ON CORRESPONDENCES OF A K3 SURFACE WITH ITSELF

Viacheslav V. Nikulin^{1}

Abstract. Let X be a K3 surface with a polarization H of degree H^{2} = 2rs, r, s ≥ 1. The moduli space of sheaves over X with the primitive isotropic Mukai vector (r, H, s) is again a K3 surface Y. We consider the problem: When Y ≅ X? In this case, we get a 2-dimensional algebraic cycle on the product X × X, and a correspondence in the sense of Mukai (of the isometry type in periods) of X with itself.

The preprint consists of two parts.

In the Part I we treat a particular case when H · N(X) = ℤ where N(X) is the Picard lattice of X. In this case results are very nice, complete and very efficient. One can consider these results as ideal.

In the Part II we consider the general case: H can be non-primitive and H·N(X) = γℤ where γ can be arbitrary. In this general case results are much more complicated, far from being complete and much less efficient.

There is a chance that the general results of Part II can be made so nice and efficient as in the particular case of Part I, after further study. It is the reason I puted both parts together in the preprint.

1Supported by Russian Fund of Fundamental Research.

Typeset by AMS-TEX 2 VIACHESLAV V. NIKULIN^{2}

ON CORRESPONDENCES OF A K3 SURFACE WITH ITSELF, I

Viacheslav V. Nikulin

Abstract. Let X be a K3 surface with a polarization H of degree H^{2} = 2rs, r, s ≥ 1. Assume H · N(X) = ℤ for the Picard lattice N(X). The moduli space of sheaves over X with the isotropic Mukai vector (r, H, s) is again a K3 surface Y.

We prove that Y ≅ X, if there exists h_{1} ∈ N(X) with (h_{1})^{2} = f(r, s), H · h_{1} ≡ 0 mod g(r, s), and h_{1} satisfies some condition of primitivity. These conditions are necessary, if X is general with rk N(X) = 2. Existence of such kind a criterion is surprising, and it also gives some geometric interpretation of elements in N(X) with negative square.

We describe all irreducible 18-dimensional components of moduli of the (X, H) with Y ≅ X. We prove that their number is always infinite.

This generalizes results of [4, 5] for r = s.

0. Introduction

Let X be a K3 surface with a polarization H of degree H^{2} = 2rs, r, s ≥ 1, and a primitive Mukai vector (r, H, s). Let Y be the moduli of sheaves (coherent) over X with the Mukai vector (r, H, s). The Mukai vector v = (r, H, s) is isotropic, then Y is a K3 surface with a natural nef element h with h^{2} = 2ab where c = g.c.d(r, s) and a = r/c, b = s/c. The surface Y is isogenous to X in the sense of Mukai. The second Chern class of the corresponding quasi-universal sheave gives a 2-dimensional algebraic cycle Z ⊂ X ×Y, and an algebraic correspondence between X and Y. See [6, 7] and also [1] about these results.

Let N(X) be the Picard lattice of X. We consider an invariant γ(H) ∈ ℕ of H ∈ N(X) defined by H · N(X) = γ(H)ℤ (clearly, γ(H)|2rs). In this paper, we assume that

γ(H) = 1, that is H · N(X) = ℤ.  (0.1)

Then H is primitive and ρ(X) = rk N(X) ≥ 2. By Mukai [6, 7], the transcendental periods (T (X), H^{2,0}(X)) and (T (Y), H^{2,0}(Y)) are isomorphic in this case. We can expect that surfaces X and Y also can be isomorphic, and we then get a cycle Z ⊂ X × X, and a correspondence of X with itself. Thus, an interesting for us question is

Question 1. When is Y isomorphic to X?

We want to answer this question in terms of Picard lattices N(X) and N(Y) of X and Y. Then our question can be reformulated as follows:

Question 2. Assume that N is a hyperbolic lattice, H̃ ∈ N an element with square 2rs and H̃ · N = ℤ. What are conditions on N and H̃ such that for any K3 surface X with Picard lattice N(X) and s polarization H ∈ N(X) the corresponding K3 surface Y is isomorphic to X, if the pairs (N(X), H) and (N, H̃) are isomorphic as abstract lattices with fixed elements?

In other words, what are conditions on (N(X), H) as an abstract lattice with a vector H which are sufficient for Y to be isomorphic to X and are necessary, if X is a general K3 surface with its Picard lattice N(X)?

We answered this question in [4] for r = s = 2 (then H^{2} = 8, and the condition (0.1) always satisfies, if H is primitive); in [5] for r = s under the condition (0.1) (then (0.1) always satisfies, if H is primitive and r = s is even). The main surprising result of [4, 5] was that Y ≅ X, if the Picard lattice N(X) has an element h_{1} with some prescribed square (h_{1})^{2} and some minor additional conditions, and these conditions are necessary for a general K3 surface X with ρ(X) = rk N(X) = 2. Here we prove similar results in general, for arbitrary r and s, under the condition (0.1).

Theorem 1. Let X be a K3 surface with a polarization H such that H^{2} = 2rs, r, s ≥ 1, the Mukai vector (r, H, s) is primitive and Y the moduli of sheaves on X with the Mukai vector (r, H, s).

Then Y ≅ X, if at least for one of signs ± there exists h_{1} ∈ N(X) such that the elements H, h_{1} are contained in a 2-dimensional sublattice N ⊂ N(X) with H · N = ℤ, and h_{1} belongs to one of a-series or b-series described below:

a-series:

h_{1}^{2} = ±2bc, H · h_{1} ≡ 0 mod bc, H · h_{1} ≢ 0 mod bcl_{1}, h_{1}/l_{2} ∉ N(X)

where l_{1} and l_{2} are any primes such that l_{1}^{2}|a and l_{2}^{2}|b; b-series:

h_{1}^{2} = ±2ac, H · h_{1} ≡ 0 mod ac, H · h_{1} ≢ 0 mod acl_{1}, h_{1}/l_{2} ∉ N(X).

where l_{1} and l_{2} are any primes such that l_{1}^{2}|b and l_{2}^{2}|a.

The conditions above are necessary for γ(H) = 1 and Y ≅ X, if ρ(X) ≤ 2 and X is a general K3 surface with its Picard lattice, i. e. the automorphism group of the transcendental periods (T (X), H^{2,0}(X)) of X is ±1.

Like in [4, 5] for r = s, we also describe all (irreducible) divisorial conditions on 19-dimensional moduli of polarized K3 surfaces (X, H) which imply γ(H) = 1 and Y ≅ X. It is sufficient to assume that ρ(X) = 2. According to [10, 11], it is equivalent to description up to isomorphisms of all possible pairs H ∈ N(X) where rk N(X) = 2. There are two defining invariants of such pairs: d = − det N(X) ∈ ℕ and \overline{μ} = \{μ, −μ\} ⊂ (ℤ/2rs)^{∗} where d ≡ μ^{2} mod 4rs. See the definition of μ in (3.1.2) and (3.1.3)).

We introduce sets:

The a-series:

D(r, s; a)\overline{μ} ; ± = \{d ∈ ℕ | d ≡ μ^{2} mod 4abc^{2}, ∃ (p, q) ∈ ℤ × ℤ : p^{2} − dq^{2} = ±4ac,
p − μq ≡ 0 mod 2ac, g.c.d(a, p, q) = 1,
p − μq ≢ 0 mod 2acl for any prime l such that l^{2}|b }.

The b-series:

D(r, s; b)\overline{μ} ; ± = \{d ∈ ℕ | d ≡ μ^{2} mod 4abc^{2}, ∃ (p, q) ∈ ℤ × ℤ : p^{2} − dq^{2} = ±4bc,
p − μq ≡ 0 mod 2bc, g.c.d(b, p, q) = 1,
p − μq ≢ 0 mod 2bcl for any prime l such that l^{2}|a }.

We have

Theorem 2. With the above notations all possible irreducible divisorial conditions on moduli of polarized K3 surfaces (X, H) with a polarization H with H^{2} = 2rs, which imply γ(H) = 1 and Y ≅ X are labelled by the set

Div(r, s) = Div(r, s; a) ⋃ Div(r, s; b),

where

Div(r, s; a) = \{(d, \overline{μ}) | \overline{μ} = \{μ, −μ\} ⊂ (ℤ/2rs)^{∗}, d ∈ D(r, s; a)\overline{μ}\},
Div(r, s; b) = \{(d, \overline{μ}) | \overline{μ} = \{μ, −μ\} ⊂ (ℤ/2rs)^{∗}, d ∈ D(r, s; b)\overline{μ}\}

and

D(r, s; a)\overline{μ} = D(r, s; a)\overline{μ} ; + ⋃ D(r, s; a)\overline{μ} ; −, D(r, s; b)\overline{μ} = D(r, s; b)\overline{μ} ; + ⋃ D(r, s; b)\overline{μ} ; −.

The sets

D(r, s; a)\overline{μ} ; ± and D(r, s; b)\overline{μ} ; ±

are infinite if they are not empty.

The set Div(r, s) is always non-empty and then infinite: i. e. at least one of sets D(r, s; a)\overline{μ} ; ± or D(r, s; b)\overline{μ} ; ± is not empty for one of \overline{μ} and ±.

See more exact and strong statements in Lemmas 4.3, 4.4 and Theorems 4.1, 4.5 and 4.6.

These two theorems give the main results of the paper. We remark that Theorem 1 gives many (if not all) known cases when Y ≅ X (e.g. see [2], [17—19] and [20]). Theorem 1 is also interesting because it gives a very clear geometric interpretation of elements of N(X) with negative square (when the sign is −). For K3 surfaces it is well-known only for elements δ ∈ N(X) with δ^{2} = −2: then δ or −δ is effective.

In Section 5 we formulate similar results for the surface Y. They are very similar to the results for X. We only formulate these results since proofs are very similar to the corresponding proofs for X, but sometimes require additional non-trivial considerations. Thus, it is surprising that the results for Y are so similar to the results for X. Using results for X and Y together, one can construct non-trivial compositions of correspondences of the K3 surfaces X ≅ Y with itself.

In ideas we always follow [4] where a particular case r = s = 2 had been treated in great details. But technically the results of this paper are much more complicated. One needs to analyze many details and calculations in general and find appropriate formulations. As in [4, 5], the fundamental tools to get the results above is the Global Torelli Theorem for K3 surfaces proved by I.I.Piatetsky-Shapiro and I.R.Shafarevich in [14], and results of Mukai [6, 7]. By results of [6, 7], we can calculate periods of Y using periods of X; by the Global Torelli Theorem for K3 surfaces [14], we can find out if Y is isomorphic to X.

Recently the general case of a primitive isotropic Mukai vector (r, H, s) was considered in [13] where many results of this paper were generalized to the general case when γ(H) is possibly not equal to one and H is possibly not primitive in N(X). Results of [13] are much more complicated and much less efficient. It is non-trivial to deduce results of this paper directly from results of [13].

Thus, in this paper we treat a very nice and important particular case when the results are very simple, efficient and complete. Using methods of this paper and [13], one can extend these results to more general types of primitive isotropic Mukai vectors (r, H, s). We hope to consider that in further publications.

I am grateful to Carlo Madonna for useful discussions.

1. Some notations and results about lattices and K3 surfaces

1.1. Some notations about lattices. We use notations and terminology from [11] about lattices, their discriminant groups and forms. A lattice L is a nondegenerate integral symmetric bilinear form. That is L is a free ℤ-module equipped with a symmetric pairing x · y ∈ ℤ for x, y ∈ L, and this pairing should be nondegenerate. We denote x^{2} = x · x. The signature of L is the signature of the corresponding real form L⊗ℝ. The lattice L is called even if x^{2} is even for any x ∈ L. Otherwise, L is called odd. The determinant of L is defined to be det L = det(e_{i} ·e_{j}) where {e_{i}} is some basis of L. The lattice L is unimodular if det L = ±1. The dual lattice of L is L^{∗} = Hom(L, ℤ) ⊂ L⊗ℚ. The discriminant group of L is A_{L} = L^{∗}/L. It has the order | det L|. The group A_{L} is equipped with the discriminant bilinear form b_{L} : A_{L} × A_{L} → ℚ/ℤ and the discriminant quadratic form q_{L} : A_{L} → ℚ/2ℤ if L is even. To get this forms, one should extend the form of L to the form on the dual lattice L^{∗} with values in ℚ.

For x ∈ L, we shall consider the invariant γ(x) ≥ 0 where

x · L = γ(x)ℤ.  (1.1.1)

Clearly, γ(x)|x^{2} if x ≠ 0.

We denote by L(k) the lattice obtained from a lattice L by multiplication of the form of L by k ∈ ℚ. The orthogonal sum of lattices L_{1} and L_{2} is denoted by L_{1} ⊕ L_{2}. For a symmetric integral matrix A, we denote by ⟨A⟩ a lattice which is given by the matrix A in some bases. We denote

U = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .  (1.1.2)

Any even unimodular lattice of the signature (1, 1) is isomorphic to U.

An embedding L_{1} ⊂ L_{2} of lattices is called primitive if L_{2}/L_{1} has no torsion. We denote by O(L), O(b_{L}) and O(q_{L}) the automorphism groups of the corresponding forms. Any δ ∈ L with δ^{2} = −2 defines a reflection s_{δ} ∈ O(L) which is given by the formula

x → x + (x · δ)δ,

x ∈ L. All such reflections generate the 2-reflection group W^{(−2)}(L) ⊂ O(L).

1.2. Some notations about K3 surfaces. Here we remind some basic notions and results about K3 surfaces, e. g. see [14—16]. A K3 surface X is a nonsingular projective algebraic surface over ℂ such that its canonical class K_{X} is zero and the irregularity q_{X} = 0. We denote by N(X) the Picard lattice of X which is a hyperbolic lattice with the intersection pairing x · y for x, y ∈ N(S). Since the canonical class K_{X} = 0, the space H^{2,0}(X) of 2-dimensional holomorphic differential forms on X has dimension one over ℂ, and

N(X) = {x ∈ H^{2}(X, ℤ) | x · H^{2,0}(X) = 0}  (1.2.1)

where H^{2}(X, ℤ) with the intersection pairing is a 22-dimensional even unimodular lattice of signature (3, 19). The orthogonal lattice T (X) to N(X) in H^{2}(X, ℤ) is called the transcendental lattice of X. We have H^{2,0}(X) ⊂ T (X) ⊗ ℂ. The pair (T (X), H^{2,0}(X)) is called the transcendental periods of X. The Picard number of X is ρ(X) = rk N(X). A non-zero element x ∈ N(X) ⊗ ℝ is called nef if x ≠ 0 and x · C ≥ 0 for any effective curve C ⊂ X. It is known that an element x ∈ N(X) is ample if x^{2} > 0, x is nef, and the orthogonal complement x^{⊥} to x in N(X) has no elements with square −2. For any element x ∈ N(X) with x^{2} ≥ 0, there exists an element w ∈ W^{(−2)}(N(X)) such that ±w(x) is nef; it then is ample, if x^{2} > 0 and x^{⊥} had no elements with square −2 in N(X).

We denote by V^{+}(X) the light cone of X, which is the half-cone of

V (X) = {x ∈ N(X) ⊗ ℝ | x^{2} > 0 }  (1.2.2)

containing a polarization of X. In particular, all nef elements x of X belong to \overline{V^{+}(X)}: one has x · V^{+}(X) > 0 for them.

The reflection group W^{(−2)}(N(X)) acts in V^{+}(X) discretely, and its fundamental chamber is the closure \overline{K(X)} of the Kähler cone K(X) of X. It is the same as the set of all nef elements of X. Its faces are orthogonal to the set Exc(X) of all exceptional curves r on X which are non-singular rational curves r on X with r^{2} = −2. Thus, we have

\overline{K(X)} = \{0 ≠ x ∈ \overline{V^{+}(X)} | x · Exc(X) ≥ 0 \}.  (1.2.3)

2. General results on the Mukai correspondence between K3 surfaces with primitive polarizations of degrees 2rs and 2ab which gives isomorphic K3’s

2.1. The correspondence. Let X be a smooth complex projective K3 surface with a polarization H of degree 2rs, r, s ≥ 1. Let Y be the moduli space of coherent sheaves E on X with the primitive isotropic Mukai vector v = (r, H, s). Then rk E = r, χ(E) = r + s and c_{1}(E) = H. Let

H^{∗}(X, ℤ) = H^{0}(X, ℤ) ⊕ H^{2}(X, ℤ) ⊕ H^{4}(X, ℤ)  (2.1.1)

be the full cohomology lattice of X equipped with the Mukai product

(u, v) = −(u_{0} · v_{2} + u_{2} · v_{0}) + u_{1} · v_{1}  (2.1.2)

for u_{0}, v_{0} ∈ H^{0}(X, ℤ), u_{1}, v_{1} ∈ H^{2}(X, ℤ), u_{2}, v_{2} ∈ H^{4}(X, ℤ). We naturally identify H^{0}(X, ℤ) and H^{4}(X, ℤ) with ℤ. Then the Mukai product is

(u, v) = −(u_{0}v_{2} + u_{2}v_{0}) + u_{1} · v_{1}.  (2.1.3)

Since H^{2} = 2rs, the element

v = (r, H, s) = (r, H, χ − r) ∈ H^{∗}(X, ℤ)  (2.1.4)

is isotropic, i.e. v^{2} = 0. In this case (for a primitive v), Mukai showed [6, 7] that Y is a K3 surface, and one has the natural identification

H^{2}(Y, ℤ) ≅ (v^{⊥}/ℤv)  (2.1.5)

which also gives the isomorphism of the Hodge structures of X and Y. The Y has the canonical polarization h = (−a, 0, b) mod ℤv where

c = g.c.d(r, s), a = r/c , b = s/c .  (2.1.6)

Then g.c.d.(a, b) = 1.

In particular, (2.1.5) gives an embedding

T (X) ⊂ T (Y)  (2.1.7)

of the lattices of transcendental cycles of the index

[T (Y) : T (X)] = n(v) = min |v · x|  (2.1.8)

where x ∈ H^{0}(X, ℤ) ⊕ N(X) ⊕ H^{4}(X, ℤ) and v · x ≠ 0.

In this paper, we are interested in the case when Y ≅ X. By (2.1.8), it may happen, if n(v) = 1 only.

We can introduce the invariant γ = γ(H) ∈ ℕ which is defined by

H · N(X) = γℤ.  (2.1.9)

In this paper we consider the important case when γ = γ(H) = 1. Thus, N(X)·H = ℤ. Equivalently, there exists x ∈ N(X) such that x · H = 1. From γ = γ(H) = 1, it follows that H ∈ N(X) is primitive, the Mukai vector (r, H, s) is also primitive, and the invariant n(v) = 1. Thus, we may have a hope to have cases when Y ≅ X. Since the Picard lattice N(X) is even, it also follows that the Picard number ρ(X) = rk N(X) ≥ 2 in this case. Thus:

γ = γ(H) = 1 ⟹
ρ(X) ≥ 2, H is primitive, (T (Y), H^{2,0}(Y)) ≅ (T (X), H^{2,0}(X)). (2.1.10)

The same case γ = 1 was considered in [4] for r = s = c = 2, and in [5] for r = s = c. Results of [4, 5] concerning an arbitrary Picard lattice N(X) can be generalized to the general case of γ = 1.

2.2. General results for an arbitrary Picard lattice N(X). First we remind the characteristic map defined in [4]. It is defined for any non-degenerate even lattice S and any its element P ∈ S with P^{2} = 2m ≠ 0 and γ(P) = 1, i. e. P · S = 1. Then P is primitive in S.

We denote by

K(P) = P_{S}^{⊥}  (2.2.1)

the orthogonal complement to P in S. Put P^{∗} = P/2m. Then any element x ∈ S can be written as

x = nP^{∗} + k^{∗}  (2.2.2)

where n ∈ ℤ and k^{∗} ∈ K(P)^{∗}, because

ℤP ⊕ K(P) ⊂ S ⊂ S^{∗} ⊂ ℤP^{∗} ⊕ K(P)^{∗}.

