MBM classes and contraction loci on low-dimensional hyperkähler manifolds of
K3[n] type

Ekaterina Amerik111Partially supported by the Russian Academic Excellence
Project ’5-100’., Misha Verbitsky222Partially supported by the Russian
Academic Excellence Project ’5-100’, FAPERJ E-26/202.912/2018 and CNPq -
Process 313608/2017-2. Keywords: hyperkähler manifold, Kähler cone, birational
maps 2010 Mathematics Subject Classification: 53C26, 32G13

Abstract

An MBM locus on a hyperkähler manifold is the union of all deformations of a
minimal rational curve with negative self-intersection. MBM loci can be
equivalently defined as centers of bimeromorphic contractions. In [AV3], it
was shown that the MBM loci on deformation equivalent hyperkähler manifolds
are diffeomorphic. We determine the MBM loci on a hyperkähler manifold of
K3[n]-type for small $n$ using a deformation to a Hilbert scheme of a non-
algebraic K3 surface.

###### Contents

  1. 1 Introduction
  2. 2 Hyperkähler manifolds
    1. 2.1 Holomorphically symplectic manifolds
    2. 2.2 The Bogomolov-Beauville-Fujiki form
    3. 2.3 Extremal curves
    4. 2.4 MBM classes
  3. 3 Teichmüller spaces, Torelli theorem and monodromy
    1. 3.1 Teichmüller space
    2. 3.2 Global Torelli theorem
    3. 3.3 The space $\operatorname{\sf Teich}_{\alpha}^{{\operatorname{\sf min}}}$
    4. 3.4 Monodromy for manifolds of K3[n] type
  4. 4 Birational contractions and invariance by deformation
    1. 4.1 Birational contractions
    2. 4.2 Invariance by deformation
  5. 5 MBM loci for low-dimensional manifolds of $K3^{[n]}$ type
  6. 6 Monodromy computations

##  1 Introduction

In the last twenty years or so, a considerable effort has been made to
understand the Mori cone, or, dually, the ample (or Kähler in the non-
projective case) cone of irreducible holomorphic symplectic (IHS) manifolds.
For K3 surfaces it has been known for a long time that the extremal rays
correspond to $(-2)$-curves (that is, irreducible curves of square $-2$,
automatically smooth and rational), so that the ample (or Kähler) classes are
characterized, among positive classes, as those which are positive on all
$(-2)$-curves. In higher dimension, though, until recently the only general
result, due to Huybrechts and Boucksom, was that one could still test the
Kähler property on rational curves. However, no boundedness statement for such
curves was available. A boundedness statement (equivalent to the Morrison-
Kawamata cone conjecture) was obtained in [AV2].

In [HT1] Hassett and Tschinkel conjectured a description of the Mori cone for
projective IHS manifolds which are deformation equivalent to the Hilbert
square of a K3 surface. Such manifolds are also called holomorphic symplectic
fourfolds of K3[n] type. The conjecture by Hassett and Tschinkel (Conjecture
3.1 of [HT1]) stated that the extremal rays are generated by the classes with
some particular numerical properties. For the Hilbert square of a K3 surface
containing a smooth rational curve $C$, these numerics are either those of the
rational curves contracted by the Hilbert-Chow morphism, or of the rational
curves arising from $C$, or of the line in the projective plane which is the
symmetric square of such a $C$. Hassett and Tschinkel have subsequently proved
this conjecture in [HT2] (in the projective case).

For high-dimensional holomorphic symplectic manifolds of K3[n] type the
analogous initial conjecture by Hassett and Tschinkel (Conjecture 1.2 of [HT3]
turned out to be false. The description of the Mori cone was provided by Bayer
and Macri [BM] using advanced technology around Bridgeland’s stability
conditions.

Another conjecture by Hassett and Tschinkel (Conjecture 3.6 of [HT1], and
still a conjecture 24 in [HT2]) was concerned with the geometry of the loci
covered by the minimal rational curves according to the numerical properties
of the corresponding extremal rays (by Kawamata base-point-free theorem, one
can interpret those loci as exceptional sets of birational contractions). As
it is natural, one expects that such a locus is either a lagrangian plane or a
${\mathbb{P}}^{1}$-bundle over a K3 surface, the former covered by curves with
Beauville-Bogomolov square $-5/2$, the latter either $-1/2$ (the type of the
exceptional divisor of the Hilbert-Chow map) or $-2$ (the type of the divisor
of subschemes with some support on a $-2$-curve $C$ on the K3-surface). One
can also expect a similar description in higher dimension. Hassett and
Tschinkel have given lists of possible types of extremal rays and
corresponding contraction loci in [HT3] and then completed a few missing types
following Bayer and Macri in [HT4].

Bayer and Macri’s technology allows, in principle, to describe contraction
loci as fibrations in projective spaces over K3-related manifolds (up to
birational equivalence). Those manifolds arise as moduli spaces of sheaves on
a K3 surface in question, and the projective spaces are projectivisations of
Ext-groups between the sheaves. But this approach is not very explicit,
whereas at list in low dimension it seems possible and useful to describe the
extremal rays and the subvarieties covered by the corresponding curves “by
hand”.

The purpose of this note is to provide a particularly simple description of
possible extremal rays and contraction loci for holomorphic symplectic
manifolds of K3[n] type in low dimension, particularly four and six (i.e. for
manifolds which are deformation equivalent to the Hilbert square or the
Hilbert cube of a K3 surface). Rather than studying extremal rays, we work
with a slightly different notion of MBM classes. This notion was introduced in
[AV1] to put things in the deformation-invariant context, and indeed the
crucial point in our argument is the deformation to the Hilbert scheme of a
non-projective K3 surface.

The MBM classes are the classes which correspond to extremal rays of the Mori
cone on some deformation of $X$, or, equivalently, the classes of extremal
rays up to monodromy and birational equivalence (the name derives from
“monodromy birationally minimal”). From a uniruled subvariety $Z$ of an IHS
$X$ we obtain an MBM class as follows: let $C$ be a rational curve which is
minimal among the rational curves in $Z$ (in the sense that $C$ does not bend-
and-break). If its class is Beauville-Bogomolov negative, then it is MBM.

The parameter space $\operatorname{\sf Teich}_{\alpha}^{min}$ for complex
structures where an MBM class $\alpha$ is represented by a minimal rational
curve coincides with the whole space where $\alpha$ stays of type $(1,1)$ up
to inseparability issues. Such curves can always be contracted ([AV3,
Corollary 5.3]).

In [AV3] we proved some deformation-invariance statements about the
corresponding contraction loci (i.e. exceptional sets of morphisms contracting
the class $\alpha$). We have, in particular, shown that the diffeomorphism
type of such a contraction locus is constant over $\operatorname{\sf
Teich}_{\alpha}^{min}$. Moreover, so is the biholomorphism type of the fibers
of the rational quotient fibration of the exceptional locus, with a possible
exception of the points of $\operatorname{\sf Teich}_{\alpha}^{min}$
corresponding to the complex structures with maximal Picard number.

The idea now is as follows. Let $\alpha$ be a negative integral $(1,1)$-class
on a hyperkähler fourfold $X$ of K3[n] type. Using Markman’s description of
the monodromy group, we prove that $X$ can be deformed to a non-projective
manifold $X^{\prime}\cong\operatorname{Hilb}^{2}(S^{\prime})$ where
$S^{\prime}$ is a K3 surface with cyclic Picard group generated by a negative
class, and in such a way that $\alpha$ stays of type $(1,1)$. Since rational
curves on $X^{\prime}$ are easy to describe, we obtain the classification of
subvarieties which can be covered by rational curves in the class $\alpha$, by
deformation-invariance.

