Olivier Debarre

Bodo Lass

Monomial transformations of the projective space

Keywords:

We prove that, over any field, the dimension of the indeterminacy locus of a rational map f : P n P n defined by monomials of the same degree d with no common factors is at least (n -2)/2, provided that the degree of f as a map is not divisible by d. This implies upper bounds on the multidegree of f and in particular, when f is birational, on the degree of f -1 .

Introduction

We denote by P n the n-dimensional projective space over a fixed field. A monomial transformation of P n is a rational map f : P n P n whose components f 0 , . . . , f n are monomials (of the same positive degree d( f ) and with no common factors) in the variables x 0 , . . . , x n .

Monomial transformations are of course very special among all rational transformations, but also much easier to study. For this reason, they have recently attracted some attention. In particular, there is a description of all birational monomial transformations f with d( f ) = 2 in [CS], §2, from which it follows that d( f -1 ) is then at most equal to n ([CS], Theorem 2.6). Extensive computer calculations were then performed by Johnson in [J] and led him to suggest that the largest possible value for d( f -1 ) should be (d( f )-1) n -1

d( f )-2 when d( f ) ≥ 3.
Olivier Debarre Département de Mathématiques et Applications -CNRS UMR 8553, École normale supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France, e-mail: olivier.debarre@ens.fr Bodo Lass Institut Camille Jordan -CNRS UMR 5208, Université Claude Bernard -Lyon 1, 69622 Villeurbanne Cedex, France, e-mail: lass@math.univ-lyon1.fr These values should be compared with the optimal bound d(g -1 ) ≤ d(g) n-1 for all birational transformations g of P n . This maximal value for d(g -1 ) is attained if and only if the indeterminacy locus of g is finite (see §4), hence one is led to think that the indeterminacy locus of a monomial map should be rather large. This is what we prove in Theorem 1: the dimension of the indeterminacy locus of a monomial map f : P n P n is at least (n -2)/2, provided that the degree of f as a map is not divisible by d( f ).

We show in §4 that this implies a bound on d( f -1 ) for all birational monomial transformations f , which is however not as good as the one suggested by Johnson.

Monomial transformations

We represent a monomial transformation f : P n P n , with components f 0 , . . . , f n , by the (n + 1) × (n + 1) matrix A = (a i j ) 0≤i, j≤n whose ith row lists the exponents of

f i . With this notation, one has f A • f B = f AB .
The following proposition is elementary ([GSP]; [SV] Lemma 1.2; [J]).

Proposition 1. With the notation above, we have

| det(A)| = d( f ) deg( f ).
In particular, f is birational if and only if

| det(A)| = d( f ).
Proof. The condition that all monomials f 0 , . . . , f n have the same degree d := d( f ) means that in each row of A, the sum of the entries is d. Adding all columns to the 0th column, then subtracting the first row from all other rows we obtain

det(A) = d a 01 • • • a 0n . . . . . . . . . d a n1 • • • a nn = d 1 a 01 • • • a 0n . . . . . . . . . 1 a n1 • • • a nn = d det(M),
where M := (m i j ) 1≤i, j≤n is defined by m i j := a i ja 0 j . If T ≃ (C * ) n ⊂ P n is the torus defined by x 0 • • • x n = 0, the map f induces a morphism f T : T → T given by

f T (x 1 , . . . , x n ) = (x m 11 1 • • • x m 1n n , . . . , x m n1 1 • • • x m nn n ).
The induced map f T : T → T between algebraic character groups (where T is the free abelian group Z n ) is given by the transposed matrix M T : Z n → Z n . Performing elementary operations on M amounts to composing f T with monomial automorphisms, so we can reduce to the case where M is diagonal, in which case it is obvious that the degree of the morphism f T (which is the same as the degree of

f ) is | det(M)|. ⊓ ⊔ Corollary 1. With the notation above, f is birational if and only if | det(A)| = d( f ).
Its inverse is then also a monomial transformation.

Proof. It is clear from the proof above that f is birational if and only if | det(M)| = 1, i.e., if and only if M ∈ GL n (Z) or, equivalently, if and only if f T is an isomorphism, whose inverse is then given by the matrix M -1 . It is therefore a monomial transformation. ⊓ ⊔

Indeterminacy locus

In this section, f : P n P n is again a monomial transformation. We assume that its components f 0 , . . . , f n have no common factors. In terms of the matrix A defined in §2, this means that each column of A has at least one 0 entry.

The indeterminacy locus B of f is then the subscheme of P n defined by the equations f 0 , . . . ,

f n . Its blow-up X → X is the graph Γ f → P n of f ([D], §1.4).
For each nonempty subset J {0, . . . , n} such that the (n + 1) × |J| matrix A J constructed from the columns of A corresponding to the elements of J has no zero rows, we obtain a linear space contained in B by setting x j = 0 for all j ∈ J. Its codimension in P n is |J|. Moreover, B red is the union of all such linear spaces.

Theorem 1. Let f : P n P n be a dominant transformation defined by monomials of degree d with no common factors. If the degree of f is not divisible by d, the dimension of the indeterminacy locus of f is at least (n -2)/2. The condition on the degree is necessary, as shown by the morphism (x 0 , . . . , x n ) → (x d 0 , . . . , x d n ) of degree d n (whose indeterminacy locus is empty).

Proof. Since the determinant of the matrix A is nonzero we may assume, upon permuting its rows and columns, that we have a ii = 0 for all i ∈ {0, . . . , n}.

We then define an oriented graph on the set of vertices {0, . . . , n} by adding an oriented edge from i to j whenever a i j = 0. We then say that a vertex x is equivalent to a vertex y if and only if there exists an oriented path from x to y and an oriented path from y to x. This defines a partition of the set {0, . . . , n} into equivalence classes (note that x is equivalent to x since a xx = 0).

Say that an equivalence class X is greater than or equal to an equivalence class Y if there is an oriented path from an element of X to an element of Y (there exists then an oriented path from any element of X to any element of Y ). This defines a partial order on the set of equivalence classes.

Choose a class X minimal for this order. Entries of A in a row corresponding to an element of X which are not in a column corresponding to an element of X are then 0 (otherwise, at least one oriented edge should come out of a X to element not in X, contradicting the minimality of X). It follows that the determinant of the submatrix A X of A corresponding to rows and columns of X divides the determinant of A. The sum of all entries in a row of A X is d hence, by the same reasoning used in the proof of Proposition 1, the determinant of A X is nonzero, divisible by d.

Because of the condition d ∤ deg( f ) and Proposition 1, the determinant of A is not divisible by d 2 . In particular, our partial order has a unique minimal element X.

Without loss of generality, we may assume 0 ∈ X. Every other vertex then has an oriented path to 0. In particular, we may define an acyclic function ϕ : {1, . . . , n} → {0, 1, . . . , n} such that (x, f (x)) is an oriented edge of our graph for all x ∈ {1, . . . , n} ("acyclic" means that for all x ∈ {1, . . . , n}, there exists k > 0 such that f k (x) = 0).

We keep only the n edges of the type (x, f (x)); since a x f (x) = 0, they correspond to n nonzero entries, off the diagonal, in each row 1, . . . , n. Since our new graph on {0, . . . , n} has n edges and no cycles, we may color its vertices in black or white in such a way that x and f (x) have different colors, for all x ∈ {0, . . . , n}.

We select the vertices of the color which has been used less often (if both colors have been used the same number of times, we select the vertices with the same color as 0). If 0 is not selected, we add it to the selection. We end up with at most (n + 2)/2 selected vertices which are all on one of our n edges or the loop at 0.

Consider the submatrix of A formed by the ≤ (n + 2)/2 columns corresponding to the selected vertices. In each row, there is a nonzero entry: in the row 0, because 0 was selected and a 00 = 0; in any other row x because either x was selected and a xx = 0, or f (x) was selected and a x f (x) = 0. This proves the theorem. ⊓ ⊔ Example 1. The bound in the theorem is sharp: for d ≥ 2, one easily checks that the indeterminacy locus of the birational automorphism ([J], Example 2)

f n,d : (x 0 , . . . , x n ) → (x d 0 , x d-1 0 x 1 , x d-1 1 x 2 , . . . , x d-1 n-1 x n ) (1)
of P n has dimension exactly ⌈(n -2)/2⌉. But there are many other examples of birational monomial automorphisms of P n with indeterminacy locus of dimension exactly ⌈(n -2)/2⌉, such as monomial maps defined by matrices

A =                 d 0 • • • • • • • • • • • • • • • • • • 0 d -1 1 0 • • • • • • • • • • • • • • • 0 0 d -1 1 0 • • • • • • • • • • • • 0 a 30 a 31 a 32 1 0 • • • • • • • • • 0 0 0 0 d -1 1 0 • • • • • • 0 a 50 a 51 a 52 a 53 a 54 1 0 • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 1                
where, for each odd i, we choose a i0 = 0 and ∑ i-1 j=0 a i j = d -1. The (reduced) indeterminacy locus is then defined by the equations

x 0 = x 1 x 2 = x 3 x 4 = • • • = 0. It has dimension n -1 -⌊n/2⌋ = ⌈(n -2)/2⌉.
Another set of examples is provided by matrices of the form

                  1 1 1 d -3 0 • • • • • • • • • • • • 0 0 1 1 d -2 0 • • • • • • • • • • • • 0 d -1 0 1 0 0 • • • • • • • • • • • • 0 a 30 a 31 0 a 33 0 • • • • • • • • • • • • 0 a 40 a 41 a 42 a 43 1 0 • • • • • • • • • 0 0 0 0 0 d -1 1 0 • • • • • • 0 a 60 a 61 a 62 a 63 a 64 a 65 1 0 • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 1                  
where a 30 + a 31 + a 33 = d ≥ 3, a 31 a 33 = 0, and, for each even i, we choose a i0 a i2 = 0 and ∑ i-1 j=0 a i j = d -1.

Degrees of a monomial map

Let g : P n P n be a rational map. One defines the ith degree d i (g) as the degree of the image by g of a general P i ⊂ P n (more precisely, d i (g) := P n-i • f * P i ). One has d 0 (g) = 1, d n (g) = deg(g), and d 1 (g) is the integer d(g) defined earlier (i.e., the common degree of the components g 0 , . . . , g n of g, provided they have no common factors). An alternative definition of the d i (g) is as follows:

if Γ g ⊂ P n × P n is the graph of g, d i (g) = Γ g • p * 1 P i • p * 2 P n-i . (2)
The sequence d 0 (g), . . . , d n (g) is known to be a log-concave sequence: it satisfies ∀i ∈ {1, . . . , n -1}

d i (g) 2 ≥ d i+1 (g)d i-1 (g)
(this is a direct consequence of the Hodge Index Theorem; [D], (1.6)). This implies

d i (g) ≤ d 1 (g) i .
Proposition 2. Let f : P n P n be a dominant map defined by monomials of degree d with no common factors. Set c := ⌊n/2⌋ + 1. If the degree of f is not divisible by d, we have, for all i ∈ {c, . . . , n},

d i ( f ) ≤ (1 -d -c ) i-1 c-1 d i .
Proof. The degrees of f can be expressed in terms of the Segre class of its indeterminacy locus B. In particular, if c ′ := codim(B), one has ([D], Proposition 2.3.1) By log-concavity, this implies that for i ≥ c, one has

d i ( f ) = d i for i < c ′ , d i -deg s (B) for i = c ′ , (3)
d i ( f ) ≤ d c ( f ) i-1 c-1 d 1-i-1 c-1 ≤ (d c -1) i-1 c-1 d c-i c-1 = (1 -d -c ) i-1 c-1 d i .
This proves the proposition. ⊓ ⊔ When g is birational, i.e., when d n (g) = 1, it follows from (2) that d i (g -1 ) = d n-i (g) for all i ∈ {0, . . ., n}. In particular,

d(g -1 ) = d n-1 (g) ≤ d(g) n-1 .
By (3), equality occurs exactly when the indeterminacy locus of g is finite.

When f is a monomial birational transformation of P n , Proposition 2 gives the stronger bound:

d( f -1 ) ≤ (1 -d -c ) n-2 c-1 d n-1 = d n-1 - n -2 ⌊n/2⌋ d ⌊(n-3)/2⌋ + O(d -2 ), (4)
where d := d( f ). However, as mentioned in the introduction, this is not optimal. When d( f ) = 2, the set of possible values for d( f -1 ) is {2, . . . , n} and the maximal value n is obtained only (up to permutation of the factors) for the birational map f n,2 of (1) ([CS], Theorem 2.6). In particular, the other degrees of f are then fixed.

