# An elementary description of nef cone for irreducible holomorphic symplectic
manifolds

Anastasia V. Vikulova  Steklov Mathematical Institute of Russian Academy of
Sciences, 8 Gubkin str., Moscow, 119991, Russia Laboratory of Algebraic
Geometry, National Research University Higher School of Economics, 6 Usacheva
str., Moscow, 119048, Russia.
<EMAIL_ADDRESS>
###### Abstract.

We describe MBM classes for irreducible holomorphic symplectic manifolds of
$\mathrm{K3}$ and Kummer types. These classes are the monodromy images of
extremal rational curves which give the faces of the nef cone of some
birational model. We study the connection between our results and A. Bayer and
E. Macrì’s theory. We apply the numerical method of description due to E.
Amerik and M. Verbitsky in low dimensions to the $\mathrm{K3}$ type and Kummer
type cases.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Lattices of $\mathrm{K3}$ and Kummer types manifolds
  4. 4 Description of nef cone for $\mathrm{K3}$ type manifolds
  5. 5 Description of nef cone for Kummer type manifolds

##  1\. Introduction

There is a great interest in studying ample cones in algebraic geometry in the
context of minimal model program. Much had been done to find ample or Kähler
cone of irreducible holomorphic symplectic manifolds. The simplest example of
an irreducible holomorphic symplectic manifolds is a $\mathrm{K3}$ surface.
The self-intersection of smooth rational irreducible curves on $\mathrm{K3}$
surfaces is $-2.$ The Kähler cone of a $\mathrm{K3}$ surface consists of the
classes of type $(1,1)$ in the second cohomologies with real coefficients,
which have positive self-intersection and which positively intersect with
$(-2)$-curves. The ample cone of a projective $K3$ surface is characterized in
the same way in the Néron-Severi group with real coefficients. So orthogonal
hyperplanes to the classes of square $-2$ in the positive cone give the
chamber decomposition of the positive cone, where the Kähler cone (or the
ample cone) is one of the chambers (see, for example, the paper [18, Corollary
3.4]).

However, it is not so easy to describe the ample or Kähler cone for other
irreducible holomorphic symplectic manifolds. The recent results of A. Bayer
and E. Macrì use the difficult technique of the theory of Bridgeland stability
conditions. There is a connection between wall-crossing for stability
conditions and walls of the positive cone of a moduli space of $H$-Gieseker
stable sheaves thanks to a natural morphism from the set of chambers to the
Néron-Severi group, constructed by A. Bayer and E. Macrì in the paper [5].
This technique was applied to all known examples of holomorphic symplectic
manifolds (see [6],[9],[22],[28]). A. Bayer and E. Macrì described the chamber
decomposition of the positive cone of the moduli space of semistable sheaves
on a $\mathrm{K3}$ surface with primitive Mukai vector. In fact, such a moduli
space is deformation equivalent to the Hilbert scheme of points on a
$\mathrm{K3}$ surface. We mention two theorems from the papers [6] and [28],
which describe the ample cones of $\mathrm{K3}$ and Kummer types manifolds.

Before we formulate the theorems we first introduce the following notations:

######  Definition 1.1.

Let $X$ be a smooth projective surface. Denote by
$H^{*}_{\text{alg}}(X,\mathbb{Z})$ the algebraic part of the ring of even
cohomologies, i.e.

$H^{*}_{\text{alg}}(X,\mathbb{Z})\,=\,H^{0}(X,\mathbb{Z})\oplus NS(X)\oplus
H^{4}(X,\mathbb{Z}).$

For an element $v\in H^{*}_{\text{alg}}(X,\mathbb{Z})$ denote by $v^{\perp}$
the subspace of elements in $H^{*}_{\text{alg}}(S,\mathbb{Z})$ which are
orthogonal to $v$ with respect to the Mukai pairing given by

$((r_{1},\alpha_{1},s_{1}),(r_{2},\alpha_{2},s_{2}))\,=\,\alpha_{1}\cdot\alpha_{2}-r_{1}\cdot
s_{2}-r_{2}\cdot s_{1},$

where $(r_{i},\alpha_{i},s_{i})\in H^{*}_{\text{alg}}(X,\mathbb{Z})$ with
$r_{i}\in H^{0}(X,\mathbb{Z}),$ $\alpha_{i}\in NS(X)$ and $s_{i}\in
H^{4}(X,\mathbb{Z}).$

######  Theorem 1.2 ([6, Theorem 12.1]).

Let $S$ be a $\mathrm{K3}$ surface, $v\in H^{*}_{\text{alg}}(X,\mathbb{Z})$ be
a fixed primitive Mukai vector and $M_{\sigma}(v)$ be the moduli space of
$\sigma$-semistable sheaves on $S$ with the Mukai vector $v$ and the stability
condition $\sigma\in\mathrm{Stab}(S)$. Consider the set $W$ of elements $a\in
H^{*}_{\text{alg}}(S,\mathbb{Z})$ such that $a^{2}\geqslant-2$ and
$0\leqslant(v,a)\leqslant\frac{v^{2}}{2}$, where the element $a$ is identified
with its image under the isometry

$\theta:v^{\perp}\xrightarrow{\sim}NS(M_{\sigma}(v))$

which is defined in [29, (1.6)]. Then the set of $a^{\perp}$ with $a\in W$
defines a chamber decomposition of the positive cone of $M_{\sigma}(v).$ The
ample cone $\mathrm{Amp}(M_{\sigma}(v))$ is one of the chambers:
$\eta\in\mathrm{Amp}(M_{\sigma}(v))$ if and only if $(\eta,a)>0$ for all $a\in
W.$

######  Theorem 1.3 ([28]).

Let $A$ be an abelian surface, $v\in H^{*}_{\text{alg}}(A,\mathbb{Z})$ be a
fixed primitive Mukai vector, $M_{\sigma}(v)$ be the moduli space of
$\sigma$-semistable sheaves on $A$ with the Mukai vector $v$ and
$\mathrm{Kum}_{\sigma}(v)$ be the fiber of the Albanese map

$\displaystyle M_{\sigma}(v)$ $\displaystyle\longrightarrow A\times A^{*}$
$\displaystyle\mathcal{F}$
$\displaystyle\longmapsto\left(\mathrm{Alb}(c_{2}(\mathcal{F})),\det(\mathcal{F})\right)$

over zero with $A^{*}\,=\,\mathrm{Pic}^{0}(A),$ where
$\mathrm{Alb}:\mathrm{CH}_{0}(A)\to A$ is a summation map. Consider the set
$W$ of elements $a\in H^{*}_{\text{alg}}(A,\mathbb{Z})$ such that
$a^{2}\geqslant 0$ and $0<(v,a)\leqslant\frac{v^{2}}{2}$, where the element
$a$ is identified with its image under the isometry

$\theta:v^{\perp}\xrightarrow{\sim}NS(\mathrm{Kum}_{\sigma}(v))$

which is defined in [29, (1.6)]. Then the set of $a^{\perp}$ with $a\in W$
defines a chamber decomposition of the positive cone of
$\mathrm{Kum}_{\sigma}(v).$ The ample cone
$\mathrm{Amp}(\mathrm{Kum}_{\sigma}(v))$ is one of the chambers:
$\eta\in\mathrm{Amp}(\mathrm{Kum}_{\sigma}(v))$ if and only if $(\eta,a)>0$
for all $a\in W.$

######  Remark 1.4 (cf. [7, 19]).

If $S$ is a $\mathrm{K3}$ surface then the Mukai vector $v=(1,0,0)$ defines
the moduli space $M_{\sigma}(v)\simeq S$ for any $\sigma.$ If $A$ is a complex
two-dimensional torus then the Mukai vector $v=(1,0,-2)$ defines the Kummer
surface $\mathrm{Kum}_{\sigma}(v)$ isomorphic to the blowup of the quotient of
$A$ by the natural involution for any $\sigma$.

For a $\mathrm{K3}$ surface $S$ and Mukai vector $v=(1,0,-n+1)$ the moduli
space $M_{\sigma}(v)$ is isomorphic to the Hilbert scheme of $n$ points
$\mathrm{Hilb}^{n}(S)$ for any $\sigma.$ For a complex two-dimensional torus
$A$ and Mukai vector $v=(1,0,-n-1)$ the variety $\mathrm{Kum}_{\sigma}(v)$ is
isomorphic to the Kummer variety $\mathrm{Kum}^{n}(A)$ for any $\sigma.$
Recall that the Kummer variety for the two-dimensional torus $A$ is the fiber
over $0\in A$ of the summation map $\mathrm{Hilb}^{n+1}(A)\to A.$

All these theorems work for projective surfaces. In the non-algebraic case the
ample cone does not exist. However, the Kähler cone does always exist for
irreducible holomorphic symplectic manifolds and we have
$Amp(X)\,=\,\mathcal{K}(X)\cap NS(X)_{\mathbb{R}},$ where $Amp(X)$ is the
ample cone, $\mathcal{K}(X)$ is the Kähler cone and $NS(X)$ is the Néron-
Severi group. The Kähler cone of irreducible holomorphic symplectic manifolds
was studied by E. Amerik, D. Huybrechts, E. Markman, M. Verbitsky, and others
(see, for instance, [2, 15, 20]).

The structure of the ample cone for Hilbert scheme of points on $\mathrm{K3}$
surface was originally conjectured in the papers of B. Hassett and Y.
Tschinkel (for instance, see [16]). A refinement of their conjecture is
established by A. Bayer and E. Macrì. In the paper of E. Amerik and M.
Verbitsky [2] there is more elementary description of the boundary of the
Kähler cone for $\mathrm{K3}$ type varieties (i.e. deformation of Hilbert
scheme of points on a $\mathrm{K3}$ surface) for small dimensions. This means
that they describe MBM classes, i.e. the classes which correspond to the
extremal rays of the Mori cone on some deformation of the variety (see Section
2). The point is that these classes are easy to characterise numerically
without appealing to Bayer-Macrì’s machinery. In this paper we try to provide
a similar elementary description of the Kähler cone for $\mathrm{K3}$ and
Kummer types varieties of arbitrary dimension which are deformations of
Hilbert scheme of points on a $\mathrm{K3}$ surface or Kummer variety,
respectively. So the goal of this paper is to give a simple and self-contained
proof of the following theorems about the walls of the Kähler cone:

######  Theorem 1.5.

Let $X$ be an IHSM of $\mathrm{K3}$ type. Let $\alpha\in H^{2}(X,\mathbb{Z})$
be a primitive class and denote by $\widehat{\alpha}\in H_{2}(X,\mathbb{Z})$
the dual homology class of $\alpha.$ Denote by $q$ the Beauville–Bogomolov
form on the lattice $H^{2}(X,\mathbb{Z})$ and its extension form on the dual
lattice $H_{2}(X,\mathbb{Z})\hookrightarrow
H^{2}(X,\mathbb{Z})\otimes\mathbb{Q}.$ Let $d(\alpha)$ be the divisibility of
$\alpha$ and $\delta(\alpha)$ be the image of $\alpha/d(\alpha)$ in the
discriminant group of the lattice $H^{2}(X,\mathbb{Z}).$ Then $\alpha$ is MBM
if and only if one of the following holds:

  1. (1)

$q(\widehat{\alpha})=-\frac{1}{2(n-1)}$ and $\delta(\alpha)=1;$

  2. (2)

there is an integer $b\in[0,n-1]$ such that
$q(\widehat{\alpha})=2a-\frac{b^{2}}{2(n-1)},$ where $a$ is an integer number
satisfied the inequalities

$-2\leqslant 2a<\frac{b^{2}}{2(n-1)},$

and $\delta(\alpha)=b.$

In the proof of the theorem we obtain as a byproduct the following corollary:

######  Corollary 1.6.

Under the conditions of Theorem 1.5 we have:

If (1)_holds then $\alpha$ is the class of half of the exceptional divisor on
some deformation $Y=\mathrm{Hilb}^{n}(S)$ of $X$ with Picard group
$\mathrm{Pic}(Y)=\mathrm{Pic}(S)\oplus\mathbb{Z}e,$ where $S$ is a
$\mathrm{K3}$ surface and $e$ is half of exceptional divisor on $Y$._

_If_ (2) _holds then the cohomological class $\alpha$ up to a rational
multiple can be represented by_

$2(n-1)\cdot x-be$

_on some deformation $Y=\mathrm{Hilb}^{n}(S)$ of $X$ with Picard group_

$\mathrm{Pic}(Y)=\mathrm{Pic}(S)\oplus\mathbb{Z}e,$

_such that $\mathrm{Pic}(S)=\mathbb{Z}x,$ $x^{2}=2a\geqslant-2$ and $e$ is
half of the exceptional divisor on $Y.$ _

######  Theorem 1.7.

Let $X$ be an IHSM of Kummer type of dimension $2n\geqslant 4$. Assume that
$\alpha\in H^{2}(X,\mathbb{Z})$ is a primitive class and denote by
$\widehat{\alpha}\in H_{2}(X,\mathbb{Z})$ the dual homological class of
$\alpha.$ Denote by $q$ the dual Beauville–Bogomolov form on the lattice
$H^{2}(X,\mathbb{Z})$ and its extension form on the dual lattice
$H_{2}(X,\mathbb{Z})\hookrightarrow H^{2}(X,\mathbb{Z})\otimes\mathbb{Q}.$ Let
$d(\alpha)$ be the divisibility of $\alpha$ and $\delta(\alpha)$ be the image
of $\alpha/d(\alpha)$ in the discriminant group of the lattice
$H^{2}(X,\mathbb{Z}).$ Then $\alpha$ is MBM if and only if one of the
following holds:

  1. (1)

$q(\widehat{\alpha})=-\frac{1}{2(n+1)}$ and $\delta(\alpha)=1;$

  2. (2)

there is an integer number $b\in[1,n+1]$ such that
$q(\widehat{\alpha})=2a-\frac{b^{2}}{2(n+1)},$ where $a$ is an integer number
satisfying the inequalities

$0\leqslant 2a<\frac{b^{2}}{2(n+1)},$

and $\delta(\alpha)=b.$

Again, as in the $\mathrm{K3}$ case, in the proof of this theorem we also
obtain as a byproduct the following corollary:

######  Corollary 1.8.

Under the conditions of Theorem 1.7 we have:

If (1) _holds then $\alpha$ is the class of half of the exceptional divisor on
some deformation $Y=\mathrm{Kum}^{n}(A)$ of $X$ with Picard group
$\mathrm{Pic}(Y)=\mathrm{NS}(A)\oplus\mathbb{Z}e,$ where $A$ is a complex two-
dimensional torus and $e$ is half of the exceptional divisor on $Y$. _

_If_ (2) _holds then the cohomological class $\alpha$ up to a rational
multiple can be represented by_

$2(n+1)\cdot x-be$

_on some deformation $Y=\mathrm{Kum}^{n}(A)$ of $X$ with Picard group_

$\mathrm{Pic}(Y)=\mathrm{NS}(A)\oplus\mathbb{Z}e,$

_such that $\mathrm{NS}(A)=\mathbb{Z}x,$ $x^{2}=2a\geqslant 0$ and $e$ is half
of the exceptional divisor on $Y$._

We also explain by direct computations how our results agree with that of A.
Bayer and E. Macrì, and K. Yoshioka.