If $X$ is an IHS of K3[n] type of dimension six, it turns out that $X$ still
deforms to the Hilbert cube of a non-projective K3 surface $S$, but its Picard
group can now be generated by an element with negative or zero Beauville-
Bogomolov square. The geometry in this last case is also well-understood.
Namely, $S$ is an elliptic $K3$ surface with singular fibers which are nodal
rational curves, and there is no other curve at all. All rational curves (up
to deformation) on the symmetric cube of $S$ are obtained from linear systems
$g_{k}^{1},k\leqslant 3$ on the fibers,111As defined in [Har, §IV.5],
$g_{k}^{d}$ denotes a linear system of dimension $d$ and degree $k$ on a
curve. In particular, $g_{k}^{1}$ correspond to $k$-sheeted maps to
${\mathbb{P}}^{1}$. so again easy to describe. We obtain a table of five types
of MBM classes, their squares and the types of the corresponding contraction
loci, which agrees with the data by Hassett and Tschinkel (one may notice here
that the MBM class initially overlooked in [HT3] and exhibited in [HT4] while
working out Bayer and Macri’s formulae on this example, comes in our picture
from the Hilbert cube of a surface with zero intersection form on the Picard
group).

The same calculation works in fact for eightfolds (again bringing into the
picture a class originally overlooked in [HT3]) and tenfolds, but not for
higher-dimensional IHS of K3[n] type. It would be interesting to see whether
the present elementary method could be adapted to give a reasonable
classification in all dimensions.

##  2 Hyperkähler manifolds

In this section we recall some definitions concerning IHS manifolds and
rational curves. For more details see [Bea] and [AV1].

###  2.1 Holomorphically symplectic manifolds

Definition 2.1: A holomorphically symplectic manifold is a complex manifold
equipped with a nowhere degenerate holomorphic two-form.

Definition 2.2: For the rest of this paper, a hyperkähler manifold is a
compact, Kähler, holomorphically symplectic manifold.

Definition 2.3: A hyperkähler manifold $M$ is called simple, or IHS, or of
maximal holonomy if $\pi_{1}(M)=0$, $H^{2,0}(M)={\mathbb{C}}$.

The terminology originates from Bogomolov’s decomposition theorem stating that
any hyperkähler manifold admits a finite covering which is a product of a
torus and several IHS, [Bo1]. This decomposition also corresponds the de Rham-
Berger local decomposition of a Riemannian manifold into a product of
manifolds with irreducible Riemannian holonomy. Then, the IHS manifolds
correspond to the hyperkähler manifolds which have irreducible holonomy.
Further on, all hyperkähler manifolds are assumed to be of maximal holonomy.

Example 2.4: The basic series of examples is that of punctual Hilbert schemes
of K3 surfaces. As observed by Beauville in [Bea], for any K3 surface $S$, the
Hilbert scheme $S^{[n]}$ of ideal sheaves of colength $n$ on $S$ is an IHS. An
IHS which is a deformation of $S^{[n]}$ for a K3 surface $S$ is said to be of
K3[n] type. There is one more series (generalized Kummer manifolds) and two
sporadic families (O’Grady’s examples) known, and we don’t know whether there
is anything else.

###  2.2 The Bogomolov-Beauville-Fujiki form

Theorem 2.5: (Fujiki, [F]) Let $\eta\in H^{2}(M)$, and $\dim M=2n$, where $M$
is hyperkähler. Then $\int_{M}\eta^{2n}=cq(\eta,\eta)^{n}$, for some primitive
integral quadratic form $q$ on $H^{2}(M,{\mathbb{Z}})$, and $c>0$ a rational
number.

Definition 2.6: This form is called Bogomolov-Beauville-Fujiki form. It is
defined by the Fujiki’s relation uniquely, up to a sign. The sign is found
from an explicit formula due to Beauville and Bogomolov, using integration
against exterior powers of the holomorphic symplectic form $\Omega$ and its
conjugate, see e.g. [Bea].

Remark 2.7: The form $q$ has signature $(3,b_{2}-3)$. It is negative definite
on primitive forms, and positive definite on
$\langle\Omega,\overline{\Omega},\omega\rangle$, where $\omega$ is a Kähler
form.

Remark 2.8: The BBF form is in general not unimodular. We shall use it to
embed $H_{2}(M,{\mathbb{Z}})$ into $H^{2}(M,{\mathbb{Q}})$ as an overlattice
of $H^{2}(M,{\mathbb{Z}})$. This allows us to view the classes of curves on
$M$ as rational $(1,1)$-classes and compute the BBF square of a curve which is
a rational number (cf. [HT1]).

Example 2.9: If $M$ is a K3 surface, the BBF form is just the intersection
form. If $M=S^{[n]}$ for a K3 surface $S$, we have an orthogonal direct sum
decomposition of lattices
$H^{2}(M,{\mathbb{Z}})=H^{2}(S,{\mathbb{Z}})\oplus{\mathbb{Z}}e$, where
$H^{2}(S,{\mathbb{Z}})$ is equipped with the intersection form and
$e^{2}=-2(n-1)$. In fact $e$ is one half of the class of the diagonal, that
is, the divisor of non-reduced subschemes of length $n$ in $S$ (this divisor
is also the exceptional set of the Hilbert-Chow morphism for $S^{[n]}$. This
decomposition holds for IHS of K3[n] type (but the class $e$ may cease to be
algebraic).

###  2.3 Extremal curves

Let $Z\subset M$ be an irreducible uniruled subvariety of a compact Kähler
manifold $(M,\omega)$. We call a rational curve $C\subset Z$ minimal if it is
of minimal degree with respect to the Kähler form, i.e. $\int_{C}\omega$ is
minimal, among the curves passing through a general point of $Z$.

Remark 2.10: Thanks to Mori’s bend-and-break lemma, there is the following
necessary condition for minimality: for general $x,y\in Z$, the space of
deformations $Z_{x,y}$ of $C$ passing through $x,y$ is 0-dimensional. Indeed,
if $\dim Z_{x,y}>0$, we can use Mori lemma to deform $C$ to a reducible curve
which contains $x,y$. The area $\int_{C^{\prime}}\omega$ of each component of
this union is strictly smaller than that of $C$.

If $X$ is IHS, minimal rational curves on $X$ have the following remarkable
properties (see for instance [AV1]).

Proposition 2.11:  Let $X$ be IHS of dimension $2n$, then a minimal rational
curve $C$ (with respect to some maximal uniruled subvariety $Z$) deforms
together with $X$ whenever its cohomology class remains of type $(1,1)$. The
space of deformations of $C$ within $X$ is of dimension $2n-2$. The subvariety
$Z$ is coisotropic and can be of any dimension between $n$ and $2n-1$. Its
codimension in $X$ is equal to the relative dimension of its rational quotient
fibration.

Definition 2.12: A rational curve is called MBM if it is minimal and its
cohomology class has negative self-intersection.

One reason why we are especially interested in negative self-intersection
rational curves is that they determine the Kähler cone $\operatorname{Kah}(M)$
(or the ample cone in the projective case) inside the positive cone. The
latter is defined as the connected component of the set

$\\{\eta\in H^{1,1}(M,{\mathbb{R}})\ \ |\ \ (\eta,\eta)>0\\}$ (2.1)

which contains the Kähler cone $\operatorname{Kah}(M)$. Note that the BBF form
on $H^{1,1}(M,{\mathbb{R}})$ has signature $(1,b_{2}-3)$ and therefore the set
(2.1) has two connected components.

Theorem 2.13: (Huybrechts, Boucksom, see [Bou]) The Kähler cone of $M$ is the
set of all $\eta\in\operatorname{Pos}(M)$ such that $(\eta,C)>0$ for all
rational curves $C$.

Clearly we may consider only negative rational curves, as by the Hodge index
theorem the positive classes never have zero intersection with curves of
positive square. If moreover the class of such a curve cannot be written as a
sum of two other such classes, it is called an extremal rational curve. The
Kähler cone is locally polyhedral (with some round pieces in the boundary),
and its faces are orthogonal complements to the classes of extremal curves.
Note that an MBM curve as in 2.3 is extremal on a generic deformation, as only
its class, up to proporionality, survives as a $(1,1)$-class there.

