# Duality and polyhedrality of cones for Mori dream spaces

Maria Chiara Brambilla  Università Politecnica delle Marche, via Brecce
Bianche, I-60131 Ancona, Italia<EMAIL_ADDRESS>,  Olivia
Dumitrescu  1\. University of North Carolina at Chapel Hill, 340 Phillips Hall
CB 3250 NC 27599-3250 and 2. Max-Planck Institute for Mathematics, Bonn,
Vivatsgasse 7 53111 Bonn, Germany<EMAIL_ADDRESS>,  Elisa Postinghel
Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14
I-38123 Povo di Trento (TN<EMAIL_ADDRESS>and  Luis José Santana
Sánchez  Departamento de Álgebra, Análisis Matemático, Geometría y Topología
Paseo de Belén, 7, 47011 Valladolid<EMAIL_ADDRESS>
###### Abstract.

Our goal is twofold. On one hand we show that the cones of divisors ample in
codimension $k$ on a Mori dream space are rational polyhedral. On the other
hand we study the duality between such cones and the cones of $k$-moving
curves by means of the Mori chamber decomposition of the former. We give a new
proof of the weak duality property (already proved by Payne and Choi) and we
exhibit an interesting family of examples for which strong duality holds.

###### Key words and phrases:

Mori dream space, Mori chamber decomposition, stable base locus decomposition,
weak duality, strong duality

###### 2020 Mathematics Subject Classification:

Primary: 14C20 Secondary: 14E05, 14E30, 14J45

##  1\. Introduction

Let $X$ be a normal $\mathbb{Q}$-factorial variety of dimension $n$ and with
irregularity zero, defined over an algebraically closed field of
characteristic zero.

For any integer $0\leq k\leq n-1$, we consider the cone $D_{k}$ of numerical
classes of divisors on $X$ whose stable base locus has codimension larger than
$k$ and we set $\mathcal{D}_{k}$ to be the closure of $D_{k}$ in the Néron-
Severi space $\operatorname{N}^{1}(X)_{\mathbb{R}}$. Following [Payne], we
call $\mathcal{D}_{k}$ the cone of divisors that are _ample in codimension
$k$_. Clearly $\mathcal{D}_{0}$ is the pseudo-effective cone
$\overline{\operatorname{Eff}(X)}$, $\mathcal{D}_{1}$ is the cone of movable
divisors, and $\mathcal{D}_{n-1}$ is the nef cone by Kleiman’s theorem. When
$X$ is a Mori dream space, the cones $\mathcal{D}_{0}$, $\mathcal{D}_{1}$ and
$\mathcal{D}_{n-1}$ are rational polyhedral, as shown in [Hu-Keel], moreover
$\mathcal{D}_{0}$ and $\mathcal{D}_{1}$ admit a finite _Mori chamber
decomposition_ of which $\mathcal{D}_{n-1}$ is a chamber. In this article we
prove that each cone $\mathcal{D}_{k}$ inherits the chamber decomposition and
in particular it is rational polyhedral (and $D_{k}$ is closed).

We have the obvious filtration of cones

$\operatorname{Nef}(X)=\mathcal{D}_{n-1}\subseteq\cdots\subseteq\mathcal{D}_{1}\subseteq\mathcal{D}_{0}=\overline{\operatorname{Eff}(X)};$

on the other hand we shall consider an analogous filtration of cones of
pseudo-effective curves:

$\overline{\operatorname{NE}(X)}=\mathcal{C}_{n-1}\supseteq\cdots\supseteq\mathcal{C}_{1}\supseteq\mathcal{C}_{0},$

where $\mathcal{C}_{k}$ is the closure of the cone in
$\operatorname{N}_{1}(X)_{\mathbb{R}}$ generated by classes of $k$-_moving_
curves, i.e. classes of irreducible curves moving in a family that sweeps out
an $(n-k)$-dimensional subvariety of X. Corresponding ends of the above
filtrations are dual to each other with respect to the standard intersection
pairing. Indeed the dual of the Mori cone $\mathcal{C}_{n-1}^{\vee}$ is the
nef cone and the cone $\mathcal{C}_{0}$ is dual to the pseudo-effective cone
$\overline{\operatorname{Eff}(X)}$, as shown in [BDPP].

Since duality of the cones $\mathcal{D}_{k}$ and $\mathcal{C}_{k}$ for $1\leq
k\leq n-2$ is not to be expected in general, it is a natural question to ask
under which conditions and for which integers $1\leq k\leq n-2$, strong
duality holds, that is $\mathcal{D}_{k}^{\vee}=\mathcal{C}_{k}$.

In this paper, we show that the answer is positive, for every $k$, for all
blow-ups of $\mathbb{P}^{n}$ at $s$ points in general position that are Mori
dream spaces.

The answer to the duality question is not positive in general but a weaker
version holds, as proved in [Payne] for toric varieties and in [ChoiCONES] for
Mori dream spaces: for a Mori dream space $X$, $\mathcal{D}_{k}^{\vee}$ is
generated by curves that move in families sweeping out the birational image,
on a small modification of $X$, of $(n-k)$-dimensional subvarieties of X. In
other terms, $k$-moving curves on $X$ are not enough to reconstruct, by
duality, the cone $\mathcal{D}_{k}$; instead one needs to consider $k$-moving
curves on all small modifications of $X$, which are of finite number. We will
call this property _weak duality_.

The first main theorem of this paper, Theorem 3.3 describes the combinatorial
properties of the cones $\mathcal{D}_{k}$ for Mori dream spaces, providing a
different perspective on the study of these cones and setting the bases for a
new proof of the weak duality statement. As a crucial tool we use the stable
base locus decomposition of the effective and movable cones, which is refined
by the Mori chamber decomposition, and we prove that it is compatible with the
filtration of cones $\mathcal{D}_{k}$.

######  Theorem 1.1.

If $X$ is a Mori dream space, then the cones $\mathcal{D}_{k}$ are unions of
nef chambers, hence rational polyhedral, and the cones $D_{k}$ are closed,
i.e. $D_{k}=\mathcal{D}_{k}$, for every $0\leq k\leq n-1$.

The fact that the cones $\mathcal{D}_{k}$ are rational polyhedral in the case
of toric varieties was proved by Payne in [Payne] where, furthermore, a
constructive description is presented in terms of the boundary divisors.

Rational polyhedrality of the cones $\mathcal{D}_{k}$ for simplicial toric
varieties can also be derived from work of Broomhead, Ottem and Prendergast-
Smith [BOP-S] via a completely different perspective, which we briefly explain
here. Ampleness in codimension $k$ relates to different notions of
cohomological ampleness for divisors. These notions were studied by Andreotti-
Grauert and Sommese (naive ampleness), by Demailly-Peternell-Schneider
(uniform ampleness) and by Totaro ($T-$ampleness) and they turned out to all
agree in characteristic zero, see [Totaro] and references therein. A divisor
$D\in\operatorname{N}^{1}(X)_{\mathbb{R}}$ is cohomologically $k$-ample if for
any coherent sheaf $\mathcal{F}$ on $X$,
$H^{i}(X,\mathcal{O}(mD)\otimes\mathcal{F})=0$, for every $i\geq k$ and $m\gg
0$, and one can consider the corresponding (not necessarily convex) cones. In
[BOP-S] the authors studied cones of cohomologically $k-$ample divisors in the
case of simplicial toric varieties, showing that their closures are unions of
rational polyhedral cones. As Choi noticed, [Choi14], a big divisor on a Mori
dream space is cohomological $(n-k-1)-$ample if and only if it is ample in
codimension $k$ and therefore his work, together with the results of [BOP-S],
implies that if $X$ is a simplicial toric variety, then all $\mathcal{D}_{k}$
cones are rational polyhedral. Choi asked if the $\mathcal{D}_{k}$’s are
rational polyhedral also for Mori dream spaces, see [ChoiCONES, Question 4.4]
and [Choi14, p. 1177]. A version of this question was previously posed for not
necessarily big divisors on Fano varieties in characteristic zero in [Totaro,
Question 11.1]. The first statement of our Theorem 1.1 answers affirmatively
the question for big divisors on Mori dream spaces.