Johnson's calculations. When d := d( f ) > 2, Johnson's computer calculations in [J] suggest that the maximal possible value for d( f -1 ) should be

d( f -1 n,d ) = (d -1) n -1 d -2 = d n-1 -(n -2)d n-2 + O(d n-3 )
and that equality should only be attained when (up to permutation of the factors) f = f n,d . More precisely, Johnson checks that when n = 4 and 3 ≤ d ≤ 5, one has

d( f -1 ) ≤ d( f -1 n,d ) -d + 1 if (up to permutation of the factors) f = f n,d .
There are also further gaps in the list of possible values for d( f -1 ).

Mixed volumes. The degrees d i ( f ) of a monomial map f can be interpreted in terms of mixed volumes of polytopes in R n as follows. Let ∆ ⊂ R n be the standard n-dimensional simplex conv(0, e 1 , . . . , e n ). Let f : P n P n be a monomial map with associated matrix A = (a i j ) 0≤i, j≤n , and let ∆ f ⊂ R n be the simplex which is the convex hull of the points a i = (a i1 , . . . , a in ) ∈ N n , for i ∈ {0, . . . , n}. Then ([D], §3.5) d i ( f ) = MV(∆ , . . . , ∆ n-i times , ∆ f , . . . , ∆ f i times

).

The right-hand side of this equality is a mixed volume: if the n-dimensional volume is normalized so that vol(∆ ) = 1/n!, this is (ni)!i! times the coefficient of u n-i v i in the polynomial vol(u∆ + v∆ f ), where u∆ + v∆ f is the Minkowski sum {ux + vy | x ∈ ∆ , y ∈ ∆ f }.

Although mixed volumes are notoriously difficult to compute, there are computer programs such as PHCpack (available on Jan Verschelde's webpage) that can do that. We should also mention the article [A], which expresses the degrees of a monomial rational transformation in terms of integrals over an associated Newton region.

  where deg s (B) is the sum of the degrees of the top-dimensional components of B, counted with their Samuel multiplicity (this is larger than the " usual" multiplicity ([F], Examples 4.3.4 and 4.3.5.(c)); in particular deg s (B) ≥ deg(B)). Since in our case c ′ ≤ c = n -⌈(n -2)/2⌉ (Theorem 1), it follows from log-concavity that we have d c ( f ) < d c .


Olivier Debarre
email: e.macri@northeastern.edu

ON THE PERIOD MAP FOR POLARIZED HYPERK ÄHLER FOURFOLDS

Keywords: 2010 Mathematics Subject Classification. 14C34, 14E07, 14J50, 14J60 Hyperkähler fourfolds, Birational isomorphisms, Torelli Theorem, Period domains, Noether-Lefschetz loci, Cones of divisors

We study smooth projective hyperkähler fourfolds that are deformations of Hilbert squares of K3 surfaces and are equipped with a polarization of fixed degree and divisibility. They are parametrized by a quasi-projective irreducible 20-dimensional moduli space and Verbitksy's Torelli theorem implies that their period map is an open embedding.

Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. We also prove that infinitely many Heegner divisors in a given period space have the property that their general points correspond to fourfolds which are isomorphic to Hilbert squares of a K3 surfaces, or to double EPW sextics.

In two appendices, we determine the groups of biregular or birational automorphisms of various projective hyperkähler fourfolds with Picard number 1 or 2.

Introduction

We consider smooth projective hyperkähler fourfolds X which are deformations of Hilbert squares of K3 surfaces (one says that X is of K3 [2] -type). The abelian group H 2 (X, Z) is free of rank 23 and it is equipped the Beauville-Bogomolov-Fujiki form q X , a non-degenerate Zvalued quadratic form of signature (3,20) [START_REF] Beauville | Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF]Théorème 5]). A polarization H on X is the class of an ample line bundle on X that is primitive (i.e., non-divisible) in the group H 2 (X, Z). The square of H is the positive even integer 2n := q X (H) and its divisibility is the integer γ ∈ {1, 2} such that H • H 2 (X, Z) = γZ (the case γ = 2 only occurs when n ≡ -1 (mod 4)).

Smooth polarized hyperkähler fourfolds (X, H) of K3 [2] -type of degree 2n and divisibility γ admit an irreducible quasi-projective coarse moduli space M (γ) 2n of dimension 20. The period map (see Section 3.2)

℘ (γ) 2n : M (γ) 2n -→ P (γ) 2n
is algebraic and it is an open embedding by Verbitsky's Torelli Theorem 3.2. Our main result is that the image of ℘ (γ) 2n is the complement of a finite union of Heegner divisors (this can also be deduced from the general results in [AV]) which can be explicitly listed (Theorem 6.1).

The main ingredient in the proof is the explicit determination of the nef and movable cones of smooth projective hyperkähler fourfolds of K3 [2] -type (see Theorem 5.1). This is a simple consequence of previous results by Markman ([M2]), ), Bayer-Hassett-Tschinkel ([BHT]), and Mongardi ([Mo]).

The Noether-Lefschetz locus is the inverse image by the period map in M (γ) 2n of the union of all Heegner divisors. Its irreducible components were shown in [START_REF] Bergeron | The Noether-Lefschetz conjecture and generalizations[END_REF]Theorem 1.5] to generate (over Q) the Picard group of M (γ) 2n . As an application of our Theorem 5.1, we study in Section 7 birational isomorphisms between some of these components. In particular, we show that points corresponding to Hilbert squares of K3 surfaces are dense in the moduli spaces M (γ) 2n .

In the two appendices, we collect results on biregular and birational automorphisms of certain projective hyperkähler fourfolds with Picard number 1 or 2. These results are needed in some of the arguments in Section 7.

Since the nef and movable cones can be described in all dimensions, many of our results extend with some modifications to smooth projective hyperkähler manifolds of K3 [n] -type. More details in the higher dimensional case will appear in [D2].

Acknowledgements. We would like to thank Ekaterina Amerik, Arend Bayer, Samuel Boissière, Christian Lehn, Eyal Markman, Kieran O'Grady, and Emmanuel Ullmo for useful discussions and suggestions.

Lattices

A lattice is a free abelian group Λ of finite rank endowed with a Z-valued non-degenerate quadratic form q. It is even if q only takes even values. We extend q to a Q-valued quadratic form on Λ ⊗ Q, hence also on the dual

Λ ∨ := Hom Z (Λ, Z) = {x ∈ Λ ⊗ Q | ∀y ∈ Λ x • y ∈ Z}.
The discriminant group of Λ is the finite abelian group

D(Λ) := Λ ∨ /Λ.
The lattice Λ is unimodular if the group D(Λ) is trivial. If x is a non-zero element of Λ, we define the integer div Λ (x) (the divisibility of x) as the positive generator of the subgroup x • Λ of Z. We also consider x/div Λ (x), a primitive (i.e., non-zero and non-divisible) element of Λ ∨ , and its class x * = [x/div Λ (x)] in the group D(Λ), an element of order div Λ (x).

If t is a non-zero integer, we let Λ(t) be the lattice (Λ, tq). We let I 1 be the lattice Z with the quadratic form q(x) = x 2 and we let U (the hyperbolic plane) be the even unimodular lattice Z 2 with the quadratic form q(x 1 , x 2 ) = 2x 1 x 2 . There is a unique positive definite even unimodular lattice of rank 8, which we denote by E 8 .

Assume now that the lattice Λ is even. Following [Ni], we define a quadratic form q : D(Λ) → Q/2Z by setting q([x]) := q(x) ∈ Q/2Z. The stable orthogonal group O(Λ, q) is the kernel of the canonical map O(Λ, q) -→ O(D(Λ), q). This map is surjective when Λ is indefinite and its rank is at least the minimal number of generators of the finite abelian group D(Λ) plus 2 ( [START_REF] Nikulin | Integral symmetric bilinear forms and some of their applications[END_REF]Theorem 1.14.2]).

We will use the following classical result (see [START_REF]Moduli spaces of irreducible symplectic manifolds[END_REF]Lemma 3.5]).

Theorem 2.1 (Eichler's criterion). Let Λ be an even lattice that contains at least two orthogonal copies of U. The O(Λ, q)-orbit of a primitive vector x ∈ Λ is determined by the integer q(x) and the element x * of D(Λ).

3. Moduli spaces, period spaces, and period maps 3.1. Moduli spaces. Let X be a (smooth) hyperkähler (also called irreducible symplectic) fourfold of K3 [2] -type (see [START_REF]Moduli of K3 surfaces and irreducible symplectic manifolds[END_REF]Section 3] for the main definitions). The lattice (H 2 (X, Z), q X ) defined in the introduction is isomorphic to the even lattice

Λ K3 [2] := U ⊕3 ⊕ E 8 (-1) ⊕2 ⊕ I 1 (-2)
with signature (3, 20) and discriminant group Z/2Z. The divisibility of a primitive element is therefore 1 or 2, and O(Λ

K3 [2] ) = O(Λ K3 [2] ).
As recalled in the introduction, hyperkähler fourfolds X of K3 [2] -type with a polarization of fixed (positive) square 2n and divisibility γ in the lattice (H 2 (X, Z), q X ) have a quasiprojective coarse moduli space M (γ) 2n which is irreducible and 20-dimensional when γ = 1, or when γ = 2 and n ≡ -1 (mod 4) ([GHS3, Remark 3.17]).

3.2. Period spaces and period maps. By Eichler's criterion (Theorem 2.1), primitive elements of the lattice Λ K3 [2] with fixed positive square 2n and fixed divisibility γ ∈ {1, 2} form a single O(Λ K3 [2] )-orbit. We fix one such element h 0 .

If γ = 1, we have

(1) h ⊥ 0 ≃ U ⊕2 ⊕ E 8 (-1) ⊕2 ⊕ I 1 (-2) ⊕ I 1 (-2n) =: Λ (1) K3 [2]
,2n , a lattice with discriminant group Z/2nZ × Z/2Z, with q(1, 0) = -1 2n and q(0, 1) = -1 2 (see the proof of Proposition 4.1).

If γ = 2, we have n ≡ -1 (mod 4) and

(2)

h ⊥ 0 ≃ U ⊕2 ⊕ E 8 (-1) ⊕2 ⊕ -2 -1 -1 -n+1 2 =: Λ (2) K3 [2] ,2n ,
a lattice with discriminant group Z/nZ, with q(1) = -2 n (see the proof of Proposition 4.1). We now describe the period map for polarized hyperkähler fourfolds of K3 [2] -type. The complex variety

Ω h 0 := {x ∈ P(Λ K3 [2] ⊗ C) | x • h 0 = 0, x • x = 0, x • x > 0}
has two connected components, interchanged by complex conjugation, which are Hermitian symmetric domains of type IV. It is acted on by the group

O(Λ K3 [2] , h 0 ) := {Φ ∈ O(Λ K3 [2] ) | Φ(h 0 ) = h 0 }.
By results of Baily-Borel and Griffiths, the quotient

P (γ) 2n := O(Λ K3 [2] , h 0 )\Ω h 0 is
an irreducible quasi-projective variety and the period map

(3) ℘ (γ) 2n : M (γ) 2n -→ P (γ) 2n
is algebraic. Alternatively, one has

Ω h 0 ≃ {x ∈ P(Λ (γ) K3 [2] ,2n ⊗ C) | x • x = 0, x • x > 0}
and the group O(Λ K3 [2] , h 0 ) can be identified with the stable orthogonal group O(Λ [START_REF]Moduli spaces of irreducible symplectic manifolds[END_REF]Proposition 3.12 and Corollary 3.13]).

(γ) K3 [2] ,2n ) ([

The full orthogonal group O(Λ

(γ) K3 [2] ,2n ) also acts on Ω h 0 , hence the quotient group O(Λ (γ) K3 [2] ,2n )/ O(Λ (γ) K3 [2] ,2n ) ≃ O(D(Λ (γ) K3 [2] ,2n
)) acts on the period space P (γ) 2n (where -Id acts trivially). We determine this group and describe this action in the next proposition, assuming for simplicity that n is odd. For any non-zero integer r, we denote by ρ(r) the number of prime factors of r.

Proposition 3.1. Assume that n is odd. The period space P (γ) 2n is acted on generically freely by the following groups:

• if n ≡ 1 (mod 4) (so that γ = 1), by the group (Z/2Z) max{ρ(n),1} ;

• if n ≡ -1 (mod 4), by the group (Z/2Z) ρ(n)-1 . Proof. Case γ = 1. Since n is odd, (1) implies D(Λ (γ) K3 [2] ,2n ) ≃ Z/nZ × Z/2Z × Z/2Z
. This decomposition is still orthogonal for q and the values of q at the points of order 2 are q(0, 0, 1) = -1 2 , q(0, 1, 0) = -n 2 , and q(0, 1, 1) = -n+1 2 in Q/2Z. When n ≡ -1 (mod 4), these three values are all distinct and any isometry Φ of (D(Λ (1) K3 [2] ,2n ), q) must therefore be the identity on both Z/2Z factors, hence must preserve their orthogonal Z/nZ. Write Φ(1, 0, 0) = (a, 0, 0); since q(1, 0, 0) = -2 n , we have

2 n = 2a 2 n (mod 2Z), hence a 2 = 1 (mod n). Since n is odd, the group O(D(Λ (1) K3 [2] ,2n
)) is therefore isomorphic to (Z/2Z) ρ(n) . The proposition follows since only -Id acts trivially.

