\Gamma$ coming from such pairs. If there
is $z\in G(\Sigma)$ such that $\pi(z)$ is outside the cone
$\langle\pi(x),\pi(y)\rangle$, then we show that $z=x^{\prime}$ or
$z=y^{\prime}$. Indeed, the case $\pi(z)\in\langle-\pi(x),-\pi(y)\rangle$ is
covered by 4.18; $\pi(z)\in\langle\pi(x)\rangle$ or
$\pi(z)\in\langle\pi(y)\rangle$ cannot happen by an argument similar to 4.1;
and in the remaining case, $\pi(z)$ is in $\langle\pi(x),-\pi(y)\rangle$ or
$\langle-\pi(x),\pi(y)\rangle$, then we use maximality of the cone generated
by $\pi(x)$ and $\pi(y)$.

Let $v$ be such that $\pi(v)$ is in the open half plane determined by
$\operatorname{Span}\pi(x)$ not containing $\pi(y)$. Up to relabelling of $x$,
$y$, we can assume that such a $v$ exists. If $v=x^{\prime}$, then
$x+x^{\prime}=y^{\prime}$, since $\pi(x+x^{\prime})$ is non-zero and is
outside of $\langle\pi(y),\pi(x)\rangle.$ So

$\langle\pi(x),\pi(y)\rangle\subset\langle-\pi(y),-\pi(x^{\prime})\rangle\cup\langle-\pi(x^{\prime}),-\pi(y^{\prime})\rangle\cup\langle-\pi(y^{\prime}),-\pi(x)\rangle.$

It follows from 4.18 that $G(\Sigma)=P\cup\\{x,y,x^{\prime},y^{\prime}\\}$,
which concludes this case.

If now $v=y^{\prime}$, in case $\pi(v)\in\langle-\pi(x),-\pi(y)\rangle$, we
notice that $y+y^{\prime}=x^{\prime}$ as above, and in case
$\pi(v)\in\langle\pi(x),-\pi(y)\rangle$, by completeness, we find $w$ such
that $\pi(w)\in\langle-\pi(x),\pi(y)\rangle\cup\langle-\pi(x),-\pi(y)\rangle$
and notice $w=x^{\prime}$. In either case, we get

$\mathbb{Q}^{2}=\langle-\pi(x),-\pi(y)\rangle\cup\langle-\pi(y),-\pi(x^{\prime})\rangle\cup\langle-\pi(x^{\prime}),-\pi(y^{\prime})\rangle\cup\langle-\pi(y^{\prime}),-\pi(x)\rangle,$

and now 4.18 gives $G(\Sigma)=P\cup\\{x,y,x^{\prime},y^{\prime}\\}$. ∎

##  5\. Toric Fano manifolds with $m(X)=n-2$

In this section, we classify all $n$-dimensional toric Fano manifolds $X$ with
$m(X)=n-2$ and $n\geq 6$ (1.4). By 1.3, we know that $\rho(X)\leq 3$. 5.1 and
5.2 classify $n$-dimensional toric Fano manifolds $X$ with $m(X)=n-2$, $n\geq
5$, and Picard rank $\rho(X)=2$ and $3$, respectively. Together, these results
yield a classification of toric Fano manifolds with $m(X)=n-2$ and $n\geq 6$.

######  Theorem 5.1.

Let $X$ be a toric Fano manifold of dimension $n\geq 5$, $m(X)=n-2$ and
$\rho(X)=2$. Then $X\simeq\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{E})$, where
$\mathcal{E}$ is one of the following vector bundles on $\mathbb{P}^{2}$:

  * •

$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-2}$,

  * •

$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-3}$,

  * •

$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}(2)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-2}$.

###### Proof.

Let $X$ be an $n$-dimensional toric Fano manifold with $m(X)=n-2$ and
$\rho(X)=2$. We recall that toric manifolds with Picard rank $2$ are
classified by [Kle88]: they are projective space bundles over projective
spaces. The assumption $m(X)=n-2$ implies that $X$ is a
$\mathbb{P}^{n-2}$-bundle over $\mathbb{P}^{2}$.