###  2.4 MBM classes

A minimal rational curve may split, and thus cease to be minimal, under a
deformation. To put the things into a deformation-invariant context, we have
introduced the notion of MBM classes in [AV1]. These are, up to a rational
multiple, monodromy transforms (see next subsection for the definition of the
monodromy) of the classes of extremal rational curves on IHS birational models
of $M$. Note that if $M$ and $M^{\prime}$ are two birational IHS, or, more
generally, two birational manifolds with trivial canonical bundle, then
$H^{2}(M)$ is naturally identified with $H^{2}(M^{\prime})$. The following is
a reformulation of [AV1, Theorem 1.17].

Theorem 2.14: Let $\eta$ be the cohomology class of an extremal curve on a
hyperkähler manifold, and $(M_{1},\eta)$ be obtained as a deformation of
$(M,\eta)$ in a family such that $\eta$ remains of type $(1,1)$ for all fibers
of this family. Then $M_{1}$ is birational to a hyperkähler manifold
$M_{1}^{\prime}$ such that $\gamma(\eta)$ is a class of an extremal curve on
$M_{1}^{\prime}$ for some $\gamma$ in the monodromy group.

Remark 2.15: Birational hyperkähler manifolds are deformation equivalent by a
result of Huybrechts. In fact a more precise version of the theorem above (see
[AV3]) states that an extremal rational curve remains extremal on an
$M_{1}^{\prime}$ unseparable from $M_{1}$ in the Teichmüller space (see next
section for preliminaries on Teichmüller space and more details).

MBM classes can be defined in several equivalent ways. The simplest one to
formulate is as follows.

Definition 2.16: A class $\eta\in H^{2}(M,{\mathbb{Z}})$ which, up to a
rational multiple, can be represented by an extremal curve on a deformation on
$M$ is called an MBM class.

Remark 2.17: Another equivalent definition ([AV1]): An MBM class $\eta\in
H^{2}(M)$ is a Beauville-Bogomolov negative class such that $\lambda\eta$ can
be represented by a rational curve in some $(M,I)$ when $(M,I)$ is a non-
algebraic deformation of $M$ with
$\operatorname{Pic}(M,I)_{\mathbb{Q}}=\langle\eta\rangle$. In that case it can
be represented by a rational curve in all such $(M,I)$.

The set of MBM classes is thus the same for a given deformation type. If a
complex structure $I$ is fixed, some of them are of type $(1,1)$, and among
those, some correspond to extremal curves (that is, determine the Kähler
cone), others correspond to extremal curves on birational models and their
monodromy transforms. The following result further explains the connection
between MBM classes and the Kähler cone.

Theorem 2.18: ([AV1] Let $(M,I)$ be a hyperkähler manifold and $S$ the set of
all MBM classes of type (1,1) on $(M,I)$. Let $S^{\bot}$ the union of all
orthogonal complement to all $s\in S$. Then $\operatorname{Kah}(M,I)$ is a
connected component of $\operatorname{Pos}(M)\backslash S^{\bot}$.

Following E. Markman, we call these connected components the Kähler-Weil
chambers.

##  3 Teichmüller spaces, Torelli theorem and monodromy

###  3.1 Teichmüller space

This subsection and the next one follow [V1].

Definition 3.1: Let $M$ be a hyperkähler manifold, and $\operatorname{\sf
Diff}_{0}(M)$ a connected component of its diffeomorphism group (the group of
isotopies). Denote by $\operatorname{\sf Comp}$ the space of complex
structures of Kähler type on $M$, and let $\operatorname{\sf
Teich}:=\operatorname{\sf Comp}/\operatorname{\sf Diff}_{0}(M)$. We call it
the Teichmüller space.

Remark 3.2: The space $\operatorname{\sf Teich}$ is a finite-dimensional
complex manifold ([Cat]), not necessarily Hausdorff.

Definition 3.3: Let $\operatorname{\sf Diff}(M)$ be the group of
diffeomorphisms of $M$. We call $\Gamma:=\operatorname{\sf
Diff}(M)/\operatorname{\sf Diff}_{0}(M)$ the mapping class group.

The quotient $\operatorname{\sf Teich}/\Gamma$ is identified with the set of
equivalence classes of complex structures. However the notion of the moduli
space usually does not make sense as the action of $\Gamma$ on
$\operatorname{\sf Teich}$ is “too bad”.

By a result of Huybrechts, the quotient $\operatorname{\sf
Comp}/\operatorname{\sf Diff}_{0}(M)$ has only finitely many connected
components. To simplify notations we denote by $\operatorname{\sf Teich}$ the
one which contains our given complex structure, that is, a parameter space for
the deformations of $M$. The group $\Gamma$ shall mean the subgroup of the
mapping class group which preserves $\operatorname{\sf Teich}$.

Definition 3.4: The monodromy group $\operatorname{\sf Mon}$ is the image of
the representation of $\Gamma$ on $H^{2}(M,{\mathbb{Z}})$.

Definition 3.5: Let $\operatorname{\sf Per}:\;\operatorname{\sf
Teich}{\>\longrightarrow\>}{\mathbb{P}}H^{2}(M,{\mathbb{C}})$ map a complex
structure $J$ to the line $H^{2,0}(M,J)\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})$.
The map $\operatorname{\sf Per}:\;\operatorname{\sf
Teich}{\>\longrightarrow\>}{\mathbb{P}}H^{2}(M,{\mathbb{C}})$ is called the
period map. The map $\operatorname{\sf Per}$ maps $\operatorname{\sf Teich}$
to an open subset of a quadric

$\operatorname{{\mathbb{P}}\sf er}:=\\{l\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})\
\ |\ \ q(l,l)=0,q(l,\overline{l})>0\\}.$

It is called the period space of $M$.

Remark 3.6: The space $\operatorname{{\mathbb{P}}\sf er}$ can be naturally
identified with

$SO(b_{2}-3,3)/SO(2)\times
SO(b_{2}-3,1)=\operatorname{Gr}_{++}(H^{2}(M,{\mathbb{R}}))$

(the Grassmannian of Beauville-Bogomolov positive, oriented planes in
$H^{2}(M,{\mathbb{R}})$).

Indeed, the group $SO(H^{2}(M,{\mathbb{R}}),q)=SO(b_{2}-3,3)$ acts
transitively on $\operatorname{{\mathbb{P}}\sf er}$, and $SO(2)\times
SO(b_{2}-3,1)$ is the stabilizer of a point. More explicitly, we can associate
to each complex line a real plane generated by its real and imaginary part.

###  3.2 Global Torelli theorem

Definition 3.7: Let $M$ be a topological space. We say that $x,y\in M$ are
non-separable (denoted by $x\sim y$) if for any open sets $V\ni x,U\ni y$,
$U\cap V\neq\emptyset$.

Definition 3.8: The space $\operatorname{\sf Teich}_{b}:=\operatorname{\sf
Teich}/\sim$ is called the birational Teichmüller space of $M$. Since
$\operatorname{\sf Teich}_{b}$ is obtained by gluing together all non-
separable points, it is also called the Hausdorff reduction of
$\operatorname{\sf Teich}$.

Theorem 3.9: (Global Torelli theorem for hyperkähler manifolds)
The period map $\operatorname{\sf
Teich}_{b}\stackrel{{\scriptstyle\operatorname{\sf
Per}}}{{{\>\longrightarrow\>}}}\operatorname{{\mathbb{P}}\sf er}$ is a
diffeomorphism, for each connected component of $\operatorname{\sf
Teich}_{b}$.

Huybrechts in [H1] observed that unseparable points of $\operatorname{\sf
Teich}$ correspond to birational IHS, up to the action of the mapping class
group. The following precise description of unseparable points is due to
Markman.