Our Theorem 1.1, if we restrict to the toric case, gives yet another proof via
a new approach to the rational polyhedrality of the cones $\mathcal{D}_{k}$
and extends this to the entire class of Mori dream spaces.

Our second main result, proved in Theorems 5.5 and 5.8, provides a first
family of examples for which strong duality holds.

######  Theorem 1.2.

The strong duality holds for all Mori dream blow-ups of projective spaces at
points in general position.

We remark that the geometry of these spaces and their chamber decompositions
are completely encoded by _Weyl cycles_. We will investigate more deeply the
interplay between the birational geometry and the action of the Weyl group in
further work, continuing the study started in [BraDumPos3, BDP-Ciro].

Moreover, in future work we will seek criteria for strong duality to hold for
general Mori dream spaces.

The article is organised as follows. In Section 2 we recall some preliminaries
on cones of curves and divisors and on chamber decompositions. Section 3 is
dedicated to our first main result, Theorem 1.1, that established the
polyhedrality of the cones $\mathcal{D}_{k}$ for all Mori dream spaces.
Section 4 contains our proof of the weak duality property for
$\mathcal{D}_{k}$ and the cones of curves $\mathcal{C}_{k}$ for Mori dream
spaces, by means of the Mori chamber decomposition of the effective cone. We
also include the discussion of a toric example for which strong duality does
not hold. In Section 5 we prove our second main result, Theorem 1.2, which
concerns a family of examples of Mori dream spaces satisfying strong duality.

### Acknowledgements

We would like to thank Artie Prendergast-Smith for several useful comments.

The first and third authors are members of INdAM-GNSAGA. The second author
would like to express her gratitude to the Max-Planck Institute for
Mathematics, Bonn, for their generosity, hospitality and stimulating
environment. During the time of this project, she was also supported by the
NSF-FRG grant DMS 2152130 and the Simons Foundation collaboration grant
855897. The fourth author was partially supported by the ULL funded research
project MACACO.

##  2\. Preliminaries

In this section we recall some preliminaries. As a general reference we use
[Laz1].

###  2.1. Rational polyhedral cones

Let $V\subset\mathbb{R}^{n}$ be a convex cone. A subcone $W\subset V$ is
extremal if $u+v\in W$ implies $u,v\in W$. A one-dimensional subcone is called
a ray. A cone is polyhedral if it is the intersection of a finite number of
half-spaces through the origin. By the Minkowski-Weyl theorem, a cone is
polyhedral if and only if it is finite, i.e. it is the convex conical hull of
a finite number of vectors. In particular a polyhedral cone is closed. A
convex cone is rational polyhedral if it is generated by a finite set of
integral vectors.

###  2.2. Curve classes and divisor classes

Let $X$ be a complex normal projective variety of dimension $n$. Let
$\textrm{Pic}(X)$ be the Picard group and let $A_{k}(X)$ be the group of
$k$-cycles on $X$, for $1\leq k\leq n-1$. We denote with
$\operatorname{N}^{1}(X)_{\mathbb{R}}=(\operatorname{Pic}(X)/\equiv)\otimes\mathbb{R}$
the Néron-Severi space of $X$ and with
$\operatorname{N}_{1}(X)_{\mathbb{R}}=(A_{1}(X)/\equiv)\otimes\mathbb{R}$ its
dual space with respect to the standard intersection pairing. The cones of
divisors
$\operatorname{Nef}(X)\subseteq\overline{\operatorname{Eff}(X)}\subset\operatorname{N}^{1}(X)_{\mathbb{R}}$
are, respectively, the nef and the pseudo-effective cone. The former is dual
to the Mori cone
$\overline{\operatorname{NE}(X)}\subset\operatorname{N}_{1}(X)_{\mathbb{R}}$
which is the closure of the cone of effective $1$-cycles; the latter is dual
the the closure of the cone of moving curves, that are numerical curve classes
such that the irreducible representatives of the class pass through a general
point of $X$, see [BDPP, Theorem 0.2] or [Laz1, 11.4.C].

###  2.3. Stable Base Locus Decomposition

Given an effective divisor $D$ on $X$, we denote by $\operatorname{Bs}(D)$ the
base locus of the linear system $|D|$. We recall that the stable base locus of
$D$ is the Zariski closed set

$\mathbb{B}(D)=\bigcap_{m>0}\operatorname{Bs}(mD).$

Under the assumption $q(X)=h^{1}(X,\mathcal{O}_{X})=0$, linear and numerical
equivalence of $\mathbb{Q}$-divisors coincide. Thus the stable base locus is
well defined for numerical equivalence classes of $\mathbb{Q}$-divisors on
$X$, [ELMNP].

The Stable base locus decomposition is a wall-and-chamber decomposition of the
pseudo-effective cone $\overline{\operatorname{Eff}(X)}$ of $X$, where the
stable base locus is constant on the interior of the chambers , see e.g. ,
[Huizenga, Sect 4.1.3] and [LMR] for more details.

We will call stable base locus subvariety of $X$ any irreducible and reduced
subvariety which appears as stable base locus of some effective divisor of
$X$.

A divisor $D$ in $X$ is called movable if its stable base locus has
codimension at least two in $X$. The movable cone of $X$ is the convex cone
$\operatorname{Mov}(X)$ generated by classes of movable divisors. We have the
following obvious inclusions of cones:
$\operatorname{Nef}(X)\subseteq\overline{\operatorname{Mov}(X)}\subseteq\overline{\operatorname{Eff}(X)}$.

###  2.4. Mori dream spaces and Mori Chamber Decomposition

We recall the following definition from [Hu-Keel, Definition 1.2.11].

######  Definition 2.1.

A normal projective $\mathbb{Q}$-factorial variety $X$ is called a Mori dream
space if the following conditions hold:

  1. (1)

$\operatorname{Pic}(X)$ is finitely generated, that is
$h^{1}(X,\mathcal{O}_{X})=0$,

  2. (2)

$\operatorname{Nef}(X)$ is generated by the classes of finitely many semi-
ample divisors,

  3. (3)

there is a finite collection of small $\mathbb{Q}$-factorial modifications
$f_{i}:S\dashrightarrow X_{i}$, such that each $X_{i}$ satisfies (2), and

$\operatorname{Mov}(X)=\bigcup f_{i}^{*}(\operatorname{Nef}(X_{i})).$

Mori dream spaces include several classes of interesting varieties such as
toric varieties, $\mathbb{Q}$-factorial spherical varieties [Brion], Fano
varieties and varieties of Fano type [BCHM] among others.

Recall that a small $\mathbb{Q}$-factorial modification (SQM) $f:X\dasharrow
X^{\prime}$ is a contracting birational map that is an isomorphism in
codimension $1$, with $X^{\prime}$ a normal projective and
$\mathbb{Q}$-factorial variety. If $X$ is a Mori dream space, then every SQM
of $X$ factors as a finite sequence of flips and a regular contraction.

If $X$ is a Mori dream space, then the effective cone $\operatorname{Eff}(X)$
is rational polyhedral, and in particular closed, see e.g. [okawa]. The
section ring $R(X,D)$ of a divisor $D$ on $X$ is finitely generated and it
induces a natural rational map

$f_{D}:X\dashrightarrow X_{D}\subseteq\textrm{Proj}(R(X,D)),$

which is regular away from $\mathbb{B}(D)$. Two $\mathbb{Q}$-divisors $D_{1}$
and $D_{2}$ are _Mori equivalent_ if $f_{D_{1}}$ and $f_{D_{2}}$ have the same
Stein factorisation. Given an effective divisor $D$ on $X$, the _Mori chamber_
of $D$ is the closure of the set of all divisors that are Mori equivalent to
$D$. Mori chambers are convex cones and the set of such cones form a fan, the
_Mori fan_ of $X$. The maximal cones of this fan form a wall-and-chamber
decomposition of the effective cone, called the _Mori chamber decomposition_.
Such decomposition is a refinement of the stable base locus decomposition, see
[Hu-Keel, okawa] for details.

###  2.5. Cones of $k$-moving curves

We recall the notion of $k$-moving curve, for $0\leq k\leq n-1$, cf. [Payne].