Since γ(P) = 1, the map nP^{∗} + [P] → k^{∗} + K(P) gives an isomorphism of the groups ℤ/2m ≅ [P^{∗}]/[P] ≅ [u^{∗}(P) + K(P)]/K(P) where u^{∗}(P) + K(P) has order 2m in A_{K(P)} = K(P)^{∗}/K(P). It follows,

S = [ℤP, K(P), P^{∗} + u^{∗}(P)].  (2.2.3)

The element u^{∗}(P) is defined canonically mod K(P). Since P^{∗} + u^{∗}(P) belongs to the even lattice S, it follows

(P^{∗} + u^{∗}(P))^{2} = \frac{1}{2m} + u^{∗}(P)^{2} ≡ 0 mod 2.  (2.2.4)

Let \overline{P^{∗}} = P^{∗} mod [P] ∈ [P^{∗}]/[P] ≅ ℤ/2m and \overline{k^{∗}} = k^{∗} mod K(P) ∈ A_{K(P)}. Then

S/[P, K(P)] = (ℤ/2m)(\overline{P^{∗}} + \overline{u^{∗}(P)}) ⊂ (ℤ/2m)\overline{P^{∗}} + K(P)^{∗}/K(P).  (2.2.5)

Also S^{∗} ⊂ ℤP^{∗} + K(P)^{∗} since P + K(P) ⊂ S, and for n ∈ ℤ, k^{∗} ∈ K(P)^{∗} we have x = nP^{∗} + k^{∗} ∈ S^{∗}, if and only if

(nP^{∗} + k^{∗}) · (P^{∗} + u^{∗}(P)) = \frac{n}{2m} + k^{∗} · u^{∗}(P) ∈ ℤ.

It follows,

\begin{aligned}S^{∗} & = \{nP^{∗} + k^{∗} | n ∈ ℤ, k^{∗} ∈ K(P)^{∗}, n ≡ −2m (k^{∗} · u^{∗}(P)) mod 2m\} ⊂ \\ & ⊂ ℤP^{∗} + K(P)^{∗}, \end{aligned}

(2.2.6)

and

S^{∗}/[P, K(P)] = \{−2m(\overline{k^{∗}} · \overline{u^{∗}(P)}) \overline{P^{∗}} + \overline{k^{∗}}\} | \overline{k^{∗}} ∈ A_{K(P)}\} ⊂ (ℤ/2m)\overline{P^{∗}} + A_{K(P)}.

(2.2.7)

We introduce the characteristic map of P ∈ S:

κ(P) : K(P)^{∗} → A_{K(P)}/(ℤ/2m)(u^{∗}(P) + K(P)) → A_{S}  (2.2.8)

where for k^{∗} ∈ K(P)^{∗} we have

κ(P)(k^{∗}) = −2m(k^{∗} · u^{∗}(P))P^{∗} + k^{∗} + S ∈ A_{S}.  (2.2.9)

It is epimorphic and its kernel is (ℤ/2m)(u^{∗}(P) + K(P)). It gives the canonical isomorphism

\overline{κ(P)} : A_{K(P)}/(ℤ/2m)(u^{∗}(P) + K(P)) ≅ A_{S}.  (2.2.10)

For the corresponding discriminant forms we have

κ(k^{∗})^{2} mod 2 = (k^{∗})^{2} + 2m(k^{∗} · u^{∗}(P))^{2} mod 2.  (2.2.11)

Below we shall consider the characteristic map for elements P ∈ N(X) with γ(P) = 1. It will be denoted as κ(P).

We define m(a, b) mod 2ab by

m(a, b) ≡ −1 mod 2a, and m(a, b) ≡ 1 mod 2b.  (2.2.12)

The m(a, b) mod 2ab was introduced by Mukai [8].

The same calculations as in [4] and also [5] (the key calculations were done in [4]) prove the fundamental for further results Theorem 2.2.1 below. Its complete proof in the general case (for any γ and possibly non-primitive H) is contained in [13]. The Global Torelli Theorem for K3 surfaces [14] and Mukai’s result (2.1.5) are fundamental for the proof.

Theorem 2.2.1. Assume that X is a K3 surface with a polarization H with H^{2} = 2rs, r, s ≥ 1, and γ(H) = 1 for H ∈ N(X). Let Y be the moduli space of coherent sheaves on X with the Mukai vector v = (r, H, s) (it is clearly primitive).

Then the surface Y is isomorphic to X, if the following conditions (i) and (ii) hold:

(i) there exists h̃ ∈ N(X) of the degree h̃^{2} = 2ab, γ(h̃) = 1, and such that there exists an embedding

f : K(H) → K(h̃)

of negative definite lattices such that

f^{∗}(K(h̃)) = [K(H), 2abcu^{∗}(H)],
f^{∗}(u^{∗}(h̃)) + f^{∗}(K(h̃)) = m(a, b)c u^{∗}(H) + f^{∗}(K(h̃)).  (2.2.13)

(ii) For the h̃ and f in (i), there exists a choice of ± such that

κ(h̃)(k^{∗}) = ±κ(H)(f^{∗}(k^{∗}))  (2.2.14)

for any k^{∗} ∈ K(h̃)^{∗}.

The conditions (i) and (ii) are necessary for a K3 surface X with ρ(X) ≤ 19 which is general for its Picard lattice N(X) in the following sense: the automorphism group of the transcendental periods (T (X), H^{2,0}(X)) is ±1. If ρ(X) = 20, then always Y ≅ X.

For instance, let c = 1. Then one can take h̃ = H and f = id in Theorem 2.2.1, if m(a, b) ≡ 1 mod 2ab. This is valid, only if either a = 1 or b = 1. Thus, Y ≅ X, if γ(H) = 1, c = 1 and one of a, b is equal to 1. According to Mukai [8], always Y ≅ X, if c = 1 and one of a, b is equal to 1. One does not need the additional condition γ(H) = 1.

3. The case of Picard number 2

3.1. Main results for ρ(X) = 2. Here we apply results of Sect. 2 to X and Y with Picard number 2. A general K3 surface X with γ(H) = 1 has ρ(X) = 2.

First, we consider an arbitrary K3 surface X with Picard number 2 and a primitive polarization H of degree H^{2} = 2rs where r, s ≥ 1. Like above, c = g.c.d.(r, s) and r = ca, s = cb. Additionally, we assume that γ(H) = 1 for H ∈ N(X). Let

K(H) = H_{N(X)}^{⊥} = ℤδ  (3.1.1)

and δ^{2} = −t where t > 0 is even. The δ ∈ N(X) is defined uniquely up to ±δ. Since γ(H) = 1, we have

N(X) = [ℤH, ℤδ, μH^{∗} + \frac{δ}{2rs}]  (3.1.2)

where H^{∗} = H/2rs and g.c.d(μ, 2rs) = 1. The element

±μ mod 2rs ∈ (ℤ/2rs)^{∗}  (3.1.3)

is the invariant of the pair (N(X), H) up to isomorphisms of lattices with a primitive vector H of H^{2} = 2rs and γ(H) = 1. If δ changes to −δ, then μ mod 2rs changes to −μ mod 2rs. We have

(μH^{∗} + \frac{δ}{2rs})^{2} = \frac{1}{2rs}(μ^{2} − \frac{t}{2rs}) ≡ 0 mod 2.  (3.1.4)

Then t = 2rsd, for some d ∈ ℕ and μ^{2} ≡ d mod 4rs. Thus, d mod 4rs ∈ (ℤ/4rs)^{∗ 2}. We have −d = det(N(X)). Any element z ∈ N(X) can be written as z = (xH + yδ)/2rs where x ≡ μy mod 2rs. In these considerations, one can replace H by any primitive element of N(X) with square 2rs and γ(H) = 1. Thus, we have:

Proposition 3.1.1. Let X be a K3 surface with Picard number ρ = 2 equipped with a primitive polarization H of degree H^{2} = 2rs, r, s ≥ 1, and γ(H) = 1.

The pair (N(X), H) has the invariants d ∈ ℕ and ±μ mod 2rs ∈ (ℤ/2rs)^{∗} such that μ^{2} ≡ d mod 4rs. We have: det N(X) = −d, and K(H) = H_{N(X)}^{⊥} = ℤδ where δ^{2} = −2rsd. Moreover,

N(X) = [H, δ, \frac{(μH + δ)}{2rs}],  (3.1.5)
N(X) = \{z = \frac{(xH + yδ)}{2rs} | x, y ∈ ℤ and x ≡ μy mod 2rs\}.  (3.1.6)

We have z^{2} = (x^{2} − dy^{2})/2rs.

For any primitive element H′ ∈ N(X) with (H′)^{2} = H^{2} = 2rs and the same invariant ±μ, there exists an automorphism ϕ ∈ O(N(X)) such that ϕ(H) = H′.

From Proposition 3.1.1 and g.c.d(d, 2rs) = g.c.d(d, 2ab) = 1 where det N(X) = −d, we get

Proposition 3.1.2. Under conditions and notations of Propositions 3.1.1, all elements h′ = (xH + yδ)/(2rs) ∈ N(X) with (h′)^{2} = 2ab are in one to one correspondence with integral solutions (x, y) of the equation

x^{2} − dy^{2} = 4a^{2}b^{2}c^{2}  (3.1.7)

such that x ≡ μy mod 2abc^{2}.

For k ∈ ℕ and k > 1 the element h′ is divisible by k in N(X) and then it is not primitive, if and only if k | (x, y) and x ≡ μy mod 2abc^{2}k. Then k^{2}|ab. We call such a pair (x, y) as μ-divisible by k. If (x, y) is not μ-divisible by k > 1, we call such a pair (x, y) as μ-primitive. It is enough to consider only prime k. Thus, h′ is primitive, if and only if the corresponding pair (x, y) is μ-primitive. If h′ is primitive, then γ(h′) = 1.

The crucial statement is Theorem 3.1.3. Let X be a K3 surface with ρ(X) = 2, and H its polarization of the degree H^{2} = 2rs with γ(H) = 1. Let Y be the moduli space of sheaves on X with the isotropic Mukai vector v = (r, H, s) and the canonical nef element h = (−a, 0, b) mod ℤv.

As in Proposition 3.1.1, we consider the invariants ±μ mod 2rs ∈ (ℤ/2rs)^{∗} and d ∈ ℕ of (N(X), H). Thus, we have

γ(H) = 1, det N(X) = −d where μ^{2} ≡ d mod 4rs.  (3.1.8)

With notations of Propositions 3.1.1, then all elements h̃ = (xH + yδ)/2rs ∈ N(X) with square h̃^{2} = 2ab satisfying Theorem 2.2.1 are in one to one correspondence with integral solutions (x, y) of the equation

x^{2} − dy^{2} = 4a^{2}b^{2}c^{2}  (3.1.9)

which satisfy the conditions (i), (ii), (iii), (iv) below:

(i) x ≡ μy mod 2rs;

(ii) x ≡ ±2abc mod d;

(iii) (x, y) belongs to one of a-series or b-series of solutions defined below:

a-series: b|(x, y) and x − μy ≡ 0 mod 2rsa where μ^{2} ≡ d mod 4rsa, b-series: a|(x, y) and x − μy ≡ 0 mod 2rsb where μ^{2} ≡ d mod 4rsb; (iv) the pair (x, y) is μ-primitive: there does not exist a prime l such that l|(x, y)

and x − μy ≡ 0 mod 2rs l (we always have l^{2}|ab for such l), i. e.

g.c.d (x, y, \frac{x − μy}{2rs} ) = 1.  (3.1.10)

In particular, by Theorem 2.2.1, for a general X with ρ(X) = 2, we have Y ≅ X, if and only if the equation x^{2} − dy^{2} = 4a^{2}b^{2}c^{2} has an integral solution (x, y) satisfying conditions (i)—(iv) above. Moreover, a nef primitive element h̃ = (xH + yδ)/2rs with h̃^{2} = 2ab defines the pair (X, h̃) which is isomorphic to the (Y, h), if and only if (x, y) satisfies the conditions (ii) and (iii) (it satisfies conditions (i) and (iv) since it corresponds to a primitive element of N(X)).

Proof. We denote

H^{∗} = \frac{H}{2rs}, δ^{∗} = \frac{δ}{2rsd}  (3.1.11)

where K(H) = ℤδ = H^{⊥} in N(X). Since μH^{∗} + δ/(2rs) ∈ N(X), then H^{∗} + μ^{−1}δ/(2rs) ∈ N(X) where μ^{−1} is defined mod 2rs. It follows that

u^{∗}(H) = \frac{μ^{−1}}{2rs} δ + K(H) = μ^{−1}dδ^{∗} + ℤδ.  (3.1.12)

Let

h̃ = \frac{xH + yδ}{2rs} ∈ N(X)  (3.1.13)

satisfies conditions of Theorem 2.2.1. Then h̃^{2} = 2ab. This is equivalent to x, y ∈ ℤ, x ≡ μy mod 2rs and x^{2} − dy^{2} = 4a^{2}b^{2}c^{2}. We get (i). The h̃ is primitive, if and only if (x, y) is μ-primitive. We get (iv).

Consider K(h̃) = h̃^{⊥} in N(X). If k = a_{1}H + b_{1}δ ∈ K(h̃), then (k, h̃) = a_{1}x − b_{1}yd = 0 and (a_{1}, b_{1}) = λ(yd, x), λ ∈ ℚ. We have

(λ(ydH + xδ))^{2} = λ^{2}(2rsy^{2}d^{2} − 2rsdx^{2}) = 2rsλ^{2}d(y^{2}d − x^{2}) = −2rsλ^{2}d(4c^{2}a^{2}b^{2}).

For λ = 1/2rs we get

g = (ydH + xδ)/2rs.  (3.1.14)

The element g ∈ N(X) since yd, x are integers and yd − μx ≡ μ(μy − x) ≡ 0 mod 2rs. We have g^{2} = −2abd. If h̃ is primitive and γ(h̃) = 1, then K(h̃) = ℤg. Then N(X) = [h̃, g, (νh̃ + g)/2ab] where ν is a similar invariant for h̃ as μ for H. We denote

h̃^{∗} = \frac{h̃}{2ab} and g^{∗} = \frac{g}{2abd}.  (3.1.15)

Then

u^{∗}(h̃) = \frac{ν^{−1}g}{2ab} = ν^{−1}dg^{∗} + K(h).  (3.1.16)

There exists a unique (up to ±1) embedding f : K(H) = ℤδ → K(h̃) = ℤg of one-dimensional lattices. It is given by f(δ) = ±cg. Its dual is defined by f^{∗}(g^{∗}) = ±cδ^{∗}. We have f^{∗}(K(h)) = ℤf^{∗}(g) = ℤ(δ/c) = ℤ2abcu^{∗}(H) because of (3.1.12). This gives the first part of (2.2.13).

Moreover,

f^{∗}(u^{∗}(h̃)) + f^{∗}(K(h)) = ν^{−1}df^{∗}(g^{∗}) + f^{∗}(K(h)) =
±ν^{−1}cdδ^{∗} + f^{∗}(K(h)) = ±ν^{−1}cu^{∗}(H) + [K(H), 2abcu^{∗}(H)].

Thus, the second part of (2.2.13) is ±ν^{−1} ≡ m(a, b)μ^{−1} mod 2ab. Equivalently,

ν ≡ ±m(a, b)μ mod 2ab.  (3.1.17)

Thus, for ν given by (3.1.17) one has

νh̃^{∗} + g^{∗} = (((νx + dy)H + (νy + x)δ) /2rs) /2ab ∈ N(X).  (3.1.18)

This is equivalent to

\{ \begin{matrix}νx + dy ≡ 0 mod 2ab \\ νy + x ≡ 0 mod 2ab \\ νx + dy ≡ μ(νy + x) mod 2rs2ab \end{matrix} .  (3.1.19)

For example, assume that ν ≡ m(a, b)μ mod 2ab. Then ν ≡ −μ mod 2a and ν ≡ μ mod 2b.

From (3.1.19), we then get mod 2a that −μx + yμ^{2} ≡ 0 mod 2a, −μy + x ≡ 0 mod 2a and 2μx ≡ y(d + μ^{2}) mod 2rs2a. First two congruences are valid since x ≡ μy mod 2rs. The last one gives

μ\frac{x − μy}{2rs} ≡ \frac{d − μ^{2}}{4rs} y mod a .  (3.1.20)

We have d ≡ μ^{2} mod 4rs. Since μ mod 2rs is invertible and a|r, there exists the unique its lifting μ mod 2rsa such that d ≡ μ^{2} mod 4rsa. Then (3.1.20) is equivalent to x − μy ≡ 0 mod 2rsa.

Now let us consider (3.1.19) mod 2b. First two congruences in (3.1.19) give μx + yd ≡ 0 mod 2b, μy + x ≡ 0 mod 2b. Since x ≡ μy mod 2b, they are equivalent to x ≡ y ≡ 0 mod b. Third congruence in (3.1.19) gives yd ≡ μ^{2}y mod 2rs2b. It is valid since b|y and μ^{2} ≡ d mod 4rs. Thus, (x, y) belongs to the a-series in (iii). If ν ≡ −m(a, b)μ mod 2ab, then similarly (x, y) belongs to the b-series in (iii). Thus, condition (2.2.13) is equivalent to (iii).

By (2.2.9),

κ(H)(δ^{∗}) = −2rs(δ^{∗} · u^{∗}(H))H^{∗} + δ^{∗} + K(H) =

− 2rs(δ/(2rsd) · μ^{−1}δ/(2rs))H^{∗} + δ^{∗} + K(H) = μ^{−1}H^{∗} + δ^{∗} + K(H).

Thus, we have

κ(H)(δ^{∗}) = μ^{−1}H^{∗} + δ^{∗} + K(H).  (3.1.21)

Similarly,

κ(h̃)(g^{∗}) = ν^{−1}g^{∗} + K(h̃) = ±m(a, b)μ^{−1}g^{∗} + K(h̃).  (3.1.22)

Here we use (3.1.17).

The condition (2.2.14) gives then

κ(h̃)(g^{∗}) = ±cκ(H)(δ^{∗}).  (3.1.23)

From (3.1.21) and (3.1.22) we then get

\begin{matrix}κ(h̃)(g^{∗}) = ±m(a, b)μ^{−1} \frac{h̃}{2ab} + \frac{g}{2abd} + N(X) = \\ ± m(a, b)μ^{−1} \frac{xH + yδ}{2rs2ab} + \frac{ydH + xδ}{2rs2abd} + N(X) = \end{matrix}

\frac{±m(a, b)μ^{−1}x + y}{2ab} H^{∗} + \frac{±m(a, b)μ^{−1}dy + x}{2ab}δ^{∗} + N(X) .  (3.1.24)

We have A_{N(X)} = N(X)^{∗}/N(X) ≅ (ℤ/dℤ). The κ(H) gives an epimorphism of ℤδ^{∗} onto A_{N(X)} with the kernel which is dδ^{∗}. It follows that κ(h̃)(g^{∗}) = ±cκ(H)(δ^{∗}) is equivalent to

\frac{±m(a, b)μ^{−1}dy + x}{2ab} ≡ ±c mod d.  (3.1.25)

Thus, ±m(a, b)μ^{−1}yd + x ≡ ±2abc mod 2abd. Since g.c.d(2abc, d) = 1, it is equivalent to x ≡ ±2abc mod d and ±m(a, b)μ^{−1}yd + x ≡ 0 mod 2ab. The last congruence is valid because of d ≡ μ^{2} mod 2ab, (3.1.17) and (3.1.19).

This finishes the proof.

Now let us consider the a-series of solutions of (3.1.9) satisfying the main congruence (ii). Since b|(x, y) and g.c.d(b, d) = 1, we have (x, y) = (bx_{1}, by_{1}) where (x_{1}, y_{1}) satisfy

x_{1}^{2} − dy_{1}^{2} = 4a^{2}c^{2} and x_{1} ≡ ±2ac mod d.  (3.1.26)

Let us find all solutions of (3.1.26). We apply the main trick of [4] (see also [5]). Considering ±(x_{1}, y_{1}), we can assume that x_{1} ≡ 2ac mod d. Then x_{1} = 2ac − kd where k ∈ ℤ. We have 4a^{2}c^{2} − 4ackd + k^{2}d^{2} − dy_{1}^{2} = 4a^{2}c^{2}. Thus,

d = \frac{y_{1}^{2} + 4ack}{k^{2}} .  (3.1.27)

Let l be prime. Like in [4], it is easy to see that if l^{2t+1}|k and l^{2t+2} |̸k, then l|4ac. It follows that k = −αq^{2} where α|2ac and α is square-free, q ∈ ℤ. It follows that (3.1.27) is equivalent to

α|2ac is square-free, p^{2} − dq^{2} = \frac{4ac}{α}.  (3.1.28)

Then k = −αq^{2}, y = αpq. Here α can be negative. Thus, (integral) solutions α, (p, q) of (3.1.28) give all solutions

(x_{1}, y_{1}) = ±(2ac + αdq^{2}, αpq)  (3.1.29)

of (3.1.26). We call them associated solutions. Thus, all solutions (x_{1}, y_{1}) of (3.1.26) are associated solutions (3.1.29) to all solutions α, (p, q) of (3.1.28). If one additionally assumes that q ≥ 0, then (x, y) and α, (p, q) are in one to one correspondence, by our construction.