######  Proposition 1.9.

Let $X$ be a $\mathrm{K3}$ type manifold and $v$ be a Mukai vector. There is a
one-to-one correspondence between the set of orthogonal hyperplanes to MBM
classes in the positive cone $\mathrm{Pos}(X),$ which were described in
Theorem 1.5 and the set of walls in Theorem 1.2, given by

$a\in H^{*}_{alg}(X,\mathbb{Z})\mapsto\mathrm{pr}_{v^{\perp}}(a)$

where $pr_{v^{\perp}}$ is a projection on $v^{\perp}.$

######  Proposition 1.10.

Let $X$ be a Kummer type manifold and $v$ be a Mukai vector. There is a one-
to-one correspondence between the set of orthogonal hyperplanes to MBM classes
in the positive cone $\mathrm{Pos}(X),$ which were described in Theorem 1.7
and the set of walls in Theorem 1.3, given by

$a\in H^{*}_{alg}(X,\mathbb{Z})\mapsto\mathrm{pr}_{v^{\perp}}(a)$

where $pr_{v^{\perp}}$ is a projection on $v^{\perp}.$

Notation. Let $X$ be a Kähler manifold. If we need to fix some complex
structure $I$ on $X$ then we write $(X,I)$ to define the manifold $X$ with
complex structure $I.$ By $H^{p,q}(X,I)$ we denote $(p,q)$-th component of
Hodge decomposition of $H^{p+q}(X,\mathbb{C})$ with respect to the complex
structure $I.$

Acknowledgment. The author is grateful to E. Yu. Amerik for suggesting this
problem, for a lot of discussions, for attention to this paper and for her
patience.

This work was performed at the Steklov International Mathematical Center and
supported by the Ministry of Science and Higher Education of the Russian
Federation (agreement no. 075-15-2022-265). The author is a winner of the all-
Russia mathematical August Moebius contest of graduate and undergraduate
student papers and thanks the jury and the board for the high praise of his
work. The author was partially supported by Theoretical Physics and
Mathematics Advancement Foundation “BASIS” and by the HSE University Basic
Research Program.

##  2\. Preliminaries

In this section we recall some theory about irreducible holomorphic symplectic
manifolds.

######  Definition 2.1.

A smooth compact Kähler manifold $X$ is called _an irreducible holomorphic
symplectic manifold_ (or just _IHSM_) if it has trivial $\pi_{1}(X)$ and

$H^{0}(X,\Omega_{X}^{2})\simeq\mathbb{C}\sigma$

with a non-degenerate holomorphic 2-form $\sigma$.

A few examples of such manifolds are known. Two series of examples were
described by A. Beauville in [7]. They are the Hilbert scheme of $n$ points on
a $\mathrm{K3}$ surface and the generalised Kummer variety which is defined as
follows.

######  Definition 2.2.

Let $A$ be a complex two-dimensional torus. Then a Kummer variety of dimension
$n$ is the fiber over zero of the summation map

$s:\mathrm{Hilb}^{n+1}(A)\to A.$

We denote the Kummer variety by $\mathrm{Kum}^{n}(A).$

Other two examples were found by O’Grady in [25] and [26]. All mentioned
examples are not pairwise birational because they have different second Betti
numbers.

### Beauville–Bogomolov form and Mukai lattice.

The IHS manifolds have an interesting feature which, in general, only surfaces
have. If $X$ is an IHSM, then there is a quadratic form $q$ on the second
integral cohomology $H^{2}(X,\mathbb{Z})$ which is called _Beauville–Bogomolov
form_. This form is of signature $(3,b_{2}(X)-3)$ and its restriction on
$H^{1,1}(X,\mathbb{R})$ is of signature $(1,b_{2}(X)-3).$ This form is
uniquely up to a sign defines by the following theorem.

######  Theorem 2.3 ([12, Theorem 4.7]).

Let $X$ be a IHSM of dimension $2n.$ Denote by $q$ the Beauville–Bogomolov
form. Then there is a constant $c\in\mathbb{R}_{>0}$ depending only on the
manifold $X$ such that for $\alpha\in H^{2}(X,\mathbb{Z})$ we get
$\int_{X}\alpha^{2n}\,=\,cq(\alpha)^{n}$.

In fact, this form is uniquely defined by Theorem 2.3 and signature on
$H^{1,1}(X,\mathbb{R}).$ This means that the quadratic form $q$ depends only
on topology of the manifold (see [15, Remark 23.15]).

######  Remark 2.4.

The second cohomology group of an IHSM equipped with Beauville–Bogomolov form
is a lattice. We have the embedding $H_{2}(X,\mathbb{Z})\hookrightarrow
H^{2}(X,\mathbb{Z})\otimes\mathbb{Q}$ which is given by the lattice duality.

Let us give the description of the Beauville–Bogomolov form for IHSM
$\mathrm{Hilb}^{n}(S)$ and $\mathrm{Kum}^{n}(A),$ where $S$ and $A$ are a
$\mathrm{K3}$ surface and a complex two-dimensional torus, respectively. First
of all, we recall the definition and some properties of the Mukai lattice,
which will be extremely important below. Let

$L\,=\,H^{0}(X,\mathbb{Z})\oplus H^{2}(X,\mathbb{Z})\oplus
H^{4}(X,\mathbb{Z})$

be the lattice of even cohomologies of a surface $X$ with the quadratic form
which is called Mukai pairing and given by

$((r_{1},\alpha_{1},s_{1}),(r_{2},\alpha_{2},s_{2}))\,=\,\alpha_{1}\cdot\alpha_{2}-r_{1}\cdot
s_{2}-r_{2}\cdot s_{1},$

where $(r_{i},\alpha_{i},s_{i})\in L$ such that $r_{i}\in
H^{0}(X,\mathbb{Z}),$ $\alpha_{i}\in H^{2}(X,\mathbb{Z})$ and $s_{i}\in
H^{4}(X,\mathbb{Z}).$ And let

$H^{*}_{alg}(X,\mathbb{Z})\,=\,H^{0}(X,\mathbb{Z})\oplus NS(X)\oplus
H^{4}(X,\mathbb{Z}).$

It has the following geometric meaning: for the Grothendieck group of the
category of coherent sheaves $\mathcal{K}_{0}(X)$ we have a surjective
homomorphism

$v:\mathcal{K}_{0}(X)\twoheadrightarrow H^{*}_{alg}(X)$

which maps the class of sheaf $\mathcal{F}$ to the Mukai vector
$v(\mathcal{F})\,=\,ch(\mathcal{F})\sqrt{td(X)},$ where $ch(\mathcal{F})$ is
the Chern character and $td(X)$ is the Todd class of tangent bundle of $X$. In
the case of a $\mathrm{K3}$ surface we get

$v(\mathcal{F})\,=\,(rk(\mathcal{F}),c_{1}(\mathcal{F}),\chi(\mathcal{F})-rk(\mathcal{F}))$

and for a complex two-dimensional torus we have

$v(\mathcal{F})\,=\,(rk(\mathcal{F}),c_{1}(\mathcal{F}),\chi(\mathcal{F})),$

where $rk(\mathcal{F})$ is the rank, $c_{1}(\mathcal{F})$ is the first Chern
class and

$\chi(\mathcal{F})\,=\,\sum_{i}(-1)^{i}\dim\mathrm{Ext}^{i}(\mathcal{F},\mathcal{F})$

is the Euler characteristic.

The Hilbert scheme of $n$ points on the $\mathrm{K3}$ surface $S$ is a moduli
space of ideal sheaves of $n$ points on $S$ with respect to a stability
condition $\sigma$. These are sheaves with Mukai vector $v\,=\,(1,0,-n+1).$
The orthogonal complement of $v$ with respect to Mukai quadratic form is
isomorphic to the Néron-Severi group of $M_{\sigma}(v).$ So we have

$v^{\perp}\simeq NS(S)\oplus\mathbb{Z}e,$

where $e\,=\,(1,0,n-1)$ and $e^{2}\,=\,-2(n-1).$ This means that the
restriction of the Beauville–Bogomolov form for $\mathrm{Hilb}^{n}(S)$ on
$\mathrm{NS}(S)$ coincides with the intersection form on $NS(S),$ and $e$ is
orthogonal to $NS(S).$

The same description works for a complex two-dimensional torus $A$. In this
case we consider the moduli space $M_{\sigma}(v)$ for the Mukai vector
$v\,=\,(1,0,-n-1).$ Every sheaf $\mathcal{F}\in M_{\sigma}(v)$ can be written
as $\mathcal{F}=\mathcal{L}\otimes I_{T},$ where
$\mathcal{L}\in\mathrm{Pic}^{0}(A)=A^{*}$ and $T\in\mathrm{Hilb}^{n+1}(A).$
The generalized Kummer variety $\mathrm{Kum}^{n}(A)$ is the fiber over zero of
the Albanese map

$\displaystyle M_{\sigma}(v)$ $\displaystyle\longrightarrow A\times A^{*}$
$\displaystyle\mathcal{F}$
$\displaystyle\longmapsto\left(\mathrm{Alb}(T),\mathcal{L}\right)$

where $\mathrm{Alb}:\mathrm{Hilb}^{n+1}(A)\to A$ is a summation map. We have
the isomorphism

$v^{\perp}\simeq NS(A)\oplus\mathbb{Z}e,$

where $e=(1,0,n+1)$ and $e^{2}=-2(n+1).$ This means the restriction of the
Beauville–Bogomolov form for $\mathrm{Kum}^{n}(A)$ on $\mathrm{NS}(A)$
coincides with the intersection form on $NS(A)$ and $e$ is orthogonal to
$NS(A).$

The vector $e$ has the following geometric description in both $\mathrm{K3}$
and Kummer cases.

######  Proposition 2.5.

Let $\Delta$ be the exceptional divisor of the Hilbert–Chow morphism

$\mathrm{Hilb}^{n}(B)\to\mathrm{Sym}^{n}(B),$

where $B$ is either $\mathrm{K3}$ surface, or a complex two-dimensional torus.
If $B$ is a $\mathrm{K3}$ surface then the class of $\Delta$ is $2e.$ If $B$
is a complex two-dimensional torus then the class of
$\Delta\cap\mathrm{Kum}^{n-1}(B)$ is $2e.$

###### Proof.

The proposition follows directly from [7, Remark after Proposition 6] for
$\mathrm{K3}$ case and [10, Chapter 2, §1, Remark 5] for Kummer case. ∎

######  Definition 2.6.

We denote by $\widehat{e}$ the class of a general fiber on the exceptional
divisor of the Hilbert–Chow morphism in $H_{2}(X,\mathbb{Z}).$ It satisfies
the equation

$q(e,\widehat{e})=-1.$

######  Remark 2.7.

Note, that if $X=\mathrm{Hilb}^{n}(B)$ (respectively, $X=\mathrm{Kum}^{n}(B)$)
for a $\mathrm{K3}$ surface (respectively, for a complex two-dimensional torus
$B$) then we have the following isomorphisms:

(2.1) $\displaystyle H^{2}(X,\mathbb{Z})\simeq
H^{2}(B,\mathbb{Z})\oplus\mathbb{Z}e;$ (2.2)
$\displaystyle\mathrm{Pic}(X)\simeq\mathrm{NS}(B)\oplus\mathbb{Z}e;$ (2.3)
$\displaystyle\mathrm{N}_{1}(X)\simeq\mathrm{N}_{1}(B)\oplus\mathbb{Z}\widehat{e},$

where by $\mathrm{N}_{1}$ we denote the group of $1$-cycles modulo numerical
equivalence.

### The space of deformations.

Now we recall some definitions and facts of deformation theory.

######  Definition 2.8.

Let $X_{1}$ and $X_{2}$ be complex manifolds. We say that they are deformation
equivalent if there is a smooth proper holomorphic morphism $\mathcal{X}\to B$
such that $B$ is connected and there are two points $b_{1},b_{2}\in B$ such
that $\mathcal{X}_{b_{1}}\simeq X_{1}$ and $\mathcal{X}_{b_{2}}\simeq X_{2}.$
Let us fix the complex manifold $X.$ Here the variety $B$ is called a space of
deformations.

Now we give the description of the deformation space of IHSM which will be
convenient for us.

######  Definition 2.9.

Let $X$ be an IHSM, $\text{Diff}_{0}(X)$ be a connected component of its
diffeomorphism group, Comp be the space of complex structures of the Kähler
type on $X.$ Then $\text{Teich}\,=\,\text{Comp}/\text{Diff}_{0}(X)$ is the
Teichmüller space and

$\text{Teich}_{b}=\text{Teich}/\sim.$

Note that the Teichmüller space is non-Hausdorff. However, there is an
important result which is given in [17].

######  Theorem 2.10 ([17, Theorem 4.3]).

Let $X$ and $X^{\prime}$ be IHSMs which are diffeomorphic. Assume that they
are inseparable in the space $\text{Teich}.$ Then $X$ and $X^{\prime}$ are
birational.