When n ≡ 1 (mod 4), there are extra isometries given by (0, 1, 0) ↔ (0, 0, 1), (1, 0, 0) → (a, 0, 0), where a 2 ≡ 1 (mod n) (and when n = 1, we have -Id = Id).

Case γ = 2 (hence n ≡ -1 (mod 4)). We have D(Λ

(γ) K3 [2] ,2n ) ≃ Z/nZ, with q(1) = -2
n , and we proceed as in the first case. This proves the proposition.

3.3. The Torelli theorem. Verbitsky's Torelli theorem takes the following form ( [V], [START_REF]Moduli of K3 surfaces and irreducible symplectic manifolds[END_REF]Theorem 3.14]).

Theorem 3.2 (Verbitsky). For each positive integer n and each divisibility γ ∈ {1, 2}, the period map

℘ (γ) 2n : M (γ) 2n -→ P (γ) 2n
is an open embedding.

In particular, the commuting involutions of P (γ) 2n described in Proposition 3.1 induce rational involutions on the moduli space M (γ) 2n . In the case n = γ = 1 (double EPW sextics; see Example 6.3), the unique non-trivial involution was described geometrically in [O1] in terms of projective duality.

Special polarized hyperkähler fourfolds

A hyperkähler fourfold corresponding to a very general point of M (γ) 2n has Picard number 1. The Noether-Lefschetz locus (or special locus) is the subset of M (γ) 2n corresponding to hyperkähler fourfolds with Picard number at least 2. It can be described as follows.

Let K be a primitive, rank-2, signature-(1, 1) sublattice of Λ K3 [2] containing the class h 0 chosen in Section 3.2. The codimension-2 subspace P(K ⊥ ⊗ C) in P(Λ K3 [2] ⊗ C) cuts out an irreducible hypersurface in Ω h 0 whose image in P (γ) 2n will be denoted by D

D (γ) 2n,d := disc(K ⊥ )=-d D (γ) 2n,K ⊂ P (γ) 2n
of Heegner divisors is finite, hence it is either empty or of pure codimension 1. Following Hassett, we say that the polarized hyperkähler fourfolds whose period point is in

D (γ) 2n,d are special of discriminant d (the lattice K ⊥ has signature (2, 19), hence d is positive). We use the notation C (γ) 2n,d := (℘ (γ) 2n ) -1 (D (γ) 2n,d ) ⊂ M (γ) 2n .
We now describe the irreducible components of the loci D (γ) 2n,d (the case n = 3 and γ = 2 was originally studied by Hassett in [H] and the case n = γ = 1 in [DIM]).

2n,2e is irreducible, except when n ≡ 1 (mod 4) and e ≡ 1 (mod 4), or when n ≡ -1 (mod 4) and e ≡ 0 (mod 4), in which cases D

(1) 2n,2e has two irreducible components.

(2) Assume moreover n ≡ -1 (mod 4). (1) 2n,2e is reducible, its two components are exchanged by one of the involutions of the period space described in Proposition 3.1 when n ≡ 1 (mod 4), but not when n ≡ -1 (mod 4) (in that case, these involutions are in fact trivial when n is prime).

Proof of Proposition 4.1. Case γ = 1. Let (u, v) be a standard basis for a hyperbolic plane U contained in Λ K3 [2] and let ℓ be a basis for the I 1 (-2) factor. We may take h 0 := u + nv (it has the correct square and divisibility), in which case

h ⊥ 0 = Z(u -nv) ⊕ Zℓ ⊕ M, where M := {u, v, ℓ} ⊥ = U ⊕2 ⊕ E 8 (-1) ⊕2 is unimodular. The discriminant group D(h ⊥ 0 ) ≃ Z/2Z ×
Z/2nZ is generated by ℓ * = ℓ/2 and (u -nv) * = (u -nv)/2n, with q(ℓ * ) = -1/2 and q((u -nv) * ) = -1/2n. Let κ be a generator of K ∩ h ⊥ 0 . We write κ = a(u -nv) + bℓ + cw, where w ∈ M is primitive. Since K has signature (1, 1), we have κ 2 < 0 and the formula from [START_REF]Moduli of K3 surfaces and irreducible symplectic manifolds[END_REF]Lemma 7.5] reads (4)

d = disc(K ⊥ ) = κ 2 disc(h ⊥ 0 ) s 2 = 8n(na 2 + b 2 + mc 2 ) s 2 ≡ 8n(na 2 + b 2 ) s 2 (mod 8n),
where m := -1 2 w 2 and s := gcd(2na, 2b, c) is the divisibility of κ in h ⊥ 0 . If s | b, we obtain d ≡ 2 2na s 2 (mod 8n), which is the first case of (1)(a): d is even and e := d/2 is a square modulo 4n. Assume s ∤ b and, for any non-zero integer x, write x = 2 v 2 (x) x odd , where x odd is odd. One has then ν 2 (s) = ν 2 (b) + 1 and

d ≡ 2 2na s 2 + 2n b odd s odd 2 ≡ 2 2na s 2 + 2n (mod 8n),
which is the second case of (1)(a): d is even and d/2 -n is a square modulo 4n. It is then easy, taking suitable integers a, b, c, and vector w, to construct examples that show that these necessary conditions on d are also sufficient, thereby proving (1)(a). We now prove (1)(b) and (1)(c). Given a lattice K containing h 0 with disc(K ⊥ ) = -2e, we let as above κ be a generator of K ∩ h ⊥ 0 . By Eichler's criterion (Theorem 2.1), the group O(h ⊥ 0 ) acts transitively on the set of primitive vectors κ ∈ h ⊥ 0 of given square and fixed κ * ∈ D(h ⊥ 0 ). Since κ and -κ give rise to the same lattice K (obtained as the saturation of Zh 0 ⊕ Zκ), the locus D

(1) 2n,2e will be irreducible (when non-empty) if we show that the integer e determines κ 2 , and κ * up to sign.

We write as above κ = a(u-nv)+bℓ+cw ∈ h ⊥ 0 , with gcd(a, b, c) = 1 and s = div h ⊥ 0 (κ) = gcd(2na, 2b, c). From (4), we get (5) κ 2 = -es 2 /2n = -2(na 2 + b 2 + mc 2 ) and κ * = (2na/s, 2b/s) ∈ Z/2nZ × Z/2Z.

If s = 1, we have e ≡ 0 (mod 4n) and κ * = 0. If s = 2, the integer c is even and a and b cannot be both even (because κ is primitive). We have e = n(na 2 + b 2 + mc 2 ) and   

  e ≡ n 2 (mod 4n) and κ * = (n, 0) if b is even (and a is odd); e ≡ n (mod 4n) and κ * = (0, 1) if b is odd and a is even; e ≡ n(n + 1) (mod 4n) and κ * = (n, 1) if b and a are odd.

Assume now that n is square-free and n | e. From (4), we get n | 2na s 2 , hence s 2 | 4na 2 , and s | 2a because n is square-free. This implies s = gcd(2a, 2b, c) ∈ {1, 2}.

When n is even (i.e., n ≡ 2 (mod 4)), we see from the discussion above that both s (hence also κ 2 ) and κ * are determined by e, so the corresponding hypersurfaces D

(1) 2n,2e are irreducible.

If n is odd, there are coincidences:

• when n ≡ 1 (mod 4), we have n ≡ n 2 (mod 4n), hence D

(1) 2n,2e is irreducible when e ≡ 0 or 2n (mod 4n), has two irreducible components (corresponding to κ * = (n, 0) and κ * = (0, 1)) when e ≡ n (mod 4n), and is empty otherwise; • when n ≡ -1 (mod 4), we have n(n + 1) ≡ 0 (mod 4n), hence D

(1) 2n,2e is irreducible when e ≡ -n or n (mod 4n), has two irreducible components (corresponding to κ * = 0 and κ * = (n, 1)) when e ≡ 0 (mod 4n), and is empty otherwise. This proves (1)(b).

We now assume that n is prime and prove (1)(c). Since s | 2n, we have s ∈ {1, 2, n, 2n}; the cases s = 1 and s = 2 were explained above. If s = n (and n is odd), we have n | b, n | c, n ∤ a, and e ≡ 4a 2 (mod 4n) and κ * = (2a, 0).

If s = 2n, the integer c is even, a and b cannot be both even, n | b, and n ∤ a. We have

    
e ≡ a 2 (mod 4n) and κ * = (a, 0) if 2n | b (hence a is odd); e ≡ a 2 + n (mod 4n) and κ * = (a, 1) if b is odd (and n is odd); e ≡ a 2 + 2 (mod 8) and κ * = (a, 1) if 4 ∤ b is odd and n = 2.

When n = 2, one checks that the class of e modulo 8 (which is in {0, 1, 2, 3, 4, 6}) completely determines s, and κ * up to sign. The corresponding divisors D

(1) 4,2e are therefore all irreducible.

When n ≡ 1 (mod 4), we have n ≡ n 2 (mod 4n) and a 2 ≡ (n -a) 2 + n (mod 4n) when a is odd (in which case a 2 ≡ 1 (mod 4)). When n ≡ -1 (mod 4), we have n(n + 1) ≡ 0 (mod 4n) and a 2 ≡ (n -a) 2 + n (mod 4n) when a is even (in which case a 2 ≡ 0 (mod 4)). Together with changing a into -a (which does not change the lattice K), these are the only coincidences: the corresponding hypersurfaces D

(1) 2n,2e therefore have two components and the others are irreducible. This proves (1)(c).

Case γ = 2 (hence n ≡ -1 (mod 4)). We may take h

0 := 2 u + n+1 4 v + ℓ, in which case h ⊥ 0 = Zw 1 ⊕ Zw 2 ⊕ M, with w 1 := v + ℓ and w 2 := -u + n+1 4 v. The matrix of the intersection form on Zw 1 ⊕ Zw 2 is -2 -1 -1 - n+1 2
as in (2) and the discriminant group

D(h ⊥ 0 ) ≃ Z/nZ is generated by (w 1 -2w 2 ) * = (w 1 -2w 2 )/n, with q((w 1 -2w 2 ) * ) = -2/n. Let (h 0 , κ ′ ) be a basis for K, so that disc(K) = 2nκ ′2 -(h 0 • κ ′ ) 2 . Since div(h 0 ) = γ = 2, the integer h 0 • κ ′ is even and since κ ′2 is also even (because Λ K3 [2] is an even lattice), we have 4 | disc(K) and -disc(K)/4 is a square modulo n. Since the discriminant of Λ K3 [2] is 2, the integer d = | disc(K ⊥ )| is either 2 | disc(K)| or 1 2 | disc(K)|, hence it is even and e = d/2 is a square modulo n, as desired.
Conversely, it is easy to construct examples that show that these necessary conditions on d are also sufficient. This proves (2)(a).

We now prove (2)(b) and (2)(c). To prove that the loci D

(2) 2n,2e are irreducible (when non-empty), we need to show that e determines κ 2 , and κ * up to sign (where κ is a generator of K ∩ h ⊥ 0 ).

With the notation above, we have

κ = ((h 0 • κ ′ )h 0 -2nκ ′ )/t, where t := gcd(h 0 • κ ′ , 2n) is even and κ 2 = 2n t 2 disc(K). Formula (4) then gives 2e = disc(K ⊥ ) = κ 2 disc(h ⊥ 0 ) div h ⊥ 0 (κ) 2 = 2n 2 disc(K) t 2 div h ⊥ 0 (κ) 2 .
Since n is odd and t is even, and, as we saw above, disc(K) ∈ {-e, -4e}, the only possibility is disc(K) = -4e and t div h ⊥ 0 (κ) = 2n. Assume that n is square-free and n | e. Since -4e

= disc(K) = 2nκ ′2 -(h 0 • κ ′ ) 2 , we get 2n | (h 0 • κ ′ ) 2 hence
, since n is square-free and odd, 2n | h 0 • κ ′ . This implies t = 2n and div h ⊥ 0 (κ) = 1; in particular, κ * = 0 and κ 2 = -2e/n are uniquely determined. This proves (2)(b).