######  Definition 2.2.

An irreducible curve $C\subset X$ is $k$-moving if it belongs to an algebraic
family of curves, whose irreducible elements cover a Zariski open subset of an
effective cycle of dimension (at least) $n-k$.

We define $C_{k}$ to be the cone generated by the classes of $k$-moving curves
in $\operatorname{N}_{1}(X)_{\mathbb{R}}$ and we set
$\mathcal{C}_{k}=\overline{C}_{k}$ to be the closure in
$\operatorname{N}_{1}(X)_{\mathbb{R}}$.

If $c$ is a class in $\operatorname{N}_{1}(X)_{\mathbb{R}}$, we will say that
$c$ sweeps out a subvariety $V\subset X$ if there is a family of irreducible
curves of class $c$ covering a Zariski open subset of $V$.

######  Example 2.3.

Let $p_{1},\dots,p_{s}\in\mathbb{P}^{n}$ be points in general position and let
$X^{n}_{s}$ be to the blow of $\mathbb{P}^{n}$ at these points. For every
three distinct indices $i,j,k$, the strict transform on $X^{n}_{s}$ of an
irreducible conic through $p_{i},p_{j},p_{k}$ sweeps out the strict transform
of the linear span of the points and hence it is an $(n-2)-$moving curve.
Similarly, the strict transform of an irreducible curve $C$ of degree $r$
through $r+1$ points is an $(n-r)-$moving curve since it sweeps out the strict
transform of the $r-$dimensional linear span of the points.

It follows from the definitions that a $k$-moving curve class is also
$(k+1)$-moving, for $0\leq k\leq n-2$. We obtain the following filtration of
cones of curves:

(2.1)
$\overline{\operatorname{NE}(X)}=\mathcal{C}_{n-1}\hookleftarrow\mathcal{C}_{n-2}\hookleftarrow\cdots\hookleftarrow\mathcal{C}_{1}\hookleftarrow\mathcal{C}_{0}.$

Clearly $\mathcal{C}_{n-1}=\overline{\operatorname{NE}(X)}$ is the Mori cone
of curves (dual to $\operatorname{Nef}(X)$). Furthermore, $\mathcal{C}_{0}$ is
the closed cone of the moving curve classes (dual to
$\overline{\operatorname{Eff}(X)}$).

###  2.6. Cones $\mathcal{D}_{k}$ of divisors that are ample in codimension
$k$

There is a stratification of the pseudoeffective cone of divisors as follows.

######  Definition 2.4.

Let $0\leq k\leq n-1$. We define $D_{k}$ to be the cone generated by the
classes of effective divisors with no component of the stable base locus of
dimension $\geq n-k$; we denote with $\mathcal{D}_{k}$ the closure of $D_{k}$
in $\overline{\operatorname{Eff}}(X)$.

Using the same terminology as Payne [Payne], we call such divisors ample in
codimension $k$, indeed $\mathcal{D}_{k}$ is also the closure of the cone of
numerical classes of divisors D such that there is an open set U, with
$\mathcal{O}(D)_{|U}$ ample, whose complement has codimension greater than
$k$. We have the following obvious inclusions:

(2.2)
$\operatorname{Nef}(X)=\mathcal{D}_{n-1}\hookrightarrow\mathcal{D}_{n-2}\hookrightarrow\cdots\hookrightarrow\mathcal{D}_{1}\hookrightarrow\mathcal{D}_{0}=\overline{\operatorname{Eff}(X)}.$

Recall that if $X$ is a Mori dream space, then $\operatorname{Eff}(X)$ is
rational polyhedral, hence $D_{0}=\mathcal{D}_{0}=\operatorname{Eff}(X)$.
Moreover, also $\mathcal{D}_{n-1}=\operatorname{Nef}(X)$ and
$\mathcal{D}_{1}=\operatorname{Mov}(X)$ are rational polyhedral. Choi posed
the natural question as to whether all cones $\mathcal{D}_{k}$ are rational
polyhedral for Mori dream spaces, see [ChoiCONES, Question 4.4] or [Choi14, p.
1177]. In Theorem 3.3 below (cf. Theorem 1.1) we answer this question
affirmatively.

This property is not true in general for non-Mori dream spaces, as the
following example shows for $k=n-1$.

######  Example 2.5.

Let $X=\operatorname{Bl}_{9}(\mathbb{P}^{2})$ be the blow up of the projective
plane at nine general points, which is not a Mori dream space.

The anticanonical curve $F$, which is the strict transform on $X$ of a plane
cubic through the nine points, is nef. In fact the Mori cone of $X$ is
generated by the $(-1)$-curves and by $F$, and we have $F^{2}=0$ and $F\cdot
E=1$ for every $(-1)$-curve $E$, see [nagata].

However $F\not\in D_{1}$. In fact, since there is a unique plane cubic through
nine points, the linear system $|F|$ has dimension $0$. Even more:
$|mF|=\\{mF\\}$, for every integer $m\geq 1$, and this goes back to the
classical work of Castelnuovo [Castelnuovo91]. This implies that $F$ is the
stable base locus of its own linear system $|F|$.

##  3\. The geometry of the cones $\mathcal{D}_{k}$

In this section we show that the stable base locus decomposition of the
pseudo-effective cone of a normal projective variety $X$ with irrationality
zero is compatible with the filtration of cones (2.2), that is, every cone
$\mathcal{D}_{k}$ decomposes as a union of stable base locus chambers.
Moreover if $X$ is a Mori dream space, since each stable base locus chamber is
a union of nef chambers, we obtain that $\mathcal{D}_{k}$ is a union of such
chambers. Furthermore, it is possible to characterise the set of Mori chambers
whose union is $\mathcal{D}_{k}$, for every $1\leq k\leq n-1$.

######  Lemma 3.1.

Let $X$ be a normal projective variety with $h^{1}(X,\mathcal{O}_{X})=0$.
Then, for every $k$, the cone ${D}_{k}$ is a union of stable base locus
chambers.

###### Proof.

We shall prove that the stable base locus chamber decomposition of the pseudo-
effective cone of $X$ induces a chamber decomposition on each cone $D_{k}$, by
showing that every stable base locus chamber is contained in the difference
set $D_{k-1}\setminus D_{k}$, for some $1\leq k\leq n-1$.

In fact, let $B$ be a stable base locus chamber, and let $D$ be an effective
divisor that lies in $B$. Let $n-k$ be the dimension of $\mathbb{B}(D)$ or,
equivalently, the maximum dimension of the components of $\mathbb{B}(D)$. The
inequality $\dim\mathbb{B}(D)\geq n-k$ implies that $D$ does not lie in
${D}_{k}$, while since $\dim\mathbb{B}(D)\leq n-k$, $D$ lies in ${D}_{k-1}$,
that is, $D\in D_{k-1}\setminus D_{k}$. Since $B$ is a stable base locus
chamber, by definition the same holds for any other divisor in $B$. Thus,
$B\subset D_{k-1}\setminus D_{k}$. ∎

In the next theorem, we propose a detailed description of the nef chamber
decomposition of the cones $\mathcal{D}_{k}$ for Mori dream spaces.

######  Notation 3.2.

We set $F_{1}:=\\{\textrm{Id}:X\to X\\}$. Let $2\leq r\leq n-1$ and let
${F}_{r}$ be the set of all small $\mathbb{Q}$-factorial modifications
$f_{i}:X\dashrightarrow X_{i}$ given by sequences of flips of subvarieties
whose birational preimage on $X$ has dimension $<r$.

######  Theorem 3.3.

Let $X$ be a Mori dream space and fix $1\leq k\leq n-1$.

  1. (1)

The cone $\mathcal{D}_{k}$ is rational polyhedral and the following holds:

${\mathcal{D}_{k}}=\bigcup_{f_{i}\in{F}_{n-k}}{f_{i}^{*}}(\operatorname{Nef}(X_{i})).$

  2. (2)

The cone $D_{k}$ is closed, that is: $D_{k}=\mathcal{D}_{k}$.