It is convenient to consider the birational Teichmüller space, where $X\sim
X^{\prime}$ if and only if they are inseparable in $\mathrm{Teich}.$ The
birational Teichmüller space is the Hausdorff reduction of the Teichmüller
space. By Theorem 2.10 every point of $\mathrm{Teich}$ over any
$x\in\text{Teich}_{b}$ corresponds to some birational model of $X.$

######  Definition 2.11.

Let $\mathrm{Diff}(X)$ be the group of diffeomorphisms of the manifold $X$ and
$\mathrm{Diff}^{0}(X)$ be its connected component containing the identity,
i.e. the group of isotopies. Denote by $\Gamma$ the mapping class group
$\mathrm{Diff}(X)/\mathrm{Diff}^{0}(X).$

### Cones associated to an irreducible holomorphic symplectic manifold.

Now let us recall some facts about the positive cone for IHSM.

######  Definition 2.12.

Let $X$ be an IHSM. Both the set of $\alpha\in\mathrm{NS}(X)_{\mathbb{R}}$ for
projective $X$ and the set of $\alpha\in H^{1,1}(X)_{\mathbb{R}}$ for Kähler
$X,$ such that $q(\alpha)>0,$ have two connected components. The positive cone
is the component that contains ample or Kähler classes, respectively. We
denote both of them by $\mathrm{Pos}(X)$ when there is no confusion, otherwise
we denote by $\mathrm{Pos}_{\text{amp}}(X)$ and
$\mathrm{Pos}_{\text{K\"{a}h}}(X)$ the positive cones of ample and Kähler
classes, respectively.

######  Remark 2.13.

Note that $\mathrm{Pos}_{\text{amp}}(X)$ and
$\mathrm{Pos}_{\text{K\"{a}h}}(X)$ are different, because in general the
inclusion

$\mathrm{NS}(X)_{\mathbb{R}}=H^{1,1}(X,\mathbb{Z})_{\mathbb{R}}\subset
H^{1,1}(X,\mathbb{R})$

is strict. However, the intersection of $\mathrm{Pos}_{\text{K\"{a}h}}(X)$
with $\mathrm{NS}(X)_{\mathbb{R}}$ gives us $\mathrm{Pos}_{\text{amp}}(X)$.
This is the reason why we will mainly deal with the positive cone
$\mathrm{Pos}_{\text{K\"{a}h}}(X).$

######  Definition 2.14.

Let $X$ be a Kähler manifold. The Kähler cone

$\mathcal{K}(X)\subset H^{1,1}(X,\mathbb{R})$

is an open convex cone of Kähler classes.

######  Definition 2.15.

Let $X$ be a projective manifold. The ample cone

$\mathrm{Amp}(X)\subset\mathrm{NS}(X)_{\mathbb{R}}$

is the open cone of finite sums $\sum_{i}a_{i}L_{i},$ where
$a_{i}\in\mathbb{R}$ and $L_{i}$ are ample classes in $\mathrm{NS}(X).$ The
nef cone $\mathrm{Nef}(X)\subset\mathrm{NS}(X)_{\mathbb{R}}$ is the set of
classes $\mathrm{NS}(X)_{\mathbb{R}}$ such that their intersections with
curves on $X$ is non-negative.

The closure of the ample cone is the nef cone, i.e. we have

$\mathrm{Amp}(X)=\mathrm{Int}\mathrm{Amp}(X)\subset\mathrm{Nef}(X)=\overline{\mathrm{Amp}(X)},$

where $\mathrm{Int}\mathrm{Amp}(X)$ is the interior of the ample cone of $X.$
We have the following result.

######  Theorem 2.16 ([8]).

The Kähler cone of an IHSM $X$ consists of all elements $\alpha$ in
$\mathrm{Pos}(X)$ such that $q(\alpha,C)\,>\,0$ for all rational curves $C$.

According to Remark 2.4, one may view homology classes as cohomology classes
with rational coefficients, so that $q(\alpha,C)$ is a well-defined rational
number. Theorem 2.16 is a generalization of the description of the Kähler cone
for a $\mathrm{K3}$ surface. In the two-dimensional case the positivity of the
intersection with (-2)-curves defines ample or Kähler classes. However, in the
higher-dimensional case the situation is more complicated. By Theorem 2.16,
the walls of the Kähler cone are given by the orthogonal elements to the
classes of rational curves, but now the value of the quadratic form can be
different on different rational curves.

Nevertheless, it is clear that for $\alpha\in\mathrm{Pos}(X)$ to check that it
lies in the Kähler cone we only need to check the positivity of $q(\alpha,C)$
on rational curves $C$ with negative square. Indeed, as Beauville–Bogomolov
form has signature $(1,b_{2}-3)$ on $(1,1)$-classes we get that the orthogonal
hyperplane to a curve with non-negative square does not intersect the positive
cone.

### MBM classes.

A rational curve $C$ is called extremal if its cohomological class $[C]$
cannot be written as $[C]=Z_{1}+Z_{2},$ where $Z_{1}$ and $Z_{2}$ are two non-
proportional effective cycles in $\mathrm{NS}(X).$ By Theorem 2.16, the walls
of the Kähler cone are given by the orthogonal hyperplanes to the extremal
rational curves. It turns out that the chamber decomposition of the positive
cone is given by the orthogonal hyperplane to the class with negative
Beauville–Bogomolov square represented by a rational curve on some deformation
of $X.$ We recall some definitions and results about these classes.

######  Definition 2.17 ([21, Definition 1.5]).

Let $X$ be an IHSM. An automorphism $g$ of $H^{2}(X,\mathbb{Z})$ is called a
monodromy operator, if there exists a family

$\pi:\mathcal{X}\to B$

of IHSM such that $\psi:X\xrightarrow{\sim}\mathcal{X}_{b_{0}}$ for some
$b_{0}\in B$ and $g$ belongs to the image of $\pi_{1}(B,b_{0})$ under the
monodromy representation. In other words, the automorphism

$g\in\mathrm{Aut}(H^{2}(X,\mathbb{Z}))$

is a monodromy operator if there is a loop $\gamma:[0,1]\to B$ such that
$\gamma(0)=\gamma(1)=b_{0}$ and the parallel transport in the local system
$R^{2}\pi_{*}\mathbb{Z}$ along $\gamma$ induces the homomorphism

$\psi_{*}\circ\pi\circ\psi^{*}:H^{2}(X,\mathbb{Z})\to H^{2}(X,\mathbb{Z})$

which coincides with $g.$

The monodromy group $\mathrm{Mon}(X)$ is a subgroup of
$\mathrm{Aut}(H^{2}(X,\mathbb{Z}))$ generated by all monodromy operators. By
[27, Theorem 7.2] the subgroup $\mathrm{Mon}(X)$ coincides with the natural
image of the mapping class group $\Gamma$ in
$\mathrm{Aut}(H^{2}(X,\mathbb{Z})).$

######  Definition 2.18 ([1, Definition 1.13]).

A negative rational class

$\alpha\in H^{1,1}(X,I)\cap H^{2}(X,\mathbb{Q})$

on $(X,I)$ is called _MBM_ if for some $\gamma\in\mathrm{Mon}(X)$ the subspace
$\gamma(\alpha)^{\perp}\subset H^{1,1}(X,I)$ contains a wall of the Kähler
cone of a birational model $(X,I^{\prime})$ of $(X,I).$

By Remark 2.4 it makes sense to speak of $(1,1)$-classes represented by
curves.

######  Proposition 2.19 ([1, Corollary 5.14]).

The class $\alpha\in H^{1,1}(X,I)\cap H^{2}(X,\mathbb{Q})$ is an MBM class if
and only if it is represented, up to rescaling, by a rational curve on some
deformation of $X.$

######  Remark 2.20.

By [1, proof of Theorem 5.10] the equivalent definition of an MBM class is the
following: a negative class $\alpha\in H^{1,1}(X,I)\cap H^{2}(X,\mathbb{Z})$
is MBM if and only if it is represented, up to rescaling, by an extremal
rational curve on some deformation of $X.$ Note that if $\alpha$ is
represented by a negative rational curve, then it is represented by an
extremal rational curve on a deformation $(X,I^{\prime})$ of $(X,I),$ where

$\mathrm{Pic}(X,I^{\prime})=\mathbb{Z}\alpha.$

######  Proposition 2.21 ([1, Corollary 5.13]).

A negative class $\alpha\in H^{2}(X,\mathbb{Z})$ is MBM or not simultaneously
in all complex structures, where it is of type (1,1).

######  Remark 2.22.

By definition an MBM class $\alpha$ on an IHSM $X$ is represented by a curve
on a sufficiently general deformation $X^{\prime}$ of $X.$ This means that it
must be represented by a possibly reducible curve on all deformations.

######  Definition 2.23.

Let $X$ be an IHSM and let us fix the Kähler class $\omega.$ Let $U\subset X$
be an irreducible uniruled subvariety. A rational curve $C\subset U$ is called
minimal if $\int_{C}\omega$ is minimal among all curves in $U$ passing through
a general point of $U.$ A rational curve $C$ is called MBM if it is minimal
and its cohomology class $[C]\in H^{2}(X,\mathbb{Z})$ has negative
Beauville–Bogomolov square.

######  Definition 2.24.

Let $X$ be an IHSM. Let $C$ be an MBM curve and let $B$ be the space of all
deformations of $X.$ We call the MBM locus of $C$ the union of all curves
parametrized by $B.$

######  Definition 2.25.

Let $X$ be an IHSM. Let $\alpha\in H^{2}(X,\mathbb{Z})$ be an MBM class. Then
we call the MBM locus of the class $\alpha$ the union of all MBM loci of MBM
curves $C$ such that $[C]$ is proportional to $\alpha.$

Now we recall the theorem which states that MBM loci on the deformation
equivalent IHSM are real analytic isomorphic.

######  Theorem 2.26 ([3, Theorem 1.8]).

Let $X$ be an IHSM of $\mathrm{K3}$ or Kummer type and let $\alpha\in
H^{2}(X,\mathbb{Q})$ be MBM and $T$ be the MBM locus on $X$ of the class
$\alpha.$ Let $X^{\prime}$ be a deformation of $X$ in $\mathrm{Teich}$ such
that $\alpha$ is $(1,1)$ and is the class of a minimal rational curve on both
$X$ and $X^{\prime}.$ Let $T^{\prime}$ be an MBM locus on $X^{\prime}$ of
$\alpha.$ Then $T$ and $T^{\prime}$ are real analytic isomorphic.

Now we recall the definition of the period map and the theorem about chamber
decomposition of positive cone.

######  Definition 2.27.

Let $X$ be an IHSM. Denote by Per the map

$\mathrm{Per}:\mathrm{Teich}\to\mathbb{P}H^{2}(X,\mathbb{C})$

such that $\text{Per}(I)=H^{2,0}(X,I)\in\mathbb{P}H^{2}(X,\mathbb{C})$ for a
complex structure $I\in\text{Teich}.$ The image of Per is

$\mathbb{P}\text{er}=\\{l\in\mathbb{P}H^{2}(X,\mathbb{C})\mid
q(l)=0,\;q(l+\bar{l})>0\\}.$

######  Definition 2.28.

Let $\alpha$ be an MBM class. The space $\text{Teich}_{\alpha}$ consists of
complex structures $I$ such that $\alpha$ is of type (1,1) with respect to
$I.$ We have

$\text{Teich}_{\alpha}=\text{Per}^{-1}(\alpha^{\perp}).$

######  Theorem 2.29 ([1, Theorem 1.19], [20, Theorem 1.4]).

Let $X$ be an IHSM and $I$ be its complex structure. Then the points in
$\textup{Per}^{-1}\left(\textup{Per}(I)\right)$ are in one-to-one
correspondence with the Kähler chambers, i.e. the connected components of

$\mathcal{BK}_{X}=\mathrm{Pos}(X)\setminus\bigcup_{z\in\textup{MBM
classes}}z^{\perp}.$

This means that the connected components (or chambers) of $\mathrm{Pos}(X)$
are Kähler (or ample) cones of $X$ and its birational models, as well as their
monodromy images (see [2] and other papers of the authors). Here we use the
following theorem:

######  Theorem 2.30 ([20, Theorem 1.3]).

Let $X$ and $Y$ be IHSM which are deformation equivalent. Then they are
bimeromorphic if and only if there is a parallel transport operator
$f:H^{2}(X,\mathbb{Z})\to H^{2}(Y,\mathbb{Z})$ which is an isomorphism of
integral Hodge structures.

So our main goal is to find numerical characterization for these MBM classes
for Beauville examples. For more details about the Teichmüller space see [2,
Section 3].

######  Remark 2.31.

Let $X$ be a $2n$-dimensional IHSM. We call it a $\mathrm{K3}$ type manifold
if $X$ is deformation equivalent to $\mathrm{Hilb}^{n}(S)$ for some
$\mathrm{K3}$ surface $S.$ We call it a Kummer type manifold if $X$ is
deformation equivalent to $\mathrm{Kum}^{n}(A)$ for some complex two-
dimensional torus $A.$

### Torelli theorem.

For IHSM there is a generalization of Global Torelli Theorem for $\mathrm{K3}$
surfaces, namely

######  Theorem 2.32 ([27, Theorem 1.18]).

For each connected component of the birational Teichmüller space
$\mathrm{Teich}_{b}^{0},$ the period map
$\textup{Per}:\textup{Teich}_{b}^{0}\to\mathbb{P}\textup{er}$ is a
diffeomorphism.

######  Corollary 2.33.

Let $(X,I)$ be an IHSM with a complex structure $I.$ Consider the lattice
$L=H^{2}(X,\mathbb{Z})$ equipped with the Beauville–Bogomolov form. Suppose
that

$\alpha\in L\cap H^{1,1}(X,I)$

is a negative vector, i.e. $q(\alpha)<0,$ and $\alpha\in L^{\prime}\subset L,$
where $L^{\prime}$ is a primitive sublattice of $L$ and
$(L^{\prime\perp})_{\mathbb{R}}$ contains a positive $2$-plane. Then there is
a deformation $(X,I^{\prime})$ of $(X,I)$ in $\mathrm{Teich}_{\alpha}$ such
that $\mathrm{Pic}(X,I^{\prime})=L^{\prime}.$

###### Proof.

Let $M$ be a positive $2$-plane in $L^{\prime\perp}.$ It is not hard to see
that we can find a vector $l\in M\otimes\mathbb{C}$ such that $q(l)=0$ and
$q(l+\bar{l})>0.$ By Theorem 2.32 there is a complex structure
$I^{\prime}\in\mathrm{Teich}_{\alpha}$ such that

$\mathrm{Per}(I^{\prime})=l\in\mathbb{P}H^{2}(X,\mathbb{C}).$

If such an element $l$ is generic in $M\otimes\mathbb{C},$ then

$L\cap H^{1,1}(X,I^{\prime})=L^{\prime},$

but as $\mathrm{Pic}(X,I^{\prime})=L\cap H^{1,1}(X,I^{\prime}),$ we get the
desired.