Theorem 3.10:  The points in $\operatorname{\sf Per}^{-1}(\operatorname{\sf
Per}I)$ correspond to the Kähler-Weil chambers (that is, to the connected
components of $\operatorname{Pos}(M)\backslash\cup z^{\bot}$ where $z$ runs
through the MBM classes of type $(1,1)$). Each chamber is the Kähler cone of
the corresponding point in $\operatorname{\sf Per}^{-1}(\operatorname{\sf
Per}I)$.

In particular, as a general hyperkähler manifold has no curves (the ones which
have curves belong to a countable union of divisors in $\operatorname{\sf
Teich}$), $\operatorname{\sf Teich}$ is generically Hausdorff.

###  3.3 The space $\operatorname{\sf Teich}_{\alpha}^{{\operatorname{\sf
min}}}$

Let now $\alpha$ be an MBM class, we are interested in the complex structures
$I$ from $\operatorname{\sf Teich}$ such that $\alpha$ is of type $(1,1)$ in
$I$. Denote the subspace of such structures in $\operatorname{\sf Teich}$ by
$\operatorname{\sf Teich}_{\alpha}$. Clearly, $\operatorname{\sf
Teich}_{\alpha}=\operatorname{\sf Per}^{-1}(\alpha^{\bot})$. From 3.2 it is
clear that $\operatorname{\sf Teich}_{\alpha}$ is not Hausdorff even
generically (indeed each complex structure $I$ in $\operatorname{\sf
Teich}_{\alpha}$ has at least two Kähler-Weil chambers in which
$\alpha^{\bot}\subset H^{1,1}(I)$ divides the positive cone). On the other
hand, $\operatorname{\sf Teich}_{\alpha}$ is naturally divided in two halves,
according to whether ${\alpha}$ is negative or positive on the Kähler classes.
In the “positive” half $\operatorname{\sf Teich}_{\alpha}^{+}$ take the part
formed by the complex structures whose Kähler cones are adjacent to
${\alpha}^{\bot}$.

Definition 3.11: The space thus defined is called $\operatorname{\sf
Teich}_{\alpha}^{{\operatorname{\sf min}}}$. In other words $\operatorname{\sf
Teich}_{\alpha}^{{\operatorname{\sf min}}}$ parameterizes the complex
structures such that ${\alpha}$ is the class of a minimal rational curve, up
to a positive multiple.

Remark 3.12: It is clear from this description that $\operatorname{\sf
Teich}_{\alpha}^{{\operatorname{\sf min}}}$ is generically Hausdorff and that
$\operatorname{\sf Per}$ is an isomorphism from $\operatorname{\sf
Teich}_{\alpha,b}^{{\operatorname{\sf min}}}$ to
$\alpha^{\bot}\subset\operatorname{{\mathbb{P}}\sf er}$.

###  3.4 Monodromy for manifolds of K3[n] type

The monodromy group is very important for understanding the geometry of IHS
manifolds, but in the general (“classification-free”) case its structure
remains somewhat mysterious. One knows, thanks to Markman [Ma2], that the
reflections in the classes of uniruled divisors are in $\operatorname{\sf
Mon}$, and this already has nontrivial consequences. But it is not clear how
to decide whether a given MBM class is “divisorial” or not (that is, whether
the corresponding rational curves cover a divisor or a subvariety of higher
codimension).

A lot of work has been done on the known examples of IHS. For manifolds of
K3[n] type, Markman proved the following.

Theorem 3.13: ([Ma3, Theorem 1.2]) The monodromy group $\operatorname{\sf
Mon}$ is the subgroup of $O(H^{2}(M,{\mathbb{Z}}))$ generated by the
reflections in the classes of square $-2$ and and the antireflections in those
of square $2$, that is, by the transformations
$\rho_{u}(x)=\frac{-2x}{q(u)}+q(x,u)u$, where the Beauville-Bogomolov square
$q(u)=\pm 2$, and the same notation $q$ is used for the bilinear form
associated to $q$. Up to $\pm$ sign, it also coincides with the stable
orthogonal group $\widetilde{O}(H^{2}(M,{\mathbb{Z}}))$ of isometries acting
trivially on the discriminant group.

Here the discriminant group of a lattice $L$ is the quotient $L^{*}/L$ where
$L^{*}$ is the dual lattice and $L$ is naturally embedded into $L^{*}$. If $L$
is $H^{2}(M,{\mathbb{Z}})$ for $M$ a $2n$-dimensional IHS of K3[n] type, the
discriminant group is cyclic of order $2n-2$ (see 2.2). We shall often view it
as a subgroup of ${\mathbb{Q}}/{\mathbb{Z}}$, like in 3.4 just below.

Remark 3.14: When $n-1$ is a power of a prime number, one can show that
$\operatorname{\sf Mon}$ coincides with the orientation-preserving part of the
orthogonal group itself ([Ma3], lemma 4.2). We shall not need this as the
group $\widetilde{O}$ has also been studied. Notably we shall use the
following.

Theorem 3.15: (see e.g. [GHS], Lemma 3.5) Let $(L,q)$ be a lattice containing
two orthogonal isotropic planes. The orbit of $l\in L$ under
$\widetilde{O}(L)$ is determined by $q(l)$ and the image of $l/d(l)$ in the
discriminant group. Here we view the dual lattice as an overlattice of $L$ in
$L\otimes{\mathbb{Q}}$, and $d(l)$ is the divisibility of $l$, that is, the
pairing of $l$ with $L$ gives the ideal $d(l){\mathbb{Z}}$.

##  4 Birational contractions and invariance by deformation

###  4.1 Birational contractions

In this section we recall the Kawamata base point free theorem which implies
that extremal rational curves on projective IHS manifolds are contractible, as
well as a more recent observation by Bakker and Lehn concerning the
nonprojective case.

Definition 4.1: A line bundle $L$ on a compact complex k-fold $M$ is nef if
$c_{1}(L)$ lies in the closure of the Kähler cone, big if $H^{0}(M,L^{\otimes
N})=O(k^{N})$ for $N$ sufficiently big and sufficiently divisible, and
semiample if $L^{\otimes N}$ is generated by its global sections for such $N$.

Theorem 4.2: (Kawamata base point free theorem, [K1, Theorem 4.3])
Let $L$ be a nef line bundle on a projective $M$ such that $L^{\otimes
N}\otimes{\cal O}(-K_{M})$ is nef and big for some $N$. Then $L$ is semiample.

For manifolds with trivial canonical bundle this just means that big and nef
bundles are semiample.

It follows that the sections of a suitable power of a big and nef bundle $L$
on such manifolds define a birational contraction (that is, a birational
morphism with connected fibers) $\varphi:M{\>\longrightarrow\>}Y$.

Suppose now that $M$ is a projective IHS manifold, $\alpha$ is an MBM class
and the given complex structure $I$ on $M$ is such that $I\in\operatorname{\sf
Teich}_{\alpha}^{{\operatorname{\sf min}}}$, so that $\alpha$ is in fact the
class of an extremal curve: the hyperplane $\alpha^{\bot}$ in
$H^{1,1}(M,{\mathbb{R}})$ contains a wall of the Kähler cone. Since $M$ is
projective, the interior of this wall contains an integral $(1,1)$-class which
by Lefschetz theorem is $c_{1}(L)$ for a line bundle $L$. This line bundle is
nef, and also big. Indeed its BBF square is positive, and hence so is
$c_{1}(L)^{2n}$. Therefore a power of $L$ defines a birational morphism
contracting exactly the curves of class proportional to $\alpha$.

It turns out that extremal curves can be contracted also in the non-projective
case. This is essentially due to Bakker and Lehn in [BL]. More precisely,
putting together some results in [AV1], [BL] and [AV3] gives the following.

Theorem 4.3: ([BL, Proposition 4.10], [AV3, Corollary 5.3])
Let $M$ be an IHS manifold and $\alpha$ the class of an extremal rational
curve on $M$. Then there exists a birational contraction
$\varphi:M{\>\longrightarrow\>}Y$ contracting to points the curves whose class
is proportional to $\alpha$. Moreover, for every component $Z$ of the
exceptional set, the fibers of $\varphi|_{Z}$ are rationally connected, and
$\varphi|_{Z}$ is the rational quotient fibration of $Z$.