We can write $X=\mathbb{P}(\mathcal{E})$, where
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}(a_{0})\oplus\mathcal{O}_{\mathbb{P}^{2}}(a_{1})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{2}}(a_{n-2})$
and $a_{0}\geq a_{1}\geq\dots\geq a_{n-2}=0$. The Fano assumption on $X$ is
equivalent to saying that $\sum_{i=0}^{n-2}a_{i}\leq 2$. Thus we have the
following cases:

  1. (1)

$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}^{\oplus n-1}$ and
$X\simeq\mathbb{P}^{n-2}\times\mathbb{P}^{2}$,

  2. (2)

$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-2}$,

  3. (3)

$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-3}$,

  4. (4)

$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{2}}(2)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-2}$.

In Case (1), we have $m(X)=2$. Cases (2)—(4) provide the complete list of
toric Fano manifolds of Picard rank $2$ and $m(X)=n-2$. ∎

######  Theorem 5.2.

Let $X$ be a toric Fano manifold of dimension $n\geq 5$, $m(X)=n-2$ and
$\rho(X)=3$. Then one of the following holds:

  1. (1)

$X=\mathbb{P}_{S}(\mathcal{E})$ is a $\mathbb{P}^{n-2}$-bundle over a toric
surface $S$, where $(S,\mathcal{E})$ is one of the following:

     * •

$S=\mathbb{P}^{1}\times\mathbb{P}^{1}$ and
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,1)\oplus\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}^{\oplus
n-2}$,

     * •

$S=\mathbb{P}^{1}\times\mathbb{P}^{1}$ and
$\mathcal{E}=\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,0)\oplus\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(0,1)\oplus\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}^{\oplus
n-3}$,

     * •

$S=\mathbb{F}_{1}$ and
$\mathcal{E}=\mathcal{O}_{\mathbb{F}_{1}}(e+f)\oplus\mathcal{O}_{\mathbb{F}_{1}}^{\oplus
n-2}$, where $e\subset\mathbb{F}_{1}$ is the $-1$-curve, and
$f\subset\mathbb{F}_{1}$ is a fiber of $\mathbb{F}_{1}\to\mathbb{P}^{1}$.

  2. (2)

Let
$Y\simeq\mathbb{P}_{\mathbb{P}^{2}}\big{(}\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-2}\big{)}$ be the blowup of $\mathbb{P}^{n}$ along a linear subspace
$L=\mathbb{P}^{n-3}$, and denote by $E\subset Y$ the exceptional divisor. Then
$X$ is the blowup of $Y$ along a codimension $2$ center $Z\subset Y$, where:

     * •

$Z$ is the intersection of $E$ with the strict transform of a hyperplane of
$\mathbb{P}^{n}$ containing the linear subspace $L$, or

     * •

$Z$ is the intersection of the strict transforms of two hyperplanes of
$\mathbb{P}^{n}$, one containing the linear subspace $L$, and the other one
not containing it.

###### Proof.

We apply Batyrev’s classification of toric manifolds with $\rho(X)=3$, stated
in 2.17. We adopt the same notation as in 2.17, and treat separately the cases
when the number of primitive collections is $l=3$ and $l=5$.

##### Case $l=3$.

As $G(\Sigma)=X_{0}\sqcup X_{1}\sqcup X_{2}$ and $m(X)=n-2$, we have
$X_{0}=\\{x_{0},\dots,x_{n-2}\\}$, $X_{1}=\\{v_{1},v_{2}\\}$ and
$X_{2}=\\{z_{1},z_{2}\\}$. The corresponding primitive relations are all
extremal by 2.17. By 2.13, $X$ is a $\mathbb{P}^{n-2}$-bundle over a surface
$S$. Up to relabelling in $X_{0}$, $X_{1}$ and $X_{2}$, three possible choices
for the remaining two primitive relations are (noting that $m(X)=n-2>1$):