###### Proof.

Recall that, for a Mori dream space, every closed stable base locus chamber is
a finite union of nef chambers, see Section 2.4. By Lemma 3.1, $D_{k}$ is a
union of stable base locus chambers and hence its closure $\mathcal{D}_{k}$ is
a finite union of nef chambers. We shall now characterise which are the nef
chambers appearing in the union as follows. Let $f_{i}:X\dashrightarrow X_{i}$
be a small $\mathbb{Q}$-factorial modification of $X$ and let
$f_{i}^{\ast}(\operatorname{Nef}(X_{i}))$ be the corresponding nef chamber.
Let $D\in f_{i}^{\ast}(\operatorname{Nef}(X_{i}))$ be any pull-back of an
ample class on $X_{i}$: we have that $D$ is movable on $X$. The stable base
locus of $D$ equals the indeterminacy locus of $f_{i}$ (see e.g. [Hu-Keel,
okawa]). Hence, the latter is a union of subvarieties of $X$ of dimension
$<n-k$ if and only if $D$ lies in $D_{k}$. This argument is independent of $D$
but rather depends on its base locus, and it applies to any pull-back on $X$
of an ample divisors on $X_{i}$. Therefore, the interior of the nef chamber
$f_{i}^{\ast}(\operatorname{Nef}(X_{i}))$ is contained in $D_{k}$. Finally,
taking closures, we have that
$f_{i}^{\ast}(\operatorname{Nef}(X_{i}))\subseteq\mathcal{D}_{k}$ if any only
if $f_{i}\in F_{n-k}$.

Since $X$ is a Mori dream space, there are finitely many nef chambers and each
is a rational polyhedral cone. Therefore the above argument shows that for
every $k$, the convex cone $\mathcal{D}_{k}$ is a union of finitely many
rational polyhedral cones. Since $\mathcal{D}_{k}$ is a cone, it is a rational
polyhedral cone. This proves (1).

We shall prove (2), i.e. that $\mathcal{D}_{k}=D_{k}$. In order to do so, we
show that every extremal ray of $\mathcal{D}_{k}$ lies in $D_{k}$. Let $E$ be
an effective divisor on $X$ spanning an extremal ray. Then $E$ spans an
extremal ray of some nef chamber
$f_{j}^{\ast}(\operatorname{Nef}(X_{j}))\subseteq\mathcal{D}_{k}$ and hence
the pull-back of a semi-ample divisor $E_{j}$ on $X_{j}$, where
$f_{j}:X\dasharrow X_{j}$, with $f_{j}\in F_{n-k}$, by (1). Let $m\geq 1$ be
an integer such that $mE_{j}$ is basepoint free. The stable base locus of
$mE=f_{j}^{\ast}(mE_{j})$ is therefore the indeterminacy locus of $f_{j}$
which is a union of subvarieties of $X$ of dimension strictly less than $n-k$.
Since $\mathbb{B}(E)=\mathbb{B}(mE)$, we conclude that $E$ lies in $D_{k}$.

∎

######  Remark 3.4.

For $k=1$, Theorem 3.3 is just property (3) of the definition of Mori dream
spaces. For $k=n-1$ it is always true by Kleiman’s theorem.

For non-Mori dream spaces, none of the statements of Theorem 3.3 hold in
general; in fact the properties may hold or fail as we vary $k$. The following
is an instance of this.

######  Example 3.5.

Let $X=X^{5}_{9}$ be the blow-up of $\mathbb{P}^{5}$ at nine points in general
position. Recall that $X$ is not a Mori dream space, in particular
$\mathcal{D}_{0}=\overline{\operatorname{Eff}(X)}$ is not finitely generated.
On the other hand, the Mori cone $\mathcal{C}_{4}$ is finitely generated by
lines, see e.g. [DP-positivityI, Theorem 3.2], hence its dual,
$\operatorname{Nef}(X)$, is rational polyhedral. Moreover the following holds:
$\mathcal{D}_{4}=D_{4}=\operatorname{Nef}(X)$. Indeed, the nef cone
$\operatorname{Nef}(X)$ is generated by the following classes:
$H,H-E_{i},2H-\sum_{i\in I}E_{i}$, for every $3\leq|I|\leq 9$ as one can
easily check with the help of any computer software that can compute cone
duals, such as _magma_ [Magma]. It is an easy exercise to show that each of
them is basepoint free, hence it lies in $D_{4}$.

It would be interesting to determine whether or not the intermediate cones
$\mathcal{D}_{1}$, $\mathcal{D}_{2}$ and $\mathcal{D}_{3}$ are finitely
generated.

##  4\. Weak duality

Duality of the cones $\mathcal{D}_{k}$ and $\mathcal{C}_{k}$ is not to be
expected in general, but a weaker version for Mori dream spaces holds, that
takes into account $k$-moving curves not just on $X$ but also on its small
modifications. We will refer to this as _weak duality_ , as opposed to _strong
duality_ that holds whenever $\mathcal{C}_{k}=\mathcal{D}_{k}^{\vee}$ for
every $k$. This will be the subject of this section.

It is easy to show that
$\operatorname{Mov}(X)\subseteq\mathcal{C}_{1}^{\vee}$. In fact if $D$ is in
the dual of $\mathcal{C}_{1}$ then $D\cdot C\geq 0$ for every $1$-moving curve
$C$. If it is not in the dual, then there is a $1$-moving curve $C$ such that
$D\cdot C<0$, and so the divisor ($(n-1)$-cycle) spanned by the algebraic
family of $C$ is a fixed component of the base locus of $D$. If $n=2$, then it
holds that $\operatorname{Mov}(X)=\mathcal{C}_{1}^{\vee}$. In fact the first
cone is the nef cone, the second cone is the Mori cone.

A similar inclusion of cones is valid in general:

######  Remark 4.1.

If $C\in\mathcal{C}_{k}$ we know that it sweeps out an effective cycle $V$ of
dimension $n-k$. If $C\not\in\mathcal{D}^{\vee}_{k}$, then there exists an
effective divisor $D$ such that $C\cdot D<0$ and whose stable base locus does
not have any component of dimension $\geq n-k$. But then the cycle $V$ is in
the stable base locus of $D$ which is a contradiction. This proves that

$\mathcal{C}_{k}\subseteq\mathcal{D}_{k}^{\vee}.$

We adopt the definition of $k$-moving curves on small modifications proposed
by Payne in [Payne].

######  Definition 4.2.

We say that a curve class $c\in\operatorname{N}_{1}(X)_{\mathbb{R}}$ is _bir-
$k$-moving_ on $X$ if there exists a small modification
$f_{i}:X\dashrightarrow X_{i}$, such that there is a curve $C\subset X_{i}$,
in the class corresponding to $c$, which sweeps out a cycle $f_{i}(V)$
birational to a cycle $V\subset X$ of codimension $k$.

We denote by $\mathcal{C}_{k}^{bir}$ the cone generated in
$\operatorname{N}_{1}(X)_{\mathbb{R}}$ by all the bir-$k$-moving classes on
$X$.

Applying the same idea as in Remark 4.1, one can easily see that the inclusion
$\mathcal{C}^{bir}_{k}\subseteq\mathcal{D}_{k}^{\vee}$ holds. Using the
results of Section 3, we prove that the latter inequality holds with equality,
hence recovering a result that was first proved by Payne for complete
$\mathbb{Q}-$factorial toric varieties [Payne] and later generalised by Choi
for log Fano varieties and for Mori dream spaces [ChoiCONES, Corollary 4.3],
running the log minimal model programme.

The main ingredient of our proof is the Mori chamber decomposition of the
effective cone of $X$, and of $\mathcal{D}_{k}$, proved in Theorem 3.3.

######  Theorem 4.3 (Weak duality theorem for Mori dream spaces).

If $X$ is a Mori dream space then weak duality holds, that is

$\mathcal{C}_{k}^{Bir}=\mathcal{D}_{k}^{\vee}.$

###### Proof.

We only have to prove that the following inclusion holds:
$\mathcal{D}_{k}^{\vee}\subseteq\mathcal{C}^{bir}_{k}.$ Every facet
$\mathcal{F}$ of $\mathcal{D}_{k}$ corresponds, by duality, to an extremal ray
of $\mathcal{D}_{k}^{\vee}$, spanned by a curve class
$c=c(\mathcal{F})\in\overline{\textrm{NE}(X)}$. We shall prove that
$c\in\mathcal{C}^{bir}_{k}$. We consider the following cases separately.