We now assume that n is prime. Since t div h ⊥ 0 (κ) = 2n and t is even,

• either (t, div h ⊥ 0 (κ), κ 2 ) = (2n, 1, -2e/n) and n | e; • or (t, div h ⊥ 0 (κ), κ 2 ) = (2, n, -2ne) and n ∤ e (because n ∤ h 0 • κ ′ and d = -1 2 disc(K) ≡ 1 2 (h 0 • κ ′ ) 2 (mod n))
. Given e = a 2 + nn ′ , the integer κ 2 is therefore uniquely determined by e:

• either n | a, κ 2 = -2e/n, and κ * = 0;

• or n ∤ a, κ 2 = -2ne, κ * = κ/n, and q(κ * ) = -2a 2 /n (mod 2Z). In the second case, κ * = ±a(w 1 -2w 2 ) * ; it follows that in all cases, κ * is also uniquely defined, up to sign, by e. This proves (2)(c).

5.

The nef cone of a projective hyperkähler fourfold of K3 [2] -type Cones of divisors on projective hyperkähler manifolds of K3 [n] -type were described in [START_REF] Bayer | Mori cones of holomorphic symplectic varieties of K3 type[END_REF][START_REF]MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations[END_REF][START_REF]A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF][START_REF] Mongardi | A note on the Kähler and Mori cones of hyperkähler manifolds[END_REF]. When n = 2, these results take a very special form.

Let X be a projective hyperkähler fourfold of K3 [2] -type. The positive cone

Pos(X) ⊂ Pic(X) ⊗ R is the connected component of the open subset {x ∈ Pic(X) ⊗ R | x 2 > 0} containing the class of an ample divisor. The movable cone Mov(X) ⊂ Pic(X) ⊗ R
is the (not necessarily open nor closed) convex cone generated by classes of movable divisors (i.e., those divisors whose base locus has codimension at least 2). We have inclusions Int(Mov(X)) ⊂ Pos(X) of the interior of the movable cone into the positive cone, and Amp(X) ⊂ Mov(X) of the ample cone into the movable cone. We set

Div X := {a ∈ Pic(X) | a 2 = -2}, Flop X := {a ∈ Pic(X) | a 2 = -10, div H 2 (X,Z) (a) = 2}.
Given a divisor class a ∈ Pic(X) ⊗ R, we denote by H a the hyperplane Proof. Statement (a) follows from the general result [START_REF]A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Lemma 6.22]. We sketch instead the proof of (b).

H a := {x ∈ Pic(X) ⊗ R | x • a = 0}. Theorem 5.1. Let X be a hyperkähler fourfold of K3 [2] -type. (a) The interior Int(Mov(X)) of the movable cone is the connected component of Pos(X)
There is an extension H 2 (X, Z) ⊂ Λ X of lattices and weight-2 Hodge structures, where the lattice Λ X is isomorphic to the lattice U ⊕4 ⊕E 8 (-1) ⊕2 and the orthogonal H 2 (X, Z) ⊥ ⊂ Λ X is generated by a primitive vector v X of square 2 ([M2, Section 9], [BHT, Section 1]). We denote by Λ alg,X the algebraic (i.e., (1, 1)-type) part of Λ X , so that Pic(

X) = v ⊥ X ∩ Λ alg,X . Finally, we set Flop ′ X := {ã ∈ Λ alg,X | ã2 = -2, ã • v X = 1}. The dual statement to [BHT, Theorem 1] is then the following: the ample cone Amp(X) is the connected component of Int(Mov(X)) ã∈Flop ′ X H ã
containing the class of an ample divisor, where the hyperplane H ã is defined as before by

H ã := {x ∈ Pic(X) ⊗ R | x • ã = 0}.
We notice that the actual statement of [START_REF] Bayer | Mori cones of holomorphic symplectic varieties of K3 type[END_REF]Theorem 1] says that we need to exclude the hyperplanes H a , where a 2 ≥ -2 and |a • v X | ≤ 1. We may in fact only consider classes with a 2 = -2, as explained in [START_REF]MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations[END_REF]Sections 12 and 13].

Given a class ã ∈ Flop ′ X , we let a := 2ã-v X . Then a ∈ Flop X and H a = H ã. Conversely, given a ∈ Flop X , we let b := a + v X ∈ Λ alg . Since div H 2 (X,Z) (a) = 2, we have b = 2ã, and ã ∈ Flop ′ X with H ã = H a .
This proves (b). Remark 5.2. We can make the description in Theorem 5.1 more precise. (a) As explained in [M2, Section 6], it follows from [M3] that there is a group of reflections W Exc acting on Pos(X). Using the Zariski decomposition ( [Bo]), one shows ([M2, Lemma 6.22]) that W Exc acts faithfully and transitively on the set of connected components of Pos(X) a∈Div X H a .

In particular, Mov(X) ∩ Pos(X) is a fundamental domain for the action of W Exc on Pos(X). (b) By [START_REF] Matsushita | On almost holomorphic Lagrangian fibrations[END_REF]Proposition 2.1] (see also [START_REF]Moving and ample cones of holomorphic symplectic fourfolds[END_REF]Theorem 7]), each connected component of

Int(Mov(X))

a∈Flop X H a corresponds to the ample cone of a hyperkähler fourfold X ′ of K3 [2] -type via a birational map X X ′ which is a composition of Mukai flops with respect to numerically equivalent Lagrangian planes ([WW, Theorem 1.1]). (c) By [A,Proposition 4] (generalized to the twisted case in [Hu2, proof of Proposition 4.1]) and [START_REF] Bayer | Projectivity and birational geometry of Bridgeland moduli spaces[END_REF][START_REF]MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations[END_REF], if Div X = ∅, the fourfold X is isomorphic to a moduli space M of stable sheaves on a possibly twisted K3 surface (S, α). The moduli space M is birational to the Hilbert square of a K3 surface if there exists a ∈ Div X with div H 2 (X,Z) (a) = 2; otherwise, M is birational to a moduli space of possibly twisted rank-2 torsion-free sheaves. (d) Similarly, if there exists a class w ∈ Pic(X) with w 2 = 0, the fourfold X is birational to a moduli space M of torsion sheaves on a (possibly twisted) K3 surface (S, α). If the divisor class w is also nef and primitive, X is actually isomorphic to such an M and the Beauville integral system f :

X ∼ → M → P 2 is a Lagrangian fibration on X such that w = [f * O P 2 (1)].
Before discussing a few examples of Theorem 5.1 when Pic(X) has rank 2, we briefly review Pell-type equations (see [N, Chapter VI]). Given non-zero integers e and t with e > 0, we denote by P e (t) the equation ( 6) ). Let (S, L) be a polarized K3 surface such that Pic(S) = ZL and L 2 =: 2e. Then Pic(S [2] ) = ZL 2 ⊕ Zδ, where L 2 is the class on S [2] induced by L and 2δ is the class of the divisor in S [2] that parametrizes non-reduced length-2 subschemes of S ([B1, Remarque, p. 768]). In the lattice (H 2 (S [2] , Z), q S [2] ), we have the following products

a 2 -eb 2 = t,
L 2 2 = 2e , δ 2 = -2 , L 2 • δ = 0.
Cones of divisors on S [2] can be described as follows.

(a) The extremal rays of the (closed) movable cone Mov(S [2] ) are spanned by L 2 and L 2 -µ e δ, where

• if e is a perfect square, µ e = √ e;

• if e is not a perfect square and (a 1 , b 1 ) is the minimal solution of the equation P e (1), µ e = e b1 a 1 . (b) The extremal rays of the nef cone Nef(S [2] ) are spanned by L 2 and L 2 -ν e δ, where

• if the equation P 4e (5) is not solvable, ν e = µ e ;

• if the equation P 4e (5) is solvable and (a 5 , b 5 ) is its minimal solution, ν e = 2e b 5 a 5 . 2

Example 5.4. Let n be a positive integer such that n ≡ -1 (mod 4). Let (X, H) be a polarized hyperkähler fourfold of K3 [2] -type with H of divisibility 2 and Pic(X) = ZH ⊕ ZL, with intersection matrix 2n 0 0 -2e ′ . Since any two embeddings of the lattice

K = I 1 (2n) ⊕ I 1 (-2e ′ ) into Λ K3 [2]
for which the image of a generator of I 1 (2n) has divisibility 2 differ by an isometry of Λ K3 [2] , 3 they represent very general elements of one component of the special divisor C

(2) 2n,2e ′ n (we will prove in Theorem 6.1 that they exist if and only if n > 0 and e ′ > 1). We assume in the rest of this example that n is square-free. The hypersurface C

(2) 2n,2e ′ n is then irreducible by Proposition 4.1(2)(b) and very general elements of C

(2) 2n,2e ′ n are of the type described above. Cones of divisors on X can be described as follows (we set e := e ′ n).

(a) The extremal rays of the closure of the movable cone Mov(X) are spanned by ) is its minimal solution, ν n,e = a -5n 2e ′ b -5n . To prove these statements, it is enough to notice that, in the notation of Theorem 5.1, a class in Div X corresponds to a solution to the equation P e (-n); similarly, a class in Flop X corresponds to a solution to the equation P 4e (-5n). The description of the cones of divisors on X then follows from Theorem 5.1 by a direct computation.

The image of the period map

The description of the cones of divisors for hyperkähler fourfolds of K3 [2] -type given in Section 5 easily implies our main result on the images of their period maps. Theorem 6.1. Let n be a positive integer and let γ ∈ {1, 2}. The image of the period map

℘ (γ) 2n : M (γ) 2n -→ P (γ) 2n
is exactly the complement of the union of finitely many Heegner divisors. More precisely, these Heegner divisors are

• if γ = 1,
some irreducible components of the hypersurface D 3 In the notation of the second part of the proof of Proposition 4.1 (case γ = 2), a generator of I 1 (2n) can be sent to the class h 0 ; a generator of I 1 (-2e ′ ) is then sent to the class κ ′ = κ. We have t := gcd(h 0 • κ ′ , 2n) = 2n and the formula t div h ⊥ 0 (κ) = 2n implies div h ⊥ 0 (κ) = 1, i.e., κ * = 0 in D(K ⊥ ). We then apply Eichler's criterion again in K ⊥ and conclude by using the isomorphism

O(Λ K3 [2] , h 0 ) ≃ O(K ⊥ ).
• if γ = 2 (and n ≡ -1 (mod 4)), one irreducible component of the hypersurface D

(2) 2n,2n . Remark 6.2. Assume that n is square-free (so in particular n ≡ 0 (mod 4)). We proved in Proposition 4.1 that

• the hypersurface D (1)
2n,2n has two components if n ≡ 1 (mod 4), one component otherwise;

• the hypersurface D

(1) 2n,8n has two components if n ≡ -1 (mod 4), one component otherwise;

• the hypersurface D

(1) 2n,10n has two components if n ≡ 1 (mod 4), one component otherwise;

• the hypersurface D

(1) 2n,2n/5 has two components if n ≡ 1 (mod 4), one component otherwise;

• the hypersurface D

(2) 2n,2n is irreducible (when n ≡ -1 (mod 4)). Furthermore, it follows from the proof of the theorem that when moreover n ≡ ±5 (mod 25),

• the hypersurface D

(1) 2n,2n/5 is irreducible.

Proof of Theorem 6.1. Take a point x ∈ P (γ) 2n . Since the period map for smooth compact (not necessarily projective) hyperkähler fourfolds is surjective ([Hu1, Theorem 8.1]), there exists a compact hyperkähler fourfold X ′ with the given period point x. Since the class h 0 is algebraic and has positive square, X ′ is projective by [START_REF] Huybrechts | Compact hyperkähler manifolds: Basic results[END_REF]Theorem 3.11]. Moreover, the class h 0 corresponds to the class of an integral divisor H in the positive cone of X ′ . By Remark 5.2(a), we can let an element in the group W Exc act and assume that the pair (X ′ , H), representing the period point x and the class h 0 , is such that H is in Mov(X ′ ) ∩ Pos(X ′ ). By Remark 5.2(b), we can find a projective hyperkähler fourfold X which is birational to X ′ (hence still has period x), such that the divisor H, with class h 0 , is nef and big on X and has divisibility γ. Note that, since X ′ is birational to X, it is deformation equivalent to X ([Hu1, Theorem 4.6]), hence still of K3 [2] -type.

To summarize, the point x is in the image of the period map ℘ (γ) 2n if and only if H is actually ample on X. We now use Theorem 5.1: H is ample if and only if it is not orthogonal to any algebraic class either with square -2, or with square -10 and divisibility 2.