$\begin{cases}v_{1}+v_{2}=x_{0},\\\ z_{1}+z_{2}=v_{1};\end{cases}\quad\hskip
12.0pt\begin{cases}v_{1}+v_{2}=x_{0},\\\
z_{1}+z_{2}=x_{0};\end{cases}\quad\quad\begin{cases}v_{1}+v_{2}=x_{0},\\\
z_{1}+z_{2}=x_{1}.\end{cases}$

It follows that $S$ is isomorphic to $\mathbb{F}_{1}$ when $r(X_{1})$ and
$r(X_{2})$ are as in the first column above, or to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ in two other cases.

##### Case $l=5$.

We denote by $l_{i}=|X_{i}|\geq 1$ the cardinality of $X_{i}$.

If $(\overline{c},\overline{b})=(\overline{0},\overline{0})$, then both
$r_{2}\colon\overline{1}\cdot X_{2}+\overline{1}\cdot X_{3}=0$ and
$r_{4}\colon\overline{1}\cdot X_{4}+\overline{1}\cdot X_{0}=0$ are centrally
symmetric primitive relations. By the assumption $m(X)=n-2$, we have $2n-2\leq
l_{2}+l_{3}+l_{4}+l_{0}\leq|G(\Sigma)|-1=n+2$, which implies $n\leq 4$, a
contradiction.

So we have $(\overline{c},\overline{b})\neq(\overline{0},\overline{0})$, and
$P\coloneqq X_{2}\sqcup X_{3}$ is the only centrally symmetric primitive
collection, so $l_{2}+l_{3}=n-1$ and $l_{0}+l_{1}+l_{4}=4$, with
$l_{0},l_{1},l_{4}\in\\{1,2\\}$. As $\deg(r_{0})>0$, we get the inequality

$3\geq l_{0}+l_{1}>\sum c_{i}+\sum b_{j}+l_{3},$

which is only satisfied when $l_{3}=1$, $l_{0}+l_{1}=3$ and exactly one entry
in

$(c_{1},c_{2},\dots,c_{l_{2}},b_{1})$

equals one, while the others are all zero. Up to relabelling, there are two
cases: $c_{1}=1$ or $b_{1}=1$.

We then have $l_{4}=4-(l_{0}+l_{1})=1$. From $\deg(r_{3})>0$, we get

$l_{3}+l_{4}>l_{1},$

which means $2>l_{1}$, and this ensures $l_{1}=1$, hence
$(l_{0},l_{1},l_{4})=(2,1,1)$.

To sum up, we denote $X_{0}=\\{v_{1},v_{2}\\}$, $X_{1}=\\{y\\}$,
$X_{2}=\\{x_{1},\dots,x_{n-2}\\}$, $X_{3}=\\{x_{0}\\}$ and $X_{4}=\\{z\\}$,
and get the following two possibilities for $X$:

  * •

$b_{1}=1$ and $c_{j}=0$ for every $j$:

$\displaystyle r_{0}$ $\displaystyle\colon\quad v_{1}+v_{2}+y=2x_{0},$
$\displaystyle r_{1}$ $\displaystyle\colon\quad y+x_{1}+\cdots+x_{n-2}=z,$
$\displaystyle r_{2}$ $\displaystyle\colon\quad x_{0}+\cdots+x_{n-2}=0,$
$\displaystyle r_{3}$ $\displaystyle\colon\quad x_{0}+z=y,$ $\displaystyle
r_{4}$ $\displaystyle\colon\quad z+v_{1}+v_{2}=x_{0}.$

  * •

$c_{1}=1$, $b_{1}=0$ and $c_{j}=0$ for every $j>1$:

$\displaystyle r_{0}$ $\displaystyle\colon\quad v_{1}+v_{2}+y=x_{1}+x_{0},$
$\displaystyle r_{1}$ $\displaystyle\colon\quad y+x_{1}+\cdots+x_{n-2}=z,$
$\displaystyle r_{2}$ $\displaystyle\colon\quad x_{0}+\cdots+x_{n-2}=0,$
$\displaystyle r_{3}$ $\displaystyle\colon\quad x_{0}+z=y,$ $\displaystyle
r_{4}$ $\displaystyle\colon\quad z+v_{1}+v_{2}=x_{1}.$

By 2.13, the open $U=X\setminus\big{(}V(y)\cup V(z)\big{)}$ has a
$\mathbb{P}^{n-2}$-bundle structure. The relation $r_{3}$ corresponds to an
extremal curve class by 2.20, which induces a smooth blow-down $h\colon
X\rightarrow Y$. By 2.16, the primitive relations in $Y$ in cases $b_{1}=1$
and $c_{1}=1$ are, respectively:

$\begin{cases}x_{0}+\cdots+x_{n-2}=0,\\\
z+v_{1}+v_{2}=x_{0};\end{cases}\quad\quad\begin{cases}x_{0}+\cdots+x_{n-2}=0,\\\
z+v_{1}+v_{2}=x_{1}.\end{cases}$

It follows that
$Y\simeq\mathbb{P}\left(\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}^{\oplus
n-2}\right)$, i.e., $Y$ is the blowup of $\mathbb{P}^{n}$ along a linear
subspace $L=V(z,v_{1},v_{2})\simeq\mathbb{P}^{n-3}\subset\mathbb{P}^{n}$, with
exceptional divisor $E=V(x_{0})\subset Y$ in the first case, and
$E=V(x_{1})\subset Y$ in the second case. The center of the blowup $h\colon
X\rightarrow Y$ is $V(x_{0},z)\subset Y$, yielding the two varieties described
in (2). ∎

##  Appendix A Code for computing primitive collections

by Will Reynolds

Let $\Sigma$ be the fan of a toric manifold of dimension $n$. For each integer
$k\in\\{1,\dots,n\\}$, we denote by $\Sigma(k)$ the subset of $\Sigma$
consisting of the $k$-dimensional cones. To determine whether a given subset
$P\subseteq G(\Sigma)$ is a primitive collection, we proceed in two steps.
First make sure there does not exist $\sigma\in\Sigma(n)$ with $P\subseteq
G(\sigma),$ and if there does, stop; otherwise, for each $v\in P$, make sure
there exist $\sigma\in\Sigma(n)$ with $P\setminus\\{v\\}\subseteq G(\sigma)$.
In the special case $\lvert P\rvert=2$ it suffices to check that there does
not exist $\sigma\in\Sigma(n)$ with $P\subseteq G(\sigma)$.

Note that, by definition, if $P$ is a primitive collection and $P\subseteq Q,$
then $Q$ is not a primitive collection. Therefore, when looking for primitive
collections, we go through subsets of $G(\Sigma)$ in increasing cardinality.

Assuming an implementation of the above basic algorithm, a reasonably
efficient way to list all of the primitive collections of a fan is to arrive
at such a list by eliminating $P\subseteq G(\Sigma)$ which are not primitive
collections. The first step is to remove any $P$ with $\lvert P\rvert=1.$ Then
for each $P$ we check whether it is a primitive collection. If it is, we keep
it and remove all sets containing it. If it is not, we remove it. One way of
implementing this method of listing primitive collections is implemented in
pseudocode in Algorithm 1.

This algorithm is impractical if $G(\Sigma)$ is too large. In practice, on a
modern laptop, it works reasonably well up to about
$\lvert{G(\Sigma)}\rvert=17,$ partly because of the combination of the
following factors: first, eliminating all of the supersets of any $P$ with
$\lvert P\rvert=2$ cuts down the remaining search space significantly, and
second, the relative abundance of primitive collections of size 2, at least
among toric Fano varieties. For example, the 124 toric Fano 4-folds altogether
have 785 primitive collections, of which 566 have cardinality 2.