∎

######  Corollary 2.34.

Let $(X,I)$ be a $\mathrm{K3}$ (respectively, Kummer) type manifold and let

$L=H^{2}(X,\mathbb{Z})$

be the lattice equipped with the Beauville–Bogomolov form. Suppose that there
is a primitive $\alpha\in L$ such that $\alpha\in H^{1,1}(X,I)$ and
$q(\alpha)<0.$ Then the manifold $X$ can be deformed to $Y$ in
$\textup{Teich}_{\alpha}$ such that $Y=\mathrm{Hilb}^{n}(S)$ (respectively,
$Y=\mathrm{Kum}^{n}(A)$), where $S$ is a $\mathrm{K3}$ surface with cyclic
Picard group (respectively, $A$ is a complex two-dimensional torus with cyclic
Néron-Severi group).

###### Proof.

First of all, note that the lattice $\langle\alpha,e\rangle,$ where $e$ is a
class of half of the exceptional divisor, is not positive definite. Thus,
applying Corollary 2.33 to the lattice $\langle\alpha,e\rangle$ we obtain a
deformation $Y^{\prime}$ of $X,$ such that

$\mathrm{Pic}(Y^{\prime})=\langle\alpha,e\rangle.$

Such a manifold $Y^{\prime}$ has the same period point in
$\mathbb{P}\mathrm{er}$ as the Hilbert scheme of points on a $\mathrm{K3}$
surface (respectively, the Kummer variety). Indeed, the period points of
Hilbert schemes (respectively, Kummer varieties) in $\mathbb{P}\mathrm{er}$
form the union of the orthogonal hyperplanes to $e$ and to its monodromy
images. Hence $Y^{\prime}$ is unseparable from Hilbert scheme
$Y=\mathrm{Hilb}^{n}(S)$ for some $\mathrm{K3}$ surface $S$ (respectively,
unseparable from Kummer variety $Y=\mathrm{Kum}^{n}(A)$ for some complex two-
dimensional torus $A$) in the Teichmüller space. In other words, we can deform
$Y^{\prime}$ to $Y$ which is the Hilbert scheme of points on a $\mathrm{K3}$
surface (respectively, the Kummer variety) such that $Y$ and $Y^{\prime}$ have
the same Picard group.

∎

### Strategy of the proof of Theorems 1.5 and 1.7.

Let us consider the Teichmüller space of $\mathrm{K3}$ or Kummer type variety.
Now we briefly explain our strategy for proof of Theorems 1.5 and 1.7.

$\text{Teich}_{\alpha}$$\text{Teich}_{e}$$\text{Teich}_{\gamma(e)}$$X$$Y$$Z$Teich
Figure 1. Teichmüller space of IHSM $X.$

Let $X$ be an IHSM of $\mathrm{K3}$ (respectively, Kummer) type. Let

$\alpha\in H^{1,1}(X,I)\cap H^{2}(X,\mathbb{Q})$

be a class of negative Beauville–Bogomolov square and $e\in
H^{2}(X,\mathbb{Z})$ be the class of half of the exceptional divisor. Figure 1
shows the Teichmüller space of $X.$ The points correspond to complex
structures in $\text{Teich}.$ The point $X$ corresponds to the complex
structure of the original $X.$

In the first step of the proof by Corollary 2.34 we deform $X$ to
$Y=\mathrm{Hilb}^{n}(S^{\prime})$ for a $\mathrm{K3}$ surface $S^{\prime}$ in
the $\mathrm{K3}$ case or to $Y=\mathrm{Kum}^{n}(A^{\prime})$ for a complex
two-dimensional torus $A^{\prime}$ in the Kummer case. In the picture the
manifold $Y$ corresponds to some point which lies in the intersection of
$\text{Teich}_{\alpha}$ and $\text{Teich}_{e}.$

A general point of $\text{Teich}_{\alpha}\cap\text{Teich}_{e}$ does not lie on
other $\text{Teich}_{\beta}$ for $\beta\in H^{2}(X,\mathbb{Z}),$ thus, we have
$\text{Pic}(Y)=\langle\alpha,e\rangle.$ Hence we get that
$\mathrm{Pic}(S^{\prime})$ (respectively, $\mathrm{Pic}(A^{\prime})$)
generated by the unique element.

In the second step of the proof we deform $Y$ to $Z$ such that $Z$ lies in the
intersection $\text{Teich}_{\alpha}\cap\text{Teich}_{\gamma(e)}$ and does not
lie on other $\mathrm{Teich}_{\beta}$ for $\beta\in H^{2}(X,\mathbb{Z}),$ and
the element $\gamma\in\text{Mon}(X)$ is chosen with certain restrictions which
will be explained below. The manifold $Z$ is diffeomorphic to the variety
$\gamma^{-1}(Z)$ with Picard group
$\mathrm{Pic}(\gamma^{-1}(Z))=\langle\gamma^{-1}(\alpha),e\rangle.$ By
Corollary 2.32 there is the Hilbert scheme of points on a $\mathrm{K3}$
surface (respectively, Kummer variety) with the same Picard lattice as
$\gamma^{-1}(Z)$ has. We will do all calculations on the Hilbert scheme on
points on $\mathrm{K3}$ surface (respectively, Kummer variety) as MBM property
is invariant under deformations.

Then we find an explicit MBM curve in the class $\gamma^{-1}(\alpha)$ on $Z$
if a good numerical property is satisfied for $\alpha,$ and show that $Z$ does
not carry any non-exceptional curve otherwise. Let us remark that if there is
no curves on $Z$ in the class $\alpha,$ then $\alpha$ is not MBM by Remark
2.22.

##  3\. Lattices of $\mathrm{K3}$ and Kummer types manifolds

In this section we collect some basic facts about lattices.

######  Definition 3.1.

Let $L$ be a lattice and $L^{*}$ be its dual lattice, i.e.

$L^{*}=\\{l^{*}\in L\otimes\mathbb{Q}\mid(l^{*},l)\in\mathbb{Z}\;\text{for
all}\;l\in L\\}.$

Then the quotient group $L^{*}/L$ is called the discriminant group. For an
element $l\in L$ denote by $d(l)$ the divisibility of $l,$ i.e. the positive
integer number such that

$(l,L)=d(l)\mathbb{Z}.$

For $l\in L$ denote by $\delta(l)$ the image of $l/d(l)$ in the discriminant
group $L^{*}/L.$

######  Definition 3.2.

Let $L$ be a lattice. If $\alpha\in L$ is a primitive element then the element

$\widehat{\alpha}=\frac{\alpha}{d(\alpha)}\in L^{*}$

is called the dual element to $\alpha.$

######  Example 3.3.

The lattice of the second cohomology group for $\mathrm{K3}$ type manifolds of
dimension $2n$ is $L=U^{\oplus 3}\oplus E(-8)^{\oplus
2}\oplus\langle-2(n-1)\rangle.$ Thus, we have

$L^{*}/L\simeq\mathbb{Z}_{2(n-1)}.$

######  Example 3.4.

Similarly, the lattice of the second cohomology group for Kummer type
manifolds of dimension $2n$ is $L=U^{\oplus 3}\oplus\langle-2(n+1)\rangle.$
Thus, we have

$L^{*}/L\simeq\mathbb{Z}_{2(n+1)}.$

Recall that the lattice $(L,q)$ of the second cohomology of IHSM with
Beauville–Bogomolov form is the indefinite lattice of signature $(3,b_{2}-3),$
where $b_{2}$ is the second Betti number.

Denote by $O(L^{*}/L)$ the subgroup of $\mathrm{Aut}(L^{*}/L)$ consisting of
multiplication by all elements of $t\in L^{*}/L$ (viewed as
$\mathbb{Z}/2k\mathbb{Z}$ for natural number $k$) such that $t^{2}=1.$ Let
$\mathrm{Gr}^{+}(L)$ be the Grassmannian parametrizing maximal positive
definite subspaces in $L_{\mathbb{R}}$ and let
$\mathrm{Gr}^{+,\mathrm{or}}(L)$ be the oriented Grassmannian parametrizing
oriented maximal positive definite subspaces in $L_{\mathbb{R}}.$ Note that
$\mathrm{Gr}^{+,\mathrm{or}}(L)$ is a double cover of $\mathrm{Gr}^{+}(L).$
For the Grassmannian $\mathrm{Gr}^{+}(L)$ we have the following proposition.

######  Proposition 3.5.

Let $(L,q)$ be an indefinite lattice. Then $\mathrm{Gr}^{+}(L),$ parametrizing
maximal positive definite subspaces in $L_{\mathbb{R}},$ is contractible.

###### Proof.

Let $L_{\mathbb{R}}=V_{+}\oplus V_{-}$ be a decomposition of the vector space
$L_{\mathbb{R}}$ in the sum of vector subspaces such that $q|_{V_{+}}$ is
positive definite and $q|_{V_{-}}$ is negative definite. Assume that
$V_{+}\perp V_{-}$ with respect to $q.$ Let $U\subset L_{\mathbb{R}}$ be a
maximal positive subspace. Then $U$ can be seen as a graph of some linear map
$V_{+}\to V_{-}.$ Indeed, the map $f:U\to V_{+}\times V_{-}$ can be defined as

$U\ni u\mapsto(\mathrm{pr}_{+}(u),\mathrm{pr}_{-}(u)),$

where $\mathrm{pr}_{+}$ is a projection to the subspace $V_{+}$ and
$\mathrm{pr}_{-}$ is a projection to the subspace $V_{-}.$ Obviously, the
image is isomorphic to $V_{+}$ because the kernel of the composition
$\mathrm{pr}_{+}\circ f$ is zero since $U\cap V_{-}=0$ and $\dim U=\dim
V_{+}.$

So let $U$ be the graph of the linear map $\phi.$ Then $U$ is positive
definite if and only if for any non-zero $v\in V_{+}$ we get

(3.1) $q(v+\phi(v))>0.$

As 3.1 holds for any $t\phi,$ where $t\in[0,1],$ we obtain the contraction of
$\mathrm{Gr}^{+}(L).$

∎

Therefore, since $L$ is indefinite, by Proposition 3.5 we get that
$\mathrm{Gr}^{+}(L)$ is contractible and, thus, the Grassmannian
$\mathrm{Gr}^{+,\mathrm{or}}(L)$ has two connected components. Denote by
$O^{+}(L)$ the subgroup of $O(L)$ acting trivially on
$\pi_{0}\left(\mathrm{Gr}^{+,\mathrm{or}}(L)\right)$ and by $SO^{+}(L)$ the
subgroup of $O^{+}(L)$ which consists of the element with determinant equal to
$1.$ Let

$\widetilde{O(L)}\,=\,\\{g\in O(L)\mid g(l^{*})\equiv l^{*}\mod
L\quad\text{for all}\;l^{*}\in L^{*}\\}$

and let $\widetilde{SO^{+}(L)}$ be the subgroup of $\widetilde{O(L)}$ which
consists of the element with determinant equal to $1$ and acting trivially on
$\pi_{0}\left(\mathrm{Gr}^{+,\mathrm{or}}(L)\right).$

Now let us recall a basic theorem about lattices, which we shall use next.

######  Theorem 3.6 ([14, Lemma 3.5] and [24, Lemma 2.6]).

Let $L$ be a lattice containing two orthogonal isotropic planes. Then the
$\widetilde{SO^{+}(L)}$-orbit of a primitive vector $l\in L$ is uniquely
determined by two invariants: $q(l)$ and $\delta(l).$

######  Lemma 3.7.

Let $L=H^{2}(X,\mathbb{Z})$ be a lattice of second cohomologies with
Beauville–Bogomolov form, where $X$ is $\mathrm{K3}$ or Kummer type manifolds.
Then the monodromy orbit of a primitive vector $l\in L$ is uniquely determined
by two invariants: $q(l)$ and $\pm\delta(l).$

###### Proof.

First of all, let us consider $\mathrm{K3}$ type case. Recall that by Example
3.3 the lattice of the second cohomology is

$L=U^{\oplus 3}\oplus E(-8)^{\oplus 2}\oplus\langle-2(n-1)\rangle.$

Consider

$\pi:O^{+}(L)\to O(L^{*}/L),$

Then, according to [20, Lemma 9.2], the group $\mathrm{Mon}(X)$ is equal to
$\pi^{-1}(\\{1,-1\\}).$ Let $m,l\in L$ lie in the same monodromy orbit. This
means that there is $\gamma\in\mathrm{Mon}(X)$ such that $m=\gamma(l).$
Therefore, by the definition of $\gamma$ we get that $q(l)=q(\gamma(l))$ and
$\delta(l)=\pm\delta(\gamma(l)).$

In the opposite direction, if there are primitive elements $l$ and $m$ which
satisfy

$q(m)=q(l)\quad\text{and}\quad\delta(m)=\pm\delta(l),$

then if $\delta(m)=\delta(l),$ by Theorem 3.6 there is
$\gamma\in\widetilde{SO^{+}(L)}$ such that $m=\gamma(l)$ and we are done.
Otherwise, if $\delta(m)=-\delta(l),$ then
$\delta(m)=\delta\left(\mathrm{R}_{e}(l)\right)$ and by Theorem 3.6 there is
$\gamma\in\widetilde{SO^{+}(L)}$ such that $m=\gamma(\mathrm{R}_{e}(l))$ and
we are done.