If H is orthogonal to an algebraic class κ with square -2, the Picard group of X contains a rank-2 lattice K with intersection matrix 2n 0 0 -2 ; the fourfold X is therefore special of discriminant 2e := -disc(K ⊥ ) (its period point is in the hypersurface D (γ) 2n,K ). If γ = 1, the divisibility s := div h ⊥ 0 (κ) is either 1 or 2. By (5), we have es 2 = -2nκ 2 = 4n, hence

• either s = 1, e = 4n, and κ * = 0: the period point is then in one irreducible component of the hypersurface D

(1) 2n,8n ; • or s = 2, e = n, and either κ * = (0, 1); -or κ * = (n, 0) and n ≡ 1 (mod 4); -or κ * = (n, 1) and n ≡ 0 (mod 4).

The period point x is in one irreducible component of the hypersurface D

(1) 2n,2n if n ≡ 2 or 3 (mod 4), or in the union of two such components otherwise.

If γ = 2, we have e = -disc(K)/4 = n, t = 2n disc(K)/κ 2 = 2n, and div(κ) = 2n/t = 1, hence κ * = 0: the period point x is in one irreducible component of the hypersurface D

(2) 2n,2n . If H is orthogonal to an algebraic class with square -10 and divisibility 2, the Picard group of X contains a rank-2 lattice K with intersection matrix 2n 0 0 -10 , hence X is special of discriminant 2e := -disc(K ⊥ ). Again, we distinguish two cases, keeping the same notation.

If γ = 1, the divisibility s := div K ⊥ (κ) is even (because the divisibility in H 2 (X, Z) is 2) and divides κ 2 = -10, hence it is either 2 or 10 . Moreover, es 2 = -2nκ 2 = 20n, hence

• either s = 2, e = 5n, and κ * = (0, 1): the period point x is then in one irreducible component of the hypersurface D

(1) 2n,10n ; • or s = 10 and e = n/5: the period point is then in the hypersurface D

(1) 2n,n/5 . In the second case, since the divisibility of κ in H 2 (X, Z) is 2, a and c are even, so that b is odd and κ * = (a, 1). We have 10 = s = gcd(2na, 2b, 2c), hence b and c are divisible by 5, but not a, because gcd(a, b, c) = 1. We have e ≡ a 2 + n (mod 4n), hence e ≡ a 2 ≡ ±1 (mod 5).

In general, there are many possibilities for a = 2a ′ , with a ′2 ≡ e (mod 5e). However, if n is square-free, e divides a ′ and (a ′ /e) 2 ≡ 1 (mod 5), so that a ≡ ±2e (mod 2n). It follows that ±a (hence also ±κ * ) is well determined (modulo 2n), so we have a single component of D

(1) 2n,n/5 .

If γ = 2, we have e = -disc(K)/4 = 5n and t 2 = 2n disc(K)/κ 2 = n 2 /10, which is impossible.

Conversely, in each case described above, it is easy to construct a class κ with the required square and divisibility which is orthogonal to H.

Example 6.3 (Double EPW sextics: n = γ = 1). Double EPW sextics were defined in [O2] as ramified double covers of certain singular sextic hypersurfaces in P 5 . When smooth, they are hyperkähler fourfolds of K3 [2] -type with a polarization of degree 2. They fill out a dense open subset U

∪ H

(1) 2 . Example 6.4 (Varieties of lines on cubic fourfolds: n = 3 and γ = 2). If W ⊂ P 5 is a smooth cubic fourfold, the variety F (W ) of lines contained in W is a hyperkähler fourfold and its Plücker polarization has square 6 and divisibility 2 ( [BD], [START_REF] Hassett | Special cubic fourfolds[END_REF]Proposition 2.1.2]). These

4 O'Grady's hypersurfaces S ′ 2 ∪ S ′′ 2 , S 4 , S ⋆ 2 , are our D (1) 2,2 , D (1)
2,4 , D

2,8 .

fourfolds fill out a dense open subset U

(2) 6 of M

(2) 6

whose complement contains an irreducible hypersurface H

(2) 6 whose general points correspond to pairs (S [2] , 2L 2 -δ), where (S, L) is a polarized K3 surface of degree 2 (see Proposition 7.9). Theorem 6.1 and Remark 6.2 say that the image of M

(2) 6 in the period space is the complement of the irreducible hypersurface D

(2) 6,6 . This (and much more) was first proved by Laza in [La, Theorem 1.1], together with the fact that M

(2) 6 = U (2) 6 ∪ H (2) 6 ; since H (2) 6 maps onto D (2) 6,2 , the image of U (2) 6 is the complement of D (2) 6,2 ∪ D (2)
6,6 .

Unexpected isomorphisms between hyperkähler fourfolds

In this section, we study birational isomorphisms between components of various Noether-Lefschetz loci induced by "unexpected" isomorphisms between hyperkähler fourfolds. We treat first the case of Hilbert squares. 7.1. Special hyperkähler fourfolds isomorphic to Hilbert squares of K3 surfaces. If a polarized hyperkähler fourfold is isomorphic to the Hilbert square of a K3 surface, it is special in the sense defined in Section 4. We use standard notation for cohomology classes on a Hilbert square (see Example 5.3). The slope ν e was defined in the same example and the special loci C If K 2e is the moduli space of polarized K3 surfaces of degree 2e, the rational map

̟ : K 2e M (γ) 2n (S, L) -→ (S [2] , bL 2 -aδ),
where γ = 2 if b is even, and γ = 1 if b is odd, induces a birational isomorphism onto an irreducible component of C (γ) 2n,2e .

Proof. If (S, L) is a polarized K3 surface of degree 2e and

K := ZL 2 ⊕ Zδ ⊂ H 2 (S [2] , Z), the lattice K ⊥ is the orthogonal in H 2 (S, Z) of the class L. Since the lattice H 2 (S, Z) is unimodular, K ⊥ has discriminant -2e, hence S [2] is special of discriminant 2e.
The class H := bL 2 -aδ has divisibility γ and square 2n. It is primitive, because gcd(a, b) = 1, and ample on S [2] when Pic(S) = ZL because of the inequality in (7). Therefore, the pair (S [2] , H) corresponds to a point of C (γ) 2n,2e . The map ̟ therefore sends a very general point of K 2e to C (γ) 2n,2e . To prove that ̟ is generically injective, we assume to the contrary that there is an isomorphism ϕ : L) and (S ′ , L ′ ) are not isomorphic. It is straightforward to check that this implies ϕ * δ ′ = δ and that the extremal rays of the nef cone of S [2] are spanned by the primitive classes L 2 and ϕ * L ′ 2 . Comparing this with the description of the nef cone given in Example 5.3, we see that e is not a perfect square,

S [2] ∼ → S ′[2] such that ϕ * (bL ′ 2 -aδ ′ ) = bL 2 -aδ, although (S,
ϕ * L ′ 2 = a 1 L 2 -eb 1 δ and ϕ * (a 1 L ′ 2 -eb 1 δ ′ ) = L 2
, where (a 1 , b 1 ) is the minimal solution to the Pell equation P e (1).

The same proof as that of Theorem B.1 implies e > 1, the equation P e (-1) is solvable and the equation P 4e (5) is not. By Theorem B.1 again, S [2] has a non-trivial involution σ and (ϕ • σ) * (L ′ 2 ) = L 2 and (ϕ • σ) * (δ ′ ) = δ. This implies that ϕ • σ is induced by an isomorphism (S, L) ∼ →(S ′ , L ′ ), which contradicts our hypothesis. The map ̟ is therefore generically injective and since K 2e is irreducible of dimension 19, its image is a component of C (γ) 2n,2e . Remark 7.2. Assume that n is prime. The locus C

(2) 2n,2e is irreducible by Proposition 4.1. Therefore, under the assumptions of Proposition 7.1 and when b is even, we have a birational isomorphism K 2e ∼ C

(2) 2n,2e . When e > 61, the varieties K 2e are known to be of general type ( [START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF]), hence so is C

(2) 2n,2e . More precise results on the geometry of the varieties C

(2) 6,2e can be found in [Nu, TV, L].

Example 7.3. Assume n = 1. Under the assumptions of Proposition 7.1, b is odd. The locus C

(1) 2,2e has either one or two components, according to whether e is even or odd (Proposition 4.1). If e is odd, we have e > 1 and one checks that the image of ̟ is the component of C

(1) 2,2e denoted by C

(1)′′ 2,2e in [START_REF]Periods of double EPW-sextics[END_REF]Section 4.3]. Therefore, we have birational isomorphisms

̟ : K 2e ∼ C (1) 2,2e if e is even; C (1)′′ 2,2e
if e is odd. dominated by K 2e are actually isomorphic to a Hilbert square S [2] ; however, some generality condition on X is needed: the varieties of lines of some smooth cubic fourfolds of discriminant 14 (n = 3, γ = 2, e = 7) are not isomorphic to the Hilbert square of a K3 surface.

We deduce from Proposition 7.1 a characterization of Hilbert squares of general polarized K3 surfaces that are isomorphic to double EPW sextics.

Corollary 7.6. Let e be an integer such that e ≥ 3 and let (S, L) be a general polarized K3 surface of degree 2e. The following conditions are equivalent:

(i) the equation P e (-1) is solvable and the equation P 4e (5) is not;

(ii) the equation P e (-1) has a positive solution (a, b) such that a b < ν e ; (iii) the Hilbert square S [2] is isomorphic to a double EPW sextic of discriminant 2e; (iv) the variety S [2] has a non-trivial automorphism. When these conditions are realized, S [2] then has a non-trivial involution σ, the quotient S [2] /σ is an EPW sextic Y ⊂ P 5 , and the complete linear system |bL 2 -aδ| defines a morphism which factors as S [2] ։ S [2] /σ = Y ֒→ P 5 . Proof. The equivalence (i) ⇔ (iv) is Theorem B.1. The implication (iv) ⇒ (ii) comes from the facts that the equation P e (-1) has a minimal solution (a -1 , b -1 ) and, if σ is the nontrivial automorphism of S [2] (Theorem B.1), the class b -1 L 2 -a -1 δ is positively proportional to L 2 + σ * L 2 , hence ample. The implication (ii) ⇒ (iii) is Proposition 7.1 and Example 6.3. The implication (iii) ⇒ (iv) is obvious. The consequences stated at the end follow from [O4, Section 4], which explains why dim(|H|) = 5, where H is the canonical polarization on the double EPW sextic.

Remark 7.7. When e = 2, all the conditions of Corollary 7.6 hold except for (iii). The fourfold S [2] carries the non-trivial Beauville involution σ (Example B.2) and the complete linear system |L 2 -δ| defines a morphism which factors as S [2] ։ S [2] /σ 3:1 ։ Gr(2, 4) ֒→ P 5 . This fits with the fact that 3 Gr(2, 4) is a (degenerate) EPW sextic ([O5, Claim 2.14]).

Example 7.8. When e = 13, the equivalent conditions of Corollary 7.6 are satisfied, hence the Hilbert square of a general polarized K3 surface (S, L) of degree 26 is a double EPW sextic, with canonical involution σ. Moreover, two positive solutions, (7, 2) and (137, 38), of the equation P 13 (-3) satisfy the conditions (7) of Proposition 7.1 with b even. It follows that S [2] is also isomorphic to a general element of C

(2) 6,26 , i.e., to the variety of lines F (W ) on a special cubic hypersurface W ⊂ P 5 of discriminant 26 (the two isomorphisms S [2] ≃ F (W ) differ by σ, and σ * (2L 2 -7δ) = 38L 2 -137δ).

We now show that given any positive integer n, Proposition 7.1 applies to infinitely many integers e. Proposition 7.9. Let n be a positive integer. There are infinitely many distinct hypersurfaces in the moduli spaces M

(1) 2n , and M

(2) 2n if n ≡ -1 (mod 4), whose general points correspond to Hilbert squares of K3 surfaces. In both cases, the union of these hypersurfaces is dense in the moduli space for the euclidean topology. Sketch of proof. When m > 0, the pair (m, 1) is a solution of the equation P e (-n), with e = m 2 + n, and one easily checks that the inequality m < ν e holds when (n, m) = (1, 2).

When m ≥ 0, the pair (2m + 1, 2) is a solution of the equation P e (-n), with e = m 2 + m + n+1 4 and one easily checks that the inequality m + 1 2 < ν e holds when (n, m) = (3, 1). Finally, the density statement follows from a powerful result of Clozel and Ullmo (Theorem 7.10 below).

Theorem 7.10 (Clozel-Ullmo). The union of infinitely many Heegner divisors in any moduli space M (γ) 2n is dense for the euclidean topology.