In particular the uniruled subvarieties of $M$ covered by deformations of an
extremal rational curve are irreducible components of the exceptional sets of
birational contractions. We sometimes refer to these as MBM loci.

###  4.2 Invariance by deformation

The following is the main result of [AV3].

Theorem 4.4:  Let $\alpha$ be an MBM class of type $(1,1)$ on a hyperkähler
manifold $M$ with $b_{2}(M)>5$. Assume that $M$ and its deformation
$M^{\prime}$ are both in $\operatorname{\sf
Teich}_{\alpha}^{{\operatorname{\sf min}}}$, so that $\alpha$ is a class of an
extremal rational curve on both. Then the corresponding contraction centers on
$M$ and $M^{\prime}$ are sent one to another by a diffeomorphism $\Psi:M\to
M^{\prime}$ preserving the fibers of the contraction. If moreover neither $M$
nor $M_{1}$ has maximal Picard rank, $\Psi$ can be chosen identifying the
contracted extremal curves and thus induces the biholomorphism of the fibers
provided that they are normal.

Remark 4.5: Clearly, $H^{1,1}(M)$ is obtained as orthogonal complement to the
2-dimensional space
$\langle\operatorname{Re}\Omega,\operatorname{Im}\Omega\rangle$, where
$\Omega$ is the cohomology class of the holomorphic symplectic form. Then
$\operatorname{Pic}(M)=H^{1,1}(M)\cap H^{2}(M,{\mathbb{Z}})$ has maximal rank
only if the plane
$\langle\operatorname{Re}\Omega,\operatorname{Im}\Omega\rangle$ is rational.
There is at most countable number of such $M$.

Remark 4.6: Existence of $\Psi$ follows from ergodicity of mapping class group
action, global Torelli theorem, Bakker-Lehn contractibility result, and Thom-
Mathers stratification of proper real analytic maps.

##  5 MBM loci for low-dimensional manifolds of $K3^{[n]}$ type

The diffeomorphism theorem allows one to give an explicit description of
birational contraction centers and MBM loci in terms of the period spaces and
lattices. The easiest illustration is the (well-known, cf. Hassett and
Tschinkel’s works, but seemingly with no direct proof ever written)
description of the MBM loci on an IHS fourfold of K3[2] type.

Theorem 5.1:  On an IHS fourfold of K3[2] type, there are only three types of
extremal rational curves $l$ (notation: $\alpha=[l]$, $Z$ is the MBM locus of
$l$):

(a)

$q(\alpha)=-5/2$, $2\alpha$ is integral, $Z\cong{\mathbb{P}}^{2}$

(b)

$q(\alpha)=-2$, $\alpha$ is integral, $Z$ is a ${\mathbb{P}}^{1}$-bundle over
a K3 surface;

(c)

$q(\alpha)=-1/2$, $2\alpha$ is integral, $Z$ is a ${\mathbb{P}}^{1}$-bundle
over a K3 surface.

Taking into account 4.2, this is a consequence of the following precision:

Theorem 5.2:  Let $\alpha$ be the class of an extremal rational curve on an
IHS fourfold $M$ of K3[2] type and $Z$ the corresponding MBM locus. Then there
exists a deformation $M^{\prime}$ of $M$ within $\operatorname{\sf
Teich}_{\alpha}$, isomorphic to the Hilbert square of a non-projective K3
surface $S$ with cyclic Picard group generated by a negative class $z$, such
that one of the following holds:

(a)

$S$ contains a smooth rational curve $C$ and $Z$ deforms to its Hilbert
square, isomorphic to ${\mathbb{P}}^{2}$; $\alpha$ is a line in this
${\mathbb{P}}^{2}$;

(b)

$S$ contains a smooth rational curve $C$ and $Z$ deforms to $Z_{C}\subset
M^{\prime}$, the divisor of all length-two subschemes of $S$ with support
intersecting $C$; $Z_{C}$ is a ruled divisor singular along
$C^{[2]}\cong{\mathbb{P}}^{2}$ and $\alpha$ is the class of the ruling.

(c)

$Z$ deforms to the exceptional divisor of the Hilbert-Chow map
$S^{[2]}\to\operatorname{Sym}^{2}(S)$, and $\alpha$ is the class of the
ruling.

Remark 5.3: In the second case $\alpha$ is clearly not a class of an extremal
rational curve, that is, $M^{\prime}$ is not in $\operatorname{\sf
Teich}_{\alpha}^{min}$. As we know, this means that some complex structure
unseparable from $M$ is in $\operatorname{\sf Teich}_{\alpha}^{min}$. In fact,
to render $\alpha$ extremal, one performs a flop in $C^{[2]}$, which also
resolves the singularities of $Z_{C}$ and makes it a
${\mathbb{P}}^{1}$-bundle.

Once the existence of a deformation $M^{\prime}\cong S^{[2]}$,
$\operatorname{Pic}(S)=\langle z\rangle$, $q(z)<0$, is shown, the rest is
easy. Indeed one has the following description of rational curves on
$M^{\prime}$.

Proposition 5.4:  Let $S$ be a K3 surface with the Picard group
$\operatorname{Pic}(S)$ of rank 1 generated by a vector $z\in
H^{1,1}(S,{\mathbb{Z}})$ with negative square, and $Z$ an MBM locus on
$M^{[2]}$. Then $Z$ is described by (a), (b) or (c) of 5.

Proof: If $S$ has no curves, the symmetric square $\operatorname{Sym}^{2}S$
also has no curves, and the bimeromorphic contraction
$S^{[2]}{\>\longrightarrow\>}\operatorname{Sym}^{2}S$ contracts the
exceptional divisor $E$ to the singular set of $\operatorname{Sym}^{2}S$. This
means that $E=Z$, case (b) of 5.

If $S$ has a curve, it is a smooth (-2)-curve $C$ which is necessarily unique.
Indeed, any curve with negative self-intersection is a smooth rational curve
with square $-2$ by adjunction, and there cannot be two different curves in
the proportional classes, again because $(z,z)<0$. Then the (-2)-curve $C$ can
be blown down to an ordinary double point. Let $S_{1}$ be the result of this
blow-down. Then $\operatorname{Sym}^{2}(S_{1})$ has no curves, and the natural
map $S^{[2]}{\>\longrightarrow\>}\operatorname{Sym}^{2}(S_{1})$ blows down all
curves in $S^{[2]}$. The corresponding exceptional set $V$ is a union of two
divisors, the exceptional divisor $E$ and the set $W\subset S^{[2]}$ of all
ideal sheaves of colength two $I\subset{\cal O}_{S}$ with support of $I$
intersecting $C$. Clearly, these two divisors intersect in $P^{2}=C^{[2]}$,
and outside of $C^{[2]}$ each of these divisors is covered by a unique family
of rational curves, hence the classification.

As for Beauville-Bogomolov squares, we recall the computations by Hassett and
Tschinkel and others for convenience. We have the decomposition into the
orthogonal direct sum
$H^{2}(M,{\mathbb{Z}})=H^{2}(S,{\mathbb{Z}})\oplus{\mathbb{Z}}e$ where $e$ is
half of the class of the diagonal $E$, and $e^{2}=-2$.

  * •

If $\alpha$ is the class of the ruling of the exceptional divisor,
$q(e,\alpha)=-1$ since the exceptional divisor restricts as ${\cal O}(-2)$ to
the ruling. Hence $\alpha=e/2$.

  * •

If $\alpha$ is the class of the ruling of $Z_{C}$, then $\alpha$ is the class
of $C$.