Now let us consider the Kummer case. For generalized Kummer variety the
lattice is equal to $L=U^{\oplus 3}\oplus\langle-t\rangle.$ Let
$\pi:O^{+}(L)\to O(L^{*}/L).$ According to [23, Theorem 2.3] the monodromy
group consists of two types of elements, namely,

$\pi^{-1}(1)\cap
SO^{+}(L)\quad\text{and}\quad\pi^{-1}(-1)\cap(O^{+}(L)\setminus SO^{+}(L)).$

Let $m,l\in L$ lie in the same monodromy orbit. This means that there is
$\gamma\in\mathrm{Mon}(X)$ such that $m=\gamma(l).$ Again by the definition of
$\gamma$ it is obvious that

$q(l)=q(\gamma(l))\;\;\text{and}\;\;\delta(l)=\pm\delta(\gamma(l)).$

In the opposite direction, if there are primitive elements $l$ and $m$ which
satisfy

$q(m)=q(l)\quad\text{and}\quad\delta(m)=\pm\delta(l),$

then if $\delta(m)=\delta(l),$ by Theorem 3.6 there is
$\gamma\in\widetilde{SO^{+}(L)}$ such that $m=\gamma(l)$ and we are done.
Otherwise, if $\delta(m)=-\delta(l),$ then
$\delta(m)=\delta\left(\mathrm{R}_{e}(l)\right)$ and by Theorem 3.6 there is
$\gamma\in\widetilde{SO^{+}(L)}$ such that $m=\gamma(\mathrm{R}_{e}(l))$ and
we are done, because

$\gamma\mathrm{R}_{e}\in O^{+}(L)\setminus
SO^{+}(L)\quad\text{and}\quad\det\left(\gamma(\mathrm{R}_{e}(l))\right)=1.$

∎

##  4\. Description of nef cone for $\mathrm{K3}$ type manifolds

In this section we prove Theorem 1.5. Let $X$ is a $\mathrm{K3}$ type
manifold. Denote by $e\in H^{2}(X,\mathbb{Z})$ the class of half of the
exceptional divisor on a deformation of $X$ which is the Hilbert scheme of $n$
points on some $\mathrm{K3}$ surface. We identify $H^{2}(X,\mathbb{Z})$ and
$H^{2}(Y,\mathbb{Z})$ for any deformation $Y$ of $X$ in $\mathrm{Teich},$
since topologically they are the same. Half of the exceptional divisors on a
deformation of $X$ which is a Hilbert scheme of points on a $\mathrm{K3}$
surface is then $\gamma(e)$ for some $\gamma\in\mathrm{Mon}(X).$ By
$\hat{e}\,=\,\frac{e}{2(n-1)}\in H_{2}(X,\mathbb{Z})$ we denote the dual class
of $e$ and by $t$ the natural number $2(n-1)$ which is equal to $q(e).$ First
of all we need the following lemma which appears in [11, Lemma 2.1 and Remark
2.2].

######  Lemma 4.1.

Let $S$ be a $\mathrm{K3}$ surface. Let $C\subset S$ be an irreducible curve
of arithmetic genus $g$ with $\delta$ nodes. Assume that there is a linear
system $g^{1}_{k}$ on the normalization $\widetilde{C}$ of $C$ with
$1\leqslant k\leqslant n$ for some $n\in\mathbb{N}$ and all nodes are non-
neutral with respect to $g^{1}_{k}.$ Then there is a map
$f:C\to\mathrm{Hilb}^{n}(S)$ such that its image is a rational curve $R$ which
is numerically equivalent to $C-(g-\delta-1+k)\hat{e}$ in
$N_{1}(\mathrm{Hilb}^{n}(S),\mathbb{Z}).$

######  Remark 4.2.

Non-neutrality with respect to the linear system on the normalization
$\widetilde{C}$ of $C$ means that such a linear system is base point free and
the corresponding map $\widetilde{C}\to\mathbb{P}^{1}$ does not ramify over
the nodes.

###### Proof.

The lemma follows from the Riemann-Hurwitz formula. The map $f$ is induced by
a linear system $g^{1}_{k}$ on $\widetilde{C}$ and is given by

$\widetilde{C}\stackrel{{\scriptstyle
g^{1}_{k}}}{{\longrightarrow}}\mathbb{P}^{1}\subset\mathrm{Sym}^{k}(\widetilde{C})\to\mathrm{Sym}^{k}(C)\hookrightarrow\mathrm{Sym}^{k}(S)\hookrightarrow\mathrm{Sym}^{n}(S)$

The ramification points are exactly the intersection of the rational curve
$f(C)$ with the exceptional divisor. Let us write $R\equiv C-m\hat{e}$ with

$C\in N_{1}(S)\hookrightarrow N_{1}(\mathrm{Hilb}^{n}(S))$

and $m\geqslant 0.$ We have $q(e,\hat{e})\,=\,-1,$ thus, we get

$m\,=\,q(R,e)\,=\,\frac{1}{2}q(R,\Delta),$

where $\Delta\,=\,2e$ is a class of the diagonal in $\mathrm{Sym}^{n}(S).$ At
the same time, we get

$\frac{1}{2}q(R,\Delta)\,=\,\frac{1}{2}2(g-\delta-1+k)\,=\,(g-\delta-1+k),$

where the first equality is true by Riemann-Hurwitz formula.

∎

Now we are ready to prove Theorem 1.5.

###### Proof of Theorem 1.5..

Let $\alpha$ be a primitive element in $H^{2}(X,\mathbb{Z})$ such that
$q(\alpha)<0.$ First of all, assume that the class $\alpha\in
H^{2}(X,\mathbb{Z})$ has the same square and the image in the discriminant
group as $e$, then according to Lemma 3.7 they are in the same monodromy
orbit. The class $e$ is clearly an MBM class because it corresponds to the
general fibre of the exceptional divisor over the diagonal in
$\mathrm{Sym}^{n}(S).$ Therefore, the class $\alpha$ is also an MBM class.
Thus, by Theorem 2.32 there is a deformation of $X$ preserving $e,$ which is
isomorphic to $\mathrm{Hilb}^{n}(S)$ with

$\mathrm{Pic}(\mathrm{Hilb}^{n}(S))\,=\,\langle e\rangle,$

i.e. $S$ does not have curves. We have

$q(\widehat{e})=-\frac{1}{2(n-1)}=q(\widehat{\alpha}).$

Now assume that $\alpha$ and $e$ are not in the same monodromy orbit. We
divide the proof into several steps.

Step 1. _In this step we recall that there is a deformation $Y$ of $X$ within
$\textup{Teich}_{\alpha}$ with Picard lattice
$\mathrm{Pic}(Y)=\langle\alpha,e\rangle$ such that $Y$ is isomorphic to the
Hilbert scheme of $\mathrm{K3}$ surface._ This immediately follows from
Corollary 2.34. So $Y=\mathrm{Hilb}^{n}(S^{\prime})$ for a $\mathrm{K3}$
surface $S^{\prime}$ with

$\mathrm{Pic}(S^{\prime})=\langle
y\rangle=e^{\perp}\;\text{in}\;\langle\alpha,e\rangle.$

Step 2. _In this step we deform $Y$ to $Z$ in $\textup{Teich}_{\alpha}$ such
that $\mathrm{Pic}(Z)=\langle\alpha,\gamma^{-1}(e)\rangle$ and
$\gamma\in\mathrm{Mon}(X)$ is chosen to normalize $\gamma(\alpha)$ in a
certain way._ Let us prove the following lemma.

######  Lemma 4.3.

Let $Y$ be a Hilbert scheme of $n$ points $\mathrm{Hilb}^{n}(S^{\prime})$ on a
$\mathrm{K3}$ surface $S^{\prime}$ such that
$\mathrm{Pic}(Y)=\langle\alpha,e\rangle,$ where $\alpha=ay+ce$ for
$y\in\mathrm{Pic}(S^{\prime}).$ Then there is an element
$\gamma\in\textup{Mon}(Y),$ such that $\gamma(\alpha)=rx-be$ with primitive
$x\in\mathrm{Pic}(S^{\prime}),$ and with integer numbers $b\in[0,\frac{r}{2}]$
and $r=\gcd(a,t).$

###### Proof.

Let us consider the element $\alpha=ay+ce$ and let

$r\,=\,\gcd(a,t)\quad a\,=\,rm\quad\text{and}\quad t\,=\,rs.$

Note that as the lattice of the second cohomologies of a $\mathrm{K3}$ surface
is unimodular, every even number can be realized as a square of some primitive
element in the lattice, so we take some primitive $y^{\prime}\in
H^{2}(S^{\prime},\mathbb{Z})$ with $q(y^{\prime})\,=\,m^{2}q(y).$ Let us first
prove that the class $\alpha\,=\,ay+ce$ and
$\widetilde{\alpha}\,=\,ry^{\prime}+ce$ have the same squares and images of
$\delta$ (see Definition 3.1). The equality
$q(\alpha)\,=\,q(\widetilde{\alpha})$ is obvious, because

$q(\alpha)\,=\,a^{2}q(y)-c^{2}t$

and

$q(\widetilde{\alpha})\,=\,r^{2}q(y^{\prime})-c^{2}t\,=\,(rm)^{2}q(y)-c^{2}t\,=\,a^{2}q(y)-c^{2}t.$

The discriminant group is equal to $\mathbb{Z}/t\mathbb{Z}.$ This means that

$\delta(\alpha)\,=\,\frac{\alpha}{r}\mod L,$

because $d(\alpha)\,=\,\gcd(a,t)\,=\,r$ and thus,

$\delta(\alpha)\,=\,cs\mod\,t.$

For $\widetilde{\alpha}$ we get

$\delta(\widetilde{\alpha})\,=\,\frac{\widetilde{\alpha}}{r}\mod L,$

because $d(\widetilde{\alpha})\,=\,\gcd(r,t)\,=\,r$ and thus,

$\delta(\widetilde{\alpha})\,=\,cs\mod\,t.$

Therefore, according to Lemma 3.7 we obtain
$\widetilde{\gamma}\in\text{Mon}(Y)$ such that
$\widetilde{\gamma}(\alpha)\,=\,\widetilde{\alpha}.$

Now we prove that there is $\widehat{\gamma}\in\text{Mon}(Y)$ such that

$\widehat{\gamma}(\widetilde{\alpha})=rx-be$

with $0\leqslant b\leqslant\frac{t}{2}$ and a primitive $x\in
H^{1,1}(Y,\mathbb{Z})$ orthogonal to $e.$ This means that we can find an
element $\widehat{\alpha}\in H^{2}(Y,\mathbb{Z})$ such that

$\widetilde{\alpha}\,=\,ry^{\prime}+ce\quad\text{and}\quad{\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\alpha}$}}}}\,=\,rx+(c-lr)e$

for all $l\in\mathbb{Z}$ are in the same monodromy orbit. Indeed, they have
the same image in the discriminant group, as

$\delta(\widetilde{\alpha})\,=\,cs\mod\,t\;\;\text{and}\;\;\delta({\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\alpha}$}}}})\,=\,(c-lr)s\mod\,t,$

and their squares are the same if we choose $x\in H^{1,1}(Y,\mathbb{Z})$
correctly. Let us choose it. We get

$q(\widetilde{\alpha})\,=\,r^{2}q(y^{\prime})-c^{2}t\quad\text{and}\quad
q({\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\alpha}$}}}})\,=\,r^{2}q(x)-c^{2}t-l^{2}r^{2}t+2clrt.$

Thus,
$q(\widetilde{\alpha})\,=\,q({\mathchoice{\widetilde{\hbox{\set@color$\displaystyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\textstyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\scriptstyle\widetilde{\alpha}$}}}{\widetilde{\hbox{\set@color$\scriptscriptstyle\widetilde{\alpha}$}}}})$
if and only if

(4.1) $q(x)\,=\,q(y^{\prime})+tl^{2}-2cls,$

So we can choose $b^{\prime}=c-lr$ in the segment
$[-\frac{r}{2},\frac{r}{2}].$ Also we can choose primitive $x\in
H^{1,1}(Y,\mathbb{Z})$ satisfying (4.1). Moreover, by Lemma 3.7 we can choose
negative $b^{\prime}.$ Thus, there is $\gamma\in\text{Mon}(Y)$ such that
$\gamma(\alpha)=rx-be,$ where $r$ divides $t,$ an integer $b$ belongs to the
segment $[0,\frac{r}{2}]$ and $\gcd(r,b)=1,$ because $\gcd(a,c)=1.$

∎

So we get the desired element $\gamma\in\mathrm{Mon}(Y)$ such that we deform
$Y$ to $Z$ in $\textup{Teich}_{\alpha}$ with
$\mathrm{Pic}(Z)=\langle\alpha,\gamma^{-1}(e)\rangle$ and $\gamma(\alpha)$
satisfies the condition of Lemma 4.3. Note that in any monodromy orbit there
is only one element which satisfies the conditions of Lemma 4.3.