Assume first that $\mathcal{F}$ is a facet of
$\mathcal{D}_{0}=\overline{\textrm{Eff}(X)}$. Then $c$ is a moving curve class
and the statement follows.

Now, let $\mathcal{F}$ be a facet of $\mathcal{D}_{k}$ but not a facet of
$\mathcal{D}_{0}$. Then there is a Mori chamber inside $\mathcal{D}_{k}$ and
adjacent to $\mathcal{F}$, that means that a facet of the chamber is contained
or equal to $\mathcal{F}$. Let
$f_{-}^{*}(\operatorname{Nef}(X_{-}))\subset\mathcal{D}_{k}$ be one such
chamber, for $f_{-}:X\dashrightarrow X_{-}$ a small modification of $X$. It
follows that
$c\in\operatorname{N}_{1}(X)_{\mathbb{R}}\cong\operatorname{N}_{1}(X_{-})_{\mathbb{R}}$
generates an extremal ray of the Mori cone of $X_{-}$.

Let $V_{-}\subset X_{-}$ be the subvariety of $X_{-}$ swept out by $c$, or let
$V_{-}:=C$ if $V_{-}$ is $1$-dimensional.

Now, let $f_{+}^{*}(\operatorname{Nef}(X_{+}))\ast\textrm{ex}(f_{+})$ be a
Mori chamber outside $\mathcal{D}_{k}$, and adjacent to $\mathcal{F}$, that
has a common facet with $f_{-}^{*}(\operatorname{Nef}(X_{-}))$ contained in
$\mathcal{F}$. The contracting birational map $f_{+}:X\dashrightarrow X_{+}$
can be decomposed as $f_{+}=g\circ f_{-}$, where $g:X_{-}\dashrightarrow
X_{+}$ corresponds to crossing the wall $\mathcal{F}$, and the indeterminacy
locus of $g$ is $V_{-}$. Take $D\in
f_{+}^{*}(\operatorname{Nef}(X^{+}))\ast\textrm{ex}(f_{+})$ a divisor on $X$
whose ray lies in the relative interior of the Mori chamber. Since
$D\notin\mathcal{D}_{k}$, then $D$ must have a component of its stable base
locus of codimension $\leq k$. But the stable base locus of $D$, as a divisor
on $X$, is the indeterminacy locus of $f_{+}$, which is the union of the
indeterminacy locus of $f_{-}$ and $V:=f_{-}^{-1}(V_{-})$. Since by Theorem
3.3, the indeterminacy locus of $f_{-}$ has codimension $>k$, then $V$ must
have codimension $p\leq k$. In particular, $V$ is not in the indeterminacy
locus of $f_{-}$ and therefore $V$ and $V_{-}$ are birational. We conclude
that $c\in\mathcal{C}_{p}^{bir}\subseteq\mathcal{C}_{k}^{bir}$. ∎

###  4.1. Losev-Manin’s moduli $3$-space

Strong duality is a rare property even for toric varieties, and it is not
preserved under small modification for dimension three or higher. An explicit
example of a toric threefold for which strong duality does not hold was given
by Payne in [Payne, Example 1].

In this section we discuss another interesting smooth threefold for which
strong duality holds, i.e. such that $\mathcal{D}_{1}^{\vee}=\mathcal{C}_{1}$,
but for which it fails, i.e.
$\mathcal{D}_{1}^{\vee}\supsetneq\mathcal{C}_{1}$, as soon as we flop one or
more curves. Let $X$ be the $3$-dimensional Losev-Manin moduli space
introduced in [LM] as a toric compactification of the moduli space of rational
curves with six marked points. It can be interpreted as the blow-up of
$\mathbb{P}^{3}$ at its four coordinate points and, subsequently, along the
strict transforms of the six coordinate lines. We denote by $H$ the class of
the pull-back on $X$ of a hyperplane of $\mathbb{P}^{3}$, by $E_{i},\
i=1,\dots,4$, the strict transforms of the exceptional divisors of the points
and by $E_{ij},\ 1\leq i<j\leq 4$, the exceptional divisors of the lines,
which together generate the Picard group of $X$. Similarly, we denote by
$h=H^{2},e_{i}=-E_{i}^{2},e_{ij}=HE_{ij}=E_{i}E_{ij}$ the classes of a general
line, a general line in $E_{i}$ and a vertical fibre of $E_{ij}$ respectively:
they generate the Chow group of $1-$cycles. The matrix of the intersection
pairing
$\operatorname{N}^{1}(X)_{\mathbb{R}}\times\operatorname{N}_{1}(X)_{\mathbb{R}}\to\mathbb{R}$,
with respect to the above bases, is diagonal and defined by:

$H\cdot h=1,\ E_{i}\cdot e_{i}=E_{ij}\cdot e_{ij}=-1.$

We denote by $E_{ijk}=H-E_{i}-E_{j}-E_{k}-E_{ij}-E_{ik}-E_{jk}$ the strict
transform of a fixed hyperplane of $\mathbb{P}^{3}$ spanned by three
coordinate points. The following divisors generate the extremal rays of the
effective cone $\mathcal{D}_{0}$ of $X$:

$E_{i},E_{ij},E_{ijk},$

for all distinct $i,j,k$.

Now, using [ADHL, Prop. 3.3.2.3], we can describe the movable cone of $X$ via
its defining inequalities. In other terms, one can show that the extremal rays
of the dual cone $\mathcal{D}_{1}^{\vee}$ are generated by the following
effective curve classes, up to permutations of indices:

  * •

classes of curves sweeping out $E_{i}$: $e_{i}-e_{ij}$, and
$2e_{i}-e_{ij}-e_{ik}-e_{il}$,

  * •

classes of curves sweeping out $E_{ij}$: $e_{ij}$ and $h-e_{i}-e_{j}+e_{ij}$,

  * •

classes of curves sweeping out $E_{ijk}$: $h-e_{i}-e_{jk}$ and
$h-e_{ij}-e_{ik}-e_{jk}$.

This computation can be checked using any computer software that can compute
cone duals, such as _Magma_ [Magma]. While we are guaranteed that the
effective cones $\mathcal{D}_{0}$ and the Mori cone $\mathcal{C}_{0}$ are dual
to each other for all small $\mathbb{Q}-$ factorial modifications of $X$, the
same need not happen for the movable cone $\mathcal{D}_{1}$ and the cone of
$1$-moving curves $\mathcal{C}_{1}$; whether or not it does, it depends on the
model. In fact we will now discuss a small modification of $X$ for which
$\mathcal{D}_{1}$ and $\mathcal{C}_{1}$ are not dual cones.

In order to do this, it is convenient to describe $X$, and its small
modifications, using the language of toric varieties. For a general reference
on toric variety, see [toric-book] or [Fulton]. Since $X$ is obtained from
$\mathbb{P}^{3}$ by the iterated blow-up of torus invariant subvarieties, then
$X$ is toric. Let $N\cong\mathbb{Z}^{3}$ be a three-dimensional lattice; the
rays of $\Sigma_{X}$ are generated by the following primitive vectors:

$\displaystyle v_{1}=(1,1,1),v_{2}=(0,0,-1),v_{3}=(0,-1,0),v_{4}=(-1,0,0),$
$\displaystyle v_{12}=(1,1,0),v_{13}=(1,0,1),v_{14}=(0,1,1),$ $\displaystyle
v_{23}=(0,-1,-1),v_{24}=(-1,0,-1),v_{34}=(-1,-1,0),$ $\displaystyle
v_{123}=(1,0,0),v_{124}=(0,1,0),v_{134}=(0,0,1),v_{234}=(-1,-1,-1).$

The ray generated by the vector $v_{I}$, which we will denote with $\rho_{I}$,
corresponds to the boundary divisor $E_{I}$, for every index set
$I\subset\\{1,\dots,4\\}$ of cardinality $1\leq|I|\leq 3$. The set of maximal
cones, which are all simplicial, is the following:

$\textrm{Cone}(v_{i},v_{ij},v_{ijk}),\forall\\{i\\}\subset\\{i,j\\}\subset\\{i,j,k\\}\subset\\{1,\dots,4\\}.$

The fan of $X$ is dual to a $3$-dimensional permutohedron, which is a polytope
with $8$ hexagonal faces (corresponding to the boundary divisors $E_{ijk}$ and
$E_{i}$) and six quadrilateral faces (corresponding to the $E_{ij}$’s). Figure
1 displays a section of a portion of the fan of $X$ lying in the plane of
$N\otimes_{\mathbb{Z}}\mathbb{R}\cong\mathbb{R}^{3}$ of equation $x+y+z=a$,
with $a>0$.