  * •

If $\alpha$ is the class of the line in the Lagrangian plane $C^{[2]}$, then
$\alpha=[C]-e/2$ ([HT3], Example 4.11; for $S^{[n]}$ this would be the same,
indeed $q(e,\alpha)=n-1$)

We postpone the existence of $M^{\prime}$ until the next section and deal with
the next example of an IHS sixfold of K3[3] type. There are five types of
extremal rational curves/MBM classes/contraction loci. Four of them appear in
the tables of [HT3] and are described geometrically exactly in the same way as
those on a fourfold: on such an $M$ which is the Hilbert cube of a K3 surface
$S$ containing a (-2)-curve $C$, we have four natural subvarieties covered by
deformations of rational curves in BBF-negative classes, that is, the
exceptional divisor of the Hilbert-Chow map (ruled over the big diagonal), the
ruled divisor of subschemes with some support on $C$ (ruled over the symmetric
square of $S$), the fourfold of subschemes with at least length-two part
supported on $C$ (birational to a ${\mathbb{P}}^{2}$-bundle over $S$), and
${\mathbb{P}}^{3}=C^{[3]}$.

After Bayer and Macri have given their description of the Mori cone in [BM],
Hassett and Tschinkel observed in [HT4] that one case was actually missing
from their table in dimension 6. The cohomology class of the extremal curve in
question is half-integer and has square $-1$. We would like to present the
following simple way to find this MBM class.

Theorem 5.5:  Let $\alpha$ be a class of an extremal rational curve on an IHS
$M$ of K3[n] type of dimension at most ten and let $Z$ be the corresponding
MBM locus. Then there exists a deformation $M^{\prime}$ of $M$ within
$\operatorname{\sf Teich}_{\alpha}$, isomorphic to the Hilbert scheme of a
non-projective K3 surface $S$ with cyclic Picard group (thus generated by a
non-positive class $z$).

Again, let us describe the geometry we get in dimension six before going to
the proof.

First case: If $\operatorname{Pic}S$ is generated by a negative class, the
rational curves on $M^{\prime}$ are described as in 5: one has the four types
listed above.

Second case: If $\operatorname{Pic}S$ is generated by a class of square zero,
then one deduces from Riemann-Roch theorem that $S$ carries an elliptic pencil
with some degenerate fibers which are rational curves with a singularity, and
no other curves. For Hilbert cube this means that the extremal rational curves
arise from linear systems of degree $k\leqslant 3$ on the (eventually
normalized) curves from the elliptic pencil: one fixes $3-k$ points on $S$ and
lets the remaining $k$-uplet run through a pencil in the system. Of course,
not all curves arising in this way are extremal, and some of them are already
on the list obtained in the negative case.

Note that the computation of the BBF square is particularly easy since only
the intersection with $e$ counts. Indeed the intersection form is zero on its
orthogonal complement (which is naturally identified with
$\operatorname{Pic}S$).

For instance, the curve obtained by fixing two points and letting the third
one run through the singular fiber is of BBF square zero, so not extremal.
Considering a rational curve $R$ coming from a $g^{1}_{3}$ on a smooth
elliptic curve, we see that $q(R)=-9/4$ and $R$ is quarter-integer ($e$ comes
with coefficient $-3/4$), by the following well-known lemma (see [CK] for a
version which also applies to singular curves).

Lemma 5.6:  The class of a rational curve coming from a $g^{1}_{k}$ on a curve
$C$ of genus $g$ is $C-\frac{g+k-1}{2(n-1)}e$.

The idea is that the intersection with the diagonal is computed by the Hurwitz
formula, as this is roughly speaking the ramification of the morphism to the
projective line defined by the $g^{1}_{3}$.

Such an extremal class (quarter-integer with square $-9/4$) is already
obtained by Hassett-Tschinkel’s computation when $S$ contains a $(-2)$-curve
$C$. It appears as the class of a line in the projective plane $C^{[2]}$. By
the description of monodromy orbits, the two classes are conjugate by a
monodromy transform, so we get no new geometry here.

On the contrary, fixing one point on $S$ and considering a $g^{1}_{2}$ on an
elliptic curve, one gets a half-integer class of square $-1$, exactly the one
omitted in the table of [HT3].

The following corollary results from the above computations.

Corollary 5.7:  There are 5 types of MBM classes on an IHS sixfold of K3[3]
type:

(a)

$q(\alpha)=-3$, $2\alpha$ is integral, $Z\cong{\mathbb{P}}^{3}$

(b)

$q(\alpha)=-9/4$, $4\alpha$ is integral, $Z$ is birationally a
${\mathbb{P}}^{2}$-bundle over a K3 surface;

(c)

$q(\alpha)=-2$, $\alpha$ is integral, $Z$ is birationally a
${\mathbb{P}}^{1}$-bundle over the Hilbert square of a K3 surface;

(d)

$q(\alpha)=-1/4$, $4\alpha$ is integral, $Z$ is birationally a
${\mathbb{P}}^{1}$-bundle over a product of two K3 surfaces;

(e)

$q(\alpha)=-1$, $2\alpha$ is integral, $Z$ is birationally a
${\mathbb{P}}^{1}$-bundle over a product of two K3 surfaces.

It could be interesting to see whether one can make this more precise with the
help of flops, as in the fourfold case, though there is no hope that one could
suppress the word “birationally” altogether. Indeed the diagonal divisor (case
(d)) is a contraction locus but not a ${\mathbb{P}}^{1}$-bundle, the fibers
over the small diagonal are more complicated (cones over twisted cubics, [T]).

In the next section we perform a monodromy computation and use it to prove
theorems 5 and 5. In dimension 8 and 10, one derives a similar description of
MBM classes and their loci. 5 does not hold beyond dimension ten. Nevertheless
one can pursue the explicit description of extremal curves along these lines.

##  6 Monodromy computations

Theorem 6.1:  Let $S$ be a K3 surface and let $\Lambda$ be the lattice
$H^{2}(S^{[k]},{\mathbb{Z}})=H^{2}(S,{\mathbb{Z}})\oplus{\mathbb{Z}}e$ with
its BBF form. If $k=2$, then for any primitive $z\in\Lambda$ with $q(z,z)<0$,
there exists $\gamma\in\widetilde{O}(\Lambda)$ such that the rank 2 lattice
$\langle\gamma(z),e\rangle$ is negative definite, unless $z$ and $e$ belong to
the same $\widetilde{O}(\Lambda)$-orbit. If $3\leqslant k\leqslant 5$, there
exists $\gamma\in\operatorname{\sf Mon}$ such that the rank 2 lattice
$\langle\gamma(z),e\rangle$ is negative semidefinite.

Remark 6.2: In this statement $\widetilde{O}(\Lambda)$ can be replaced by
“monodromy”, we just want to stress that the statement is about lattices.

This implies 5 and 5 because of the following proposition.

Proposition 6.3: Let $Z\subset X$ be a birational contraction center on a
deformation $X$ of the $n$-th Hilbert scheme of a K3 surface. Suppose that 6
holds for its second cohomology lattice. Then the pair $(X,Z)$ can be deformed
to $(S^{[n]},Z^{\prime})$, where $S$ is a non-algebraic K3 surface with
(necessarily BBF nonpositive) cyclic Picard group.

Proof: Let $\eta$ be the MBM class associated with $Z$ and $\operatorname{\sf
Teich}_{\eta}$ the Teichmüller space of deformations of $X$ such that $\eta$
remains of type (1,1). From the global Torelli theorem it follows that any
lattice $\Lambda_{1}\subset\Lambda$ with $\Lambda_{1}^{\bot}$ containing a
positive 2-plane and $\Lambda_{1}\ni\eta$ can be realized as a Picard lattice
of $I\in\operatorname{\sf Teich}_{\eta}$. Moreover all structures unseparables
from $I$ (the points of $\operatorname{\sf Per}^{-1}(\operatorname{\sf
Per}(I))$ have this Picard lattice, and the corresponding manifolds are all
birational. Applying this to the lattice $\langle\eta,\gamma^{-1}e\rangle$
from 6 , we obtain deformations of $(X,Z)$ with Picard lattice
$\langle\eta,\gamma^{-1}e\rangle$. On such deformations, $\eta$ remains of
type $(1,1)$. Finally, some of them are Hilbert schemes of K3 surfaces: indeed
the period points of Hilbert schemes of K3 surfaces in
$\operatorname{{\mathbb{P}}\sf er}$ form the union of the orthogonal
hyperplanes to the monodromy transforms of $e$ (each point of this union is a
period point by Torelli theorem for K3 surfaces).