This last factor makes the computation of the value of $m(X)$ for a given
toric Fano variety $X$ easier as well. Of the toric Fano varieties of a given
dimension $n$ (for $n\leq 6$) those $X$ with $m(X)=1$ make up an overwhelming
majority. This means that computing $m(X)$ is usually extremely fast, even in
the most straightforward way. The following table summarizes the data for
$\dim(X)\in\\{4,5,6\\}.$

$\dim(X)$ | # Fanos | #(m=1) | #(m=2) | #(m=3) | #(m=4) | #(m=5) | #(m=6)
---|---|---|---|---|---|---|---
4 | 124 | 107 | 15 | 1 | 1 |  |
5 | 866 | 744 | 112 | 8 | 1 | 1 |
6 | 7622 | 6333 | 1174 | 105 | 8 | 1 | 1

Input: fan $\Sigma$;

$n\leftarrow\dim{\Sigma}$;

$PC\leftarrow\\{P\subseteq G(\Sigma):\lvert P\rvert>1\\}$;

for  _$P\in PC$ satisfying $\lvert P\rvert=2$_ do

if  _there exists $\sigma\in\Sigma(\text{n})$ such that $P\subseteq
G(\sigma)$_ then

$PC\leftarrow PC\setminus\\{P\\}$;

else

$PC\leftarrow PC\setminus\\{Q\in G(\Sigma):P\subsetneq Q\\}$;

end if

end for

for  _$i\in\\{3,\dots,n\\}$_ do

for  _$P\in PC$ satisfying $\lvert P\rvert=i$_ do

if  _there exists $\sigma\in\Sigma(n)$ such that $P\subseteq G(\sigma)$_ then

$PC\leftarrow PC\setminus\\{P\\}$;

else

$b\leftarrow\textbf{True}$;

for  _$v\in P$_ do

if  _there does not exist $\sigma\in\Sigma(n)$ with
$P\setminus\\{v\\}\subseteq G(\sigma)$_ then

$b\leftarrow\textbf{False}$;

end if

end for

if  _$b$_ then

$PC\leftarrow PC\setminus\\{Q\subseteq G(\Sigma):P\subsetneq Q\\}$;

else

$PC\leftarrow PC\setminus\\{P\\}$;

end if

end if

end for

end for

Output: $PC$;

Algorithm 1 List primitive collections of a given fan

For convenience, we also provide Macaulay2 code implementing the algorithm for
computing primitive collections.

    coneExistenceCheck = (S, fan) -> (
        for cone in fan do (
            if isSubset(S, cone) then (
                return true;
            );
        );
        return false;
    );

    properSubsetCheck = (S, fan) -> (
        for ray in S do (
            if coneExistenceCheck(S-set{ray}, fan) == false then (
                return false;
            );
        );
        return true;
    );

    isPrimitiveCollection = (P, Var) -> (
        if coneExistenceCheck(P, orbits(Var, 0)) then (
            return false)
        else (
            return properSubsetCheck(P, orbits(Var, 0)
        );
    );

    supsetsOfPrimColl = (E, B) -> (
        return set{for P in E-set{B} when isSubset(B, P) list P};
    );

    primitiveCollections = (Var) -> (
        n = length rays Var;
        primColls = select(subsets(toList(0..n-1)), x -> length x > 1);
        for P in subsets(toList(0..n-1), 2) do (
            if coneExistenceCheck(P, orbits(Var, 0)) == false then (
                primColls = primColls - supsetsOfPrimColl(primColls, P);)
            else (
                primColls = primColls - set{P};
            );
        );
        for i in toList(3..n) do (
            for P in subsets(toList(0..n-1), i) do (
                if member(P, primColls) == false then continue;
                if isPrimitiveCollection(P, Var) then (
                    primColls = primColls - supsetsOfPrimColl(primColls, P);
                ) else (
                    primColls = primColls - set{P};
                );
            );
        );
        return sort primColls;
    );

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