Figure 1. A section of a portion of the fan $\Sigma_{X}$ and of $\Sigma_{Y}$

Consider the cone $\textrm{Cone}(v_{1},v_{123})$, which is a common two
dimensional face of $\textrm{Cone}(v_{1},v_{12},v_{123})$ and
$\textrm{Cone}(v_{1},v_{13},v_{123})$. The corresponding invariant subvariety
is a floppable curve whose numerical class is $e_{1}-e_{12}-e_{13}$ and which
corresponds to a fixed line inside $E_{1}$. Let $X\dashrightarrow Y$ be the
flop of such curve. The fan $\Sigma_{Y}$ differs from $\Sigma_{X}$ only within
the quadrilateral cone generated by $v_{1},v_{12},v_{13},v_{123}$ and, in
particular, the image $-e_{1}+e_{12}+e_{13}$ corresponds to the cone
$\textrm{Cone}(v_{12},v_{13})\in\Sigma_{Y}$.

We argue that on $Y$ strong duality does not hold. Consider the class
$\gamma=e_{1}-e_{14}$ which is $1-$moving on $X$. The intersections
$\gamma\cdot E_{1}=-1$ and $\gamma\cdot H_{123}=1$ imply that every
representative of $\gamma$ is contained in $E_{1}$ and it intersects the
hyperplane $E_{123}$, where by abuse of notation we use the same symbols for
cycles on $X$ and their birational images on $Y$. Since the two divisors are
disjoint in $Y$, no irreducible curve with class $\gamma$ exists on $Y$. In
particular $\gamma\notin\mathcal{C}_{1}(Y)$; thus strong duality does not hold
for $Y$, that is $\mathcal{C}_{1}(Y)\subsetneq\mathcal{D}_{1}^{\vee}$.

##  5\. Strong duality for blow ups of $\mathbb{P}^{n}$

In this section we consider $X=X^{n}_{s}$, i.e. the blow-up of
$\mathbb{P}^{n}$ at $s$ points in general position, for the pairs $(n,s)$ for
which $X$ is a Mori dream space. These varieties are classified in [Mukai04,
Mukai05, CT] and they fit inthe following list:

  * •

$X^{2}_{s}$ for $s\leq 8$;

  * •

$X^{3}_{s}$ for $s\leq 7$;

  * •

$X^{4}_{s}$ for $s\leq 8$;

  * •

$X^{n}_{s}$ for $n\geq 5$ and $s\leq n+3$.

The main result of this section is the proof of Theorem 1.2. More precisely,
we show that the cone of $k$-moving curves is dual to the cone of divisor that
are ample in codimension $k$, for every Mori dream $X^{n}_{s}$. The case
$X^{n}_{\leq n+3}$ will be treated in Section 5.1, while $X^{3}_{7}$ and
$X^{4}_{8}$ will be considered in Section 5.2.

###  5.1. Blow up of $\mathbb{P}^{n}$ in $s\leq n+3$ general points

Let $X=X_{s}^{n}$ be the blow up of $\mathbb{P}^{n}$ at $s\leq n+3$ general
points. We recall some notation from [BraDumPos3]. For any subset
$I\subset\\{1,\ldots,s\\}$ we denote by $L_{I}$ the strict transform of the
linear span of the points indexed by $I$. For $s=n+3$, we denote by $C$ the
strict transform of the unique rational normal curve of degree $n$ passing
through all points and by $\sigma_{t}$ the strict transform of its $t$-secant
variety, where we set $\sigma_{1}:=C$. Let $J(L_{I},\sigma_{t})$ denote the
strict transform of the join of the linear span of the points indexed by $I$
and the $t$-secant variety of the rational normal curve. We make the following
identifications: $J(L_{I},\sigma_{t})=\emptyset$ if $t=|I|=0$,
$J(L_{I},\sigma_{t})=E_{i}$ if $t=0$ and $I=\\{i\\}$,
$J(L_{I},\sigma_{t})=L_{I}$ if $t=0$ and $J(L_{I},\sigma_{t})=\sigma_{t}$ if
$|I|=0$. It is easy to see that the join $J(L_{I},\sigma_{t})$ has dimension
$2t+|I|-1$, unless $J(L_{I},\sigma_{t})=E_{i}$.

For $s\leq n+3$, the inequalities cutting out the effective cone of
$X^{n}_{s}$ were obtained in [BraDumPos3] and a formula for the dimension of
all linear systems of divisors on $X^{n}_{s}$ was proved in [LPS]. Moreover,
by [BraDumPos3, Lemma 4.1], if $H$ denotes the general hyperplane class in
$X^{n}_{s}$ and $E_{1},\dots,E_{n+3}$ denote the exceptional divisors, then
for an effective divisor

(5.1) $D=dH-\sum_{i=1}^{n+3}m_{i}E_{i},$

the following integer

(5.2) $\kappa_{J(L_{I},\sigma_{t})}=-(|I|+(n+1)t-1)d+\sum_{i\in
I}(t+1)m_{i}+\sum_{i\notin I}tm_{i},$

when positive, is the exact multiplicity of containment of the join
$J(L_{I},\sigma_{t})$ in the base locus of $|D|$. Notice that since equation
(5.2) is linear in $d$ and in the $m_{i}$’s, then if the join
$J(L_{I},\sigma_{t})$ is in the base locus of a linear system, then it is a
component of its stable base locus.

Formulas (5.2) return the Mori chamber decomposition of the effective cone of
$X^{n}_{s}$, already studied by Mukai. This turns out to coincide with its
stable base locus decomposition. This is the content of the following result
that already appeared in [BCP, Theorem 2.60], but that we include here for the
sake of completeness.

######  Theorem 5.1.

For $X=X_{n+3}^{n}$, the Mori chamber decomposition and the stable base locus
decomposition of the effective cone coincide and they are induced by a
hyperplane arrangement whose elements correspond to the subvarieties

(5.3) $\\{J(L_{I},\sigma_{t}):0\leq t\leq\frac{n}{2},0\leq|I|\leq n-2t\\}.$

###### Proof.

The set (5.3) induces a chamber decomposition of the effective cone of
divisors of $X$, such that the divisors in any given chamber contain the same
joins in the stable base locus. This decomposition is induced by the
hyperplane arrangement

(5.4) $\\{\kappa_{J(L_{I},\sigma_{t})}=0:0\leq t\leq\frac{n}{2},0\leq|I|\leq
n-2t\\}.$

It is refined by the stable base locus decomposition of the effective cone,
because a priori effective divisors might have stable base locus that is not
of the form $J(L_{I},\sigma_{t})$. This turns out not to be the case. In fact
Mukai in [Mukai04, Mukai05] proved that the Mori chamber decomposition of the
effective cone of $X$, which is a further refinement of the stable base locus
decomposition, cf. Section 2.4, is given by the same hyperplane arrangement
(5.4). This concludes the proof. ∎

Recall that for a Mori dream space we have $\mathcal{D}_{k}=D_{k}$, by Thorem
3.3. Using Theorem 5.1, we can describe the cones $D_{k}$ of $X^{n}_{s}$
$s\leq n+3$, with their inequalities.

######  Proposition 5.2.

For any integer $k$ with $1\leq k\leq n-1$. The cone $\mathcal{D}_{k}$ of
divisors on $X^{n}_{n+3}$ that are ample in codimension $k$ is cut out by the
following inequalities:

  1. (I)

$0\leq d$ and $m_{i}\leq d$, for every $i=1,\ldots,n+3$;

  2. (II)

$-m_{i}+\sum_{j=1}^{n+3}m_{j}\leq nd$, for every $i=1,\ldots,n+3$;

  3. (IIIk)

$\kappa_{I,\sigma_{t}}\leq 0$, for every $0\leq t\leq n/2$ and $|I|=n-2t-k+1$;

  4. (IV)

$0\leq m_{i}$, for every $i=1,\ldots,n+3$.