Step 3. _In this step we prove that if $\alpha$ is an MBM class, then
$q(x)\geqslant-2.$_ Indeed, if $q(x)<-2,$ then the $\mathrm{K3}$ surface $S$
does not have any curves. Therefore, by Remark 2.22 the class $\alpha$ is not
an MBM class, because the only rational curves on the Hilbert scheme are of
class $\hat{e}.$ If $q(x)\geqslant-2,$ we use the uniform representation of
monodromy orbit from Step 2, namely, for every $\alpha\in H^{2}(X,\mathbb{Q})$
there is $\gamma\in\text{Mon}(X)$ such that

$\gamma(\alpha)=rx-be,$

where $r$ divides $t,$ the integer number $b$ belongs to the segment
$[0,\frac{r}{2}]$ and $\gcd(r,b)=1.$ Thus, we deform $X$ to $Z$ such that
$\gamma(Z)$ is bimeromorphic to $\mathrm{Hilb}^{n}(S)$ with

$\mathrm{Pic}(S)=\langle x\rangle.$

Step 4. _In this step we prove that if $q(x)\geqslant-2$ then the class of
$\widehat{\gamma(\alpha)}\in H_{2}(Z,\mathbb{Q})$ on $Z$ is represented by a
smooth rational curve. Thus, by Remark 2.20 the class $\alpha$ is MBM._ Note
that

$\widehat{\gamma(\alpha)}=\frac{\gamma(\alpha)}{d(\gamma(\alpha))}=x-\frac{b}{r}e=x-\frac{bt}{r}\widehat{e}.$

Assume that $\gamma(Z)$ is bimeromorphic to $\mathrm{Hilb}^{n}(S)$ with
$\mathrm{Pic}(S)=\langle x\rangle$ and $q(x)\geqslant-2$ such that

(4.2) $\gamma(\alpha)=rx-
be,\;\;r\;\;\text{divides}\;\;t,\;\;b\in[0,\frac{r}{2}]\;\;\text{is integer
and}\;\;\gcd(r,b)=1.$

By Lemma 4.1 we need to prove that for a smooth curve $C$ of genus $g$ on $S$
in the class $x$ there is a linear system $g^{1}_{k}$ with

(4.3) $k=\frac{bt}{r}-\frac{q(x)}{2}=\frac{bt}{r}-g+1$

such that $k\leqslant n.$ The inequality is clear, because this means that

$\frac{bt}{r}-g+1\leqslant\frac{t}{2}+1=n$

or, equivalently, $\frac{b}{r}\leqslant\frac{1}{2}+\frac{g}{t}$ which is true
according to the fact that by our conditions (4.2) we have the inequality
$0\leqslant b\leqslant\frac{r}{2}.$

So we need to prove that the gonality of $C$ is less than or equal to $k$:
then by Lemma 4.1 the linear system $g^{1}_{k}$ gives a rational curve on $Z.$
Thus, by the inequality for the gonality in [13, Chapter 2, page 261] we have
to prove that

(4.4) $\frac{g}{2}+1\leqslant k.$

We need to prove that

$\frac{g}{2}+1\leqslant\frac{bt}{r}-g+1$

or, equivalently,

(4.5) $g\leqslant\frac{2bt}{3r}.$

As $q(\gamma(\alpha))<0,$ we have the inequality

(4.6) $g<\frac{b^{2}t}{2r^{2}}+1\leqslant\frac{t}{8}+1.$

Put $w=\frac{bt}{r}.$ We have $w\in[0,\frac{t}{2}]$ is an integer number. If
$w=0,$ then by (4.6) we obtain $g=0,$ thus, the inequality (4.4) is true. So
assume that $w\geqslant 1.$

If $t=2,$ then we have $w=1.$ Thus, we have $b=1$ and $\gamma(\alpha)=2x-e.$
Here we get that $q(x)=-2,$ because by (4.6) we have $q(x)\leqslant 0$ (and if
$q(x)=0,$ then $e$ and $\gamma(\alpha)$ would have belonged to the same
monodromy orbit). The class $\gamma(\alpha)$ corresponds to a smooth rational
curve which is the image of $g^{1}_{2}$ on a rational curve $C.$

In the case $t\,=\,4$ we get $w\leqslant 2.$ Thus, by (4.6) we obtain
$g\leqslant 1.$ If $w=1,$ then $b=1$ and we have $\gamma(\alpha)=4x-e$ with
$q(x)=-2,$ as $\gamma(\alpha)$ and $e$ are not in the same monodromy orbit. In
this case we obtain $k=2$ and the inequality (4.4) is true. If $w=2,$ then by
(4.2) we obtain $b=1$ and $r=2,$ so we have $\gamma(\alpha)=2x-e.$ If
$q(x)=-2,$ then $\gamma(\alpha)$ is a class of a rational curve because the
curve $C$ with $g=0$ has a linear system $g^{1}_{3}.$ If $q(x)=0,$ then
$\gamma(\alpha)$ is a class of a rational curve because the curve $C$ with
$g=1$ has a linear system $g^{1}_{2}.$

Now assume that $t\geqslant 6$ and $w\geqslant 2.$ To prove (4.5), by (4.6) it
is enough to prove that

$\frac{w^{2}}{2t}+1\leqslant\frac{2w}{3}.$

So we need to prove that

$3w^{2}-4tw+6t\leqslant 0,$

or, equivalently,

$w\in[\frac{2}{3}t-\sqrt{\frac{4}{9}t^{2}-2t},\frac{2}{3}t+\sqrt{\frac{4}{9}t^{2}-2t}].$

One can check that

$[2,\frac{t}{2}]\subset[\frac{2}{3}t-\sqrt{\frac{4}{9}t^{2}-2t},\frac{2}{3}t+\sqrt{\frac{4}{9}t^{2}-2t}].$

Now assume that $w=1.$ Then we have $g=0.$ Indeed, we obtain $b=1$ by (4.2),
and therefore, by (4.6) the genus $g\leqslant 1.$ However, the case $g=1$ is
impossible by our assumption, because otherwise $\gamma(\alpha)=tx-e$ and $e$
are in the same monodromy orbit. So $g=0$ and the class $\gamma(\alpha)$
corresponds to a smooth rational curve which is the image of $g^{1}_{2}$ on
$C.$

Thus we get that a linear system $g^{1}_{k}$ with required properties does
exist. Hence, it gives us a rational smooth curve on $Z.$

Step 5. _In this step we prove that $\alpha\in H^{2}(X,\mathbb{Z})$ is MBM if
and only if its rational multiple satisfies_ _(1) _or_ (2)_ _from the
statement of Theorem 1.5._ Indeed, if $\alpha$ and $e$ are in the same
monodromy orbit by Lemma 3.7 we obtain (1). Recall that an element $\alpha$ is
MBM if and only if the manifold $X$ can be deformed so that on the deformed
variety the class $\widehat{\alpha}\in H_{2}(X,\mathbb{Q})$ is represented by
an extremal rational curve. By Lemma 4.3 the element $\alpha$ lies in the same
monodromy orbit as the element

$\alpha^{\prime}=rx-b^{\prime}e,$

where $r$ divides $t,$ $b^{\prime}\in[0,r]$ and $\gcd(r,b)=1.$ Let $t=rs,$
where $s\in\mathbb{N}.$ Note that by Lemma 3.7 we have
$q(\alpha)=q(\alpha^{\prime})$ and
$\delta(\alpha)=\pm\delta(\alpha^{\prime}).$ By direct calculation we get that
$\delta(\alpha^{\prime})=-b^{\prime}s.$ Therefore, we obtain

$-\delta(\alpha)=-b^{\prime}s\in[0,n-1].$

As

$q(\widehat{\alpha})=q(x-\frac{b^{\prime}}{r}e)=2a-\frac{b^{\prime
2}t}{r^{2}}=2a-\frac{b^{\prime 2}s^{2}}{t}$

for $q(x)=2a,$ where $q(x)$ is even because the lattice of the second
cohomologies of $\mathrm{K3}$ surface is even. By Step $3$ and $4$ we get that
$\alpha$ is MBM if and only if $q(x)\geqslant-2.$ This means that

$-2\leqslant 2a<\frac{b^{2}}{2(n-1)}.$

∎

######  Corollary 4.4.

Under the conditions of Theorem 1.5 we have

$g=\frac{x^{2}}{2}+1\leqslant\lceil\frac{n+3}{4}\rceil-1.$

###### Proof.

From (4.6) we get $g\,<\,\frac{n+3}{4}.$ Thus,
$g\leqslant\lceil\frac{n+3}{4}\rceil-1$ as $g$ is less or equal than the
maximal natural number strictly less than $\frac{n+3}{4}.$

∎

######  Corollary 4.5 (of the proof).

Let $\alpha\in H^{2}(X,\mathbb{Z})$ and $\widehat{\alpha}\,=\,x-\frac{b}{t}e$
be its dual with $b\in[0,\frac{t}{2}]$ on some deformation $Y$ of $X$
isomorphic to $\mathrm{Hilb}^{n}(S)$ such that $S$ is a $\mathrm{K3}$ surface.
Then $\alpha$ is MBM if and only if $q(x)\geqslant-2$ and $q(\alpha)\,<\,0.$

######  Corollary 4.6 ([4, cf. Theorem 2]).

In the condition of Corollary 4.5 we have

$q(\widehat{\alpha})\geqslant-\frac{n+3}{2}.$

Moreover, the equality holds if and only if $\alpha$ is a class of the line in
the Lagrangian space $\mathrm{Sym}^{n}(C),$ where $C$ is a smooth rational
curve on $\mathrm{K3}$ surface. In this case the MBM locus of the class
$\alpha$ on $X$ is bimeromorphic to $\mathrm{Sym}^{n}(C).$

###### Proof.

We get

$q(\widehat{\alpha})\,=\,q(x)-\frac{b^{2}}{t}\geqslant-2-\frac{t}{4}\,=\,-\frac{n+3}{2}.$

For the second part we refer to [16, Example 4.11], as the equality holds if
and only if $q(x)\,=\,-2$ and $b\,=\,\frac{t}{2}.$ Actually, this follows from
Lemma 4.1. Indeed, we have

$\widehat{\alpha}=C-(n-1)\widehat{e}$

for some rational curve $C$ on a $\mathrm{K3}$ surface. Let $R$ is a rational
curve in the class $\widehat{\alpha}.$ This means that $C$ maps to the
rational curve $R$ by the map of degree $n$ and the intersection of $R$ with
the general fiber of the exceptional divisor of the Hilbert–Chow morphism is
equal to $n-1.$ Linear systems $g^{1}_{n}$ on the rational curve $C$ are
parametrized by $\mathbb{P}^{n}$ and this gives us the third part. Therefore,
by Theorem 2.26 in this extremal case for the $\mathrm{K3}$ type manifolds the
MBM locus is bimeromorphic to $\mathbb{P}^{n}.$

∎

A similar description of MBM classes for both $\mathrm{K3}$ and Kummer cases
was given in the paper [19]. However, the mentioned paper relies on the Brill-
Noether theory, whereas our approach is elementary.

Finally we prove Proposition 1.9.

###### Proof of Proposition 1.9.

According to Bayer-Macrì theory we get that the walls are the hyperplanes in
$\mathrm{Pos}(X)$ orthogonal to $a$ with $a^{2}\geqslant-2$ ($a^{2}$ in a
sense of Mukai scalar product). The vector $a\,=\,(u,\varkappa,s)$ belongs to

$H^{0}(S,\mathbb{Z})\oplus\mathrm{NS}(S)\oplus H^{4}(S,\mathbb{Z}).$

for a $\mathrm{K3}$ surface $S.$ To pass to our situation we need to project
this vector on the subspace $v^{\perp},$ i.e. to consider
$z\,=\,v^{2}a-(v,a)v.$ Obviously, this vector is orthogonal to $v.$ After some
simple calculations we have

$z\,=\,v^{2}\varkappa-(a,e)e,$

where without loss of generality we put $v\,=\,(1,0,1-n)$ and
$e\,=\,(1,0,n-1).$ So the walls in $\mathrm{Pos}(X)$ are hyperplanes
orthogonal to $z\,=\,v^{2}\varkappa-(a,e)e.$

By Theorem 1.5 and its proof $z$ is MBM if and only if we can deform $X$ to
$Z$ such that there is $\gamma\in\mathrm{Mon}(X)$ such that $\gamma(Z)$ is
bimeromorphic to $\mathrm{Hilb}^{n}(S)$ with a $\mathrm{K3}$ surface $S.$
Moreover, the following properties hold: the Picard group is

$\mathrm{Pic}(Z)=\langle z,\gamma^{-1}(e)\rangle$

with $\gamma(z)=tx-be,$ where $b\in[0,\frac{t}{2}]$ is an integer,
$q(x)\geqslant-2$ such that

(4.7) $q(z)=q(\gamma(z))$

and $\gamma\in\text{Mon}(Z).$ As $(a,e)=-s-\frac{tu}{2},$ we have
$b=\frac{tu}{2}-s,$ because by conditions of Theorem 1.2 we have

$0\leqslant\frac{tu}{2}-s=-s-\frac{tu}{2}+tu\leqslant\frac{t}{2}.$

From (4.7) we get

$t^{2}q(\varkappa)-\left(\frac{tu^{2}}{4}+s^{2}+tus\right)t\,=\,t^{2}q(x)-\left(\frac{tu^{2}}{4}+s^{2}-tus\right)t.$

So we obtain $q(x)=\varkappa^{2}-2us=a^{2}\geqslant-2.$ Thus, $z$ is an MBM
class.

In the opposite direction assume that $\alpha$ is an MBM class. Thus, there is
a deformation $Z$ of $X$ such that $\gamma(Z)$ is bimeromorphic to
$\mathrm{Hilb}^{n}(S)$ for some $\mathrm{K3}$ surface $S,$ where

$s\gamma(\alpha)=tx-be$

for some positive integer $s$ which divides $t.$ We have
$b\in[0,\frac{t}{2}].$ Thus, let us take

$a=\left(u,\varkappa,\frac{tu}{2}-b\right)$

for some integer number $u$ and
$\varkappa^{2}=x^{2}+2u\left(\frac{tu}{2}-b\right).$ So we get that $a$
satisfies the conditions of Theorem 1.2. Indeed,

$a^{2}=\varkappa^{2}-2u\left(\frac{tu}{2}-b\right)=x^{2}\geqslant-2$

and

$(a,v)=\frac{tu}{2}-\frac{tu}{2}+b=b$

Thus, we get $0\leqslant(a,v)\leqslant\frac{t}{2}.$ Moreover, we have that

$z=t\varkappa-(a,e)e=t\varkappa-be$

is a projection of $a$ on $v^{\perp}.$

∎

##  5\. Description of nef cone for Kummer type manifolds

In this section we prove Theorem 1.7. Let $X$ be a Kummer type manifold of
dimension $2n.$ Let $e\in H^{2}(X,\mathbb{Z})$ be the class of half of the
exceptional divisor on a deformation of $X$ which is the Kummer variety of
dimension $2n$ for some complex two-dimensional torus. We identify
$H^{2}(X,\mathbb{Z})$ and $H^{2}(Y,\mathbb{Z})$ for any deformation $Y$ of $X$
in $\mathrm{Teich},$ since topologically they are the same. Half of the
exceptional divisor on a deformation of $X$ which is Kummer varieties of
dimension $2n$ for some complex two-dimensional tori is then $\gamma(e),$
where $\gamma\in\mathrm{Mon}(X).$ By

$\hat{e}\,=\,\frac{e}{2(n+1)}\in H_{2}(X,\mathbb{Z})$

we denote its dual class and by $t$ we denote natural number $2(n+1)$ which is
equal to $q(e).$ We have an analogue of Lemma 4.1:

######  Lemma 5.1.

Let $A$ be a complex two-dimensional torus. Let $C\subset A$ be an irreducible
curve of arithmetic genus $g$ with $\delta$ nodes. Assume that there is a
linear system $g^{1}_{k}$ on the normalization $\widetilde{C}$ of $C$ with
$2\leqslant k\leqslant n+1$ and all nodes are non-neutral with respect to
$g^{1}_{k}.$ Then there is a map $f:C\to\mathrm{Kum}^{n}(A)$ such that its
image is a rational curve $R$ which is numerically equivalent to
$C-(g-\delta-1+k)\hat{e}$ in $\mathrm{N}_{1}(\mathrm{Kum}^{n}(A),\mathbb{Z}).$

###### Proof.

The proof is similar to the proof of Lemma 4.1. We only need to prove that one
can translate $C$ by some point $x\in A$ such that the summation function on
every element $x_{1}+x_{2}+...+x_{n+1}$ of the image of $f$ is equal to zero.
There are maps

$\widetilde{C}\xrightarrow{f}\mathbb{P}^{1}\xrightarrow{s}A,$

where $f$ is induced by the linear system $g^{1}_{k}$ and $s$ is a summation.
The function $s$ can map $\mathbb{P}^{1}$ either in $\mathbb{P}^{1},$ or in
the point. But there is no rational curve in $A.$ So
$s:\mathbb{P}^{1}\to\text{pt}.$ Translating the curve $C,$ we get
$\text{pt}\,=\,0.$

∎

Now we are ready to prove Theorem 1.7.