Now let us consider associated solutions (3.1.29) which satisfy the additional condition bx_{1} ≡ μby_{1} mod 2rs. (It is the condition (i) of Theorem 3.1.3.) Equivalently, x_{1} ≡ μy_{1} mod 2ac^{2}. By (3.1.29), this is equivalent

2ac + αdq^{2} ≡ μαpq mod 2ac^{2}.  (3.1.30)

We also have

4ac + αdq^{2} = αp^{2}.  (3.1.31)

Taking (3.1.31) - (3.1.30), we get that (3.1.30) is equivalent to

2ac ≡ αp(p − μq) mod 2ac^{2}.  (3.1.32)

Taking 2(3.1.30) - (3.1.31), we get that (3.1.30) is equivalent to

αp^{2} + αdq^{2} − 2αμpq ≡ 0 mod 4ac^{2}.  (3.1.33)

Since d ≡ μ^{2} mod 4abc^{2}, this is equivalent to

α(p − μq)^{2} ≡ 0 mod 4ac^{2}.  (3.1.34)

Since α is square-free, it follows that

2c |p − μq.  (3.1.35)

From (3.1.32), we then get

2a ≡ αp (\frac{p − μq}{c} ) mod 2ac  (3.1.36)

where (p − μq)/c is an integer. Since α|2ac, it follows that α|2a. Thus, we get

α|2a, c|(p − μq) and \frac{2a}{α} ≡ p (\frac{p − μq}{c} ) mod \frac{2ac}{α}.  (3.1.37)

Thus, we have proved (see (3.1.34)) that the condition x_{1} ≡ μy_{1} mod 2ac^{2} is equivalent to

α|2a and (p − μq)^{2} ≡ 0 mod \frac{4ac^{2}}{α} .  (3.1.38)

Let us prove that actually α|a. Otherwise, the equation p^{2} − dq^{2} = 4ac/α has no solutions. Really, assume that 2a/α is odd. By (3.1.38) we get (p − μq)^{2} ≡ 0 mod 2c^{2}. It follows p−μq ≡ 0 mod 2c and p+μq ≡ 0 mod 2. Then p^{2} −μ^{2}q^{2} ≡ 0 mod 4c. Since μ^{2} ≡ d mod 4ac^{2}, we then get p^{2} − dq^{2} ≡ 0 mod 4c. On the other hand, if 2a/α is odd, then p^{2} − dq^{2} = 4ac/α ≡ 2c mod 4c. We get a contradiction.

Now we add the condition (iv) and the last condition of the a-series x − μy ≡ 0 mod 2rsa of Theorem 3.1.3 (here we assume that μ^{2} ≡ d mod 4rsa). For (x, y) = (bx_{1}, by_{1}) we then get

x_{1} − μy_{1} ≡ 0 mod 2a^{2}c^{2} and g.c.d(bx_{1}, by_{1}, \frac{x_{1} − μy_{1}}{2ac^{2}} ) = 1.  (3.1.39)

By (3.1.29), we then have α|(x_{1}, y_{1}, (x_{1} − μy_{1})/2ac^{2}). It follows the fundamental result:

α = ±1.  (3.1.40)

The condition x_{1} − μy_{1} ≡ 0 mod 2a^{2}c^{2} gives

2ac + αdq^{2} ≡ μαpq mod 2a^{2}c^{2}.  (3.1.41)

As always

−4ac + αp^{2} − αdq^{2} = 0.  (3.1.42)

Taking 2(3.1.41)+(3.1.42), we get that (3.1.41) is equivalent to

αdq^{2} + αp^{2} ≡ 2μαpq mod 4a^{2}c^{2}.  (3.1.43)

Since α = ±1, this is equivalent to

dq^{2} + p^{2} ≡ 2μpq mod 4a^{2}c^{2}.  (3.1.44)

Since μ^{2} ≡ d mod 4a^{2}c^{2}, the (3.1.44) is equivalent to

(p − μq)^{2} ≡ 0 mod 4a^{2}c^{2}  (3.1.45)

which is equivalent to

p − μq ≡ 0 mod 2ac.  (3.1.46)

Thus, the congruence x_{1} − μy_{1} ≡ 0 mod 2a^{2}c^{2} in (3.1.39) is equivalent to (3.1.46). Now let us analyze the condition of primitivity

g.c.d(bx_{1}, by_{1}, \frac{x_{1} − μy_{1}}{2ac^{2}} ) = 1  (3.1.47)

in (3.1.39). The condition (3.1.47) is equivalent to the fact that there does not exist a prime l such that

l|(bx_{1}, by_{1}, (x_{1} − μy_{1})/2ac^{2}) where always l^{2}|ab.  (3.1.48)

Assume l^{2}|a. Since a|(x_{1} − μy_{1})/2ac^{2}, (3.1.48) is equivalent to l|(x_{1}, y_{1}). We have (x_{1}, y_{1}) = ±(2ac+αdq^{2}, αpq) = ±(−2ac+αp^{2}, αpq). It follows l|(p, q). Thus, the l^{2}|a in (3.1.48) does not exist, if and only if l|(p, q) is impossible for l^{2}|a.

Assume l^{2}|b. Then l|(bx_{1}, by_{1}, (x_{1} − μy_{1})/2ac^{2}) is equivalent to x_{1} − μy_{1} ≡ 0 mod 2ac^{2}l. This is equivalent to x_{1} − μy_{1} ≡ 0 mod 2a^{2}c^{2}l because we know that x_{1} − μy_{1} ≡ 0 mod 2a^{2}c^{2} and g.c.d(l, a) = 1. From (3.1.29), we get

2ac + αdq^{2} ≡ μαpq mod 2a^{2}c^{2}l.  (3.1.49)

By −4ac + αp^{2} − αdq^{2} = 0 and α = ±1, we then get like above that (3.1.49) is equivalent to

dq^{2} + p^{2} − 2μpq ≡ 0 mod 4a^{2}c^{2}l.  (3.1.50)

Since μ^{2} ≡ d mod 4abc^{2}a and l|b, the (3.1.50) is equivalent to

(p − μq)^{2} ≡ 0 mod 4a^{2}c^{2} l.  (3.1.51)

The (3.1.51) is equivalent to

p − μq ≡ 0 mod 2ac l.  (3.1.52)

Thus, we finally get that (3.1.39) is equivalent to

\{ \begin{matrix}p ≡ μq mod 2ac, \\ g.c.d(a, p, q) = 1, \\ p ≢ μq mod 2acl for any prime l|b. \end{matrix}  (3.1.53)

We get the final result:

Theorem 3.1.4. With conditions of Theorem 3.1.3, for a general K3-surface X with ρ(X) = 2 and γ(H) = 1, we have Y ≅ X, if and only if for one of α = ±1 there exists an integral solution (p, q) for one of a-series or b-series:

a-series:

\{ \begin{matrix}p^{2} − dq^{2} = 4ac/α \\ p ≡ μq mod 2ac \\ g.c.d(a, p, q) = 1 \\ p ≢ μq mod 2acl for any prime l|b \end{matrix} .  (3.1.54)

A solution (p, q) of (3.1.54) gives a solution (x, y) of Theorem 3.1.3 as associated solution

(x, y) = ±(2abc + αbdq^{2}, αbpq).  (3.1.55)

For b-series, one should just change a and b places:

b-series:

\{ \begin{matrix}p^{2} − dq^{2} = 4bc/α \\ p ≡ μq mod 2bc \\ g.c.d(b, p, q) = 1 \\ p ≢ μq mod 2bcl for any prime l|a \end{matrix} .  (3.1.56)

A solution (p, q) of (3.1.56) gives a solution (x, y) of Theorem 3.1.3 as associated solution

(x, y) = ±(2abc + αadq^{2}, αapq).  (3.1.57)

All h̃ (or solutions (x, y)) of Theorem 3.1.3 are given by (3.1.55) and (3.1.57) as associated solutions.

Now let us interpret solutions (p, q) of Theorem 3.1.4 as elements of the Picard lattice N(X). Further ±1 denotes α. If there is an ambiguity with other ±, we introduce α (e.g. see (3.1.63) below).

Let us consider a solution (p, q) of the a-series (3.1.54). The congruence p ≡ μq mod 2ac is equivalent to

tp ≡ tμq mod 2abc^{2} ,  (3.1.58)

where

t = bc.  (3.1.59)

The congruence (3.1.58) is equivalent to

h_{1} = \frac{t(pH + qδ)}{2abc^{2}} ∈ N(X) and h_{1} · H ≡ 0 mod t.  (3.1.60)

We have p^{2} − dq^{2} = ±4ac is equivalent to (h_{1})^{2} = t^{2}(p^{2} − dq^{2})/(2abc^{2}) = ±2bc. Moreover, H · h_{1} ≡ 0 mod bc.

The condition g.c.d(a, p, q) = 1 can be interpreted as l does not divide p = H · h_{1}/t, if l^{2}|a. Really, if l|a and l|p, then l|q since p^{2} − dq^{2} = ±4ac. Thus, we get l|(H · h_{1})/bc is impossible for l^{2}|a.

Assume that p ≡ μq mod 2acl for a prime l|b. This is equivalent to pt/l ≡ μqt/l mod 2abc^{2}. Equivalently, h_{1}/l = ((pt/l)H + (qt/l)δ) /(2abc^{2}) ∈ N(X). Thus, such a prime l does not exist, if and only if h_{1} is not divisible by l|b in N(X). It is enough to consider l^{2}|b.

Changing a and b places, we get similar statements for the b-series.

Thus, we finally get

Theorem 3.1.5. With notations and conditions of Theorem 3.1.3, for a general K3-surface X with ρ(X) = 2 and γ(H) = 1 for H ∈ N(X), we have Y ≅ X, if and only if at least for one of signs ± (i. e. α = ±1) there exists h_{1} ∈ N(X) such that h_{1} belongs to one of a-series or b-series:

a-series:

h_{1}^{2} = ±2bc, H · h_{1} ≡ 0 mod bc, H · h_{1} ≢ 0 mod bcl_{1}, h_{1}/l_{2} ∉ N(X) (3.1.61)

where l_{1} and l_{2} are any primes such that l_{1}^{2}|a and l_{2}^{2}|b; b-series:

h_{1}^{2} = ±2ac, H · h_{1} ≡ 0 mod ac, H · h_{1} ≢ 0 mod acl_{1}, h_{1}/l_{2} ∉ N(X) (3.1.62)

where l_{1} and l_{2} are any primes such that l_{1}^{2}|b and l_{2}^{2}|a.

Important Remark. By Theorem 3.1.3 and the formulae (3.1.55), (3.1.57) for the associated solution, we get that

±h̃ = \{ \begin{matrix}−\frac{H}{c} + \frac{α(H·h_{1})h_{1}}{bc^{2}} , if h_{1} is from a-series, \\ −\frac{H}{c} + \frac{α(H·h_{1})h_{1}}{ac^{2}} , if h_{1} is from b-series \end{matrix}  (3.1.63)

belongs to N(X) and

(Y, h) ≅ (X, ±w(h̃)) for some w ∈ W^{(−2)}(N(X)).  (3.1.64)

It describes moduli (Y, h) of sheaves in terms of X where h = (−a, 0, b) mod ℤv.

Applying additionally Theorem 2.2.1, we get the following simple sufficient condition when Y ≅ X which is valid for X with any ρ(X). This is one of the main results of the paper.

Theorem 3.1.6. Let X be a K3 surface with a polarization H such that H^{2} = 2rs, r, s ≥ 1, the Mukai vector (r, H, s) is primitive and Y the moduli of sheaves on X with the Mukai vector v = (r, H, s).

Then Y ≅ X, if at least for one of signs ± (i. e. α = ±1) there exists h_{1} ∈ N(X) such that elements H, h_{1} are contained in a 2-dimensional sublattice N ⊂ N(X) with H · N = ℤ and h_{1} belongs to one of a-series or b-series described below:

a-series:

h_{1}^{2} = ±2bc, H · h_{1} ≡ 0 mod bc, H · h_{1} ≢ 0 mod bcl_{1}, h_{1}/l_{2} ∉ N(X) (3.1.65)

where l_{1} and l_{2} are any primes such that l_{1}^{2}|a and l_{2}^{2}|b; b-series:

h_{1}^{2} = ±2ac, H · h_{1} ≡ 0 mod ac, H · h_{1} ≢ 0 mod acl_{1}, h_{1}/l_{2} ∉ N(X) (3.1.66)

where l_{1} and l_{2} are any primes such that l_{1}^{2}|b and l_{2}^{2}|a.

We also have the formulae (3.1.63) and (3.1.64) for (Y, h) where h = (−a, 0, b) mod ℤv.

The conditions above are necessary for γ(H) = 1 and Y ≅ X, if ρ(X) ≤ 2 and X is a general K3 surface with its Picard lattice, i. e. the automorphism group of the transcendental periods (T (X), H^{2,0}(X)) of X is ±1.

4. Divisorial conditions on moduli of (X, H) which imply γ(H) = 1 and Y ≅ X

Further we use the following notations. We fix:

r, s ∈ ℕ, r, s ≥ 1, r = ca, s = cb where c = g.c.d.(r, s) and then g.c.d.(a, b) = 1; \overline{μ} = \{μ, −μ\} ⊂ (ℤ/2rs)^{∗}. For any choice of the sign ± (equivalently, α = ±1), we denote

D(r, s; a)\overline{μ} ; ± = \{d ∈ ℕ | d ≡ μ^{2} mod 4abc^{2}, ∃ (p, q) ∈ ℤ × ℤ : p^{2} − dq^{2} = ±4ac,
p − μq ≡ 0 mod 2ac, g.c.d(a, p, q) = 1,
p − μq ≢ 0 mod 2acl for any prime l such that l^{2}|b}.  (4.1)

The sets D(r, s; a)\overline{μ} ; ± are called the a-series.

For any choice of the sign ± we denote

D(r, s; b)\overline{μ} ; ± = \{d ∈ ℕ | d ≡ μ^{2} mod 4abc^{2}, ∃ (p, q) ∈ ℤ × ℤ : p^{2} − dq^{2} = ±4bc,
p − μq ≡ 0 mod 2bc, g.c.d(b, p, q) = 1,
p − μq ≢ 0 mod 2bcl for any prime l such that l^{2}|a}.  (4.2)

The sets D(r, s; b)\overline{μ} ; ± are called the b-series.

Let X be a K3 surface with a polarization H of the degree H^{2} = 2rs and a primitive Mukai vector (r, H, s). The moduli space Y of sheaves on X with the Mukai vector v = (r, H, s) has the canonical nef element h = (−a, 0, b) mod ℤv with h^{2} = 2ab.

If γ(H) = 1, then ρ(X) ≥ 2. Since the dimension of the moduli space of (X, H) is equal to 20 − ρ(X), it follows that describing general (X, H) with ρ(X) = 2, γ(H) = 1 and Y ≅ X, we at the same time describe all possible divisorial conditions on moduli of (X, H) which imply γ(H) = 1 and Y ≅ X. They are described by the invariants of the pairs (N(X), H) where rk N(X) = 2. Here we use results of [10,11] (see also [3]) about irreducibility of moduli of K3 surfaces with a fixed Picard lattice.

Using Theorem 3.1.4, we get

Theorem 4.1. With the above notations all possible divisorial conditions on moduli of polarized K3 surfaces (X, H) with a polarization H of the degree H^{2} = 2rs, which imply γ(H) = 1 and Y ≅ X are labelled by the set

Div(r, s) = Div(r, s; a) ⋃ Div(r, s; b),  (4.3)

where

Div(r, s; a) = \{(d, \overline{μ}) | \overline{μ} = \{μ, −μ\} ⊂ (ℤ/2rs)^{∗}, d ∈ D(r, s; a)\overline{μ}\},
Div(r, s; b) = \{(d, \overline{μ}) | \overline{μ} = \{μ, −μ\} ⊂ (ℤ/2rs)^{∗}, d ∈ D(r, s; b)\overline{μ}\}  (4.4)

and

D(r, s; a)\overline{μ} = D(r, s; a)\overline{μ} ; + ⋃ D(r, s; a)\overline{μ} ; −, D(r, s; b)\overline{μ} = D(r, s; b)\overline{μ} ; + ⋃ D(r, s; b)\overline{μ} ; −.

(4.5)

The sets

D(r, s; a)\overline{μ} ; ± and D(r, s; b)\overline{μ} ; ±  (4.6)

are infinite if they are not empty.

The set D(r, s; a)\overline{μ} ; ± is not empty, if and only if the congruence

μqt + act^{2} ≡ ±1 mod bcq^{2}  (4.7)

has a solution q ∈ ℕ, t ∈ ℤ. E.g. it is true for q = 1, if b|c^{2}. The set D(r, s; b)\overline{μ} ; ± is not empty, if and only if the congruence

μqt + bct^{2} ≡ ±1 mod acq^{2}  (4.8)

has a solution for q ∈ ℕ, t ∈ ℤ. E. g. it is true for q = 1, if a|c^{2}. See other general results of this type in Theorems 4.5 and 4.6 below.

The set Div(r, s) is always non-empty and then infinite: i. e. at least one of sets D(r, s; a)\overline{μ} ; ± or D(r, s; b)\overline{μ} ; ± is not empty for one of \overline{μ} and ±. Its subset Div(r, s; a) ⊂ Div(r, s) is non-empty and then infinite, if either abc is odd or ac is even; its subset Div(r, s; b) ⊂ Div(r, s) is non-empty and then infinite, if either abc is odd or bc is even.

Below we prove statements of Theorem 4.1 which don’t follow from Theorem 3.1.4 straightforward. These considerations are important as itself, and they give more exact and strong statements as we have formulated in Theorem 4.1.

Further we consider the a-series. The same results are valid for the b-series changing a and b places. For q ∈ ℕ such that g.c.d(a, q) = 1 we introduce

D(r, s; a)\overline{μ} ; ±(q) = \{d ∈ ℕ | d ≡ μ^{2} mod 4abc^{2}, ∃ p ∈ ℤ : p^{2} − dq^{2} = ±4ac,
p − μq ≡ 0 mod 2ac, g.c.d(a, p, q) = 1,
p − μq ≢ 0 mod 2acl for any prime l such that l^{2}|b}. (4.9)

By definition, the set D(r, s; a)\overline{μ} ; ±(q) = ∅, if g.c.d(a, q) > 1.

Obviously,

D(r, s; a)\overline{μ} ; ± = ⋃_{q∈ℕ} D(r, s; a)\overline{μ} ; ±(q).  (4.10)

We have

Lemma 4.2. For any ± and any q ∈ ℕ the set D(r, s; a)\overline{μ} ; ±(q) is either empty or infinite. It is infinite, if and only if the system of congruences

\{ \begin{matrix}p ≡ μq mod 2ac, \\ p^{2} ≡ ±4ac + μ^{2}q^{2} mod 4abc^{2}q^{2}, \\ p ≢ μq mod 2acl \end{matrix}  (4.11)

has a solution. Here l is any prime such that l^{2}|b, the condition p ≢ μq mod 2acl must satisfy for all these primes. The set D(r, s; a)\overline{μ} ; α(q) consists of all d = (p^{2} ∓ 4ac)/q^{2} where p satisfies the above system of congruences and p^{2} ∓ 4ac > 0.

Proof. It at once follows from definitions using that d = (p^{2} ∓4ac)/q^{2} > 0. Clearly, if p = p_{0} satisfies the system (4.11), then p = p_{0} + t(4abc^{2}q^{2}), t ∈ ℤ, satisfies (4.11), and p^{2} ∓4ac > 0 for |t| >> 0. It gives an infinite subset of D(r, s; a)\overline{μ} ; ±(q). It finishes the proof.

We denote

α = ±1.  (4.12)

Let us rewrite (4.11). We have p = μq + 2act, t ∈ ℤ where g.c.d(t, b) = 1. Then (μq + 2act)^{2} ≡ 4αac + μ^{2}q^{2} mod 4abc^{2}q^{2}. It follows μqt + act^{2} ≡ ±1 mod bcq^{2}.

Then g.c.d(t, b) = 1 and g.c.d(q, a) = 1. It follows that (4.11) is equivalent to the congruence

μqt + act^{2} ≡ ±1 mod bcq^{2}.  (4.13)

A solution t ∈ ℤ of (4.13) gives the infinite series of elements from D(r, s; a)\overline{μ} ; ±(q) which is

\frac{(μq + 2ac(t + bcq^{2}m))^{2} ∓ 4ac}{q^{2}} > 0, m ∈ ℤ .  (4.14)

Thus, additionally to Lemma 4.2 we get

Lemma 4.3. For q ∈ ℕ the set D(r, s; a)\overline{μ} ; ±(q) is non-empty, if and only if the congruence (4.13) has a solution t ∈ ℤ. Each such solution t generates an infinite series of elements from D(r, s; a)\overline{μ} ; ±(q) given by (4.14). Together they give all elements of D(r, s; a)\overline{μ} ; ±(q).