###### Proof.

For $k=1$ the statement is proved in [BraDumPos3, Theorem 5.3].

We proceed by induction on $k$. We want to prove that $\mathcal{D}_{k}$ is
defined by the inequalities (I), (II), (IIIk) and (IV), for $k\geq 2$. We know
that $\mathcal{D}_{k}\subseteq\mathcal{D}_{k-1}$. Hence, by the induction
hypothesis, we have that $\mathcal{D}_{k}$ is contained in the cone cut out by
the inequalities (I), (II), (IIIk-1) and (IV). Notice that the only
inequalities to be added to the list are the ones excluding divisors with
stable base locus of dimension exactly $n-k$. By Theorem 5.1, these are given
by $\kappa_{I,\sigma_{t}}\leq 0$ for all joins $J(L_{I},\sigma_{t})$ of
dimension $n-k$. Recalling that the dimension of any such join is $|I|+2t-1$,
we get that these inequalities are precisely the ones of (IIIk). We conclude
by noticing that, by definition of $\kappa_{I,\sigma_{t}}$ and the effectivity
conditions (I) and (II), the inequalities (IIIk) imply the ones of (IIIk-1),
so they become redundant and can be discarded. ∎

######  Remark 5.3.

The effective cone $\mathcal{D}_{0}$ of $X^{n}_{n+3}$ is described in
[BraDumPos3, Theorem 5.1]. The defining inequalities in this case are only
(I), (II), (IIIk=0).

######  Corollary 5.4.

For $X=X_{s}^{n}$ with $1\leq s\leq n+2$, the cone $\mathcal{D}_{k}$ is
defined by the inequalities from Proposition 5.2 as follows:

  1. (1)

if $s=n+2$, by the inequalities (I), (II), (III)k (with $t=0$) and $({IV})$;

  2. (2)

if $s\leq n+1$, then

    1. (a)

if $k\geq n-s+1$, by the inequalities (I), (III)k (with $t=0$) and $({IV})$,

    2. (b)

if $k<n-s+1$, then we have $\mathcal{D}_{k}=\mathcal{D}_{k+1}$.

###### Proof.

The statement follows from Proposition 5.2 by using the natural isomorphism
between the Néron-Severi space of $X=X^{n}_{s}$ and the linear subspace of
$\operatorname{N}^{1}(X^{n}_{n+3})$ defined by the equations $m_{i}=0$, for
$s+1\leq i\leq n+3$. ∎

With a description of the cones $\mathcal{D}_{k}$ in $X_{s}^{n}$ via their
inequalities at hand, it is immediate to compute their dual with respect to
the standard intersection pairing. Recall that
$\operatorname{N}_{1}(X_{s}^{n})$ is generated by the class of a general line
in $X_{s}^{n}$ which we denote by $h$, along with the classes $e_{i}$, for
every $i=1,\dots,n+3$, of a general line in the exceptional divisor $E_{i}$.
Thus, the intersection product of divisor
$D\in\operatorname{N}^{1}(X_{s}^{n})_{\mathbb{R}}$ of the form (5.1) and a
curve of in $\operatorname{N}_{1}(X_{s}^{n})_{\mathbb{R}}$ of class

(5.5) $c=\delta
h-\sum_{i=1}^{s}\mu_{i}e_{i}\in\operatorname{N}_{1}(X_{s}^{n})_{\mathbb{R}}$

is computed by

(5.6) $D\cdot c=d\delta-\sum m_{i}\mu_{i}.$

In particular, to every hyperplane in
$\operatorname{N}^{1}(X_{s}^{n})_{\mathbb{R}}$, there corresponds an effective
curve class in $\operatorname{N}_{1}(X_{s}^{n})_{\mathbb{R}}$.

We will show that for $X^{n}_{s}$, $s\leq n+3$, the dual of $\mathcal{D}_{k}$
is precisely $\mathcal{C}_{k}$ for $0\leq k\leq n-1$, that is: strong duality
holds for every $k$. We need to introduce another ingredient before we can
state and prove the result: the Weyl group action on
$\operatorname{N}_{1}(X_{s}^{n})_{\mathbb{R}}$, which is described in detail
in [DM2, Proposition 2.5]. For an index set $\Gamma\subseteq\\{1,\ldots,s\\}$
of length $n+1$, the standard Cremona transformation ${\rm
Cr}_{\Gamma}:\mathbb{P}^{n}\dasharrow\mathbb{P}^{n}$ based at the points of
$\mathbb{P}^{n}$ parametrized by $\Gamma$, lifts to an isomorphism of
$\operatorname{N}_{1}(X_{s}^{n})_{\mathbb{R}}$ sending the curve class
$c\in\operatorname{N}_{1}(X_{s}^{n})_{\mathbb{R}}$ of the form (5.5) to the
following curve class:

(5.7) ${\rm
Cr}_{\Gamma}(c)=(\delta-(n-1)a_{\Gamma})h-\sum_{j\in\Gamma}(\mu_{j}-a_{\Gamma})e_{j}-\sum_{i\notin\Gamma}\mu_{i}e_{i},$

where we set

$a_{\Gamma}:=\sum_{j\in\Gamma}\mu_{j}-\delta.$

A composition of Cremona transformations is called a Weyl transformation and
we say that two curve classes $c$ and $c^{\prime}$ are in the same Weyl orbit
if there is a Weyl transformation $\phi$ such that $\phi(c)=c^{\prime}$.

We are now in position to prove the following theorem.

######  Theorem 5.5 (Strong duality theorem for $X^{n}_{n+3}$).

Let $1\leq s\leq n+3$ and $X=X^{n}_{s}$, then

$\mathcal{C}_{k}=\mathcal{D}_{k}^{\vee}.$

###### Proof.

Using the intersection pairing (5.6), for every $k$, the inequalities of
Proposition 5.2 correspond to the following effective curve classes which
generate the dual cone $\mathcal{D}_{k}^{\vee}$:

  * (I)

$h$ and $h-e_{i}$, for every $i=1,\ldots,n+3$;

  * (II)

$nh-\sum_{j=1}^{n+3}e_{j}+e_{i}$, for every $i=1,\ldots,n+3$;

  * (IIIk)

$(|I|+(n+1)t-1)h-\sum_{i\in I}(t+1)e_{i}-\sum_{i\notin I}te_{i}$, for every
$0\leq t\leq n/2$ and $|I|=n-2t-k+1$;

  * (IV)

$e_{i}$, for every $i=1,\ldots,n+3$.

Since the inclusion $\mathcal{C}_{k}\subseteq\mathcal{D}_{k}^{\vee}$ is always
satisfied, in order to prove the strong duality statement, we only need to
show that each one of the above curve classes belongs to $\mathcal{C}_{k}$.

From now on we will assume that $s=n+3$. For smaller $s$ the proof is similar
and uses Corollary 5.4, so we leave the details to the reader.

First of all notice that the curves classes (I) and (II) are dual to
effectivity inequalities and hence, by duality, they are in $\mathcal{C}_{0}$
which is contained in $\mathcal{C}_{k}$ for every $k$.

The curves classes (IV) sweep out the exceptional divisors, so they belong to
$\mathcal{C}_{1}$, which is contained in $\mathcal{C}_{k}$ for every $1\leq
k\leq n-1$.

It remains to deal with the curve classes (IIIk). Given a pair $(t,I)$
satisfying the bounds in (IIIk), the curve class

(5.8) $c_{I,t}=(|I|+(n+1)t-1)h-\sum_{i\in I}(t+1)e_{i}-\sum_{i\notin I}te_{i}$

is dual to the wall $\kappa_{J(L_{I},\sigma_{t})}=0$. This wall separates the
divisors which do not contain the join $J(L_{I},\sigma_{t})$ in their stable
base locus from those that do. By Theorem 4.3, we have that the curve class
$c_{I,t}$ sweeps out a subvariety of some small modification of $X$ that is
birational to $J(L_{I},\sigma_{t})\subset X$. We claim that $c_{I,t}$ has
effective irreducible representatives on $X$, so we conclude that $c_{I,t}$
sweeps out $J(L_{I},\sigma_{t})$.