###### Proof of Theorem 1.7..

Let $\alpha$ be a primitive element in $H^{2}(X,\mathbb{Z})$ such that
$q(\alpha)<0.$ The proof of this theorem is similar to the proof of Theorem
1.5. If MBM classes $\alpha$ and $e$ are in the same monodromy orbit, then
there is a deformation of $X$ preserving $e$, which is isomorphic to
$\mathrm{Kum}^{n}(A)$ with $\mathrm{Pic}(\mathrm{Kum}^{n}(A))\,=\,\langle
e\rangle,$ i.e. $A$ does not have curves. Note that $e$ is MBM because it
corresponds to a fiber of the ruling over the diagonal on
$\mathrm{Sym}^{n+1}(A),$ which also lies on $\mathrm{Kum}^{n}(A).$ We have

$q(\widehat{e})=-\frac{1}{2(n+1)}=q(\widehat{\alpha}).$

So assume that $\alpha$ and $e$ are not in the same monodromy orbit.

Step 1. _In this step we recall that there is a deformation $Y$ of $X$ within
$\textup{Teich}_{\alpha}$ with Picard lattice
$\mathrm{Pic}(Y)=\langle\alpha,e\rangle$ such that $Y$ is a Kummer variety._
This immediately follows from Corollary 2.34. So
$Y=\mathrm{Kum}^{n}(A^{\prime})$ for a complex two-dimensional torus
$A^{\prime}$ with

$\mathrm{Pic}(A^{\prime})=\langle
y\rangle=e^{\perp}\;\text{in}\;\langle\alpha,e\rangle.$

Step 2. _In this step we deform $Y$ to $Z$ in $\textup{Teich}_{\alpha}$ such
that_

$\mathrm{Pic}(Z)=\langle\alpha,\gamma^{-1}(e)\rangle$

_and $\gamma\in\mathrm{Mon}(X)$ is chosen to normalize $\gamma(\alpha)$ in a
certain way._ So let us formulate the following lemma.

######  Lemma 5.2.

Let $Y$ be the Kummer variety $\mathrm{Kum}^{n}(A^{\prime})$ of a complex two-
dimensional torus such that $\mathrm{Pic}(Y)=\langle\alpha,e\rangle,$ where
$\alpha=ay+ce$ for $y\in\mathrm{Pic}(A^{\prime}).$ Then there is
$\gamma\in\textup{Mon}(Y),$ such that $\gamma(\alpha)=rx-be$ with primitive
$x\in\mathrm{Pic}(A^{\prime})$ and with integer numbers $b\in[1,\frac{r}{2}]$
and $r=\gcd(a,t).$

###### Proof.

The proof of this lemma is similar to the proof Lemma 4.3. However, in this
case we get $b\in[1,\frac{t}{2}],$ because if $b=0,$ then $\alpha$ lies in the
same monodromy orbit as an element of $H^{2}(A,\mathbb{Z})$ for some complex
two-dimensional torus $A.$ But smooth curves on $A$ have non-negative squares.
This contradicts with our assumptions that $\alpha$ is of negative square.

∎

So we get the desired element $\gamma\in\mathrm{Mon}(Y)$ such that we deform
$Y$ to $Z$ in $\textup{Teich}_{\alpha}$ with
$\mathrm{Pic}(Z)=\langle\alpha,\gamma^{-1}(e)\rangle$ and $\gamma(\alpha)$
satisfies the condition of Lemma 5.2. Note that in any monodromy orbit there
is only one element which satisfies the conditions of Lemma 5.2.

Step 3. _In this step we prove that if $\alpha$ is an MBM class, then
$q(x)\geqslant 0.$_ Indeed, if $q(x)<0,$ then the complex two-dimensional
torus $A$ does not have any curves. Therefore, by Remark 2.22 the class
$\alpha$ is not an MBM class, because the only rational curves on the Kummer
variety are of class $\hat{e}.$ If $q(x)\geqslant 0,$ we use the uniform
representation of monodromy orbit from Step 2, namely, for every $\alpha\in
H^{2}(X,\mathbb{Q})$ there is $\gamma\in\text{Mon}(X)$ such that

$\gamma(\alpha)=rx-be,$

where $r$ divides $t,$ the integer number $b$ belongs to the segment
$[1,\frac{r}{2}]$ and $\gcd(r,b)=1.$ Thus, we deform $X$ to $Z$ such that
$\gamma(Z)$ is bimeromorphic to $\mathrm{Kum}^{n}(S)$ with

$\mathrm{Pic}(A)=\langle x\rangle.$

Step 4. _In this step we prove that if $q(x)\geqslant 0,$ then the class of
$\widehat{\gamma(\alpha)}\in H_{2}(Z,\mathbb{Q})$ on $Z$ is represented by a
smooth rational curve. So by Remark 2.20 the class $\alpha$ is MBM._ Note that

$\widehat{\gamma(\alpha)}=\frac{\gamma(\alpha)}{d(\gamma(\alpha))}=x-\frac{b}{r}e=x-\frac{bt}{r}\widehat{e}.$

Assume that $\gamma(Z)$ is bimeromorphic to $\mathrm{Kum}^{n}(A)$ with
$\mathrm{Pic}(A)=\langle x\rangle$ and $q(x)\geqslant 0$ such that

(5.1) $\gamma(\alpha)=rx-
be,\;\;r\;\;\text{divides}\;\;t,\;\;b\in[1,\frac{r}{2}]\;\;\text{is integer
and}\;\;\gcd(r,b)=1.$

By Lemma 5.1 we need to prove that for a smooth curve $C$ of genus $g$ on $A$
in the class $x$ there is a linear system $g^{1}_{k}$ with

(5.2) $k=\frac{bt}{r}-\frac{q(x)}{2}=\frac{bt}{r}-g+1,$

such that $k\leqslant n+1.$ The inequality is clear, because this means that

$\frac{bt}{r}-g+1\leqslant\frac{t}{2}=n+1$

or, equivalently,

$\frac{b}{r}\leqslant\frac{1}{2}+\frac{g}{t}-\frac{1}{t}$

which is true because according to our conditions (5.1) we have $1\leqslant
b\leqslant\frac{r}{2}.$

So we need to prove that the gonality of $C$ is less than or equal to $k$:
then by Lemma 5.1 the linear system $g^{1}_{k}$ gives a rational curve on $Z.$
Thus, by by the inequality for the gonality in [13, Chapter 2, page 261] we
have to prove that

(5.3) $\frac{g}{2}+1\leqslant k.$

We need to prove that

$\frac{g}{2}+1\leqslant\frac{bt}{r}-g+1$

or, equivalently,

(5.4) $g\leqslant\frac{2bt}{3r}.$

As $q(\gamma(\alpha))<0,$ we have the inequality

(5.5) $g<\frac{b^{2}t}{2r^{2}}+1\leqslant\frac{t}{8}+1.$

Put $w=\frac{bt}{r}.$ We have $w\in[\frac{t}{r},\frac{t}{2}]$ is an integer
number. We have $t\geqslant 6.$ Note that the case $r=1$ is impossible,
because in this case we have $b=0,$ and therefore, $\gamma(\alpha)$ lies in
the same monodromy orbit as an elements in $H^{2}(A,\mathbb{Z}).$ Thus, we can
assume that $r\geqslant 2.$ To prove (5.4), by (5.5) it is enough to prove
that

$\frac{w^{2}}{2t}+1\leqslant\frac{2w}{3}.$

So we need to prove that

$3w^{2}-4tw+6t\leqslant 0,$

or, equivalently,

$w\in[\frac{2}{3}t-\sqrt{\frac{4}{9}t^{2}-2t},\frac{2}{3}t+\sqrt{\frac{4}{9}t^{2}-2t}].$

One can check that

$[\frac{t}{r},\frac{t}{2}]\subset[\frac{2}{3}t-\sqrt{\frac{4}{9}t^{2}-2t},\frac{2}{3}t+\sqrt{\frac{4}{9}t^{2}-2t}].$

Indeed, we need to prove that

$\frac{t}{r}\geqslant\frac{2}{3}t-\sqrt{\frac{4}{9}t^{2}-2t}.$

But for this it is enough to prove that

$2\leqslant\frac{t}{r}\left(\frac{4}{3}-\frac{1}{r}\right).$

But it is true, because we have $r\geqslant 2$ and $t\geqslant 6.$

Thus we get that a linear system $g^{1}_{k}$ with required properties does
exist. Hence, it gives us a smooth rational curve on $Z.$

Step 5. _In this step we prove that $\alpha\in H^{2}(X,\mathbb{Z})$ is MBM if
and only if its rational multiple satisfies_ _(1) _or_ (2)_ _from the
statement of Theorem 1.7._ Indeed, if $\alpha$ and $e$ are in the same
monodromy orbit by Lemma 3.7 we obtain (1). Recall that an element $\alpha$ is
MBM if and only if the manifold $X$ can be deformed so that on the deformed
variety the class $\widehat{\alpha}\in H_{2}(X,\mathbb{Q})$ is represented by
an extremal rational curve. By Lemma 5.2 the element $\alpha$ lies in the same
monodromy orbit as the element

$\alpha^{\prime}=rx-b^{\prime}e,$

where $r$ divides $t,$ $b^{\prime}\in[1,r]$ and $\gcd(r,b)=1.$ Let $t=rs,$
where $s\in\mathbb{N}.$ Note that by Lemma 3.7 we have
$q(\alpha)=q(\alpha^{\prime})$ and
$\delta(\alpha)=\pm\delta(\alpha^{\prime}).$ By direct calculation we get that
$\delta(\alpha^{\prime})=-b^{\prime}s.$ Therefore, we obtain

$-\delta(\alpha)=-b^{\prime}s\in[1,n+1].$

As

$q(\widehat{\alpha})=q(x-\frac{b^{\prime}}{r}e)=2a-\frac{b^{\prime
2}t}{r^{2}}=2a-\frac{b^{\prime 2}s^{2}}{t}$

for $q(x)=2a,$ where $q(x)$ is even because the lattice of the second
cohomologies of a complex two-dimension torus is even. By Step $3$ and $4$ we
get that $\alpha$ is MBM if and only if $q(x)\geqslant 0.$ This means that

$0\leqslant 2a<\frac{b^{2}}{2(n+1)}.$

∎

######  Corollary 5.3.

Under the conditions of Theorem 1.7 we have

$g=\frac{q(x)}{2}+1\leqslant\lceil\frac{n+5}{4}\rceil-1.$

###### Proof.

From (5.5) we get $g\,<\,\frac{n+5}{4}.$ Thus,
$g\leqslant\lceil\frac{n+5}{4}\rceil-1$ as $g$ is less or equal than the
maximal natural number strictly less than $\frac{n+5}{4}.$

∎

######  Corollary 5.4 (of the proof).

Let $\alpha\in H^{2}(X,\mathbb{Z})$ and $\widehat{\alpha}\,=\,x-\frac{b}{t}e$
be its dual with $b\in[1,\frac{t}{2}]$ on some deformation $Y$ of $X$
isomorphic to $\mathrm{Kum}^{n}(A)$ such that $A$ is a complex two-dimensional
torus. Then $\alpha$ is MBM if and only if $q(x)\geqslant 0$ and
$q(\alpha)\,<\,0.$

######  Corollary 5.5.

In the conditions of Corollary 5.4 we have
$q(\widehat{\alpha})\geqslant-\frac{n+1}{2}.$ Moreover, the equality holds if
and only if $\alpha$ is a class of the line in the Lagrangian space
$\mathbb{P}^{n}$, which is the fiber of the summation map
$\mathrm{Sym}^{n+1}(C)\to C$ over zero for an elliptic curve $C\subset A.$ In
this case the MBM locus of $\alpha$ on the variety $X$ is bimeromorphic to the
fiber of the summation map $\mathrm{Sym}^{n+1}(C)\to C$ over zero.

###### Proof.

We get

$q(\widehat{\alpha})\,=\,q(x)-\frac{b^{2}}{t}\geqslant-\frac{t}{4}\,=\,-\frac{n+1}{2}.$

In the inequality the equality holds if and only if $q(x)\,=\,0$ and
$b\,=\,\frac{t}{2}.$ Therefore, by [16, Example 6.3] we get that $\alpha$ is a
class of the line in the Lagrangian subspace. By Lemma 5.1 the extremal curves
on $\mathrm{Kum}^{n}(A)$ which correspond to the class $\alpha$ are the fiber
of the summation map $\mathrm{Sym}^{n+1}(C)\to C$ over zero for the elliptic
curve $C\subset A.$ Indeed, by Lemma 5.1 we have

$\widehat{\alpha}=C-(n+1)\widehat{e},$

where $C$ is an elliptic curve on a two-dimensional complex torus. Let $R$ be
a rational curve in the class $\widehat{\alpha}.$ This means that that $C$
maps to the rational curve $R$ by the map of degree $n+1$ and the intersection
of $R$ and the exceptional divisor on the Kummer variety is equal to $n+1.$
All linear systems on $C$ of degree $n+1$ which give the covering of a
rational curve on a Kummer variety are parametrized by the fiber of

$\mathrm{Sym}^{n+1}(C)\to C$

over zero. The fiber is isomorphic to $\mathbb{P}^{n}.$ Therefore, by Theorem
2.26 in this extremal case for the Kummer type variety $X$ the MBM locus is
bimeromorphic to the fiber of the summation map $\mathrm{Sym}^{n+1}(C)\to C$
over zero.