In particular, the set D(r, s; a)\overline{μ} ; ± is non-empty (and then it is infinite), if and only if the congruence (4.13) has a solution (q, t) for q ∈ ℕ and t ∈ ℤ.

For q = 1, the congruence (4.13) gives

μt + act^{2} ≡ α mod bc.  (4.15)

We have μt ≡ α mod c and t = μ^{−1}α + kc, k ∈ ℤ. Simple calculations show that (4.15) is equivalent to the quadratic equation

k^{2}ac^{2} + k(μ + 2αacμ^{−1}) + aμ^{−2} ≡ 0 mod b.  (4.16)

For A ≠ 0, the quadratic equation

Ax^{2} + Bx + C ≡ 0 mod b  (4.17)

is equivalent to

(2Ax + B)^{2} ≡ B^{2} − 4AC mod 4Ab.  (4.18)

It follows that (4.15) is equivalent to

(2ac^{2}k + μ + 2αacμ^{−1})^{2} ≡ (μ^{2} + 4αac) mod 4abc^{2}.  (4.19)

In particular, (4.16) (equivalently (4.19)) has a solution, if c^{2} ≡ 0 mod b. Then A = ac^{2} ≡ 0 mod b and B = μ + 2αacμ^{−1} is invertible mod b. Thus, we get

Lemma 4.4. The set D(r, s; a)\overline{μ} ; ±(1) is non-empty and then infinite, if and only if the quadratic equation (4.16) (equivalently, (4.19)) has a solution k.

In particular, this is valid, if b|c^{2}.

Using Lemma 4.4, we also get the following general result:

Theorem 4.5. The set D(r, s; a)\overline{μ} ; α(1) is non-empty for μ ≡ θ^{−1}ac−θα mod 2abc^{2}, if θ ∈ (ℤ/2abc^{2})^{∗} and g.c.d(2b, θ^{−1}ac − θα) = 1.

It follows that D(r, s; a)\overline{μ} ; α(1) is non-empty at least for one of α and \overline{μ}, if ac is even. In particular, the set Div(r, s; a) is not empty, if ac is even.

Proof. By conditions on θ, we have that μ ≡ θ^{−1}ac − θα mod 2abc^{2} ∈ (ℤ/2abc^{2})^{∗}. Then μ^{2} + 4αac ≡ ν^{2} mod 4abc^{2} where ν ≡ θ^{−1}ac + θα mod 2abc^{2}. Thus, (4.19) satisfies, if

2ac^{2}k + μ + 2α acμ^{−1} ≡ −ν mod 2abc^{2}

has a solution k. Thus, we should have

2ac^{2}k + (θ^{−1}ac − θα) + \frac{2αac}{θ^{−1}ac − θα} ≡ −(θ^{−1}ac + θα) mod 2abc^{2}.

This gives

(θ^{−1}ac − θα)2ac^{2}k ≡ −2θ^{−2}a^{2}c^{2} mod 2abc^{2}.

Equivalently,

(θ^{−1}ac − θα)k ≡ −θ^{−2}a mod b.

It has a solution k since g.c.d(θ^{−1}ac − θα, b) = 1. By Lemma 4.4, the set D(r, s; a)\overline{μ} ; α(1) is not then empty.

Assume that ac is even. We can find α such that for any prime p_{i}|2b there exists x_{i} mod p_{i} such that x_{i} ≢ 0 mod p_{i} and −αx_{i}^{2} + ac ≢ 0 mod p_{i}. Really, if p_{i} = 2, we can take x_{i} ≡ 1 mod 2. If p_{i} = 3, we can take x_{i} ≡ ±1 mod 3. If ac ≡ 0 mod 3, we can take any α = ±1. If ac ≡ 1 mod 3, we take α = −1. If ac ≡ −1 mod 3, we take α = 1. If p_{i} ≥ 5, the equation −αx^{2} + ac ≡ 0 mod p_{i} has not more then two solutions mod p_{i}, and there always exists x_{i} mod p_{i} with the above conditions.

By standard results, there exists x mod 2b such that x ≡ x_{i} mod p_{i}. It then follows that x ∈ (ℤ/2b)^{∗} and (−αx^{2}+ac, 2b) = 1. By standard results and considerations, there exists θ ∈ (ℤ/2abc^{2})^{∗} such that θ ≡ x mod 2b. Then (−αθ^{2}+ac, 2b) = 1 and also (−αθ + acθ^{−1}, 2b) = 1. For the found α and θ, the set D(r, s; a)\overline{μ} ; α(1) is not empty, if μ ≡ −αθ + acθ^{−1} mod 2abc^{2}.

This finishes the proof.

Considering Theorem 4.5 also for the b-series, we get that the set Div(r, s) = Div(r, s; a) ⋃ Div(r, s; b) is not empty, if at least one of a, b or c is even.

Thus, to prove that the set Div(r, s) is not empty, we need to consider only the case when abc is odd. Then we have

Theorem 4.6. Assume that abc is odd. Then the set D(r, s; a)\overline{μ} ; α(1) is non-empty for μ mod 2abc^{2} ∈ (ℤ/2abc^{2})^{∗}, if μ ≡ θ^{−1}ac − θα mod abc^{2} where θ ∈ (ℤ/abc^{2})^{∗} and g.c.d(b, θ^{−1}ac − θα) = 1.

It follows that D(r, s; a)\overline{μ} ; α(1) is not empty at least for one of α and \overline{μ}. In particular, the set Div(r, s; a) is not empty.

Proof. The congruence (4.19) is the identity mod 4. Then, since abc is odd, (4.19) is equivalent to

(2ac^{2}k + μ + 2αacμ^{−1})^{2} ≡ (μ^{2} + 4αac) mod abc^{2}.  (4.20)

Thus, it is enough to consider μ mod abc^{2}. Then the same considerations as for Theorem 4.5 prove the statement.

We can summarize the main results:

Theorem 4.7. For any r, s the set Div(r, s) of divisorial conditions on moduli is non-empty and then infinite.

Its subset Div(r, s; a) is non-empty and then infinite, if either abc is odd or ac is even; its subset Div(r, s; b) is non-empty and then infinite, if either abc is odd or bc is even.

5. The results from the viewpoint of Y.

It is interesting and important to consider the same problems from the viewpoint of Y. In this case, we fix a, b ∈ ℕ with g.c.d(a, b) = 1, and we fix c ∈ ℕ. Further we denote r = ac and s = bc.

We consider a K3 surface Y with a nef element h ∈ N(Y) such that h^{2} = 2ab. We ask: When (Y, h) is the moduli space M of sheaves on Y with the Mukai vector v = (r, H, s) and the canonical nef element h̃ = (−a, 0, b) mod ℤv where H ∈ N(Y) is nef, H^{2} = 2rs and γ(H) = 1? More exactly, when there exists an isomorphism ξ : M → Y such that ξ(h̃) = h?

This question is very similar to the previous ones for X, and, in principle, the corresponding results can be deduced from the results for X above. But it is simpler to reconsider the previous calculations from the viewpoint of Y. They are very similar but sometimes very non-trivial. It is surprising that the results for Y below are so similar to the results for X.

Theorem 5.1 below is a reformulation of Theorem 2.2.1. We use notations from Sect. 2 as in Theorem 2.2.1.

Theorem 5.1. Let Y be a K3 surface with a nef element h of the degree h^{2} = 2ab.

Then (Y, h) is (isomorphic to) the moduli space (M, h̃) of coherent sheaves on Y with the Mukai vector v = (r, H, s), where H^{2} = 2rs, γ(H) = 1, and the canonical nef element h̃ = (−a, 0, b) mod ℤv, if γ(h) = 1 and the conditions (i) and (ii) below hold:

(i) there exists H̃ ∈ N(Y) with H̃^{2} = 2rs, γ(H̃) = 1, and such that there exists an embedding

f : K(H̃) → K(h)

of negative definite lattices such that

f^{∗}(K(h)) = [K(H̃), 2abcu^{∗}(H̃)],
f^{∗}(u^{∗}(h)) + f^{∗}(K(h)) = m(a, b)c u^{∗}(H̃) + f^{∗}(K(h)).  (5.1)

(ii) For the h and f in (i), there exists a choice of ± such that

κ(h)(k^{∗}) = ±κ(H̃)(f^{∗}(k^{∗}))  (5.2)

for any k^{∗} ∈ K(h)^{∗}.

The condition γ(h) = 1 is necessary for any Y.

The conditions (i) and (ii) are necessary for a K3 surface Y with ρ(Y) ≤ 19 which is general for its Picard lattice N(Y) in the following sense: the automorphism group of the transcendental periods (T (Y), H^{2,0}(Y)) is ±1.

If ρ(Y) = 20 the theorem is always valid, if γ(h) = 1.

Now assume that rk N(Y) = 2 and h ∈ N(Y) has the degree h^{2} = 2ab, γ(h) = 1. Applying Proposition 3.1.1 to c = 1, we get

Proposition 5.2. Let Y be a K3 surface with rk N(Y) = 2 equipped with a primitive nef element h of the degree h^{2} = 2ab, and γ(h) = 1.

The pair (N(Y), h) has the invariants d ∈ ℕ and ±ν mod 2ab ∈ (ℤ/2ab)^{∗} such that ν^{2} ≡ d mod 4ab. We have: det N(Y) = −d, and K(h) = h_{N(Y)}^{⊥} = ℤδ_{1} where δ_{1}^{2} = −2abd. Moreover,

N(Y) = [h, δ_{1}, \frac{(νh + δ_{1})}{2ab}],  (5.3)

N(Y) = \{z = \frac{(xh + yδ_{1})}{2ab} | x, y ∈ ℤ and x ≡ νy mod 2ab\}.  (5.4)

We have z^{2} = (x^{2} − dy^{2})/2ab.

For any primitive element h′ ∈ N(Y) of the degree (h′)^{2} = h^{2} = 2ab and the same invariant ±ν, there exists an automorphism ϕ ∈ O(N(Y)) such that ϕ(h) = h′.

From Proposition 3.1.1 applied to (N(Y), H̃) where H̃^{2} = 2abc^{2} and γ(H̃) = 1, and Proposition 5.2 we get that g.c.d(d, 2rs) = g.c.d(d, 2ab) = 1, and we get similar statement as Proposition 3.1.2.

Proposition 5.3. Under conditions and notations of Propositions 5.2, all elements H̃ = (xh + yδ_{1})/(2ab) ∈ N(Y) with H̃^{2} = 2abc^{2} are in one to one correspondence with integral solutions (x, y) of the equation x^{2} − dy^{2} = 4a^{2}b^{2}c^{2} such that x ≡ νy mod 2ab.

For k ∈ ℕ and k > 1 the element H̃ is divisible by k in N(Y) and then it is not primitive, if and only if k | (x, y) and x ≡ νy mod 2abk. Then k^{2}|abc^{2}. We call such a pair (x, y) as ν-divisible by k. If (x, y) is not ν-divisible by k > 1, we call such a pair (x, y) as ν-primitive. It is enough to consider only prime k. Thus, H̃ is primitive, if and only if the corresponding pair (x, y) is ν-primitive. If g.c.d(d, c) = 1 and H̃ is primitive, then γ(H̃) = 1.

The crucial statement is the statement which is similar to Theorem 3.1.3:

Theorem 5.4. Let Y be a K3 surface with ρ(Y) = 2, and h ∈ N(Y) a nef element of the degree h^{2} = 2ab with γ(h) = 1. As in Proposition 5.2, we consider invariants d and ν ∈ (ℤ/2ab)^{∗} of the pair (N(Y), h). Thus, d ≡ ν^{2} mod 4ab. Assume that g.c.d(c, d) = 1.

With notations of Proposition 5.2, all elements H̃ = (xh + yδ_{1})/(2ab) ∈ N(Y) with H̃^{2} = 2abc^{2} satisfying Theorem 5.1 are in one to one correspondence with integral solutions (x, y) of the equation

x^{2} − dy^{2} = 4a^{2}b^{2}c^{2}  (5.5)

which satisfy the conditions (i), (ii), (iii), (iv) below:

(i) x ≡ νy mod 2ab;

(ii) x ≡ ±2abc mod d;

(iii) (x, y) belongs to one of a-series or b-series of solutions defined below:

a-series: b|(x, y) and x − νy ≡ 0 mod 2a^{2}b where ν^{2} ≡ d mod 4a^{2}b, b-series: a|(x, y) and x − νy ≡ 0 mod 2ab^{2} where ν^{2} ≡ d mod 4ab^{2}; (iv) the pair (x, y) is ν-primitive: there does not exist a prime l such that l|(x, y)

and x − νy ≡ 0 mod 2ab l (we always have l^{2}|abc^{2} for such l), i. e. g.c.d (x, y, (x − νy)/(2ab)) = 1.

In particular, by Theorem 5.1, for a general Y with ρ(Y) = 2, the pair (Y, h) is the moduli space (M, h̃) of coherent sheaves over Y with the Mukai vector v = (r, H, s), where H^{2} = 2rs, γ(H) = 1, and the canonical nef element h̃ = (−a, 0, b) mod ℤv, if g.c.d(d, c) = 1 and the equation x^{2} − dy^{2} = 4a^{2}b^{2}c^{2} has an integral solution (x, y) satisfying conditions (i)—(iv) above. Moreover, a nef primitive element H̃ = (xh + yδ_{1})/(2ab) ∈ N(Y) with H̃^{2} = 2abc^{2} defines the Mukai vector (r, H̃, s) with the moduli space (M, h̃) of coherent sheaves on Y which is isomorphic to (Y, h), if and only if (x, y) satisfies the conditions (ii) and (iii) (it satisfies the conditions (i) and (ii) since it corresponds to a primitive element of N(Y)).

We can deduce from Theorem 5.4 a similar result as Theorem 3.1.4

Theorem 5.5. With conditions and notations of Theorem 5.4, for a general Y with ρ(Y) = 2 and γ(h) = 1, we have (Y, h) is the moduli space (M, h̃) of sheaves over Y with the Mukai vector v = (r, H̃, s) where H̃^{2} = 2rs, γ(H̃) = 1, and the canonical nef element h̃ = (−a, 0, b) mod ℤv, if and only if g.c.d(c, d) = 1, for one of α = ±1 there exists an integral solution (p, q) for one of the a-series or the b-series:

The a-series:

\{ \begin{matrix}p^{2} − dq^{2} = 4ac/α \\ p ≡ νq mod 2a \\ g.c.d(l_{1}, p, q) = 1 for any prime l_{1}|ac and l_{1} |̸b \\ p ≢ νq mod 2al_{2} for any prime l_{2}|b \end{matrix} .  (5.6)

A solution (p, q) of (5.6) gives a solution (x, y) of Theorem 5.4 as the associated solution

(x, y) = ±(2abc + αbdq^{2}, αbpq).  (5.7)

The b-series:

\{ \begin{matrix}p^{2} − dq^{2} = 4bc/α \\ p ≡ νq mod 2b \\ g.c.d(l_{1}, p, q) = 1 for any prime l_{1}|bc and l_{1} |̸a \\ p ≢ νq mod 2bl_{2} for any prime l_{2}|a \end{matrix} .  (5.8)

A solution (p, q) of (5.8) gives a solution (x, y) of Theorem 5.4 as the associated solution

(x, y) = ±(2abc + αadq^{2}, αapq).  (5.9)

All H̃ (or solutions (x, y)) of Theorem 5.4 are given by (5.7) and (5.9) as associated solutions.

Like in Theorems 3.1.5 and 3.1.6 we can interpret (p, q) of Theorem 5.5 as elements h_{1} = b(ph+qδ_{1})/(2ab) ∈ N(Y) for the a-series and h_{1} = a(ph+qδ_{1})/(2ab) ∈ N(Y) for the b-series. We formulate only the statement which is analogous to Theorem 3.1.6. It contains the statement which is similar to Theorem 3.1.5.

Theorem 5.6. Let Y be a K3 surface with a nef element h such that h^{2} = 2ab and γ(h) = 1. Then (Y, h) is the moduli space (M, h̃) of sheaves over Y with the Mukai vector v = (r, H̃, s) where H̃^{2} = 2rs, γ(H̃) = 1, and the canonical nef element h̃ = (−a, 0, b) mod ℤv, if at least for one of signs ± (i. e. α = ±1) there exists h_{1} ∈ N(Y) such that elements h, h_{1} are contained in a 2-dimensional sublattice N ⊂ N(Y) with h · N = ℤ, g.c.d(det(N), c) = 1, and h_{1} belongs to one of the a-series or the b-series:

The a-series:

h_{1}^{2} = ±2bc, h · h_{1} ≡ 0 mod b, h · h_{1} ≢ 0 mod bl_{1}, h_{1}/l_{2} ∉ N(Y)  (5.10)

where l_{1} and l_{2} are any primes such that l_{1}|ac and l_{1} |̸b, and l_{2}|b.

The b-series:

h_{1}^{2} = ±2ac, h · h_{1} ≡ 0 mod a, h · h_{1} ≢ 0 mod al_{1}, h_{1}/l_{2} ∉ N(Y) (5.11)

where l_{1} and l_{2} are any primes such that l_{1}|bc and l_{1} |̸a, and l_{2}|a.

The conditions above are necessary, if ρ(Y) ≤ 2 and Y is a general K3 surface with its Picard lattice, i. e. the automorphism group of the transcendental periods (T (Y), H^{2,0}(Y)) of Y is ±1.

Important Remark. By Theorem 5.4 and the formulae (5.7) and (5.9) we get

±w(H̃) = \{ \begin{matrix}−ch + \frac{α(h·h_{1})h_{1}}{b} , if h_{1} is from a-series, \\ −ch + \frac{α(h·h_{1})h_{1}}{a} , if h_{1} is from b-series \end{matrix}  (5.12)

where w ∈ W^{(−2)}(N(Y)).

Using results of Sects 3 and 5 both, one can construct many compositions of the correspondences from Sects 3 and 5 of the K3 surfaces X ≅ Y with itself.

References

[1] Abe T., A remark on the 2-dimensional moduli spaces of vector bundles on K3 surfaces, Math. Res. Lett. 7 (2002), no. 4, 463–470.

[2] Cossec F.R., Reye Congruences, Trans. Amer. Math. Soc. 280 (1983), no. 2, 737–751.

[3] James D.G., On Witt’s theorem for unimodular quadratic forms, Pacific J. Math. 26 (1968), 303–316.

[4] Madonna C., Nikulin V.V., On a classical correspondence between K3 surfaces, Proc. Steklov Math. Inst. (2003) (to appear); (see also math.AG/0206158).

[5] Madonna C., Nikulin V.V., On a classical correspondence between K3 surfaces II, Clay Mathematics Proceedings (2003) (to appear); (see also math.AG/0206158).

[6] Mukai Sh., Symplectic structure of the moduli space of sheaves on an Abelian or K3 surface, Invent. math. 77 (1984), 101-116.

[7] Mukai Sh., On the moduli space of bundles on K3 surfaces, Vector bundles on algebraic varieties (Bombay, 1984), Tata Inst. Fund. Res. Studies in Math. no. 11, Bombay, 1987, pp. 341–413.

[8] Mukai Sh., Duality of polarized K3 surfaces, New trends in algebraic geometry (Warwick, 1996), London Math. Soc. Lecture Notes Series, 264, Cambridge Univ. Press, Cambridge, 1999, pp. 311–326.

[9] Mukai Sh., Non abelian Brill–Noether theory and Fano 3-folds,, alg-geom 9704015.

[10] Nikulin V.V., Finite automorphism groups of Kählerian surfaces of type K3, Trudy Mosk. Matem. Ob-va, 38 (1979), 75–137 (Russian); English transl. in Trans. Moscow Math. Soc. 38 (1980), no. 2, 71–135.

[11] Nikulin V.V., Integral symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177 (Russian); English transl. in Math. USSR Izv. 14 (1980).

[12] Nikulin V.V., On correspondences between K3 surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 2, 402–411 (Russian); English transl. in Math. USSR Izv. 30 (1988), no. 2 .

[13] Nikulin V.V., On correspondences of a K3 surface with itself, II, Preprint math.AG/ 0309348 (2003), 1–37 (Part II of this preprint).

[14] I.I. Pjatetckiĭ-S̆apiro and I.R. S̆afarevich, A Torelli theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), no. 3, 530–572 (Russian); English transl. in Math. USSR Izv. 5 (1971), no. 3, 547–588.

[15] Saint-Donat B., Projective models of K–3 surfaces, Amer. J. of Mathem. 96 (1974), no. 4, 602–639.