We now prove the claim. Assume that $t=0$. In this case the class of $c_{I,0}$
reads as follows:

(5.9) $(n-k)h-\sum_{|I|=n-k+1}e_{i}.$

This curve class has irreducible representatives and it sweeps out the linear
cycle $L_{I}$, see Example 2.3.

Now, for any $t\geq 0$ we show that the curve class (5.8) is in the Weyl orbit
of a curve class of type (5.9). Assume the statement is true for any pair
$(t-1,I^{\prime})$ satisfying the conditions in (IIIk), and let $c_{I,t}$ be a
curve class for a pair $(t,I)$ as in (5.8). After relabelling the indexes if
necessary we may assume that $I=\\{1,\ldots,r_{t}\\}$ where $r_{t}=n-2t-k+1$.
Now take $\Gamma=\\{1,\ldots,r_{t}\\}\cup\\{r_{t}+3,\ldots,n+3\\}$, we have
that

${\rm
Cr}_{\Gamma}(c_{I,t})=(r_{t}+2+(n+1)(t-1)-1)h-\sum_{i=1}^{r_{t}+2}te_{i}-\sum_{i=r_{t}+3}^{n+3}(t-1)e_{i},$

which is a curve class as in (IIIk) for the pair
$(t-1,I^{\prime}=I\cup\\{r_{t}+1,r_{t}+2\\})$. By the induction hypothesis,
there is a Weyl transformation $\phi^{\prime}$ taking ${\rm
Cr}_{\Gamma}(c_{I,t})$ to a curve with class as in (5.9). Thus, taking
$\phi=\phi^{\prime}\circ{\rm Cr}_{\Gamma}$ we see that $c$ is in the Weyl
orbit of some curve class sweeping out a linear cycle of dimension equal to
the dimension of $J(L_{I},\sigma_{t})$. This proves that $c_{I,t}$ has
irreducible representatives on $X$, which are the images of the irreducible
representatives of (5.9). Hence, $c$ belongs to $\mathcal{C}_{k}$.

∎

######  Remark 5.6.

We recall that an $(i)-$curve class on a smooth $n$-dimensional variety $X$ is
defined as a smooth irreducible rational curve with normal bundle isomorphic
to $\oplus_{i=1}^{n-1}\mathcal{O}(i)$, for $i\in\\{-1,0,1\\}$, see [DM2,
Definition 1.1]. The extremal rays of $\mathcal{C}_{0}$ are examples of
$(1)-$curves and $(0)-$curves. The extremal rays of the Mori cone
$\mathcal{C}_{n-1}$ of type $h-e_{i}-e_{j}$ and (for $s=n+3$) the rational
normal curve class $C$ are all the $(-1)-$curves of $X^{n}_{s}$.

###  5.2. Blow up of $\mathbb{P}^{4}$ in $8$ points and of $\mathbb{P}^{3}$ in
$7$ points

We now prove strong duality for the two remaining cases: $X^{3}_{7}$ and
$X^{4}_{8}$.

We first recall the definition of Weyl cycles in $X^{n}_{s}$ introduced in
[BDP-Ciro]. A Weyl divisor is an effective divisor
$D\in\operatorname{Pic}(X^{n}_{s})$ which belongs to the Weyl orbit of an
exceptional divisor $E_{i}$. A Weyl cycle of codimension $r$ is an irreducible
component of the intersection of pairwise orthogonal Weyl divisors, where
orthogonality is taken with respect to the Dolgachev-Mukai pairing.

######  Theorem 5.7.

For $X=X_{7}^{3}$ or $X=X_{8}^{4}$ the Mori chamber decomposition and the
stable base locus decomposition coincide. In particular they are induced by
the Weyl cycles of $X$.

###### Proof.

The proof is similar to that of Theorem 5.1 and it uses two ingredients: on
the one hand the classification of Weyl cycles and their corresponding base
locus lemmata done in [BDP-Ciro]; on the other hand the description of the
Mori chamber decomposition. For the threefold $X^{3}_{7}$ the latter is easy
to obtain and left to the reader (there are only floppable curves and
contracting divisors). Whereas obtaining the Mori chamber decomposition for
the fourfold $X^{4}_{8}$ is not trivial: it was done by Mukai [Mukai05] and by
Casagrande, Codogni and Fanelli [CCF] by means of the Gale duality between
$X^{4}_{8}$ and the Del Pezzo surface $S:=X^{2}_{8}$, a projective plane blown
up in eight general points.

In [BDP-Ciro] (see also [DM1] for an alternative approach) the Weyl cycles on
$X^{3}_{7}$ and $X^{4}_{8}$ are classified and explicitly described. For
$X^{3}_{7}$ these are $(-1)-$curves and Weyl divisors, (see [BDP-Ciro, Section
4]), for $X_{8}^{4}$ we further have five types of Weyl surfaces (see [BDP-
Ciro, Section 5] for details): they are the strict transforms of the following
surfaces of $\mathbb{P}^{4}$:

  * •

$S^{1}_{i,j,k}$ a plane through three points,

  * •

$S^{3}_{i,\widehat{j}}=J(C_{\widehat{j}},p_{i})$, a pointed cone over the
rational normal curve passing through seven points
$p_{1},\dots,\widehat{p_{j}},\dots,p_{8}$,

  * •

$S^{6}_{i,j,k}$ a sextic surface with five triple points and simple at three
points,

  * •

$S^{10}_{i,j}$ a degree$-10$ surface with two sextuple points and five triple
points,

  * •

$S^{15}_{i}$ a degree$-15$ surface with seven sextuple points and a triple
point.

For each of the above cycles, a base locus lemma is available that gives a
hyperplane in the Néron-Severi space. In particular, for a Weil curve $C$, the
hyperplane is defined by $D\cdot C=0$, for a Weyl divisor $E$, the hyperplane
is defined by $\langle D,E\rangle=0$, where $\langle\cdot,\cdot,\rangle$ is
the Dolgachev-Mukai pairing, and for the 5 surfaces the hyperplanes are given
by $D\cdot\mu=0$ where $\mu$ is the corresponding curve class:

  * •

$\mu^{1}_{i,j,k}=2h-e_{i}-e_{j}-e_{k}$,

  * •

$\mu^{3}_{i,\widehat{j}}=5h-\sum_{k=1}^{8}e_{k}-e_{i}+e_{j}$,

  * •

$\mu^{6}_{i,j,k}=8h-2\sum_{l=1}^{8}e_{l}+e_{i}+e_{j}+e_{k}$,

  * •

$\mu^{10}_{i,j}=11h-2\sum_{k=1}^{8}e_{k}-e_{i}-e_{j}$,

  * •

$\mu^{15}_{i}=14h-3\sum_{j=1}^{8}e_{j}+e_{i}$.

The dual hyperplanes to the above curve classes induce a chamber decomposition
of the effective cone which is refined by the stable base locus decomposition
which, in turn, is refined by the Mori chamber decomposition. Since the two
decompositions agree, and in particular since Weyl curves are the only
flopping curves, we conclude. ∎

######  Theorem 5.8.

Let $X=X^{3}_{7}$ or $X=X^{4}_{8}$, then

$\mathcal{C}_{k}=\mathcal{D}_{k}^{\vee}.$

###### Proof.

Following the proof of Theorem 5.5, one can prove the duality statement for
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$. Since for the movable cone
$\mathcal{D}_{1}$ the statement is easy, we just give a hint of the proof for
the case of $\mathcal{D}_{2}(X^{4}_{8})$.

Using the notation of the proof of Theorem 5.7, one first observes that
$\mu^{1}_{1,2,3}$ belongs to $\mathcal{C}_{2}(X^{4}_{8})$ (cf. also Example
2.3) and then verifies that the remaining four curve types $\mu^{1}_{1,2,3}$,
$\mu^{3}_{1,\widehat{8}}$, $\mu^{6}_{1,2,3}$, $\mu^{10}_{1,2}$ and
$\mu^{15}_{1}$ may be transformed to $\mu^{1}_{i,j,k}$ via the Weyl action,
for some indices $i,j,k$. The details are left to the reader. ∎

## References