We denote by $\delta(z)$ the image of $\frac{z}{d(z)}$ in the discriminant
group, where $d(z)$ denotes the divisibility of $z$ (see 3.4). Recall from 3.4
that $z,z^{\prime}\in\Lambda=(H^{2}(S^{[n]},{\mathbb{Z}}),q)$ belong to the
same orbit of $\widetilde{O}(\Lambda)$ if and only if $q(z)=q(z^{\prime})$ and
$\delta(z)=\delta(z^{\prime})$.

Example 6.4:  Let $S$ be a K3 surface, and
$\Lambda=(H^{2}(S^{[2]},{\mathbb{Z}}),q)$, so that
$\Lambda=H^{2}(S,{\mathbb{Z}})\oplus{\mathbb{Z}}e$, where $q(e)=-2$, and the
discriminant group of $\Lambda$ is ${\mathbb{Z}}/2{\mathbb{Z}}$. The primitive
vectors $z\in\Lambda$ are $ax+be$, where $x\in H^{2}(S,{\mathbb{Z}})$ is
primitive and $a,b\in{\mathbb{Z}}$ relatively prime. Since the lattice
$H^{2}(S,{\mathbb{Z}})$ is unimodular, $d(x)=1$ for any primitive $x\in
H^{2}(S,{\mathbb{Z}})$. Therefore, $d(z)=2$ if and only if $a$ is even, and
$\delta(z)\neq 0$ if and only if $a$ is even and $b$ odd. For such $z$, one
has $q(z)=a^{2}q(x)-2b^{2}$. More generally, if
$\Lambda=(H^{2}(S^{[n]},{\mathbb{Z}}),q)$ and $e$ denotes half the diagonal
class, then the divisibility of $ax+be$, $x\in H^{2}(S,{\mathbb{Z}})$
primitive, is equal to g.c.d. $(a,2(n-1)b)$.

Proof of 6 in the case $k=2$: We consider two subcases, $\delta(z)=0$ and
$\delta(z)\neq 0$. If $\delta(z)=0$, we can find $z^{\prime}\in
H^{2}(S,{\mathbb{Z}})\subset H^{2}(S^{[2]},{\mathbb{Z}})$ with
$q(z)=q(z^{\prime})$, and 3.4 implies that $z^{\prime}=\gamma(z)$ for some
$\gamma\in\widetilde{O}(\Lambda)$. Clearly, the lattice $\langle
z^{\prime},e\rangle$ is negative definite.

Now, let $\delta(z)\neq 0$. Then $q(z)=a^{2}q(x)-2b^{2}$, where $a=2a_{1}$ is
even, and $q(z,z)=4a_{1}^{2}q(x)-2b^{2}$ is equal to -2 modulo 8. Moreover,
any number $r=-2\mod 8$ can be realized as $q(z^{\prime})=4q(x)-2$ for
$z^{\prime}=2x^{\prime}-e$, because any even number can be realized as a
square of $x^{\prime}\in H^{2}(S,{\mathbb{Z}})$. Since $q(z)<0$, we have
$q(z)\leqslant-10$ unless $q(z)=-2$. In the latter case $z$ belongs to the
same $\widetilde{O}(\Lambda)$-orbit as $e$. In the first case $z$ belongs to
the same $\widetilde{O}(\Lambda)$-orbit as $z^{\prime}=2x^{\prime}-e$, with
$q(x^{\prime})\leqslant-2$. In this case, the determinant of the lattice
$\langle z^{\prime},e\rangle$ is equal to
$q(z^{\prime})q(e)-q(z^{\prime},e)^{2}=-2q(z^{\prime})-4\geqslant 16$, hence
the lattice $\langle z^{\prime},e\rangle$ is also negative definite.

Proof of 6 in the case $k=3$: Now $e^{2}=-4$ and the discriminant group of the
lattice is the cyclic group of order 4 which we view as a subgroup of
${\mathbb{Q}}/{\mathbb{Z}}$, and we have three possible subcases.

The first one $\delta(z)=0$ is the same as above.

If $\delta(z)=\pm{1/4}$, $z$ is conjugate to $e$ when $q(z)=-4$. Otherwise we
write $z$ as $4x+be$ where $x\in H^{2}(S,{\mathbb{Z}})$ and $b$ is odd. Since
$q(z)=16q(x)-4b^{2}$ is equal to -4 modulo 16 and is negative, we have
$q(z)\leqslant-20$. Moreover, $z$ is conjugate to some element of the form
$z^{\prime}=4x^{\prime}\pm e$, as $b^{2}-1$ is divisible by 8, and hence
$16q(x)-4b^{2}+4$ is divisible by 32. Now the lattice generated by $e$ and
$z^{\prime}$ is negative definite.

Some novelty appears when $\delta(z)=1/2$. Then one still finds a negative
definite lattice whenever $q(z)\neq-4$, but when $q(z)=-4$ then $z$ is
conjugate to $z^{\prime}$ of the form $2x^{\prime}+e$ and the determinant of
the lattice $\langle z^{\prime},e\rangle$ is zero.

Proof of 6 in the case of greater $k$: We set $t=2(k-1)$ so that $q(e)=-t$,
and we eventually replace $z$ by a multiple in order to write uniformly
$z=tx+be$, $b\in{\mathbb{Z}}$, $x$ primitive. We first remark is that one may
suppose that $|b|\leqslant t/2$. Indeed $tx+be$ has the same square as
$tx^{\prime}+(b\pm t)e$ for some $x^{\prime}\in H^{2}(S,{\mathbb{Z}})$ (the
equation $t^{2}q(x)-tb^{2}=t^{2}q(x^{\prime})-t(b^{2}\pm 2tb+t^{2})$ on
$q(x^{\prime})$ has a solution since the terms $tb^{2}$ cancel and the other
ones are divisible by $2t^{2}$), and when $x^{\prime}$ is primitive the two
have the same divisibility: indeed by 6 this divisibility is g.c.d.$(t,bt)$
resp. g.c.d. $(t,(b\pm t)t)$, and the two are clearly equal. Therefore $tx+be$
and $tx^{\prime}+(b\pm t)e$ are in the same monodromy orbit.

We know that $q(z)$ is negative. For small $t$ it means that $q(z)\leqslant-
tb^{2}$: indeed this is the case when $q(x)$ is non-positive, and when
$t\leqslant 8$ (that is, $k\leqslant 5$), already for $q(x)=2$ one has
$2t^{2}-b^{2}t\geqslant 0$ provided $|b|\leqslant t/2$. Therefore the matrix
of $q$ on $\langle z,e\rangle$ is negative semidefinite, and one concludes in
the same way as before.

Starting from $k=6$, the statement fails. Indeed, let $k=6$, so $t=10$, and
$z=2x-e$ where $x$ is primitive of square 2, then $z^{2}=-2$ but the
determinant of $q$ on $\langle z,e\rangle$ is negative and will remail
negative when we replace $z$ by a conjugate.

Remark 6.5: In the last case mentioned, that is, $k=6$, the computation above
shows that the only case when the pair $(M,z)$ cannot be deformed to the
Hilbert scheme of a nonprojective surface with cyclic Picard group is exactly
as described: $z^{2}=-2$, $z=2x-e$ with $x$ orthogonal to $e$, $x$ of square
$2$. On a generic deformation where $e$ becomes a Hodge class, $M$ becomes the
Hilbert scheme of a $K3$ surface $S$ with Picard group generated by $x$ of
square $2$. Such an $S$ is a double covering of ${\mathbb{P}}^{2}$ and the MBM
curve has class $z/2$. Up to monodromy, by 5 and the unicity statement (3.4),
it comes from a $g_{4}^{1}$ on a curve of genus 2 (the inverse image of a line
under the covering map is of genus 2). Its MBM locus is birational to a
${\mathbb{P}}^{2}$-bundle over a product of a K3 surface with the symmetric
square of a K3 surface.