[16] Shafarevich I.R. (ed.), Algebraic surfaces, Trudy Matem. Inst. Steklov, T. 75, 1965 (Russian); English transl. in Proc. Steklov Inst. Math. 75 (1965).

[17] Tyurin A.N., Special 0-cycles on a polarized K3 surface, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 1, 131 – 151 (Russian); English transl. in Math. USSR Izv. 30 (1988), no. 1, 123–143.

[18] Tyurin A.N., Cycles, curves and vector bundles on algebraic surfaces, Duke Math. J. 54 (1987), no. 1, 1 – 26.

[19] Tyurin A.N., Symplectic structures on the varieties of moduli of vector bundles on algebraic surfaces with p_{g} > 0., Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 4, 813 – 852; English transl. in Math. USSR Izv. 33 (1989), no. 1, 139–177.

[20] Verra A., The étale double covering of an Enriques’ surface, Rend. Sem. Mat. Univers. Politecn. Torino 41 (1983), no. 3, 131–167.

ON CORRESPONDENCES OF A K3 SURFACE WITH ITSELF, II

Viacheslav V. Nikulin

Abstract. Let X be a K3 surface with a polarization H of degree H^{2} = 2rs, r, s ≥ 1, and with a primitive Mukai vector (r, H, s). The moduli space of sheaves over X with the isotropic Mukai vector (r, H, s) is again a K3 surface Y.

We prove that Y ≅ X, if the Picard lattice N(X) has an element h_{1} with (h_{1})^{2} = f(r, s), and the pair (H, h_{1}) satisfies a finite number of congruence conditions modulo N_{i}(r, s). All these conditions are exactly written, and they are necessary, if X is general with rk N(X) = 2.

Existence of such kind a criterion is very surprising, and it also gives some geometric interpretation of elements in N(X) with negative square. We also describe all irreducible divisorial conditions on moduli of (X, H) which imply Y ≅ X, and we prove that their number is infinite if non-empty.

Thus, this paper treats in general problems considered in [MN1], [MN2] and [N4], when the additional condition H · N(X) = ℤ had been imposed. In general, the results are much more complicated and less efficient.

0. Introduction

Let X be a K3 surface with a polarization H of degree H^{2} = 2rs, r, s ≥ 1. Assume that the Mukai vector v = (r, H, s) is primitive.

Let Y be the moduli of coherent sheaves over X with the isotropic Mukai vector v = (r, H, s) (see [Mu1]—[Mu4] and also [A]). The Y is a K3 surface which is equipped with a natural primitive nef element h with h^{2} = 2ab where we denote c = g.c.d(r, s) and a = r/c, b = s/c. The surface Y is isogenous to X in the sense of Mukai. The second Chern class of the corresponding quasi-universal sheave gives then a 2-dimensional algebraic cycle Z ⊂ X × Y and an algebraic correspondence between X and Y.

Let H be divisible by d ∈ ℕ where H̃ = H/d is primitive in the Picard lattice N(X) of X. Primitivity of v = (r, H, s) means that g.c.d(r, d, s) = 1. We have d^{2}|rs. Let γ = γ(H̃) is defined by H̃ · N(X) = γℤ, i.e. H · N(X) = γd. Clearly, γ|(2rs/d^{2}) = H̃^{2}.

We denote

n(v) = g.c.d(r, s, dγ).

By Mukai, [Mu2], [Mu3], n(v) = [T (Y) : T (X)] where T (X) and T (Y) are transcendental lattices of X and Y. We assume that

n(v) = g.c.d(r, s, dγ) = g.c.d(c, dγ) = 1.  (0.1)

By Mukai [Mu2], [Mu3], the transcendental periods (T (X), H^{2,0}(X)) and (T (Y), H^{2,0}(Y)) are isomorphic in this case. We can expect that sometimes the surfaces X and Y are also isomorphic, and we then get a cycle Z ⊂ X×X, and a correspondence of X with itself. Thus, an interesting for us question is

Question 1. When is Y isomorphic to X?

We want to answer this question in terms of Picard lattices N(X) and N(Y) of X and Y. Then our question can be reformulated as follows:

Question 2. Assume that N is a hyperbolic lattice, H_{1} ∈ N an element with square 2rs. What are conditions on N and H_{1} such that for any K3 surface X with Picard lattice N(X) and s polarization H ∈ N(X) the corresponding K3 surface Y is isomorphic to X, if the the pairs (N(X), H) and (N, H_{1}) are isomorphic as abstract lattices with fixed elements?

In other words, what are conditions on (N(X), H) as an abstract lattice with an element H which are sufficient for Y to be isomorphic to X, and they are necessary, if X is a general K3 surface with the Picard lattice N(X)?

We answered this question in [MN1], [MN2] and [N4] under the condition γ = 1: in [MN1] for r = s = 2 and d = 1; in [MN2] for r = s and d = 1; in [N4] for arbitrary r and s and d = 1.

The main surprising result of [MN1], [MN2] and [N4] was that Y ≅ X, if the Picard lattice N(X) has an element h_{1} with some prescribed square (h_{1})^{2} and some minor additional conditions. Moreover, these conditions are necessary to have Y ≅ X for a general K3 surface X with ρ(X) = 2. Here we prove similar results in general.

We assume (0.1). Then d^{2}|ab and γ|2ab/d^{2}. Moreover, g.c.d(a, b) = 1. We have d = d_{a}d_{b} where d_{a} = g.c.d(d, a) and d_{b} = g.c.d(d, b). We define

a_{1} = \frac{a}{d_{a}^{2}} , b_{1} = \frac{b}{d_{b}^{2}}

and

a_{2} = \frac{a_{1}}{γ_{a}} , b_{2} = \frac{b_{1}}{γ_{b}} , e_{2} = \frac{2}{γ_{2}}

where γ_{a} = g.c.d(a_{1}, γ), γ_{b} = g.c.d(b_{1}, γ), γ_{2} = γ/(γ_{a}γ_{b}).

We prove (see Theorem 3.1.16)

Theorem 0.1. Assume that

g.c.d(c, dγ) = 1.

Then Y ≅ X, if at least for one square-free α | 2a_{1}b_{1} there exists h̃_{1} ∈ N(X) which belongs to the a-series or to the b-series described below (moreover, one has (3.1.153) and (3.1.154)).

The h̃_{1} belongs to the a-series, if there exists a primitive sublattice N ⊂ N(X) of the rank two containing H̃ and h̃_{1} such that H̃ · N = γℤ and

(h̃_{1})^{2} = \frac{2γ^{2}b_{1}^{2}c}{α}, H̃ · h̃_{1} ≡ 0 mod b_{1}c, f(H̃) · h̃_{1} ≡ 0 mod b_{1}c det N,

and

p = \frac{H̃ · h̃_{1}}{b_{1}c} , q = \frac{f(H̃) · h̃_{1}}{b_{1}c det N}

satisfy conditions (G) (see (3.1.113)—(3.1.117)) and (A) (see (3.1.118)—(3.1.126)).

The h̃_{1} belongs to the b-series, if there exists a primitive sublattice N ⊂ N(X) of the rank two containing H̃ and h̃_{1} such that H̃ · N = γℤ and

(h̃_{1})^{2} = \frac{2γ^{2}a_{1}^{2}c}{α}, H̃ · h̃_{1} ≡ 0 mod a_{1}c, f(H̃) · h̃_{1} ≡ 0 mod a_{1}c det N,

and

p = \frac{H̃ · h̃_{1}}{a_{1}c} , q = \frac{f(H̃) · h̃_{1}}{a_{1}c det N}

satisfy the conditions (G) and (B) (see (3.1.127)—(3.1.135)).

These conditions are necessary to have Y ≅ X, if ρ(X) ≤ 2 and X is a general X K3 surface with its Picard lattice, i. e. the automorphism group of the transcendental periods (T (X), H^{2,0}(X)) is ±1.

It seems many (if not all) known examples when Y ≅ X follow from this Theorem. E.g. see [C], [T1]—[T3] and [V].

In Theorem 0.1 ℤf(H̃) is the orthogonal complement to H̃ in N. In conditions (G), (A) and (B) we use the invariant μ ∈ (ℤ/2a_{1}b_{1}c^{2}/γ))^{∗} of the pair H̃ ∈ N (use (3.1.2) for S = N) and δ = − det N/γ.

Like in [MN1], [MN2] and [N4] we also describe irreducible divisorial conditions on moduli of polarised K3 surfaces (X, H) which imply H̃ ·N(X) = γℤ and Y ≅ X. We show that they are labelled by pairs (±μ, δ) where ±μ ∈ (ℤ/(2a_{1}b_{1}c^{2}/γ))^{∗}, δ ∈ ℕ where δ ≡ μ^{2}γ mod 2a_{1}b_{1}c^{2}/γ, moreover at least for one α|2a_{1}b_{1} the equation

p^{2} − γδq^{2} = \frac{4a_{1}b_{1}c}{α}

has an integral solution (p, q) where (p, q) satisfy either conditions (G) and (A) (the a-series) or (G) and (B) (the b-series). See Sect. 4. As a result, we prove that the number of the divisorial conditions is either empty or infinite.

We remark that similar results in [N4] for the particular case d = γ = 1 are much more exact and simple. They don’t follow from the general results of this paper. At least, now we don’t know how to get results of [N4] from main results above of this paper.

As in [MN1], [MN2] and [N4], the fundamental tools to get the results above is the Global Torelli Theorem for K3 surfaces proved by Piatetsky-Shapiro and Shafarevich in [PS], and results of Mukai [Mu2], [Mu3]. By results of [Mu2], [Mu3], we can calculate periods of Y using periods of X; by the Global Torelli Theorem for K3 surfaces [PS], we can find out if Y is isomorphic to X.

These paper treats in general problems considered in [MN1], [MN2] and [N4] where the additional condition γ = d = 1 had been imposed. It makes results of this paper much more complicated and less efficient. For instance, in these paper we don’t consider the question of non-emptiness of the divisorial conditions on moduli. It is much more complicated in the general setting of this paper. We hope to consider this problem later.

1. Preliminary notations and results about lattices and K3 surfaces

1.1. Some notations about lattices. We use notations and terminology from [N2] about lattices, their discriminant groups and forms. A lattice L is a nondegenerate integral symmetric bilinear form. I. e. L is a free ℤ-module equipped with a symmetric pairing x · y ∈ ℤ for x, y ∈ L, and this pairing should be nondegenerate. We denote x^{2} = x · x. The signature of L is the signature of the corresponding real form L⊗ℝ. The lattice L is called even if x^{2} is even for any x ∈ L. Otherwise, L is called odd. The determinant of L is defined to be det L = det(e_{i} ·e_{j}) where {e_{i}} is some basis of L. The lattice L is unimodular if det L = ±1. The dual lattice of L is L^{∗} = Hom(L, ℤ) ⊂ L⊗ℚ. The discriminant group of L is A_{L} = L^{∗}/L. It has the order | det L|. The group A_{L} is equipped with the discriminant bilinear form b_{L} : A_{L} × A_{L} → ℚ/ℤ and the discriminant quadratic form q_{L} : A_{L} → ℚ/2ℤ if L is even. To get this forms, one should extend the form of L to the form on the dual lattice L^{∗} with values in ℚ.

For x ∈ L, we shall consider the invariant γ(x) ≥ 0 where

x · L = γ(x)ℤ.  (1.1.1)

Clearly, γ(x)|x^{2} if x ≠ 0.

We denote by L(k) the lattice obtained from a lattice L by multiplication of the form of L by k ∈ ℚ. The orthogonal sum of lattices L_{1} and L_{2} is denoted by L_{1} ⊕ L_{2}. For a symmetric integral matrix A, we denote by ⟨A⟩ a lattice which is given by the matrix A in some bases. We denote

U = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .  (1.1.2)

Any even unimodular lattice of the signature (1, 1) is isomorphic to U.

An embedding L_{1} ⊂ L_{2} of lattices is called primitive if L_{2}/L_{1} has no torsion. We denote by O(L), O(b_{L}) and O(q_{L}) the automorphism groups of the corresponding forms. Any δ ∈ L with δ^{2} = −2 defines a reflection s_{δ} ∈ O(L) which is given by the formula

x → x + (x · δ)δ,

x ∈ L. All such reflections generate the 2-reflection group W^{(−2)}(L) ⊂ O(L).

1.2. Some notations about K3 surfaces. Here we remind some basic notions and results about K3 surfaces, e. g. see [PS], [S-D], [Sh]. A K3 surface X is a non-singular projective algebraic surface over ℂ such that its canonical class K_{X} is zero and the irregularity q_{X} = 0. We denote by N(X) the Picard lattice of X which is a hyperbolic lattice with the intersection pairing x · y for x, y ∈ N(S). Since the canonical class K_{X} = 0, the space H^{2,0}(X) of 2-dimensional holomorphic differential forms on X has dimension one over ℂ, and

N(X) = {x ∈ H^{2}(X, ℤ) | x · H^{2,0}(X) = 0}  (1.2.1)

where H^{2}(X, ℤ) with the intersection pairing is a 22-dimensional even unimodular lattice of signature (3, 19). The orthogonal lattice T (X) to N(X) in H^{2}(X, ℤ) is called the transcendental lattice of X. We have H^{2,0}(X) ⊂ T (X) ⊗ ℂ. The pair (T (X), H^{2,0}(X)) is called the transcendental periods of X. The Picard number of X is ρ(X) = rk N(X). A non-zero element x ∈ N(X) ⊗ ℝ is called nef if x ≠ 0 and x · C ≥ 0 for any effective curve C ⊂ X. It is known that an element x ∈ N(X) is ample if x^{2} > 0, x is nef, and the orthogonal complement x^{⊥} to x in N(X) has no elements with square −2. For any element x ∈ N(X) with x^{2} ≥ 0, there exists a reflection w ∈ W^{(−2)}(N(X)) such that the element ±w(x) is nef; it then is ample, if x^{2} > 0 and x^{⊥} had no elements with square −2 in N(X).

We denote by V^{+}(X) the light cone of X, which is the half-cone of

V (X) = {x ∈ N(X) ⊗ ℝ | x^{2} > 0 }  (1.2.2)

containing a polarization of X. In particular, all nef elements x of X belong to \overline{V^{+}(X)}: one has x · V^{+}(X) > 0 for them.

The reflection group W^{(−2)}(N(X)) acts in V^{+}(X) discretely, and its fundamental chamber is the closure \overline{K(X)} of the Kähler cone K(X) of X. It is the same as the set of all nef elements of X. Its faces are orthogonal to the set Exc(X) of all exceptional curves r on X which are non-singular rational curves r on X with r^{2} = −2. Thus, we have

\overline{K(X)} = \{0 ≠ x ∈ \overline{V^{+}(X)} | x · Exc(X) ≥ 0 \}.  (1.2.3)

2. Condition of Y ≅ X for a general K3 surface X with a given Picard lattice

2.1. The correspondence. Let X be a smooth complex projective K3 surface with a polarization (or more generally a nef element) H of degree 2rs, r, s ≥ 1.

Let H is divisible by d ∈ ℕ and H̃ = H/d is primitive in N(X). Then H̃^{2} = 2rs/d^{2} and d^{2}|rs. We denote

c = g.c.d(r, s), a = r/c, b = s/c.  (2.1.1)

We assume that the Mukai vector (r, H, s) is primitive, i. e.

g.c.d(r, s, d) = g.c.d(c, d) = 1.  (2.1.2)

Let Y be the moduli space of (coherent) sheaves E on X with the primitive isotropic Mukai vector v = (r, H, s). Then rk E = r, χ(E) = r + s and c_{1}(E) = H. Let

H^{∗}(X, ℤ) = H^{0}(X, ℤ) ⊕ H^{2}(X, ℤ) ⊕ H^{4}(X, ℤ)  (2.1.3)

be the full cohomology lattice of X equipped with the Mukai product

(u, v) = −(u_{0} · v_{2} + u_{2} · v_{0}) + u_{1} · v_{1}  (2.1.4)

for u_{0}, v_{0} ∈ H^{0}(X, ℤ), u_{1}, v_{1} ∈ H^{2}(X, ℤ), u_{2}, v_{2} ∈ H^{4}(X, ℤ). We naturally identify H^{0}(X, ℤ) and H^{4}(X, ℤ) with ℤ. Then the Mukai product is

(u, v) = −(u_{0}v_{2} + u_{2}v_{0}) + u_{1} · v_{1}.  (2.1.5)

The element

v = (r, H, s) = (r, H, χ − r) ∈ H^{∗}(X, ℤ)  (2.1.6)

is isotropic, i.e. v^{2} = 0, since H^{2} = 2rs. In this case (for a primitive v), Mukai showed [Mu2], [Mu3] that Y is a K3 surface, and one has the natural identification

H^{2}(Y, ℤ) ≅ (v^{⊥}/ℤv)  (2.1.7)

which also gives the isomorphism of the Hodge structures of X and Y. The Y has the canonical nef element h = (−a, 0, b) mod ℤv with h^{2} = 2ab.

In particular, (2.1.5) gives the embedding

T (X) ⊂ T (Y)  (2.1.8)

of the transcendental periods of the index

[T (Y) : T (X)] = n(v) = min |v · x|  (2.1.9)

where x ∈ H^{0}(X, ℤ) ⊕ N(X) ⊕ H^{4}(X, ℤ) and v · x ≠ 0 (see [Mu2], [Mu3]). In this paper, we are interested in the case when Y ≅ X. By (2.1.10), it may happen, if n(v) = 1 only.

We can introduce the invariant γ = γ(H̃) ∈ ℕ which is defined by

H̃ · N(X) = γℤ,  (2.1.10)

equivalently, H · N(X) = γdℤ. Clearly, γ|2rs/d^{2} = H̃^{2}, and

n(v) = g.c.d(r, s, γd) = g.c.d(c, γd).  (2.1.11)

Thus, n(v) = 1, and it is possible to have Y ≅ X, only if

g.c.d(r, s, γd) = g.c.d(c, γ) = g.c.d(c, d) = 1.  (2.1.12)

This is exactly the case when, according to Mukai, the transcendental periods

(T (X), H^{2,0}(X)) ≅ (T (Y), H^{2,0}(Y))  (2.1.13)

are isomorphic.

2.2. The characteristic map of a primitive element of a lattice. Let S be an even lattice and P ∈ S its primitive element with P^{2} = 2m ≠ 0 and γ(P) = γ|2m (in S), i. e. P · S = γℤ.

We want to calculate the discriminant quadratic form of S. Consider

K(P) = P_{S}^{⊥}  (2.2.1)

the orthogonal complement to P in S. Put P^{∗} = P/2m. Then any element x ∈ S can be written as

x = nγP^{∗} + k^{∗}  (2.2.2)

where n ∈ ℤ and k^{∗} ∈ K(P)^{∗}, because

ℤP ⊕ K(P) ⊂ S ⊂ S^{∗} ⊂ ℤP^{∗} ⊕ K(P)^{∗}

and P · S = γℤ. Since γ(P) = γ, the map nγP^{∗} + [P] → k^{∗} + K(P) gives an isomorphism of the groups ℤ/\frac{2m}{γ} ≅ [γP^{∗}]/[P] ≅ [u^{∗}(P) + K(P)]/K(P) where u^{∗}(P)+K(P) has order 2m/γ in A_{K(P)} = K(P)^{∗}/K(P). Clarify in [N2]. It follows,

S = [ℤP, K(P), γP^{∗} + u^{∗}(P)].  (2.2.3)

The element u^{∗}(P) is defined canonically mod K(P) by the condition that γP^{∗} + u^{∗} ∈ S. The element

u^{∗}(P) + K(P) ∈ K(P)^{∗}/K(P)  (2.2.4)

is called the canonical element of P. Since γP^{∗} + u^{∗}(P) belongs to the even lattice S, it follows

(γP^{∗} + u^{∗}(P))^{2} = \frac{γ^{2}}{2m} + u^{∗}(P)^{2} ≡ 0 mod 2.  (2.2.5)

For n ∈ ℤ and k^{∗} ∈ K(P)^{∗}, we have x = nP^{∗} + k^{∗} ∈ S^{∗}, if and only if

(nP^{∗} + k^{∗}) · (γP^{∗} + u^{∗}(P)) = \frac{nγ}{2m} + k^{∗} · u^{∗}(P) ∈ ℤ.

It follows,

\begin{aligned}S^{∗} & = \{nP^{∗} + k^{∗} | n ∈ ℤ, k^{∗} ∈ K(P)^{∗}, n ≡ −\frac{2m}{γ} (k^{∗} · u^{∗}(P)) mod \frac{2m}{γ}\} ⊂ \\ & ⊂ ℤP^{∗} + K(P)^{∗}. \end{aligned}

(2.2.6)

It gives the calculation of the discriminant group A_{S} = S^{∗}/S where S is given by (2.2.3) and S^{∗} is given by (2.2.6).