Proof. This follows from the main result of [CU]: the space M (γ) 2n is a Shimura variety and each Heegner divisor D x is a "strongly special" subvariety, hence is endowed with a canonical probability measure µ Dx . Given any infinite family (D xa ) a∈N of Heegner divisors, there exists a subsequence (a k ) k∈N , a strongly special subvariety Z ⊂ M (γ) 2n which contains D xa k for all k ≫ 0 such that (µ Dx a k ) k∈N converges weakly to µ Z ( [START_REF] Clozel | Équidistribution de sous-variétés spéciales[END_REF]th. 1.2]). For dimensional reasons,

we have Z = M (γ) 2n ; this implies that a D xa is dense in M (γ)
2n . Remark 7.11. It was proved in [MM] that Hilbert schemes of projective K3 surfaces are dense in the coarse moduli space of all (possibly non-algebraic) hyperkähler manifolds of K3 [n] -type. 7.2. Isomorphisms between various special hyperkähler fourfolds. We now apply a similar construction with the polarized hyperkähler fourfolds (X, H) studied in Example 5.4, whose notation we keep. For the sake of simplicity, we assume that n is square-free; these fourfolds then correspond to points of the irreducible hypersurface C

(2) 2n,2e ′ n . Proposition 7.12. Let n, m, and e be positive integers. Assume that n is square-free, n ≡ -1 (mod 4), n | e, and n = e. Assume further that the equation P e (nm) has a solution (na, b) with a > 0 that satisfies the conditions

(8) |b| a < ν n,e and gcd(a, b) = 1.
There is a rational map

̟ : C (2) 2n,2e M (γ) 2m (X, H) -→ (X, aH + bL),
where γ = 2 if b is even, and γ = 1 if b is odd. This map induces a birational isomorphism onto an irreducible component of

C (γ) 2m,2e .
In the proposition, the locus C

(2) 2n,2e is non-empty and irreducible by Proposition 4.1 and Theorem 6.1.

Proof. The proof is the same as that of Proposition 7.1 and is based on the fact that if (X, H) corresponds to a very general point of C

(2) 2n,2e , the class aH + bL is primitive, has square 2a 2 n -2b 2 e ′ = 2m and divisibility γ, and is ample on X because of the inequality in (8). Therefore, the pair (X, aH + bL) corresponds to a point of C (γ) 2m,2e . To prove that ̟ is generically injective, assume that there is an isomorphism ϕ : X ∼ → X ′ such that ϕ * (aH ′ + bL ′ ) = aH + bL. If ϕ * H ′ = H, the matrix of ϕ * in the bases (H ′ , L ′ ) and (H, L) is that of a non-trivial isometry with a fixed vector, hence a reflection.

As we will see during the proof of Proposition A.3, the matrix of such an isometry that extends to an isometry between H 2 (X ′ , Z) and H 2 (X, Z) must be of the form 2s 2 e ′ +1 -2e ′ rs 2nrs -(2s 2 e ′ +1) , where (nr, s) is a solution to the equation P e (n), both equations P e (-n) and P 4e (-5n) are not solvable, and e is not a perfect square. We then have ϕ * (rH ′ + sL ′ ) = rH + sL. Since ϕ * (aH ′ + bL ′ ) = aH + bL and aH ′ + bL ′ is primitive, we must have m = 1 (and a = r, b = s). In that case, by Proposition A.3, X does have an involution σ that acts as on Pic(X) as the reflection with axis spanned by rH + sL. The isomorphism ϕ • σ : X ∼ → X ′ then pulls back H ′ to H. This proves the proposition.

Example 7.13. Under the assumptions of Proposition 7.12, when m is prime and b is even, the locus C

(2) 2m,2e is irreducible by Proposition 4.1. Therefore, there is a birational isomorphism C

(2) 2n,2e ∼ C

(2) 2m,2e . Example 7.14. Assume m = 1. As in Example 7.3, we have, under the assumptions of Proposition 7.12, birational isomorphisms

̟ : C (2) 2n,2e ∼ C (1) 2,2e if e is even; C (1)′′ 2,2e
if e is odd.

Remark 7.15. Given a pair (a, b) that satisfies the conditions (8), we can construct two maps ̟ ± by sending (X, H) either to (X, aH + bL) or to (X, aH -bL). These two maps are distinct unless there exists an automorphism ϕ of X that sends aH + bL to aH -bL. One checks using the computations of the proof of Proposition A.3 that this is only possible when we are in case (a) of that proposition, ϕ * acts as a rotation 2s 2 e+1 2e ′ rs 2nrs 2s 2 e+1 on Pic(X), with r 2 -es 2 = 1, and moreover, a = r, b = ns, and m = n. The maps

̟ ± : C (2) 2n,2e ∼ -→ C (2)
2n,2e then correspond to changing the polarization by an automorphism of X: they are just particular cases of an infinite family of such maps.

As in Section 7.2, we characterize which of our special hyperkähler fourfolds are isomorphic to double EPW sextics.

Corollary 7.16. Let n and e be positive integers. Assume that n is square-free, n ≡ -1 (mod 4), n | e, and n = e. Let (X, H) be a polarized hyperkähler fourfold corresponding to a general point of C

(2) 2n,2e . Then X is isomorphic to a double EPW sextic if and only if the equation P e (n) is solvable but the equation P 4e (-5n) is not.

Under the hypotheses of the corollary, the automorphism group of X is isomorphic to Z ⋊ Z/2Z hence contains infinitely many involutions (σ m ) m∈Z (Proposition A.3). When X is very general, all the quotients X/σ m are EPW sextics.

Proof. We may assume that (X, H) is very general in C

(2) 2n,2e . If X is isomorphic to a double EPW sextic, it has a non-trivial automorphism and the conclusion follows from Proposition A.3. Conversely, if the equation P e (n) is solvable but the equation P 4e (-5n) is not, one checks that e is not a perfect square, hence X has, by Proposition A.3, a non-trivial involution σ (and in fact, countably many such involutions) that fixes a square-2 class rH + sL which is positively proportional to H + ν n,e L + σ * (H + ν n,e L), hence ample. By Proposition 7.12, the pair (X, rH +sL) is a general element of C

(1) 2,2e , hence X is a double EPW sextic by Example 6.3 (note that e ≥ 2n ≥ 6).

Remark 7.17. Assume that both equations P 4e (-5n) and P e (n) are solvable. As in Remark A.5, let X ′ be the other birational model of a general X in an irreducible component of C

(2) 2n,2e . Then X ′ is isomorphic to a double EPW sextic by the same proof as above.

Finally, we show that given any positive integer n, Proposition 7.12 applies to infinitely many integers e.

Corollary 7.18. Let n be a positive square-free integer such that n ≡ -1 (mod 4). There are infinitely many distinct hypersurfaces in the moduli space M

(2) 2n whose general points correspond to double EPW sextics. Their union is dense in M

(2) 2n .

Proof. When m > 0 and e = n(nm 2 -1), the pair (nm, 1) is a solution to the equation P e (n).

If 5 | n, we will show in the proof of Proposition A.3 that P 4e (-5n) is not solvable. If 5 ∤ n, we can choose m such that n(nm 2 -1) ≡ ±2 (mod 5) and by reducing modulo 5, we see that the equation P 4e (-5n) is then not solvable.

We can therefore apply Corollary 7.16. Since there are infinitely many such m, this concludes the proof, using Theorem 7.10 for the density statement. of X is 1, unless n ∈ {1, 3, 23}. These three cases are actual exceptions: we just saw that all fourfolds corresponding to points of M (1) 2 carry a non-trivial biregular involution; there is a 10-dimensional subfamily of M

(2) 6 whose elements consists of fourfolds that have a biregular automorphism of order 3 and whose very general elements have Picard number 1 ([BCS, Section 7.1]); there is a (unique) fourfold in M

(2) 46 with Picard number 1 and a biregular automorphism of order 23 ( [START_REF] Boissière | Isometries of ideal lattices and hyperkähler manifolds[END_REF]Theorem 1.1]).

We now turn our attention to the polarized hyperkähler fourfolds studied in Example 5.4. Proposition A.3. Let n be a positive square-free integer such that n ≡ -1 (mod 4). Let (X, H) be a polarized hyperkähler fourfold of K3 [2] -type of degree 2n and divisibility 2, such that Pic(X) = ZH ⊕ ZL, with intersection matrix 2n 0 0 -2e ′ . Set e := e ′ n. (a) If neither equations P e (-n) and P 4e (-5n) are solvable and e is not a perfect square, the groups Aut(X) and Bir(X) are equal. They are infinite cyclic, except when the equation P e (n) is solvable, in which case these groups are isomorphic to the infinite dihedral group Z ⋊ Z/2Z. (b) If the equation P e (-n) is not solvable but the equation P 4e (-5n) is, the group Aut(X) is trivial and the group Bir(X) is infinite cyclic, except when the equation P e (n) is solvable, in which case it is infinite dihedral. (c) If the equation P e (-n) is solvable or if e is a perfect square, the group Bir(X) is trivial.

Proof. We saw that the map Ψ A X : Aut(X) → O(H 2 (X, Z)) is injective. Its image consists of isometries which preserve Pic(X) and the ample cone and, since b 2 (X) -ρ(X) is odd, restrict to ± Id on Pic(X) ⊥ ([Og2, proof of Lemma 4.1]). Conversely, by the Torelli Theorem 3.2, any isometry with these properties is in the image of Ψ A X . We begin with some general remarks on the group G of isometries of H 2 (X, Z) which preserve Pic(X) and the components of the positive cone, and restrict to ε Id on Pic(X) ⊥ , with ε ∈ {-1, 1}.

The orthogonal group of the rank-2 lattice (Pic(X), q X ) ≃ I 1 (2n) ⊕ I 1 (-2e ′ ) is easily determined: if we let δ := gcd(n, e ′ ) and we write n = δn ′ and e ′ = δe ′′ , we have

O(Pic(X), q X ) = a αe ′′ b n ′ b αa a, b ∈ Z, a 2 -n ′ e ′′ b 2 = 1, α ∈ {-1, 1} .
Note that α is the determinant of the isometry and • such an isometry preserves the components of the positive cone if and only if a > 0; we denote the corresponding subgroup by O + (Pic(X)); • when e is not a perfect square, the group SO + (Pic(X)) is infinite cyclic, generated by the isometry R corresponding to the minimal solution to the equation P n ′ e ′′ (1) and the group O + (Pic(X)) is infinite dihedral; • when e is a perfect square, so is n ′ e ′′ = e/δ 2 , and O + (Pic(X)) = {Id, ( 1 0 0 -1 )}. As we saw during the proof of Proposition 4.1, there exist standard bases (u 1 , v 1 ) and (u 2 , v 2 ) for two orthogonal hyperbolic planes in Λ K3 [2] , a generator ℓ for the I 1 (-2) factor, and an isometric identification H 2 (X, Z) ∼ → Λ K3 [2] such that

H = 2u 1 + n + 1 2 v 1 + ℓ and L = u 2 -e ′ v 2 .
The elements Φ of G must then have a > 0 and satisfy

Φ(2u 1 + n+1 2 v 1 + ℓ) = a(2u 1 + n+1 2 v 1 + ℓ) + n ′ b(u 2 -e ′ v 2 ) Φ(u 2 -e ′ v 2 ) = αe ′′ b(2u 1 + n+1 2 v 1 + ℓ) + αa(u 2 -e ′ v 2 ) Φ(v 1 + ℓ) = ε(v 1 + ℓ) Φ(u 1 -n+1 4 v 1 ) = ε(u 1 -n+1 4 v 1 ) Φ(u 2 + e ′ v 2 ) = ε(u 2 + e ′ v 2 )
(the last three lines correspond to vectors in Pic(X) ⊥ ). From this, we deduce

nΦ(v 1 ) = 2(a -ε)u 1 + (a + ε) n+1 2 -ε v 1 + (a -ε)ℓ + n ′ b(u 2 -e ′ v 2 ) 2Φ(u 2 ) = 2αe ′′ bu 1 + αe ′′ b n+1 2 v 1 + αe ′′ bℓ + (ε + αa)u 2 + e ′ (ε -αa)v 2 2e ′ Φ(v 2 ) = -2αe ′′ bu 1 -αe ′′ b n+1 2 v 1 -αe ′′ bℓ + (ε -αa)u 2 + e ′ (ε + αa)v 2 .
From the first equation, we get δ | b and a ≡ ε (mod n); from the second equation, we deduce that e ′′ b and ε + αa are even; from the third equation, we get 2δ | b and a ≡ αε (mod 2e ′ ). All this is equivalent to a > 0 and

(10) 2δ | b , a ≡ ε (mod n) , a ≡ αε (mod 2e ′ ).
Conversely, if these conditions are realized, one may define Φ uniquely on Zu 1 ⊕ Zv 1 ⊕ Zu 2 ⊕ Zv 2 ⊕ Zℓ using the formulas above, and extend it by ε Id on the orthogonal of this lattice in Λ K3 [2] to obtain an element of G.