This evokes the paper [KLM] where many rational curves on Hilbert schemes of a
K3 surface and on generalized Kummer varieties are constructed from special
linear systems on (possibly nodal) curves via Brill-Noether theory. Notice
that in this particular case no Brill-Noether theory is needed, since
obviously any curve of genus 2 carries a $g_{4}^{1}$. It seems likely and
could be interesting to verify that such a description can be given in
general.

Acknowledgements: We are grateful to Brendan Hassett, Ljudmila Kamenova,
Emanuele Macri and Yuri Tschinkel for useful discussions.

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Ekaterina Amerik
Laboratory of Algebraic Geometry,
National Research University HSE,
Department of Mathematics, 7 Vavilova Str. Moscow, Russia,
<EMAIL_ADDRESS>also:
Université Paris-11,
Laboratoire de Mathématiques,
Campus d’Orsay, Bâtiment 425, 91405 Orsay, France

Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry,
National Research University HSE,
Department of Mathematics, 7 Vavilova Str. Moscow, Russia,
<EMAIL_ADDRESS>


# Occam’s Razor and Bender and Koller’s Octopus

Michael Guerzhoy
University of Toronto
<EMAIL_ADDRESS>
###### Abstract

We discuss the teaching of the discussion surrounding Bender and Koller’s
prominent ACL 2020 paper, “Climbing toward NLU: on meaning form, and
understanding in the age of data" Bender and Koller (2020).

We present what we understand to be the main contentions of the paper, and
then recommend that the students engage with the natural counter-arguments to
the claims in the paper.

We attach teaching materials that we use to facilitate teaching this topic to
undergraduate students.

Occam’s Razor and Bender and Koller’s Octopus

Michael Guerzhoy University of Toronto<EMAIL_ADDRESS>

##  1 Introduction

The claim in Bender and Koller’s argument in Bender and Koller (2020) is that
a being that only has access to the form of the communication – e.g., an
intelligent octopus that taps into only the submarine signals that encode
accounts of the events above the sea that two people on land send each other –
will not be able to “understand" what is happening above sea-level, lacking
the semantics of the Morse code that was used to communicate the events
transpiring above the seas. Koller and Bender argue that even if the octopus
can send messages based on the patterns it sees that would be understood by
the humans and the humans would be fooled into thinking they are reading
messages from another human, the shallow understanding of the octopus would
necessarily be revealed when trying to pretend to answer more complicated
queries.

Implicit in the argument is that the intelligent octopus is analogous to a
Large Language Model (LLM), akin to GPT-2 Brown et al. (2020), GPT-4
111https://openai.com/index/gpt-4-research/, Claude 3
222https://www.anthropic.com/news/claude-3-family, or Meta Llama 3
333https://llama.meta.com/llama3/, and that such LLMs would not be able to
truly understand natural language in the same way that Bender and Koller’s
octupus will not.

In this paper, we present a lecture + activity that challenges Bender and
Koller’s argument. Students will engage with Bender and Koller’s argument and
with the counterargument, and come away with their own conclusions

##  2 Building theory from data

The scientific process itself can be analogized to a B&K octopus observing
data they don’t understand.

For example, astronomers observe and try to predict the motions of heavenly
bodies, initially with no mechanistic understanding of why the stars appear to
move the way they do. Historically, astronomers came up with multiple
incorrect theories for why the heavenly bodies move the way they do (notably,
the family of geocentric models). Astronomers used “epicycles" as a way to
align predictions with their model, at the expense of parsimony Duhem (2015).

Historically, Copernicus’ first models did not predict the data as well as the
best epicycle-based geocentric models Klein and Loftus (2015).

Note that, unlike the astronomers, the octopus cannot interact with the world
– he cannot influence what observations are made (at least before he starts
communicating with the astronomers). This can influence how fast the octopus
can “converge." Historically, much of the data used by Kepler was previously
collected by Tycho Brahe.

###  2.1 Occam’s razor

Occam’s razor – the principle that, all things being equal, we should prefer
the simpler theory – can help select the better scientific theory. For
example, the B& K octopus might consider all possible theories of the world
over the sea, and settle on the simplest one that explains the communications
the octopus decodes.

##  3 Can the B& K octopus learn science

Every student can make their own conclusions, but ours is that it’s not in
principle impossible for that to happen (or if it is impossible, we don’t have
a clear reason to think so). The success of LLMs on tasks that require some
level of world-theory-building such as the addition of integers task Lee et
al. (2023), predicted to be impossible by Bender and Koller (2020) (see
Appendix B), indicates that if there are barriers to learning world models
from observational data, they are not well-understood. Our view is that the
prediction by B&K that a pure LLM could not learn to do arithmetic is due to
insufficiently accounting for the possibility of using inductive biases to
build a model of the data that corresponds to the world that the data is
describing.

##  4 Materials

We provide slides we used in class to follow up the class’s reading of Bender
and Koller (2020). We also provide the following guiding questions

  1. 1.

If the octopus observes different content in messages when it’s dark vs. when
it’s light, what can the octopus possibly conclude about language?

  2. 2.

Describe how the octopus might use tides to infer words that have to do with
tides

  3. 3.

Describe how the octopus might decode conversations about physics based on the
conversations about tides – perhaps building up from observations of tides,
stars, etc.

  4. 4.

If you assume no “cheating" such as jointly observing tides, might you imagine
conversations that involve physical and mathematical constant like $G$ and
$\pi$ playing a similar role?

  5. 5.

Explain why without Occam’s razor, the Octopus will have a practically
infinite number of theories about what the two humans could be talking about

  6. 6.

What might be some insurmountable challenges for the octopus in the quest to
understand the meaning of the cable signals? How

  7. 7.

Consider the claim from the original paper that arithmetic is not learnable by
form alone: where might that argument have gone wrong?

##  5 Additional materials

Julian Michael, To Dissect An Octopus
https://julianmichael.org/blog/2020/07/23/to-dissect-an-octopus.html provides
an excellent overview.

##  6 Conclusion

Many students in NLP would be familiar with Bender and Koller’s argument, but
have probably not engaged in the critical analysis of the arguments. We
provide materials for critically analyzing the arguments made by Bender and
Koller. We focus on the counterarguments since the argument itself is ably
presented by the original authors. We provide slides introducing the B&K
argument to the best our ability as well.

Many (though not all) students are captivated by the debate. We find that the
structure provided by the guiding questions helps in our lectures.

##  7 Teaching materials

Slides: https://github.com/guerzh/octopus

Video lecture: https://youtu.be/6QVjGF_J7I0

## References

  * Bender and Koller (2020) Emily M Bender and Alexander Koller. 2020.  Climbing towards nlu: On meaning, form, and understanding in the age of data.  In _Proceedings of the 58th annual meeting of the association for computational linguistics_ , pages 5185–5198.
  * Brown et al. (2020) Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. 2020.  Language models are few-shot learners.  _Advances in neural information processing systems_ , 33:1877–1901.
  * Duhem (2015) Pierre Duhem. 2015.  _To save the phenomena: An essay on the idea of physical theory from Plato to Galileo_.  University of Chicago Press.
  * Klein and Loftus (2015) Stanley B Klein and Judith Loftus. 2015.  The mental representation of trait and autobiographical knowledge about the self.  _The mental representation of trait and autobiographical knowledge about the self_ , pages 1–49.
  * Lee et al. (2023) Nayoung Lee, Kartik Sreenivasan, Jason D Lee, Kangwook Lee, and Dimitris Papailiopoulos. 2023.  Teaching arithmetic to small transformers.  In _The Twelfth International Conference on Learning Representations_.