∎

Now we prove Proposition 1.10.

###### Proof of Proposition 1.10.

According to Yoshioka theory we get that the walls are the hyperplanes in
$\mathrm{Pos}(X)$ orthogonal to $a$ with $a^{2}\geqslant 0$ (the square of $a$
in the sense of the Mukai scalar product). The vector $a\,=\,(u,\varkappa,s)$
belongs to

$H^{0}(A,\mathbb{Z})\oplus NS(A)\oplus H^{4}(A,\mathbb{Z}),$

where $A$ is a complex two-dimensional torus. Turning to our situation we need
to project this vector on $v^{\perp},$ i.e. consider $z\,=\,v^{2}a-(v,a)v.$
Obviously, this vector is orthogonal to $v.$ By simple calculations we have

$z\,=\,v^{2}\varkappa-(a,e)e,$

where without loss of generality we put $v\,=\,(1,0,-1-n)$ and
$e\,=\,(1,0,n+1).$ So the walls in $\mathrm{Pos}(X)$ are hyperplanes
orthogonal to $z\,=\,v^{2}\varkappa-(a,e)e.$

By Theorem 1.7 and its proof $z$ is MBM if and only if we can deform $X$ to
$Z$ such that $\gamma(Z)$ is bimeromorphic to $\mathrm{Kum}^{n}(A)$ for some
complex two-dimensional torus $A.$ Moreover, the following properties hold:
the Picard group is

$\mathrm{Pic}(Z)=\langle z,\gamma^{-1}(e)\rangle$

with $\gamma(z)=tx-be,$ where $b\in[1,\frac{t}{2}]$ is an integer,
$q(x)\geqslant 0$ such that

(5.6) $q(z)=q(\gamma(z))$

and $\gamma\in\text{Mon}(Z).$ As $(a,e)=-s-\frac{tu}{2},$ we have
$b=\frac{tu}{2}-s,$ because by conditions of Theorem 1.3 we have

$0<\frac{tu}{2}-s=-s-\frac{tu}{2}+tu\leqslant\frac{t}{2}.$

From (5.6) we get

$t^{2}q(\varkappa)-\left(\frac{tu^{2}}{4}+s^{2}+tus\right)t\,=\,t^{2}q(x)-\left(\frac{tu^{2}}{4}+s^{2}-tus\right)t.$

So we obtain $q(x)=q(\varkappa)-2us=a^{2}\geqslant 0.$ Thus, $z$ is an MBM
class.

In the opposite direction assume that $\alpha$ is an MBM class. Thus, there is
a deformation $Z$ of $X$ such that $\gamma(Z)$ is bimeromorphic to
$\mathrm{Kum}^{n}(A)$ for some complex two-dimensional torus $A,$ where

$s\gamma(\alpha)=tx-be$

for some positive integer $s$ which divides $t.$ We have
$b\in[1,\frac{t}{2}].$ Thus, let us take

$a=\left(u,\varkappa,\frac{tu}{2}-b\right)$

for some integer number $u$ and
$q(\varkappa)=q(x)+2u\left(\frac{tu}{2}-b\right).$ So we get that $a$
satisfies the conditions of Theorem 1.3. Indeed,

$a^{2}=q(\varkappa)-2u\left(\frac{tu}{2}-b\right)=q(x)\geqslant 0$

and

$(a,v)=\frac{tu}{2}-\frac{tu}{2}+b=b$

Thus, we get $0<(a,v)\leqslant\frac{t}{2}.$ Moreover, we have that

$z=t\varkappa-(a,e)e=t\varkappa-be$

is a projection of $a$ on $v^{\perp}.$

∎

Let us consider $\mathrm{Kum}^{2}(A)$ and $\mathrm{Kum}^{3}(A)$ cases for a
complex two-dimensional torus $A.$ Note that by Theorem 2.26 the MBM loci on
deformation equivalent IHSM are real analytic isomorphic. So without loss of
generality we can study MBM classes and MBM loci on Kummer varieties and by
Theorem 1.7 we have the following tables of classification of MBM classes
represented by a curve $R$ and the MBM loci of a class $[R].$ Note that in
these cases the only possible value for $q(x)$ is $0$ due to $q(R)\,<\,0$ and
small enough value of $q(e).$

For $n\,=\,2$ we have

$[R]$ | $q(x)$ | $q(R)$ | MBM locus
---|---|---|---
$\frac{e}{6}$ | 0 | $-\frac{1}{6}$ |  $\mathbb{P}^{1}$–bundle over $A$
$x-\frac{e}{3}$ | 0 | $-\frac{2}{3}$ |  $\mathbb{P}^{1}$–bundle over $A$
$x-\frac{e}{2}$ | 0 | $-\frac{3}{2}$ | $\mathbb{P}^{2}$

In the first line the MBM locus is birational to a $\mathbb{P}^{1}$–bundle
over $A.$ Indeed, for the Hilbert scheme $\mathrm{Hilb}^{3}(A)$ the
exceptional divisor is a blow up of the big diagonal $2p_{1}+p_{2}$ on
$\mathrm{Sym}^{3}(A),$ where $p_{i}\in A.$ This divisor is birationally ruled
over $A\times A$ and its intersection with $\mathrm{Kum}^{2}(A)$ is ruled over
$A,$ since the variety which is defined by $2p_{1}+p_{2}=0$ is isomorphic to
$A.$ Note that $\frac{e}{6}$ and $x-\frac{e}{6}$ with $q(x)\,=\,0$ are in the
same monodromy orbit because they have the same squares and divisibilities. In
the second line the MBM locus is also birational to a $\mathbb{P}^{1}$–bundle
over $A$ because the corresponding rational curve on the Kummer variety arises
from the linear system of degree $2$ on an elliptic curve (which gives us
$\mathbb{P}^{1}$) and the third point on the complex two-dimensional torus
(which gives us $A$ in the locus). And in the third line the MBM locus is
birational to $\mathbb{P}^{2}$ because the rational curve $R$ on the Kummer
variety arises from the linear system of degree $3$ on an elliptic curve.

For $n\,=\,3$ we get

$[R]$ | $q(x)$ | $q(R)$ | MBM locus
---|---|---|---
$\frac{e}{8}$ | 0 | $-\frac{1}{8}$ |  $\mathbb{P}^{1}$–bundle over $A\times A$
$x-\frac{e}{4}$ | 0 | $-\frac{1}{2}$ |  $\mathbb{P}^{1}$–bundle over $\mathrm{Sym}^{2}(A)$
$x-\frac{3e}{8}$ | 0 | $-\frac{9}{8}$ |  $\mathbb{P}^{2}$–bundle over $A$
$x-\frac{e}{2}$ | 0 | $-2$ | $\mathbb{P}^{3}$

Again in the first line the MBM locus is birational to a
$\mathbb{P}^{1}$–bundle over $A\times A.$ Indeed, for the Hilbert scheme
$\mathrm{Hilb}^{4}(A)$ the exceptional divisor is a blow up of the big
diagonal $2p_{1}+p_{2}+p_{3}$ on $\mathrm{Sym}^{3}(A),$ where $p_{i}\in A.$
This divisor is birationally ruled over $A\times A\times A$ and its
intersection with $\mathrm{Kum}^{3}(A)$ is ruled over $A\times A,$ since the
variety which is defined by $2p_{1}+p_{2}+p_{3}=0$ is isomorphic to $A\times
A.$ The elements $\frac{e}{8}$ and $x-\frac{e}{8}$ with $q(x)\,=\,0$ have the
same squares and divisibilities. In the second line the MBM locus is
birational to a $\mathbb{P}^{1}$–bundle over $\mathrm{Sym}^{2}(A)$ because the
corresponding rational curve on the Kummer variety arises from the linear
system of degree $2$ on an elliptic curve (which gives us $\mathbb{P}^{1}$)
and the two points on the symmetric square of the complex two-dimensional
torus (which gives us $\mathrm{Sym}^{2}(A)$ in the locus). In the third line
the MBM locus is birational to $\mathbb{P}^{2}$ over $A$ because the
corresponding rational curve on the Kummer variety arises from the linear
system on elliptic curve of degree 3 (which gives us $\mathbb{P}^{2}$) and the
point on the complex two-dimensional torus (which gives us $A$ in the locus).
In the last line the MBM locus is birational to $\mathbb{P}^{3}$ because the
rational curve $R$ on Kummer variety arises from the linear system of degree
$4$ on an elliptic curve.

For $n\,=\,4$ we get the element $R\,=\,x-\frac{e}{2}$ with $q(x)\,=\,2$ and
$q(R)\,=\,-\frac{1}{2}.$ The MBM locus in this situation is
$\mathbb{P}^{2}$–bundle over $A\times A$ because the rational curve $R$ arises
from the linear system on genus 2 curve of degree 4 (which gives
$\mathbb{P}^{2}$), a point on the complex two-dimensional torus (which gives
$A$) and the Jacobian of this curve is a complex two-dimensional torus.
Obviously, for $n\,\geqslant 4$ we do have $R$ with positive $q(x).$

## References

  * [1] E. Amerik, M. Verbitsky, _Rational curves on hyperkähler manifolds_ , Int. Math. Res. Notices 23 (2015), 13009–13045.
  * [2] E. Amerik, M. Verbitsky, _MBM classes and contraction loci on low-dimensional hyperkähler manifolds of $\mathrm{K3}$ type_, arXiv:1907.13256 [math.AG].
  * [3] E. Amerik, M. Verbitsky, _Contraction centers in families of hyperkahler manifolds_ , Sel. Math. New Ser. 27 (2021), article number: 60.
  * [4] B. Bakker, _A classification of Lagrangian planes in holomorphic symplectic varieties_ , J. Inst. Math. Jussieu 16 (2017), no. 4, 859–877.
  * [5] A. Bayer, E. Macrì, _Projectivity and birational geometry of Bridgeland moduli spaces_ , J. Amer. Math. Soc. 27 (2014), 707–752.
  * [6] A. Bayer, E. Macrì, _MMP for moduli of sheaves on $\mathrm{K3}$s via wall-crossing: nef and movable cones, Lagrangian fibrations_, Invent. Math. 198 (2014), 505–590.
  * [7] A. Beauville, _Variétés Kähleriennes dont la première classe de Chern est nulle_ , J. Differential Geom. 18 (1983), 755–782.
  * [8] S. Boucksom, _Le cône kählérien d’une variété hyperkählérienne_ , C.R. Acad. Sci. Paris Ser. I Math. 333 (2001), 935–938.
  * [9] T. Bridgeland, _Stability conditions on $\mathrm{K3}$ surfaces_, Duke Math. J. 141 (2008), 241–291.
  * [10] M. Britze, _On the cohomology of generalized Kummer varieties_ , PhD thesis, Universität zu Köln (2002).
  * [11] C. Ciliberto, A. L. Knutsen _On k-gonal loci in Severi varieties on general $\mathrm{K3}$ surfaces and rational curves on hyperkähler manifolds_, J. Math. Pures Appl. 101 (2014), 473–494.
  * [12] A. Fujiki, _On the de Rham Cohomology Group of a Compact Kähler Symplectic Manifold_ , Adv. Stud. Pure Math. 10 (1987), 105–165.
  * [13] Ph. Griffiths, J. Harris, _Principles of Algebraic geometry_ , Wiley, New York, Pure and applied mathematics (1978).
  * [14] V. Gritsenko, K. Hulek, G. Sankaran, _Moduli of $\mathrm{K3}$ Surfaces and Irreducible Symplectic Manifolds_, Handbook of moduli, Volume I, Advanced Lectures in Mathematics (2013).
  * [15] M. Gross, D. Joyce, D. Huybrechts, _Calabi–Yau Manifolds and Related Geometries_ , Universitext. Springer-Verlag, Berlin, (2003).
  * [16] B. Hassett, Y. Tschinkel, _Intersection numbers of extremal rays on holomorphic symplectic varieties_ , Asian J. Math. 14 (2010), 303–322.
  * [17] D. Huybrechtsl, _Compact hyperkähler manifolds: basic results_ , Invent. Math. 135 (1999), 63-–113.
  * [18] D. Huybrechtsl, _The Kähler cone of a compact hyperkähler manifold_ , Math. Ann. 326 (2003), 499–513.
  * [19] A. L. Knutsen, G. Mongardi, M. Lelli-Chiesa _Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds_ , Trans. Amer. Math. Soc. 371 (2019), 1403–1438.
  * [20] E. Markman, _A survey of Torelli and monodromy results for holomorphic symplectic varieties_ , Complex and differential geometry, Springer-Verlag Berlin Heidelberg, (2011).
  * [21] E. Markman, _On the monodromy of moduli spaces of sheaves on $\mathrm{K3}$ surfaces_, J. Algebraic Geom. 17 (2008), 29–99.
  * [22] C. Meachan, Z. Zhang, _Birational Geometry of Singular Moduli Spaces of O’Grady Type_ , Adv. Math. 296 (2016), 210–267.
  * [23] G. Mongardi, _On the monodromy of irreducible symplectic manifolds_ , Alg. Geom. 3 (2016), 385–391.
  * [24] G. Mongardi, A. Rapagnetta, _Monodromy and birational geometry of O’Grady’s sixfolds_ , J. Math. Pures Appl. no.9, 146 (2021), 31–68.
  * [25] K. G. O’Grady, _Desingularized moduli spaces of sheaves on a $\mathrm{K3}$_, J. reine angew. Math. 512 (1999), 49–117.
  * [26] K. G. O’Grady, _A new six-dimensional irreducible symplectic variety_ , J. Algebraic Geom. 12 (2003), 435–505.
  * [27] M. Verbitsky, _A global Torelli theorem for hyperkahler manifolds_ , Duke Math. J. 162 (2013), 2929–2986.
  * [28] K. Yoshioka, _Bridgeland’s stability and the positive cone of the moduli spaces of stable objects on an abelian surface_ , Adv. Stud. Pure Math. 69 (2016), 473–537.
  * [29] K. Yoshioka, _Moduli spaces of stable sheaves on abelian surfaces_ , Math. Ann. 321 (2001), 817–884.