We define the canonical submodule K̃(P)^{∗} ⊂ ℤ ⊕ K(P)^{∗} by the condition

K̃(P)^{∗} = \{n ∈ ℤ, k^{∗} ∈ K(P)^{∗} | n ≡ −\frac{2m}{γ} (k^{∗} · u^{∗}(P)) mod \frac{2m}{γ}\}. (2.2.7)

Now we define the characteristic map

κ(P) : K̃(P)^{∗} → A_{S},  (2.2.8)

by the condition

κ(P)(n, k^{∗}) = nP^{∗} + k^{∗} mod S.  (2.2.9)

Obviously, the characteristic map is epimorphic. Its kernel is

K̃(P)_{0}^{∗} = [(2mℤ, K), ℤ(γ, u^{∗}(P))] ≅ S.  (2.2.10)

Thus, we correspond to a primitive P ∈ S with P^{2} = 2m and γ(P) = γ the canonical triplet

(K(P), u^{∗}(P) + K(P), κ(P)).  (2.2.11)

This triplet is important because of the trivial but very important for us

Lemma 2.2.1. Let P_{1} ∈ S and P_{2} ∈ S are two primitive elements of an even lattice S with P_{1}^{2} ≠ 0 and P_{2}^{2} ≠ 0.

There exists an automorphism f ∈ O(S) such that f(P_{1}) = P_{2} and f gives ±1 on the discriminant group A_{S} = S^{∗}/S, if and only if P_{1}^{2} = P_{2}^{2}, γ(P_{1}) = γ(P_{2}), and there exists an isomorphism of lattices ϕ : K(P_{1}) → K(P_{2}) such that ϕ^{∗}(u^{∗}(P_{2}) + K(P_{2})) = u^{∗}(P_{1}) + K(P_{1}) and ϕ̃^{∗} = (id, ϕ^{∗}) : K̃(P_{2})^{∗} → K̃(P_{1})^{∗} is ± commuting with the characteristic maps κ(P_{1}) and κ(P_{2}), i. e.

κ(P_{1})ϕ̃^{∗} = ±κ(P_{2}).

Proof. Trivial.

We also mention that

det(S) = γ^{2} det K(P)/2m.  (2.2.12)

because [S : ℤP ⊕ K(P)] = 2m/γ 2.3. Relation between periods of X and Y. Here we consider the case and notations of Sect. 2.1. Thus, for the primitive isotropic Mukai vector v = (r, H, s), r, s ≥ 1, H^{2} = 2rs, we assume that for d ∈ ℕ, the element H̃ = H/d is primitive in N(X), γ(H̃) = γ (in N(X)). We remind that c = g.c.d(r, s), a = r/c, b = s/c, and n(v) = g.c.d(r, s, dγ) = (c, dγ) = 1. It follows, d^{2}|ab and γd^{2}|2ab. Thus, our data are defined by

a, b, c, d, γ ∈ ℕ, such that (a, b) = (d, c) = (d, γ) = 1, d^{2}|ab, γd^{2}|2ab, (2.3.1)

and by a primitive polarization

H̃ ∈ N(X) such that H̃^{2} = 2abc^{2}/d^{2}, γ(H̃) = γ.  (2.3.2)

Then, the Mukai vector v = (r, H, s) = (ac, dH̃, bc).

Let us denote by e_{1} the canonical generator of H^{0}(X, ℤ) and by e_{2} the canonical generator of H^{4}(X, ℤ). They generate the sublattice U in H^{∗}(X, ℤ) with the Gram matrix U. Consider Mukai vector v = (re_{1} + se_{2} + H). We have

N(Y) = v _{U⊕N(X)}^{⊥}/ℤv.  (2.3.3)

Let us calculate N(Y). Let K(H) = K(H̃) = (H)_{N(X)}^{⊥}. We denote H^{∗} =

H/(2rs) ∈ (ℤH)^{∗} = ℤH^{∗}, and H̃^{∗} = H̃/(2rs/d^{2}) = dH^{∗} ∈ (ℤH̃)^{∗} = ℤH̃^{∗}. Then we have an embedding of lattices of finite index

ℤH̃ ⊕ K(H) ⊂ N(X) ⊂ N(X)^{∗} ⊂ ℤH̃^{∗} ⊕ K(H)^{∗}  (2.3.4)

We have the orthogonal decomposition up to finite index

U ⊕ ℤH̃ ⊕ K(H) ⊂ U ⊕ N(X) ⊂ U ⊕ ℤH̃^{∗} ⊕ K(H)^{∗}.  (2.3.5)

Let f = x_{1}e_{1} + x_{2}e_{2} + yH^{∗} + z^{∗} ∈ v_{U⊕N(X)}^{⊥}, z^{∗} ∈ K(H)^{∗}. Then −sx_{1} − rx_{2} + y = 0 since f ∈ v^{⊥} and hence (f, v) = 0. Thus, y = (sx_{1} + rx_{2}) and

f = x_{1}e_{1} + x_{2}e_{2} + (sx_{1} + rx_{2})H^{∗} + z^{∗}.  (2.3.6)

Here f ∈ U ⊕ N(X) if and only if

x_{1}, x_{2} ∈ ℤ, sx_{1} + rx_{2} ≡ 0 mod d, \frac{sx_{1} + rx_{2}}{d} H̃^{∗} + z^{∗} ∈ N(X).  (2.3.7)

Equivalently, by Sect. 2.2, we have

sx_{1} + rx_{2} ≡ 0 mod dγ, z^{∗} = \frac{sx_{1} + rx_{2}}{dγ} u^{∗}(H) mod K(H).  (2.3.8)

We denote

h′ = (−a, b) ⊕ 0 ∈ U ⊕ N(X).

Clearly, h′ ∈ v^{⊥} and h = h′ mod ℤv ∈ N(Y). Thus, the orthogonal complement contains

[ℤv, ℤh′, K(H)]  (2.3.9)

where h′ = −ae_{1} + be_{2}, and (2.3.9) is a sublattice of finite index in (v^{⊥})_{U⊕N(X)}. The generators v, h′ and generators of K(H) are free, and we can rewrite f above using these generators with rational coefficients. We have

e_{1} = \frac{v − ch′ − H}{2r} , e_{2} = \frac{v + ch′ − H}{2s} .  (2.3.10)

It follows,

f = \frac{sx_{1} + rx_{2}}{2rs} v + \frac{c(−sx_{1} + rx_{2})}{2rs}h′ + z^{∗}  (2.3.11)

where x_{1}, x_{2}, z^{∗} satisfy (2.3.8). Considering mod ℤv, we finally get

N(Y) = \frac{c(−sx_{1} + rx_{2})}{2rs}h+ \frac{sx_{1} + rx_{2}}{dγ} u^{∗}(H̃)+K(H), where sx_{1}+rx_{2} ≡ 0 mod dγ.

(2.3.12)

Let us calculate the lattice K(h) = h_{N(Y)}^{⊥}. It is equal to (sx_{1}+rx_{2})/(dγ)u^{∗}(H̃)+ K(H) where sx_{1} + rx_{2} ≡ 0 mod dγ and −sx_{1} + rx_{2} = 0. It follows, x_{1} = ax, x_{2} = bx, x ∈ ℤ, and sx_{1} +rx_{2} = 2abcx ≡ 0 mod dγ, which is always true. Thus, K(h) = [2abc/(dγ)u^{∗}(H̃) + K(H)]. By Sect. 2.2, u^{∗}(H̃) + K(H) has the order 2abc^{2}/(d^{2}γ). We have (2abc/(dγ), 2abc^{2}/(d^{2}γ) = 2abc/(d^{2}γ)(d, c) where 2abc/(d^{2}γ) ∈ ℕ and (d, c) = 1. It follows, K(h) = [K(H), 2abc/(d^{2}γ)u^{∗}(H̃)], and the index [K(h) : K(H)] = c.

Let us show that d|h in N(Y) and h/d is primitive in N(Y). By (2.3.12), the primitive submodule in N(Y), generated by h, is c(−sx_{1} + rx_{2})/(2rs)h where (sx_{1} + rx_{2})/(dγ)u^{∗}(H̃) ∈ K(h) and sx_{1} + rx_{2} ≡ 0 mod dγ. From calculation of K(h) above, we then get (sx_{1} + rx_{2})/(dγ) ≡ 0 mod 2abc/(d^{2}γ). It follows, bx_{1} + ax_{2} ≡ 0 mod 2ab/d. Let d = d_{a}d_{b} where d_{a}|a and d_{b}|b. Then (a/d_{a})|x_{1} and (b/d_{b})|x_{2}. It follows, x_{1} = (a/d_{a})x̃_{1} and x_{2} = (b/d_{b})x̃_{2} where x̃_{1}, x̃_{2} ∈ ℤ and d_{b}x̃_{1} + d_{a}x̃_{2} ≡ 0 mod 2. It follows that the module c(−sx_{1} + rx_{2})/(2rs)h = (−d_{b}x̃_{1} + d_{a}x̃_{2})/(2d_{a}d_{b}) where d_{b}x̃_{1} + d_{a}x̃_{2} ≡ 0 mod 2. It follows that this module is ℤ(h/d). It proves the statement. We denote

h̃ = \frac{h}{d} .  (2.3.13)

We had proved that h̃ is primitive in N(Y) and h = dh̃. We have h̃^{2} = 2ab/d^{2}.

Let us show that γ(h̃) = γ(H) = γ. We remind that γ(h̃)ℤ = h̃ · N(Y). By Sect. 2.2 and (2.3.12), h̃^{2}/γ(h̃) is equal to the index [[(sx_{1} + rx_{2})/(dγ)u^{∗}(H̃) + K(H)] : K(h)] where sx_{1} + rx_{2} ≡ 0 mod dγ. Such elements x_{1}, x_{2} give cdγℤ. Thus, [(sx_{1} + rx_{2})/(dγ)u^{∗}(H̃) + K(H)] = [cu^{∗}(H̃) + K(H)]. We had proved that K(h) = [2abc/(d^{2}γ)u^{∗}(H̃) + K(H)]. Thus, the index is equal to 2ab/d^{2}γ. Then we get h̃^{2}/γ(h̃) = 2ab/(d^{2}γ(h̃)) = 2ab/(d^{2}γ). It follows γ(h̃) = γ. We had also proved that u^{∗}(h̃) = mcu^{∗}(H) + K(h) where m is defined mod 2ab/(d^{2}γ). We shall calculate m below.

Now we can rewrite (2.3.12) in the form (2.2.2). We denote h̃^{∗} = h̃/h̃^{2} = h̃/(2ab/d^{2}). We have

N(Y) = \frac{−bx_{1} + ax_{2}}{d} h̃^{∗} + \frac{(bx_{1} + ax_{2})}{dγ}cu^{∗}(H̃) + K(H)  (2.3.14)

where bx_{1} + ax_{2} ≡ 0 mod dγ. We have proved that the elements (−bx_{1} + ax_{2})/d give ℤγ.

Let us write d = d_{a}d_{b} where d_{a}|a and d_{b}|b. Since γ|2ab/d^{2} and g.c.d(a, b) = 1, we can write γ = γ_{2}γ_{a}γ_{b} where γ_{a} = g.c.d(γ, a/d_{a}^{2}), γ_{b} = g.c.d(γ, b/d_{b}^{2}) and γ_{2} = γ/(γ_{a}γ_{b}). Clearly, γ_{2}|2.

We define m = m(a, b, d, γ) mod 2ab/(d^{2}γ) by the conditions

m ≡ −1 mod 2a/(d_{a}^{2}γ_{a}γ_{2}) and m ≡ 1 mod 2b/(d_{b}^{2}γ_{b}γ_{2}).  (2.3.15)

The congruences (2.3.15) define m uniquely mod 2ab/(d^{2}γ). Really, assume that x ≡ 0 mod 2b/(d_{b}^{2}γ_{b}γ_{2}) and x ≡ 0 mod 2a/(d_{a}^{2}γ_{a}γ_{2}). Then x/(2/γ_{2}) ≡ 0 mod b/(d_{b}^{2}γ_{b}) and x/(2/γ_{2}) ≡ 0 mod a/(d_{a}^{2}γ_{a}). It follows that x/(2/γ_{2}) ≡ 0 mod ab/(d^{2}γ_{a}γ_{b}). Thus, x ≡ 0 mod 2ab/(d^{2}γ_{a}γ_{b}γ_{2}) where 2ab/(d^{2}γ_{a}γ_{b}γ_{2}) = 2ab/(d^{2}γ). Thus x ≡ 0 mod 2ab/(d^{2}γ). Clearly, there exists a unique m(a, b) mod 2ab defined by the condition

m(a, b) ≡ −1 mod 2a and m(a, b) ≡ 1 mod 2b.  (2.3.16)

Then

m(a, b, d, γ) ≡ m(a, b) mod \frac{2ab}{d^{2}γ} ,  (2.3.17)

and one can take (2.3.16) and (2.3.17) as definition of m(a, b, d, γ).

Let us prove that u^{∗}(h̃) + K(h) = m(a, b, d, γ)cu^{∗}(H̃) + K(h). We had proved that u^{∗}(h̃) + K(h) = mcu^{∗}(H̃) + K(h) where m is defined mod 2ab/(d^{2}γ). To find m mod 2ab/(d^{2}γ), one should put (−bx_{1} + ax_{2})/d = γ in (2.3.14) or

−bx_{1} + ax_{2} = dγ.  (2.3.18)

From (2.3.18) we have d_{a}γ_{a}|x_{1} and d_{b}γ_{b}|x_{2}. From (2.3.14) and (2.3.18), we get

m ≡ \frac{ax_{2} + bx_{1}}{dγ} ≡ 1 + \frac{2bx_{1}}{dγ} mod 2ab/(d^{2}γ),  (2.3.19)

and

m ≡ −1 + \frac{2ax_{2}}{dγ} mod 2ab/(d^{2}γ).  (2.3.20)

Since d_{a}γ_{a}|x_{1}, we get from (2.3.19) that m ≡ 1 mod 2b/(d_{b}^{2}γ_{b}γ_{2}), and from (2.3.20) that m ≡ −1 mod 2a/(d_{a}^{2}γ_{a}γ_{2}). Thus, m = m(a, b, d, γ) mod 2ab/(d^{2}γ). It proves the statement.

Since h^{2} = 2ab and H^{2} = 2abc^{2}, we can formally put h = H/c. By our construction, K(H) ⊂ K(h) is a sublattice. Thus, we can consider N(X) and N(Y) as extensions of finite index of a common sublattice ℤH̃ + K(H).

Finally, we get the very important for us

Proposition 2.3.1. The Picard lattice of X is

N(X) = [H̃, K(H), γH̃^{∗} + u^{∗}(H̃)]  (2.3.21)

where H̃ = H/d is primitive in N(X) with H^{2} = 2abc^{2} and H̃^{∗} = d^{2}H̃/(2abc^{2}), the lattice K(H) = H_{N(X)}^{⊥}, and u^{∗}(H̃) + K(H) has the order 2abc^{2}/(d^{2}γ) in K(H)^{∗}/K(H).

The Picard lattice of Y is

N(Y) = [h̃ = h/d, K(h), γh̃^{∗} + u^{∗}(h̃)],  (2.3.22)

where the element h = (−a, 0, b) mod ℤv, h^{2} = 2ab and h̃ = h/d is primitive in N(Y), the element h̃^{∗} = d^{2}h̃/(2ab), and u^{∗}(h̃) + K(h) has the order 2ab/(d^{2}γ) in K(h)^{∗}/K(h).

They are related as follows:

K(h) = h_{N(Y)}^{⊥} = [K(H), \frac{2ab}{d^{2}γ} cu^{∗}(H̃)],  (2.3.23)

and

u^{∗}(h̃) + K(h) = m(a, b)cu^{∗}(H̃) + K(h)  (2.3.24)

where m(a, b) mod 2ab is defined by m(a, b) ≡ −1 mod 2a and m(a, b) ≡ 1 mod 2b. To define u^{∗}(h̃) + K(h) above, it is enough to consider m(a, b) mod 2ab/(d^{2}γ).

We can formally put h = H/c, equivalently h̃ = H̃/c. Then N(X) and N(Y) become the extensions of a common sublattice:

N(X) ⊃ [H̃ = c h̃, K(H)] ⊂ N(Y).  (2.3.25)

Since n(v) = 1, the transcendental lattices T (X) = T (Y) are canonically identified in v^{⊥}. It follows that we have the canonical identifications

N(X)^{∗}/N(X) = T (X)^{∗}/T (X) = T (Y)^{∗}/T (Y) = N(Y)^{∗}/N(Y).  (2.3.26)

Here we use that discriminant groups of orthogonal complements in a unimodular lattice are canonically isomorphic. For example, here the identification N(X)^{∗}/N(X) = T (X)^{∗}/T (X) is given by n^{∗} + N(X) → t^{∗} + T (X), if n^{∗} + t^{∗} ∈ H^{2}(X, ℤ).

Let us calculate the identification

N(X)^{∗}/N(X) = N(Y)^{∗}/N(Y).  (2.3.27)

Obviously (from the description above), it is given by the canonical maps

N(X)^{∗} ← (U ⊕ N(X))^{∗} ⊃ (U ⊕ N(X))^{∗}_{v}^{⊥} → (v^{⊥})_{0}^{∗} ← (v^{⊥}/ℤv)^{∗}.  (2.3.28)

Here (U ⊕ N(X))^{∗}_{v}^{⊥} = {x ∈ (U ⊕ N(X))^{∗}|x · v = 0}, (v^{⊥}) = {x ∈ U ⊕ N(X)|x · v = 0} and (v^{⊥})_{0}^{∗} = {x ∈ (v^{⊥})^{∗}|x · v = 0}.

Let f^{∗} = nH̃^{∗} + k^{∗} ∈ N(X)^{∗} where n ∈ ℤ and k^{∗} ∈ K(H)^{∗}. The element f̃^{∗} = x_{1}e_{1} + x_{2}e_{2} + nH̃^{∗} + k^{∗} is its lift to (U ⊕ N(X))^{∗} where x_{1}, x_{2} ∈ ℤ. We have f̃^{∗} ∈ (U ⊕ N(X))^{∗}_{v}^{⊥}, if −sx_{1} − rx_{2} + dn = 0. It follows that n = c(bx_{1} + ax_{2})/d where bx_{1} + ax_{2} ≡ 0 mod d, and f̃^{∗} = x_{1}e_{1} + x_{2}e_{2} + (c(bx_{1} + ax_{2})/d) H̃^{∗} + k^{∗}. It follows that

f^{∗} = \frac{c(bx_{1} + ax_{2})}{d}H̃^{∗} + k^{∗}, bx_{1} + ax_{2} ≡ 0 mod d.  (2.3.29)

Like in (2.3.11), (2.3.12) and (2.3.14), we finally get that the corresponding to f^{∗} + N(X) element in N(Y)^{∗}/N(Y) is ϵ(f^{∗}) + N(Y) where

ϵ(f^{∗}) = \frac{−bx_{1} + ax_{2}}{d} h̃^{∗} + k^{∗} ∈ N(Y)^{∗}.  (2.3.30)

Thus, the identification (2.3.27) is given by f^{∗} = c ((bx_{1} + ax_{2})/d) H̃^{∗} + k^{∗} + N(X) → ϵ(f^{∗}) = (−bx_{1} + ax_{2})h̃^{∗} + k^{∗} + N(Y) where bx_{1} + ax_{1} ≡ 0 mod d. We have x_{1} = d_{a}x̃_{1} and x_{2} = d_{b}x̃_{2} where x̃_{1}, x̃_{2} ∈ ℤ. Then

f^{∗} = c ((b/d_{b})x̃_{1} + (a/d_{a})x̃_{2}) H̃^{∗} + k^{∗} → ϵ(f^{∗}) = (−(b/d_{b})x̃_{1} + (a/d_{a})x̃_{2}) h̃^{∗} + k^{∗}.

Let us denote n_{1} = (b/d_{b})x̃_{1}+(a/d_{a})x̃_{2} and n_{2} = −(b/d_{b})x̃_{1}+(a/d_{a})x̃_{2}. We have n_{1} ≡ m(a, b)n_{2} mod 2ab/d^{2}. Really, n_{1} − m(a, b)n_{2} ≡ (2a/d_{a})x̃_{2} ≡ 0 mod 2a/d_{a}^{2} because m(a, b) ≡ −1 mod 2a/d_{a}^{2}. We have n_{1} − m(a, b)n_{2} ≡ (2b/d_{b})x̃_{1} ≡ 0 mod 2b/d_{b}^{2} because m(a, b) ≡ 1 mod 2b/d_{b}^{2}. It follows the statement. Here we consider m(a, b) mod 2ab/d^{2}.