The first congruence in (10) tells us that the identity on Pic(X) extended by -Id on its orthogonal does not lift to an isometry of H 2 (X, Z). This means that the restriction G → O + (Pic(X)) is injective. Moreover, the two congruences in (10) imply a ≡ ε ≡ αε (mod δ). If δ > 1, since n, hence also δ, is odd, we get α = 1, hence the image of G is contained in SO + (Pic(X)). Assume α = 1. The relations (10) imply that a -ε is divisible by n and 2e ′ , hence by their least common multiple 2δn ′ e ′′ . We write b = 2δb ′ and a = 2δn ′ e ′′ a ′ + ε and obtain from the equality a 2 -n ′ e ′′ b 2 = 1 the relation

4δ 2 n ′2 e ′′2 a ′2 + 4εδn ′ e ′′ a ′ = 4δ 2 n ′ e ′′ b ′2 , hence δn ′ e ′′ a ′2 + εa ′ = δb ′2 .
In particular, a ′′ := a ′ /δ is an integer and b ′2 = a ′′ (ea ′′ + ε).

Since a > 0 and a ′′ and ea ′′ + ε are coprime, both are perfect squares and there exist coprime integers r and s, with r > 0, such that

a ′′ = s 2 , ea ′′ + ε = r 2 , b ′ = rs.
Since -1 is not a square modulo n, we obtain ε = 1; the pair (r, s) satisfies the Pell equation r 2 -es 2 = 1, and a = 2es 2 + 1 and b = 2δrs. In particular, either e is not a perfect square and there are always infinitely many solutions, or e is a perfect square and we get r = 1 and s = 0, so that Φ = Id. Assume α = -1. As observed before, we have δ = 1, i.e., n and e ′ are coprime. Using (10), we may write b = 2b ′ and a = 2a ′ e ′ -ε. Since 2 ∤ n and a ≡ ε (mod n), we deduce gcd(a ′ , n) = 1. Substituting into the equation a 2 -ne ′ b 2 = 1, we obtain a ′ (e ′ a ′ -ε) = nb ′2 , hence there exist coprime integers r and s, with r ≥ 0, such that b ′ = rs, a ′ = s 2 , and e ′ a ′ -ε = nr 2 . The pair (r, s) satisfies the equation nr 2 -e ′ s 2 = -ε, and a = 2e ′ s 2 -ε and b = 2rs. In particular, one of the two equations P e (±n) is solvable. Note that at most one of the equations P e (±n) may be solvable: if P e (-εn) is solvable, εe ′ is a square modulo n, while -1 is not. These isometries are all reflections and, since n ≥ 2 and e ′ ≥ 2, ( 1 0 0 -1 ) is not one of them. In particular, if e is a perfect square, G = {Id}.

We now go back to the proof of the proposition. We proved that the composition Aut(X) → G → O + (Pic(X)) is injective and so is the morphism Bir(X) → G → O + (Pic(X)) (any element of its kernel is in Aut(X)).

Under the hypotheses of (a), both slopes of the nef cone are irrational (Example 5.4). By [Og1, Theorem 1.3], the groups Aut(X) and Bir(X) are then equal and infinite. The calculations above allow us to be more precise: in this case, the ample cone is just one component of the positive cone and the groups Aut(X) and G are isomorphic. The proposition then follows from the discussions above (note that when there are involutions, the equation P e (n) has a solution (nr, s) hence, in the notation above, ε = -1 and these involutions act on H 2 (X, Z) as the symmetries about ample square-2 classes rH + sL).

Under the hypotheses of (c), the slopes of the extremal rays of the nef and movable cones are rational (Example 5.4) hence, by [START_REF] Oguiso | Automorphism groups of Calabi-Yau manifolds of Picard number two[END_REF]Theorem 1.3] again, Bir(X) is a finite group. By [Og1, Proposition 3.1(2)], any non-trivial element Φ of its image in O + (Pic(X)) is an involution which satisfies Φ(Mov(X)) = Mov(X), hence switches the two extremal rays of this cone. This means Φ(H ± µ n,e L) = H ∓ µ n,e L, hence Φ(H) = H, so that Φ = ( 1 0 0 -1 ). Since we saw that this is impossible, the group Bir(X) is trivial.

Under the hypotheses of (b), the slopes of the nef cone are both rational and the slopes of the movable cone are both irrational (Example 5.4). By [Og1, Theorem 1.3] again, Aut(X) is a finite group and Bir(X) is infinite. The same reasoning as in case (c) shows that the group Aut(X) is in fact trivial; moreover, the group Bir(X) is a subgroup of Z, except when the equation P e (n) is solvable, where it is a subgroup of Z ⋊ Z/2Z.

In the latter case, such an infinite subgroup is isomorphic either to Z or to Z⋊Z/2Z and we exclude the first case by showing that there is indeed a regular involution on a birational model of X (this generalizes the case n = 3 and e = 6 treated in [START_REF]Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces[END_REF]). We denote by (na n , b n ) the minimal solution to the equation P e (n) and set

x n := na n + b n √ e ∈ Z[ √ e].
As observed in Remark 5.2(b), the set of all positive solutions (a, b) to the equation na 2 -4e ′ b 2 = -5 (so that aH ± 2bL ∈ Flop X , or equivalently, (na, b) is a solution to the equation P 4e (-5n)) determines an infinite sequence of rays R ≥0 (2e ′ bH ± naL) in Mov(X) and the nef cones of hyperkähler fourfolds birational to X can be identified with the chambers with respect to this collection of rays. For example, if (na 5n , b 5n ) is the minimal solution to the equation P 4e (-5n), the two extremal rays of the cone Nef(X) are spanned by α 0 := 2e ′ b -5n H -na -5n L and α 1 := 2e ′ b -5n H + na -5n L. We want to describe all solutions (a, b). determine the group Aut(S [2] ) of its biregular automorphisms ([BCNS, Theorem 1.1]) and also, using the description of the nef and movable cones (Example 5.3), the group Bir(S [2] ).

Theorem B.1 (Boissière-Cattaneo-Nieper-Wißkirchen-Sarti). Let (S, L) be a polarized K3 surface of degree 2e with Picard group ZL. The variety S [2] has a non-trivial automorphism if and only if either e = 1, or the equation P e (-1) is solvable and the equation P 4e (5) is not.

The non-trivial automorphism in Theorem B.1 is then unique and an anti-symplectic involution. When e ≥ 2, this involution acts on H 2 (S [2] , Z) as the symmetry s D about the line spanned by the square-2 class D := b -1 L 2 -a -1 δ, where (a -1 , b -1 ) is the minimal solution of the equation P e (-1). When e = 1, this involution is induced by an involution of (S, L) and its acts on H 2 (S [2] , Z) as the symmetry about the plane Pic(S [2] ).

Example B.2. Theorem B.1 applies for example for e = m 2 + 1 with m = 2, or e = 13. When e = 2, the surface S is a quartic in P 3 which contains no lines nor conics, and the involution σ of S [2] is the Beauville involution: it sends a pair of points in S to the residual intersection with S of the line that they span. We have D = L 2 -δ and s D (L 2 ) = 3L 2 -4δ ([D1, Théorème 4.1], [START_REF] Boissière | The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface[END_REF]Section 6.1]); the quotient S [2] /σ is a triple cover of the Plücker quadric Gr(2, 4) ⊂ P5 . When e ≥ 3, the quotient S [2] /σ is an EPW sextic (Corollary 7.6).

Proposition B.3. Let (S, L) be a polarized K3 surface of degree 2e with Picard group ZL. The group Bir(S [2] ) is trivial except in the following cases:

• e = 1, or the equation P e (-1) is solvable and the equation P 4e (5) is not, in which cases Aut(S [2] ) = Bir(S [2] ) ≃ Z/2Z; • e > 1, and e = 5 or 5 ∤ e, and both equations P e (-1) and P 4e (5) are solvable, in which case Aut(S [2] ) = {Id} and Bir(S [2] ) ≃ Z/2Z. 5

In the second case, there is a difference between the case e = 5 and the case 5 ∤ e: when 5 ∤ e, there is a hyperkähler fourfold (in fact, a double EPW sextic) birational to S [2] on which the involution is biregular, but not when e = 5.

Proof. If ϕ ∈ Bir(S [2] ) is not biregular, ϕ * acts on the movable cone Mov(S [2] ) in such a way that ϕ * (Amp(S [2] )) ∩ Amp(S [2] ) = ∅. This implies Mov(S [2] ) = Nef(S [2] ) hence, by Example 5.3, the equation P 4e (5) has a minimal solution (a 5 , b 5 ). By Theorem B.1, the group Aut(S [2] ) is then trivial.

Moreover, ϕ * maps one extremal ray of the movable cone (spanned by L 2 ) to the other extremal ray (spanned by the primitive vector a 1 L 2 -eb 1 δ). Therefore, we have ϕ * (L 2 ) = a 1 L 2 -eb 1 δ and, by applying this relation to ϕ -1 , also ϕ * (a 1 L 2 -eb 1 δ) = L 2 . This implies that ϕ * is a completely determined involution of Pic(S [2] ). In particular, ϕ 2 is an automorphism, hence is trivial: ϕ is an involution.

The transcendental lattice Pic(S [2] ) ⊥ ⊂ H 2 (S [2] , Z) carries a simple rational Hodge structure. 6 Since the eigenspaces of the involution ϕ * of H 2 (S [2] , Z) are sub-Hodge structures, the restriction of ϕ * to Pic(S [2] ) ⊥ is ε Id, with ε ∈ {-1, 1}. On Pic(S [2] ), we saw that ϕ * has matrix a 1 b 1 -eb 1 -a 1 in the basis (L 2 , δ). The extension from Pic(S [2] ) ⊕ Pic(S [2] ) ⊥ to the overlattice H 2 (S [2] , Z) of such an involution can be studied as in the proof of Proposition A.3 (see also [START_REF] Boissière | The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface[END_REF]Lemma 5.2] when ε = -1). The conclusion is that there exist positive integers r and s such that r 2 -es 2 = ε and a 1 + b 1 √ e = (r + s √ e) 2 . The value ε = 1 would contradict the minimality of the solution (a 1 , b 1 ) to the equation P e (1). Hence we have ε = -1 and (r, s) is the minimal solution to the equation P e (-1). In particular, ϕ * is a completely determined involution of H 2 (S [2] , Z) and, since Ψ B X is injective, Bir(S [2] ) has at most 2 elements. By [START_REF] Nagell | Introduction to number theory[END_REF]Theorem 110], the solutions to the equation P 4e (5) are all given by ±x 5 x m 1 , m ∈ Z, where x 5 := a 5 + 2b 5 √ e and x 1 := a 1 + b 1 √ e. The associated positive elements of

Z[

√ e] are ordered as follows [START_REF] Nagell | Introduction to number theory[END_REF]Theorem 110],

• • • < x 5 x -2 1 ≤ x 5 x -1 1 < x 5 x -1 1 ≤ x < √ 5 < x 5 ≤ x 5 x 1 < x 5 x 1 ≤ x 5 x 2 1 < x 5 x 2 1 < • • • Still by
(a) either 5 | e, the conjugate solutions are the same, and x 5 = x 5 x 1 ; (b) or 5 ∤ e, the conjugate solutions are different, and x < x 5 x -1 1 . Set x -1 := r + s √ e, so that x 1 = x 2 -1 . In case (a), we have x 5 x -1 = -x 5 x -1 and since x 5 x -1 x 5 x -1 = -5, we obtain e = 5. In that case, there is indeed a non-trivial birational involution on S [2] (Example B.4).

In terms of the rays generated by aL 2 -2ebδ, where (a, b) is a solution to the equation P 4e (5), multiplying x = a + 2b √ e by x 1 corresponds to applying the rotation R := a 1 -b 1 -eb 1 a 1 , which sends the extremal ray R ≥0 L 2 of the movable cone to its other extremal ray R ≥0 (a 1 L 2eb 1 δ); the operation x → xx 1 therefore corresponds to applying the reflection R ( 1 0 0 -1 ) = a 1 b 1

-eb 1 -a 1 , which is the symmetry s D about the line spanned by the class D := b -1 L 2 -a -1 δ Geometrically, the situation is clear in case (b): we have exactly two rays inside Mov(S [2] ) which are symmetric about the line RD. As explained in Remark 5.2(b), the three associated chambers correspond to the hyperkähler fourfolds birational to S [2] : the "middle one" corresponds to a fourfold X whose nef cone is preserved by the involution s D . There is a biregular involution on X which induces a birational involution on S [2] .