The expression (b/d_{b})x̃_{1} + (a/d_{a})x̃_{2} gives all integers n ∈ ℤ. Thus, we have proved that if f^{∗} = cnH̃^{∗} + k^{∗} ∈ N(X)^{∗} where n ∈ ℤ, k^{∗} ∈ K(H)^{∗}, then ϵ(f^{∗}) = m(a, b)h̃^{∗} + k^{∗} ∈ N(Y). Thus, we have proved Proposition 2.3.2. In notations of Proposition 2.3.1, the canonical identification ϵ : N(X)^{∗}/N(X) ≅ N(Y)^{∗}/N(Y) of the discriminant groups given by periods:

ϵ : N(X)^{∗}/N(X) = T (X)^{∗}/T (X) = T (Y)^{∗}/T (Y) = N(Y)^{∗}/N(Y)  (2.3.31)

is given by

ϵ : cnH̃^{∗} + k^{∗} + N(X) ↦ m(a, b)nh̃^{∗} + k^{∗}  (2.3.32)

where n ∈ ℤ, k^{∗} ∈ K(H)^{∗}. Here one should consider m(a, b) mod 2ab/d^{2}.

Equivalently, the characteristic maps κ(H̃) : K̃(H) → N(X)^{∗}/N(X) and κ(h̃) : K̃(h) → N(Y)^{∗}/N(Y) are related as follows:

ϵ(κ(H̃)((cn, k^{∗}) + K̃(H)_{0}) = κ(h̃)(m(a, b)n, k^{∗}) + K̃(h)_{0}  (2.3.33)

(here one should consider m(a, b) mod 2ab/d^{2}). Here we set K̃(H) = K̃(H̃) and K̃(h) = K̃(h̃) (see (2.2.7)).

This finishes the calculation of the periods of N(Y) in terms of the periods of N(X).

Applying Lemma 2.2.1 and Propositions 2.3.1 and 2.3.2, by Global Torelli Theorem for K3 surfaces [PS], we get

Theorem 2.3.3. Assume that X is a K3 surface with a polarization H with H^{2} = 2rs, r, s ≥ 1, and a primitive Mukai vector v = (r, H, s) with the invariants

(a, b, c, d, γ)  (2.3.34)

introduced above, i.e. H̃ = H/d is primitive. Let Y be the moduli space of coherent sheaves on X with the Mukai vector v = (r, H, s). We denote by (K(H̃), u^{∗}(H̃), κ(H̃)) the invariants (2.2.11) of H̃ ∈ N(X).

The transcendental periods (T (X), H^{2,0}(X)) and (T (Y), H^{2,0}(Y)) are isomorphic, if and only if

n(v) = g.c.d(c, dγ) = 1  (2.3.35)

(this is Mukai’s result). We denote by (K(H) = K(H̃), u^{∗}(H̃), κ(H̃)) the invariants (2.2.11) of H̃ ∈ N(X).

Assume that (2.3.35) is valid. Then Y ≅ X, if the following conditions (a), (b) and (c) are valid:

(a) there exists a primitive h̃ ∈ N(X) with h̃^{2} = 2ab/d^{2} and γ(h̃) = γ.

(b) There exists an embedding ϕ : K(H) ⊂ K(h̃) of lattices such that ϕ^{∗}(K(h̃)) = [K(H), 2abc/(d^{2}γ)u^{∗}(H̃)] and ϕ^{∗}(u^{∗}(h̃)) + ϕ^{∗}(K(h̃)) = m(a, b)cu^{∗}(H̃) + ϕ^{∗}(K(h̃)). Here m(a, b) mod 2ab/(d^{2}γ) is considered.

(c) There exists a choice of ± such that κ(h̃)(m(a, b)n, z^{∗}) = ±κ(H)(cn, ϕ^{∗}(z^{∗})), if (cn, ϕ^{∗}(z^{∗})) ∈ K̃(H)^{∗} = K̃(H̃)^{∗}. Here m(a, b) mod 2ab/d^{2} is considered.

The conditions (a), (b) and (c) are necessary for a K3 surface X with ρ(X) ≤ 19 which is general for its Picard lattice N(X) in the following sense: the automorphism group of the transcendental periods (T (X), H^{2,0}(X)) is ±1. If ρ(X) = 20, then always Y ≅ X, if (2.3.35) holds.

Example 2.3.4. Let us assume that ρ(X) = 1. Thus, N(X) = ℤH̃. Assume conditions (a), (b) and (c) of Theorem 2.3.3 satisfy. Then h̃ = ±H̃. It follows c = 1. The lattices K(H) and K(h̃) are zero, and then (b) is valid. The discriminant group N(X) = ℤH̃^{∗}/ℤH̃ ≡ ℤ/(2ab/d^{2})ℤ. The condition (c) is valid, if and only if m(a, b) ≡ ±1 mod 2ab/d^{2}. This is true, if and only if (a/d_{a}^{2}) = 1 or (b/d_{b}^{2}) = 1.

Thus, X ≅ Y, if c = 1 and either a/d_{a}^{2} = 1 or b/d_{b}^{2} = 1. These conditions are necessary to have Y ≅ X, if X is a general K3 surface with ρ(X) = 1. We recover the result of Mukai from [Mu3].

Example 2.3.5. Following Example 2.3.5, let us consider the case when we can satisfy conditions of Theorem 2.3.3 taking h̃ = ±H and ϕ = ±id. Again we get c = 1. (b) satisfies, if and only if m(a, b) ≡ ±1 mod 2ab/(d^{2}γ). This is equivalent to either 2a/(d_{a}^{2}γ_{a}γ_{2}) ≤ 2 or 2b/(d_{b}^{2}γ_{b}γ_{2}) ≤ 2. (c) satisfies, if and only if m(a, b) ≡ ±1 mod 2ab/d^{2}. Thus either a/d_{a}^{2} = 1 or b/d_{b}^{2} = 1. Thus, always Y ≅ X, if c = 1 and either a/d_{a}^{2} = 1 or b/d_{b}^{2} = 1. This is just a specialization of the ρ = 1 case above. This result is also due to Mukai [Mu3]

In Sect. 3 we consider ρ = 2. We shall analyse when we can satisfy conditions of Theorem 2.3.3 in this case. By specialisation of these cases, we shall get results about K3 surfaces with any Picard number ρ ≥ 2.

3. Conditions of Y ≅ X for a general K3 surface X with ρ = 2

3.1. Main results for ρ(X) = 2. Here we apply results of Sect. 2 to X and Y with Picard number 2. Thus, we assume that ρ(X) = rk N(X) = 2.

We start with some preliminary considerations on a primitive element P ∈ S of an even hyperbolic lattice S or rk S = 2. We assume that P^{2} = 2n, n ∈ ℕ, and γ(P) = γ|2n.

Let

K(P) = P_{S}^{⊥} = ℤf(P)  (3.1.1)

and f(P)^{2} = −t where t > 0 is even. Then ±f(P) ∈ S is defined uniquely by P. Below we set f = f(P).

By elementary considerations, we have

S = [ℤP, ℤf, \frac{γ(μP + f)}{2n}]  (3.1.2)

where

g.c.d(μ, \frac{2n}{γ}) = 1.  (3.1.3)

The element

±μ mod \frac{2n}{γ} ∈ (ℤ/\frac{2n}{γ})^{∗}  (3.1.4)

is the invariant of the pair P ∈ S up to isomorphisms of lattices with a primitive vector P of P^{2} = 2n and γ(P) = γ. If f changes to −f, then μ mod 2n/γ changes to −μ mod 2n/γ.

We have (γ(μP +f)/(2n))^{2} = γ^{2}(μ^{2}−t/2n)/2n ≡ 0 mod 2. It follows 2nμ^{2}−t ≡ 0 mod 8n^{2}/γ^{2}. It follows that for some δ ∈ ℕ we have

f^{2} = −\frac{2nδ}{γ} , nδ ≡ 0 mod γ, and δ ≡ μ^{2}γ mod \frac{4n}{γ}.  (3.1.5)

We have

det S = −γδ.  (3.1.6)

Any element z ∈ S can be written as z = γ(xP + yf)/2n where x ≡ μy mod 2n/γ. We have

z^{2} = \frac{γx^{2} − δy^{2}}{(2n/γ)}  (3.1.7)

It is convenient to put

ñ = \frac{2n}{γ} .  (3.1.8)

Thus, the considered above case of a primitive P ∈ S where S is an even hyperbolic lattice of rk S = 2 is described by the invariants

ñ, γ, δ, ±μ ∈ (ℤ/ñ)^{∗},  (3.1.9)

where ñ, γ, δ ∈ ℕ. The invariants (3.1.9) must satisfy

ñγ ≡ ñδ ≡ 0 mod 2, δ ≡ μ^{2}γ mod 2ñ.  (3.1.10)

Then P^{2} = ñγ, f^{2} = −ñδ, P ⊥ f, and

S = \{\frac{xP + yf}{ñ} | x, y ∈ ℤ, x ≡ μy mod ñ\}.  (3.1.11)

We have

z^{2} = \frac{γx^{2} − δy^{2}}{ñ} .  (3.1.12)

Moreover,

det S = −δγ.  (3.1.13)

We denote

P^{∗} = \frac{P}{ñγ} , f^{∗} = \frac{f}{ñδ} .  (3.1.14)

Then

S^{∗} = {vP^{∗} + wf^{∗} | μv − w ≡ 0 mod ñ}  (3.1.15)

and

S = \{vP^{∗} + wf^{∗} | v ≡ 0 mod γ, w ≡ 0 mod δ, \frac{v}{γ} ≡ \frac{μw}{δ} mod ñ \}. (3.1.16)

Here v, w ∈ ℤ. From (3.1.15), w = μv + ñt, and

S^{∗} = v(P^{∗} + μf^{∗}) + tñf^{∗}, v, t ∈ ℤ,  (3.1.17)

and v(P^{∗} + μf^{∗}) + tñf^{∗} ∈ S, if and only if

v ≡ 0 mod γ, μv + ñt ≡ 0 mod δ, δv ≡ γμ(μv + ñt) mod δγñ. (3.1.18)

We have

u^{∗}(P) = \frac{μ^{−1}f}{ñ} + ℤf = μ^{−1}δf^{∗} + ℤf.  (3.1.19)

We remind notations we have used in Sect. 2:

a_{1} = \frac{a}{d_{a}^{2}} , b_{1} = \frac{b}{d_{b}^{2}}  (3.1.20)

where d_{a} = g.c.d(d, a) and d_{b} = g.c.d(d, b). We put

a_{2} = \frac{a_{1}}{γ_{a}} , b_{2} = \frac{b_{1}}{γ_{b}} , e_{2} = \frac{2}{γ_{2}} .  (3.1.21)

where γ_{a} = g.c.d(a_{1}, γ), γ_{b} = g.c.d(b_{1}, γ), γ_{2} = γ/(γ_{a}γ_{b}). Then γ_{2}|2.

Applying calculations above to S = N(X) and primitive P = H̃ with n = 2a_{1}b_{1}c^{2}, ñ = 2a_{1}b_{1}c^{2}/γ = e_{2}a_{2}b_{2}c^{2}, and γ(H̃) = γ, we get Proposition 3.1.1. Let X be a K3 surface with Picard number ρ = 2 equipped with a primitive polarization (or vector) H̃ ∈ N(X) of degree H̃^{2} = 2a_{1}b_{1}c^{2} and γ(H̃) = γ|2a_{1}b_{1}.

Let K(H̃) = (H̃)^{⊥} = ℤf(H̃). We have f(H̃)^{2} = −2a_{1}b_{1}c^{2}δ/γ where det N(X) = −δγ.

For some μ ∈ (ℤ/(2a_{1}b_{1}c^{2}/γ))^{∗} (the ±μ is the invariant of the pair H̃ ∈ N(X)) where δ ≡ μ^{2}γ mod 4a_{1}b_{1}c^{2}/γ, one has

N(X) = [H̃, f(H̃), \frac{(μH̃ + f(H̃))}{2a_{1}b_{1}c^{2}/γ}],  (3.1.22)
N(X) = \{z = \frac{xH̃ + yf(H̃)}{2a_{1}b_{1}c^{2}/γ} | x, y ∈ ℤ and x ≡ μy mod \frac{2a_{1}b_{1}c^{2}}{γ} \}. (3.1.23)

We have

z^{2} = \frac{γx^{2} − δy^{2}}{2a_{1}b_{1}c^{2}/γ} .  (3.1.24)

For any primitive element P ∈ N(X) with P^{2} = 2a_{1}b_{1}c^{2}, γ(P) = γ and the same invariant ±μ, there exists an automorphism ϕ ∈ O(N(X)) such that ϕ(H̃) = P.

Applying calculations above to S = N(Y) and primitive P = h̃ ∈ N(Y) with n = 2a_{1}b_{1}, ñ = 2a_{1}b_{1}/γ = e_{2}a_{2}b_{2} and γ(h̃) = γ, we get Proposition 3.1.2. Let Y be a K3 surface with Picard number ρ = 2 equipped with a primitive polarization (or vector) h̃ ∈ N(Y) of degree h̃^{2} = 2a_{1}b_{1} and γ(h̃) = γ|2a_{1}b_{1}.

Let K(h̃) = (h̃)^{⊥} = ℤf(h̃). We have f(h̃)^{2} = −2a_{1}b_{1}δ/γ where det N(Y) = −δγ. For some ν ∈ (ℤ/(2a_{1}b_{1}/γ))^{∗} (the ±ν is the invariant of the pair h̃ ∈ N(Y))

where δ ≡ ν^{2}γ mod 4a_{1}b_{1}/γ, one has

N(Y) = [h̃, f(h̃), \frac{(νh̃ + f(h̃))}{2a_{1}b_{1}/γ}],  (3.1.25)

N(Y) = \{z = \frac{xh̃ + yf(h̃)}{2a_{1}b_{1}/γ} | x, y ∈ ℤ and x ≡ νy mod \frac{2a_{1}b_{1}}{γ} \}.  (3.1.26)

We have

z^{2} = \frac{γx^{2} − δy^{2}}{2a_{1}b_{1}/γ} .  (3.1.27)

For any primitive element P ∈ N(Y) with P^{2} = 2a_{1}b_{1}, γ(P) = γ and the same invariant ±ν, there exists an automorphism ϕ ∈ O(N(Y)) such that ϕ(h̃) = P.

The crucial statement is

Theorem 3.1.3. Let X be a K3 surface, ρ(X) = 2 and H a polarization of X of degree H^{2} = 2rs, r, s ≥ 1, and Mukai vector (r, H, s) is primitive. Let Y be the moduli space of sheaves on X with the isotropic Mukai vector v = (r, H, s) and the canonical nef element h = (−a, 0, b) mod ℤv. We assume that

g.c.d(c, dγ) = 1.

With notations of Propositions 3.1.1, all elements

h̃ = \frac{xH̃ + yf(H̃)}{2a_{1}b_{1}c^{2}/γ} ∈ N(X)

with square h̃^{2} = 2a_{1}b_{1} satisfying Theorem 2.2.3 are in one to one correspondence with integral solutions (x, y) of the equation

γx^{2} − δy^{2} = 4a_{1}^{2}b_{1}^{2}c^{2}/γ  (3.1.28)

which satisfy conditions (i) — (v) below:

(i)

x ≡ μy mod 2a_{1}b_{1}c^{2}/γ ,  (3.1.29)
μγx ≡ δy mod 2a_{1}b_{1}c^{2};  (3.1.30)

(ii) (x, y) belongs to one of a-series (the sign +) or b-series (the sign −) of solutions defined below:

±m(a, b)μx + (δy/γ) ≡ 0 mod 2a_{1}b_{1}/γ ,  (3.1.31)
x ± m(a, b)μy ≡ 0 mod 2a_{1}b_{1}/γ ,  (3.1.32)
±m(a, b)μx + (δy/γ) ≡ μ(x ± m(a, b)μy) mod (2a_{1}b_{1}c^{2}/γ)(2a_{1}b_{1}/γ) ; (3.1.33)

(iii) there exists a choice of β = ±1 such that

\{ \begin{matrix}(\frac{m(a,b)γx±μγy}{2a_{1}b_{1}} − βc) ≡ 0 mod γ \\ (\frac{δm(a,b)y±μγx}{2a_{1}b_{1}} − βμc) ≡ 0 mod δ \\ δ (\frac{m(a,b)γx±μγy}{2a_{1}b_{1}} − βc) ≡ μγ (\frac{δm(a,b)y±μγx}{2a_{1}b_{1}} − βμc) mod 2a_{1}b_{1}c^{2}δ \end{matrix} (3.1.34)

and

\{ \begin{matrix}cδy ≡ 0 mod γ \\ cx − (±β)\frac{2a_{1}b_{1}c^{2}}{γ} ≡ 0 mod δ \\ δy ≡ μ (γx − (±β)2a_{1}b_{1}c) mod 2a_{1}b_{1}cδ \end{matrix}  (3.1.35)

where + is taken for a-series, and − is taken for b-series.

(iv) the pair (x, y) is μ-primitive:

g.c.d (x, y, \frac{x − μy}{2a_{1}b_{1}c^{2}/γ} ) = 1;  (3.1.36)

(v) γ(h̃) = γ, equivalently

g.c.d (γx, δy, \frac{μγx − δy}{2a_{1}b_{1}c^{2}/γ} ) = γ.  (3.1.37)

In particular (by Theorem 2.3.3), for a general X with ρ(X) = 2 we have Y ≅ X, if and only if the equation γx^{2} − δy^{2} = 4a_{1}^{2}b_{1}^{2}c^{2}/γ has an integral solution (x, y) satisfying conditions (i)—(v) above. Moreover, a nef primitive element P = (xH̃ + f(H̃))/(2a_{1}b_{1}c^{2}/γ) with P^{2} = 2a_{1}b_{1} and γ(P) = γ defines a pair (X, P) which is isomorphic to the (Y, h̃), if and only if (x, y) satisfies the conditions (ii) and (iii) (it satisfies conditions (i), (iv) and (v) since it corresponds to a primitive element of N(X) with γ(P) = γ).

Proof. We denote

H^{∗} = \frac{H̃}{2a_{1}b_{1}c^{2}} , f(H̃)^{∗} = \frac{γf(H̃)}{2a_{1}b_{1}c^{2}δ}  (3.1.38)

where K(H) = ℤf(H̃) = H^{⊥} in N(X). By (3.1.19), we have

u^{∗}(H̃) = \frac{μ^{−1}γf(H̃)}{2a_{1}b_{1}c^{2}} + ℤf(H̃) = μ^{−1}δf(H̃)^{∗} + ℤf(H̃).  (3.1.39)

Let

h̃ = \frac{xH̃ + yf(H̃)}{2a_{1}b_{1}c^{2}/γ} ∈ N(X)  (3.1.40)

satisfies conditions of Theorem 2.2.1. Then x , y ∈ ℤ and x ≡ μy mod 2a_{1}b_{1}c^{2}/γ. We get (3.1.29) in (i). Moreover, h̃ is primitive which is equivalent to (iv). We also have γ(h̃) = γ. It is equivalent to

g.c.d (H̃ · h̃, f(H̃) · h̃, (μH̃ + f(H̃))/(2a_{1}b_{1}c^{2}/γ) · h̃) = γ.

It follows (3.1.30) in (i), and (v). We have h̃^{2} = 2a_{1}b_{1}. This is equivalent to γx^{2} − δy^{2} = 4a_{1}^{2}b_{1}^{2}c^{2}/γ.

Consider K(h̃) = h̃^{⊥} in N(X). Let us denote

f(h̃) = \frac{(δy/γ)H̃ + xf(H̃)}{2a_{1}b_{1}c^{2}/γ}.  (3.1.41)

The element f(h̃) ∈ N(X) because of (i). We have f(h̃) ⊥ h̃ and f(h̃)^{2} = −2a_{1}b_{1}δ/γ. Since γ(h̃) = γ, it follows that K(h̃) = ℤf(h̃) and f(h̃) is primitive. We have

N(X) = [h̃, f(h̃), \frac{νh̃ + f(h̃)}{2a_{1}b_{1}/γ}]  (3.1.42)

where ν ∈ (ℤ/(2a_{1}b_{1}/γ))^{∗} (according to Proposition 3.1.2). We denote

h̃^{∗} = \frac{h̃}{2a_{1}b_{1}} , f(h̃)^{∗} = \frac{γf(h̃)}{2a_{1}b_{1}δ} .  (3.1.43)

By (3.1.19), we have

u^{∗}(h̃) = \frac{ν^{−1}f(h̃)}{2a_{1}b_{1}/γ} + ℤf(h̃) = ν^{−1}δf(h̃)^{∗} + ℤf(h̃) .  (3.1.44)

There exists a unique (up to ±1) embedding

ϕ : K(H̃) = ℤf(H̃) → K(h̃) = ℤf(h̃), ϕ(f(H̃)) = ±cf(h̃).  (3.1.45)

of one-dimensional lattices. Its dual is defined by ϕ^{∗}(f(h̃)^{∗}) = ±cf(H̃)^{∗}.