Example B.4 (The O'Grady involution). A general polarized K3 surface S of degree 10 is the transverse intersection of the Grassmannian Gr(2, C 5 ) ⊂ P( 2 C 5 ) = P 9 , a quadric Q ⊂ P 9 , and a P 6 ⊂ P 9 . A general point of S [2] corresponds to V 2 , W 2 ⊂ V 5 . Then

Gr(2, V 2 ⊕ W 2 ) ∩ S = Gr(2, V 2 ⊕ W 2 ) ∩ Q ∩ P 6 ∩ 2 (V 2 ⊕ W 2 )) ⊂ P 2
is the intersection of two general conics in P 2 hence consists of 4 points, including [V 2 ] and [W 2 ]. The (birational) O'Grady involution S [2] S [2] takes the pair of points ([V 2 ], [W 2 ]) to the residual two points of this intersection.

  and called a Heegner divisor. The Noether-Lefschetz locus is then the inverse image in M of irreducible hypersurfaces.For each integer d, the union

  Proposition 4.1. Let n and d be positive integers and let γ ∈ {1, 2}. If the locus D (γ) 2n,d is non-empty, the integer d is even; we set e := d/2. (1) (a) The locus D (1) 2n,2e is non-empty if and only if either e or e -n is a square modulo 4n. (b) If n is square-free and e is divisible by n and satisfies the conditions in (a), the locus D (1) 2n,2e is irreducible, except when • either n ≡ 1 (mod 4) and e ≡ n (mod 4n), • or n ≡ -1 (mod 4) and e ≡ 0 (mod 4n), has two irreducible components. (c) If n is prime and e satisfies the conditions in (a), D

  is non-empty if and only if e is a square modulo n. (b) If n is square-free and n | e, the locus D (2) 2n,2e is irreducible. (c) If n is prime and e satisfies the conditions in (a), D (2) 2n,2e is irreducible. Remark 4.2. In cases (1)(b) and (1)(c), when the hypersurface D

  class of an ample divisor. (b) The ample cone Amp(X) is the connected component of Int(Mov(X)) a∈Flop X H a that contains the class of an ample divisor.

  where a and b are integers. A solution (a, b) of this equation is called positive if a > 0 and b > 0. If e is not a perfect square, (a, b) is a solution if and only if the norm N(a + b √ e) in the quadratic number field Q( √ e) is t. The positive solution with minimal a is called the minimal solution; it is also the positive solution (a, b) for which the ratio a/b is minimal when t < 0, maximal when t > 0. Assume that e is not a perfect square. There is always a minimal solution (a 1 , b 1 ) to the Pell equation P e (1) and if x 1 := a 1 + b 1 √ e, all the solutions of the equation P e (1) correspond to the "mth powers" ±x m 1 in Z[ √ e], for m ∈ Z. Example 5.3 ([BM2, Proposition 13.1 and Lemma 13.3] 1

  (two components if n ≡ 0 or 1 (mod 4), one component if n ≡ 2 or 3 (mod 4));one irreducible component of the hypersurface D ; -and, if n ≡ ±5 (mod 25), some irreducible components of the hypersurface D

  to pairs (S[2] , L 2 -δ), where (S, L) is a polarized K3 surface of degree 4 ([O3, Section 5.3]).O'Grady proved that the image of U ([O3, Theorem 1.3] 4 ); moreover, by[START_REF] Debarre | Special prime Fano fourfolds of degree 10 and index 2, Recent Advances in Algebraic Geometry[END_REF] Theorem 8.1], this image does meet all the other components of the non-empty hypersurfaces D

  Let n and e be positive integers. Assume that the equation P e (-n) (see (6)) has a positive solution (a, b) that satisfies the conditions

  Remark 7.4. When e is a perfect square, the positive solutions (a, b) to the equation P e (-n) satisfy a-b √ e = -n ′′ and a+b √ e = n ′ , with n = n ′ n ′′ . This implies a = 1 2 (n ′ -n ′′ ) and b √ e = 1 2 (n ′ + n ′′ ), hence 0 < n ′′ < n ′ . We then have a b = n ′ -n ′′ n ′ +n ′′ √ e < √ e = ν e hence Proposition 7.1 applies to all positive solutions (a, b) of the equation P e (-n) with gcd(a, b) = 1. In particular, when n is odd and n > 1, we obtain a geometric description of the fourfolds corresponding to general points of some component of C (γ) 2n,2, where γ = 1 if n ≡ 1 (mod 4), and γ = 2 if n ≡ -1 (mod 4) (take n ′ = n and n ′′ = 1).Remark 7.5. Under the hypotheses of Proposition 7.1, one can show that all polarized hyperkähler fourfolds (X, H) with Picard number 2 which are in the component of C (γ) 2n,2e

Lemma A. 4 .

 4 The minimal solution to the Pell equation P e (1) is given by y 1 = na 2 n + e ′ b 2 n + 2a n b n √ e and all the solutions (na, b) to the equation P 4e (-5n) are given by the two disjoint families na + 2b √ e = ±x -5n y m 1 or ± x -5n y m 1 , m ∈ Z, where x -5n := na -5n + 2b -5n √ e.

  H -µ n,e L and H + µ n,e L, where • if the equation P e (-n) is not solvable, µ n,e = n/ √ e;

• if the equation P e (-n) is solvable and

(a -n , b -n ) is its minimal solution, µ n,e = a -n e ′ b -n . (b)

The extremal rays of the nef cone Nef(X) are spanned by H -ν n,e L and H + ν n,e L, where • if the equation P 4e (-5n) is not solvable, ν n,e = µ n,e ; • if the equation P 4e (-5n) is solvable and (a -5n , b -5n

Parts of the results of this example were first proved in[START_REF]Moving and ample cones of holomorphic symplectic fourfolds[END_REF] Theorem

22] and the rationality of the nef cone was also proved, by very different methods, in[START_REF] Oguiso | Automorphism groups of Calabi-Yau manifolds of Picard number two[END_REF] Corollary 5.2].2 There is a typo in[START_REF]MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations[END_REF] Lemma 13.3(b)]: one should replace d with 2d.

There are cases where both equations P e (-1) and P 4e (5) are solvable and 5 ∤ e; for example, e = 29.

This is a classical fact found for example in[START_REF]Lectures on K3 surfaces[END_REF] Lemma 3.1].

E. M. was partially supported by the NSF grants DMS-1523496, DMS-1700751, and by a Poincaré Chair from the Institut Henri Poincaré and the Clay Mathematics Institute.

Appendix A. Automorphisms of special hyperkähler fourfolds

We determine the group Aut(X) of biregular automorphisms and the group Bir(X) of birational automorphisms for some hyperkähler fourfolds X of K3 [2] -type with Picard number 1 or 2. The case of Hilbert squares of very general polarized K3 surfaces is in Appendix B.

Let X be a hyperkähler fourfold. There are natural morphisms (9) Ψ A X : Aut(X) → O(H 2 (X, Z), q X ) and Ψ B X : Bir(X) → O(H 2 (X, Z), q X ) which send a (birational) automorphism ϕ of X to its action ϕ * on cohomology (see [START_REF] Gross | Calabi-Yau manifolds and related geometries[END_REF]Proposition 25.14] for Ψ B X ). Elements of Im(Ψ A X ) preserve the nef cone Nef(X), elements of Im(Ψ B X ) preserve the movable cone Mov(X), and both preserve the Picard lattice and the Hodge structure.

The kernel of Ψ B X is contained in Aut(X), hence in the kernel of Ψ A X ([Og1, Proposition 2.4]). The group Ker(Ψ A X ) is a finite group which is invariant by smooth deformations ([HT4, Theorem 2.1]) and is trivial for the Hilbert square of a K3 surface ([B2, Proposition 10]). It follows that for any hyperkähler fourfold X of K3 [2] -type, both Ψ A X and Ψ B X are injective.

Proposition A.1. Let X be a hyperkähler fourfold corresponding to a very general point of a moduli space M (γ) 2n . The group Bir(X) of birational automorphisms of X is trivial, unless n = 1, in which case Aut(X) = Bir(X) ≃ Z/2Z.

Proof. As we saw in Section 4, the Picard group of X is generated by the class h of the polarization. Any birational automorphism leaves this class fixed, hence is in particular biregular of finite order. Let ϕ be a non-trivial automorphism of X. Since ϕ extends to small deformations of X, the restriction of ϕ * to h ⊥ is a homothety whose ratio is, by [B2, Proposition 7], a root of unity; since it is real and non-trivial (by injectivity of Ψ A X ), it must be -Id. We will prove that such an isometry of Zh ⊕ h ⊥ does not extend to an isometry Φ of H 2 (X, Z) unless h 2 = 2n = 2.

When γ = 1, we may take h = u + nv, where (u, v) is a standard basis for a hyperbolic plane U contained in H 2 (X, Z). Then, u -nv is in h ⊥ , hence the isometry Φ, if it exists, must satisfy Φ(u + nv) = u + nv and Φ(u -nv) = -u + nv, which yields 2nΦ(v) = 2u. This is possible only when n = 1. Conversely, in the case n = 1, the fourfold X is a double EPW sextic and does carry a non-trivial involution (Example 6.3). Moreover, this involution is the only non-trivial automorphism of a very general double EPW sextic (see the end of the proof of [START_REF] Debarre | Gushel-Mukai varieties: classification and birationalities[END_REF]Proposition B.9]).

When γ = 2 (so that n ≡ -1 (mod 4)), we let ℓ be an element of H 2 (X, Z) orthogonal to U and such that ℓ 2 = -2. We may take, as in the proof of Proposition 4.1, h = 2u + n+1 2 v + ℓ, and h ⊥ contains v + ℓ and u -n+1 4 v. The isometry Φ must then satisfy

Remark A.2. The conclusion of the proposition does not necessarily hold if we assume only that the Picard number of X is 1. In fact, Proposition A.1 is also proved in [START_REF] Boissière | Complex ball quotients from manifolds of K3 [n] -type[END_REF]Theorem 3.1] and the proof given there implies that Bir(X) is trivial when the Picard number Since x = yx n , this establishes a one-to-one correspondance between the solutions of the equation P e (tn) and those of P e (t). In particular, the minimal solution to the Pell equation P e (1) is given by y

The solutions to the equation P e (-5) were analyzed in [START_REF] Nagell | Introduction to number theory[END_REF]Theorem 110]: if y -5 corresponds to its minimal solution, they are all given by ±y -5 y m 1 , m ∈ Z, and their conjugates. It follows that all the solutions (na, b) to the equation P e (-5n) are given by ±y -5 x n y m 1 and their conjugates. Since the "imaginary" part of y 1 is even and its "real" part is odd, the parity of the "imaginary" parts of these solutions are all the same. Since the equation P 4e (-5n) is solvable, they are all even, and we therefore obtain all the solutions to the equation P 4e (-5n).

To prove that the conjugates provide a disjoint set of solutions, we need to check, by [START_REF] Nagell | Introduction to number theory[END_REF]Theorem 110], that 5 does not divide 4e.

Assume first 5 | e ′ . Since the equation P e (n) is solvable, we have n 5 = 1; moreover, since n ≡ -1 (mod 4), we have e ′ n = -1. The solvability of the equation P 4e (-5n) implies 5 n = e ′ n ; putting all that together contradicts quadratic reciprocity. Assume now 5 | n and set n ′ := n/5. Since the equation P e (n) is solvable, we have e ′ 5 = 1; moreover, since n ′ ≡ -1 (mod 4), we have e ′ n ′ = -1. Since 5 ∤ e ′ , the equation P n ′ ,20e ′ (-1) is solvable, hence 5e ′ n ′ = 1; again, this contradicts quadratic reciprocity.

We can reinterpret this as follows. Since gcd(n, e ′ ) = 1 (because na 2 n -e ′ b 2 n = 1), the generator R of the group SO + (Pic(X)) previously defined is

If we set α i+2 := R(α i ), the lemma means that the infinitely many rays in Mov(X) described above are the (R ≥0 α i ) i∈Z . The fact that the conjugate solutions form a disjoint family means exactly that the ray R ≥0 α 2 is "above" the ray R ≥0 α 1 ; in other words, we have an "increasing" infinite sequence of rays

It follows from the discussion above that the reflection R ( 1 0 0 -1 ) belongs to the group G and preserves the nef cone of the birational model X ′ of X whose nef cone is generated by α 1 and α 2 . It is therefore induced by a biregular involution of X ′ which defines a birational involution of X. This concludes the proof of the proposition.

Remark A.5. It follows from the proof above that in case (b), if both equations P 4e (-5n) and P e (n) are solvable, X has exactly one non-trivial birational model. It is obtained from X by a composition of Mukai flops with respect to Lagrangian planes (Remark 5.2(b)).

Appendix B. Automorphisms of Hilbert squares of very general K3 surfaces

Since the extremal rays of the movable cone of the Hilbert square S [2] of a very general K3 surface S of given degree 2e are rational (Example 5.3), its group Bir(S [2] ) of birational automorphisms is finite ([Og1, Theorem 1.3(2)]). Using the Torelli Theorem 3.2, one can