We have

ϕ^{∗}(K(h̃)) = ℤϕ^{∗}(f(h̃)) = ℤf(H̃)/c =
[K(H̃), (2a_{1}b_{1}c/γ)u^{∗}(H̃)] = [K(H̃), (2abc/(d^{2}γ))u^{∗}(H̃)]

because of (3.1.39). This gives the first part of (b) in Theorem 2.3.3.

We have

ϕ^{∗}(u^{∗}(h̃)) + ϕ^{∗}(K(h̃)) = ν^{−1}δϕ^{∗}(f(h̃)^{∗}) + ℤϕ^{∗}(f(h̃)) = ±ν^{−1}δcf(H̃)^{∗} + ℤf(H̃)/c =

±ν^{−1}δcf(H̃)^{∗} + ℤ(2a_{1}b_{1}cδ/γ)f(H̃)^{∗}.

On the other hand, by (3.1.39)

m(a, b)cu^{∗}(H̃) + ℤ(2a_{1}b_{1}cδ/γ)f(H̃)^{∗} = m(a, b)μ^{−1}δcf(H̃)^{∗} + ℤ(2a_{1}b_{1}cδ/γ)f(H̃)^{∗}.

Thus, by (3.1.44), second part of (b) in Theorem 2.3.3 is ±ν^{−1} ≡ m(a, b)μ^{−1} mod 2a_{1}b_{1}/γ. Equivalently,

ν ≡ ±m(a, b)μ mod \frac{2a_{1}b_{1}}{γ} .  (3.1.46)

Thus, for ν given by (3.1.46) one has (this is the definition of ν)

\frac{νh̃ + f(h̃)}{2a_{1}b_{1}/γ} = (\frac{νx + (δy/γ)}{2a_{1}b_{1}/γ}H̃ + \frac{x + νy}{2a_{1}b_{1}/γ} f(H̃)) /(2a_{1}b_{1}c^{2}/γ) ∈ N(X). (3.1.47)

This is equivalent for ν ≡ m(a, b)μ mod 2a_{1}b_{1}/γ to

\{ \begin{matrix}±m(a, b)μx + (δy/γ) ≡ 0 mod 2a_{1}b_{1}/γ \\ x ± m(a, b)μy ≡ 0 mod 2a_{1}b_{1}/γ \\ ±m(a, b)μx + (δy/γ) ≡ μ(x ± m(a, b)μy) mod (2a_{1}b_{1}c^{2}/γ)(2a_{1}b_{1}/γ) \end{matrix} .

(3.1.48)

This gives (ii).

Let us consider the condition (c) of Theorem 2.3.3. For a choice of β = ±1, one has

m(a, b)nh̃^{∗} + z^{∗} ≡ β(cnH̃^{∗} + ϕ^{∗}(z^{∗})) mod N(X), n ∈ ℤ, z ∈ K(h̃)^{∗} (3.1.49)

if

cnH̃^{∗} + ϕ^{∗}(z^{∗}) ∈ N(X)^{∗}.  (3.1.50)

Let z^{∗} = kf(h̃)^{∗}, k ∈ ℤ, then ϕ^{∗}(kf(h̃)^{∗}) = ±kcf(H̃)^{∗} and cnH̃^{∗} + ϕ^{∗}(z^{∗}) = cnH̃^{∗}±kcf(H̃)^{∗}. By (3.1.15), cnH̃^{∗}±kcf(H̃)^{∗} ∈ N(X)^{∗}, if and only if μcn∓kc ≡ 0 mod 2a_{1}b_{1}c^{2}/γ. This is equivalent to k ≡ ±μn mod 2a_{1}b_{1}c/γ. Like in (3.1.17), we get k = ±μn + (2a_{1}b_{1}c/γ)t where n, t ∈ ℤ. We get

cnH̃^{∗} ± kcf(H̃)^{∗} = cn(H̃^{∗} + μf(H̃)^{∗}) ± t(2a_{1}b_{1}c^{2}/γ)f(H̃)^{∗}.  (3.1.51)

Thus, it is enough to check

m(a, b)nh̃^{∗} + kf(h̃)^{∗} ≡ β(cnH̃^{∗} ± kcf(H̃)^{∗}) mod N(X),  (3.1.52)

where (n, k) = (1, ±μ) or (n, k) = (0, 2a_{1}b_{1}c/γ). Thus, one should check for one of β = ±1 that

m(a, b)h̃^{∗} ± μf(h̃)^{∗} ≡ β(cH̃^{∗} + cμf(H̃)^{∗}) mod N(X)  (3.1.53)

and

\frac{2a_{1}b_{1}cf(h̃)^{∗}}{γ} ≡ ±β \frac{2a_{1}b_{1}c^{2}f(H̃)^{∗}}{γ} mod N(X).  (3.1.54)

We have

m(a, b)nh̃^{∗} + kf(h̃)^{∗} = m(a, b)n\frac{h̃}{2a_{1}b_{1}} + k \frac{γf(h̃)}{2a_{1}b_{1}δ} =

\frac{1}{2a_{1}b_{1}δ} (δm(a, b)nh̃ + γkf(h̃)) =

\frac{1}{2a_{1}b_{1}δ} (δm(a, b)n\frac{xH̃ + yf(H̃)}{2a_{1}b_{1}c^{2}/γ} + γk \frac{(δy/γ)H̃ + xf(H̃)}{2a_{1}b_{1}c^{2}/γ}) =

\frac{1}{2a_{1}b_{1}δ} (δm(a, b)n(γxH̃^{∗} + δyf(H̃)^{∗}) + γk(δyH̃^{∗} + δxf(H̃)^{∗}) =

\frac{m(a, b)nγx + γky}{2a_{1}b_{1}} H̃^{∗} + \frac{δm(a, b)ny + γkx}{2a_{1}b_{1}}f(H̃)^{∗}.  (3.1.55)

Thus, (3.1.53) is

(\frac{m(a, b)γx ± μγy}{2a_{1}b_{1}} − βc) H̃^{∗} + (\frac{δm(a, b)y ± μγx}{2a_{1}b_{1}} − βμc) f(H̃)^{∗} ∈ N(X)

(3.1.56)

and (3.1.54) is

cyH̃^{∗} + (cx − (±β)\frac{2a_{1}b_{1}c^{2}}{γ} ) f(H̃)^{∗} ∈ N(X).  (3.1.57)

Here one should take +, if (x, y) belongs to a-series, and one should take −, if (x, y) belongs to b-series.

By (3.1.16), we can reformulate (3.1.56) as (3.1.34) in (iii), and we can reformulate (3.1.57) as (3.1.35) in (iii).

This finishes the proof.

Now we analyse conditions of Theorem 3.1.3. The most important are congruences mod δ since δ is not bounded by a constant depending on (r, s).

Let us consider the congruence cx − (±β)(2a_{1}b_{1}c^{2}/γ) ≡ 0 mod δ in (3.1.35). We know that δ ≡ γμ^{2} mod (2a_{1}b_{1}/γ)c^{2} where μ mod (2a_{1}b_{1}/γ)c^{2} is invertible, γ|2a_{1}b_{1} and g.c.d(γ, c) = 1. It follows that g.c.d(c, δ) = 1. Thus, the considered congruence is equivalent to

x ≡ ±\frac{2a_{1}b_{1}c}{γ} mod δ  (3.1.58)

since β = ±1. Later we shall see that all congruences mod δ which follow from conditions of Theorem 3.1.3 are consequences of (3.1.58). Thus, (3.1.58) is the most important condition of Theorem 3.1.3.

Let us consider all integral (x, y) which satisfy (3.1.28) and (3.1.58), i. e.

γx^{2} − δy^{2} = \frac{4a_{1}^{2}b_{1}^{2}c^{2}}{γ} and x ≡ ±\frac{2a_{1}b_{1}c}{γ} mod δ.  (3.1.59)

We apply the main trick used in [MN1], [MN2] and [N4]. Considering ±(x, y), we can assume that x ≡ 2a_{1}b_{1}c/γ mod δ, i. e. x = 2a_{1}b_{1}c/γ − kδ where k ∈ ℤ. We have γ^{2}x^{2} = 4a_{1}^{2}b_{1}^{2}c^{2} − 4a_{1}b_{1}ckδγ + γ^{2}k^{2}δ^{2} = γδy^{2} + 4a_{1}^{2}b_{1}^{2}c^{2}. It follows

δ = \frac{y^{2} + 4a_{1}b_{1}ck}{γk^{2}} .  (3.1.60)

Consider a prime l such that l |̸2a_{1}b_{1}c. Assume that l^{2t+1}|k, but l^{2t+2} |̸k. We have k|y^{2}. Then l^{2t+1}|y^{2}. Then l^{2t+2}|y^{2}. Since k^{2}|y^{2} + 4a_{1}b_{1}ck, it follows that l^{2t+2}|y^{2} + 4a_{1}b_{1}ck We then get l^{2t+2}|k. We get a contradiction. It follows that k = −αq^{2} where q ∈ ℤ (we can additionally assume that q ≥ 0), α|2a_{1}b_{1}c and α is square-free. Remark that α can be negative.

From (3.1.60), we get

δ = \frac{y^{2} − 4a_{1}b_{1}cαq^{2}}{γα^{2}q^{4}} .  (3.1.61)

It follows αq|y and y = αpq where p ∈ ℤ. From (3.1.61),

δ = \frac{p^{2} − 4a_{1}b_{1}c/α}{γq^{2}}.  (3.1.62)

Equivalently,

p^{2} − γδq^{2} = \frac{4a_{1}b_{1}c}{α}.  (3.1.63)

If integral p, q satisfy (3.1.63), we have

(x, y) = ± (\frac{2a_{1}b_{1}c}{γ} + αδq^{2}, αpq) .  (3.1.64)

satisfy (3.1.59). We call solutions (3.1.64) of (3.1.59) as associated solutions. Thus, we get the very important for us

Theorem 3.1.4. All integral solutions (x, y) of

γx^{2} − δy^{2} = \frac{4a_{1}^{2}b_{1}^{2}c^{2}}{γ} and x ≡ ±\frac{2a_{1}b_{1}c}{γ} mod δ  (3.1.65)

are associated solutions

(x, y) = ± (\frac{2a_{1}b_{1}c}{γ} + αδq^{2}, αpq)  (3.1.66)

to integral solutions α, (p, q) of

α|2a_{1}b_{1}c where α is square-free, and p^{2} − γδq^{2} = \frac{4a_{1}b_{1}c}{α} .  (3.1.67)

Any solution (x, y) of (3.1.65) can be written in the form (3.1.66) where α, (p, q) is solution of (3.1.67). Any solution of (3.1.67) gives a solution (3.1.66) of (3.1.65).

Solutions of (3.1.65) and (3.1.67) are in one-to-one correspondence if we additionally assume that q ≥ 0.

Now we can write (x, y) of Theorem 3.1.3 in the form (3.1.66) as associated solutions to (3.1.67). Putting that (x, y) to relations (i)—(v) of Theorem 3.1.3, we get some relations on α and (p, q). They are a finite number of congruences mod N_{i} where N_{i} depend only on (r, s) (or (a, b, c)). All N_{i} are bounded by functions depending only on (r, s). These congruences have a lot of relations between them and with (3.1.67). All together they give many very strong restrictions on α and (p, q) (except (3.1.67)). Unfortunately, it is hard to analyse all these relations and restrictions completely. In this paper, we were able to analyse only the most important these relations and restrictions. We present them below.

We fix

μ ∈ (ℤ/(2a_{1}b_{1}c^{2}/γ))^{∗}  (3.1.68)

and consider δ ∈ ℕ such that

δ ≡ μ^{2}γ mod \frac{4a_{1}b_{1}c^{2}}{γ}  (3.1.69)

The relation (3.1.29) in (i) of Theorem 3.1.3 is equivalent to

2a_{1}b_{1}c + αγδq^{2} ≡ γμαpq mod 2a_{1}b_{1}c^{2}.  (3.1.70)

54 V.V. NIKULIN

By (3.1.67), we have

−4a_{1}b_{1}c + αp^{2} − αγδq^{2} = 0.  (3.1.71)

Taking sum, we get

2a_{1}b_{1}c ≡ αp(p − μγq) mod 2a_{1}b_{1}c^{2}.  (3.1.72)

The relation (3.1.72) is equivalent to (3.1.29). Taking 2(3.1.70)+(3.1.71), we get

αp^{2} + αγδq^{2} ≡ 2γμαpq mod 4a_{1}b_{1}c^{2}  (3.1.73)

which is also equivalent to (3.1.29). From (3.1.69), we get

αp^{2} + αμ^{2}γ^{2}q^{2} ≡ 2γμαpq mod \frac{4a_{1}b_{1}c^{2}}{γ}  (3.1.74)

and

α(p − μγq)^{2} ≡ 0 mod \frac{4a_{1}b_{1}c^{2}}{γ} .  (3.1.75)

It is the corollary mod 4a_{1}b_{1}c^{2}/γ of the congruence (3.1.73). Since γ|2a_{1}b_{1}, it follows α(p − μγq)^{2} ≡ 0 mod 2c^{2}. Since α is square-free, it follows

c|(p − μγq).  (3.1.76)

From (3.1.72), we get

2a_{1}b_{1} ≡ αp\frac{p − μγq}{c} mod 2a_{1}b_{1}c  (3.1.77)

where (p − μγq)/c is an integer. It follows, α|2a_{1}b_{1}. Thus, we get Proposition 3.1.5. The condition (3.1.29) of Theorem 3.1.3 is equivalent to

αp^{2} + αγδq^{2} ≡ 2γμαpq mod 4a_{1}b_{1}c^{2}.  (3.1.78)

From (3.1.78) we get

α|2a_{1}b_{1}, c|(p − μγq), and α(p − μγq)^{2} ≡ 0 mod \frac{4a_{1}b_{1}c^{2}}{γ} .  (3.1.79)

The (3.1.78) is also equivalent to

2a_{1}b_{1}c ≡ αp(p − μγq) mod 2a_{1}b_{1}c^{2}.  (3.1.80)

Considering the condition (3.1.30) of Theorem 3.1.3, we similarly get Proposition 3.1.6. The condition (3.1.30) of Theorem 3.1.3 is equivalent to

μαp^{2} + μαγδq^{2} ≡ 2αδpq mod 4a_{1}b_{1}c^{2}.  (3.1.81)

Congruences (3.1.78) and (3.1.81) are equivalent mod (4a_{1}b_{1}c^{2}/γ), and (3.1.81) is equivalent to (3.1.78) together with

2αpq(δ − γμ^{2}) ≡ 0 mod 4a_{1}b_{1}c^{2}  (3.1.82)

where δ − γμ^{2} ≡ 0 mod 4a_{1}b_{1}c^{2}/γ.

Proof. To get (3.1.82), consider μ (3.1.78) - (3.1.81).

Now consider (3.1.31) of Theorem 3.1.3. Let us consider the a-series (the sign +). Since m(a, b) ≡ −1 mod 2a_{1}/(γ_{2}γ_{a}), we get −μx + (δy/γ) ≡ 0 mod 2a_{1}/(γ_{2}γ_{a}). It is satisfied because of (3.1.30). Since m(a, b) ≡ 1 mod 2b_{1}/(γ_{2}γ_{b}), we get μx + (δy/γ) ≡ 0 mod 2b_{1}/(γ_{2}γ_{b}). By (3.1.30), we get μx − (δy/γ) ≡ 0 mod 2b_{1}/(γ_{2}γ_{b}). Thus, we get 2μx ≡ 0 mod 2b_{1}/(γ_{2}γ_{b}). If γ_{2} = 1, this is equivalent to μx ≡ 0 mod b_{1}/γ_{b} and x ≡ 0 mod b_{1}/γ_{b}. If γ_{2} = 2, then b_{1}/γ_{b} is odd, and we get 2μx ≡ 0 mod (b_{1}/γ_{b}) which is equivalent to x ≡ 0 mod b_{1}/γ_{b}. Thus, at any case we get (b_{1}/γ_{b})|x, equivalently, x ≡ 0 mod (b_{1}/γ_{b}). By (3.1.66), this is equivalent to αδq^{2} ≡ 0 mod (b_{1}/γ_{b}). We have δ ≡ μ^{2}γ mod (b_{1}/γ_{b}) and μ mod (b_{1}/γ_{b}) is invertible. Thus, we get αγq^{2} ≡ 0 mod (b_{1}/γ_{b}) which is equivalent to α(γ_{b}q)^{2} ≡ 0 mod b_{1}.

Let us consider (3.1.32) of Theorem 3.1.3. We consider the a-series. We get x − μy ≡ 0 mod 2a_{1}/(γ_{2}γ_{a}). It satisfies because of (3.1.29). We get x + μy ≡ 0 mod 2b_{1}/(γ_{2}γ_{b}). Using (3.1.29), we similarly get that this is equivalent to (b_{1}/γ_{b})|x.

Thus, we finally get

Proposition 3.1.7. The conditions (3.1.31) and (3.1.32) in (ii) of Theorem 3.1.3 are equivalent to

x ≡ 0 mod \frac{b_{1}}{γ_{b}}  (3.1.83)

or to

α(γ_{b}q)^{2} ≡ 0 mod b_{1}  (3.1.84)

for a-series (the sign +), and it is equivalent to

x ≡ 0 mod \frac{a_{1}}{γ_{a}}  (3.1.85)

or to

α(γ_{a}q)^{2} ≡ 0 mod a_{1}  (3.1.86)

for b-series (the sign −).

Consider (3.1.33) in Theorem 3.1.3. We consider the a-series. We get mod 2a_{1}/(γ_{2}γ_{a}) that

2μγx ≡ (2δ + (μ^{2}γ − δ))y mod (2a_{1}b_{1}c^{2}) (2a_{1}/(γ_{2}γ_{a})) .

We get mod 2b_{1}/(γ_{2}γ_{b}) that

(δ − μ^{2}γ)y ≡ 0 mod (2a_{1}b_{1}c^{2})(2b_{1}/γ_{2}γ_{b}).

Using (3.1.66) (and (3.1.71) to make the relations homogeneous), we finally get

Proposition 3.1.8. The condition (3.1.33) of Theorem 3.1.3 is equivalent to

2μγx ≡ (δ + μ^{2}γ)y mod (2a_{1}b_{1}c^{2}) (2a_{1}/(γ_{2}γ_{a}))  (3.1.87)

and

(δ − μ^{2}γ)y ≡ 0 mod (2a_{1}b_{1}c^{2}) (2b_{1}/(γ_{2}γ_{b}))  (3.1.88)

or

μαp^{2} − α(δ + μ^{2}γ)pq + μαγδq^{2} ≡ 0 mod (2a_{1}b_{1}c^{2}) (2a_{1}/(γ_{2}γ_{a}))  (3.1.89)

and

α(δ − μ^{2}γ)pq ≡ 0 mod (2a_{1}b_{1}c^{2}) (2b_{1}/(γ_{2}γ_{b}))  (3.1.90)

for the a-series (the sign +), and it is equivalent to

2μγx ≡ (δ + μ^{2}γ)y mod (2a_{1}b_{1}c^{2}) (2b_{1}/(γ_{2}γ_{b}))  (3.1.91)

and

(δ − μ^{2}γ)y ≡ 0 mod (2a_{1}b_{1}c^{2}) (2a_{1}/(γ_{2}γ_{a}))  (3.1.92)

or

μαp^{2} − α(δ + μ^{2}γ)pq + μαγδq^{2} ≡ 0 mod (2a_{1}b_{1}c^{2}) (2b_{1}/(γ_{2}γ_{b}))  (3.1.93)

and

α(δ − μ^{2}γ)pq ≡ 0 mod (2a_{1}b_{1}c^{2}) (2a_{1}/(γ_{2}γ_{a}))  (3.1.94)

for the b-series (the sign −).

Consider the condition (3.1.35) of Theorem 3.1.3. Consider the a-series (the sign +). Second condition in (3.1.35) gives x ≡ β(2a_{1}b_{1}c/γ) mod δ. By (3.1.66), we get (x, y) = β(2a_{1}b_{1}c/γ +αδq^{2}, αpq). Then third condition in (3.1.35) is equivalent to

βδαpq ≡ μ(β2a_{1}b_{1}c + βαγδq^{2} − β2a_{1}b_{1}c) mod 2a_{1}b_{1}cδ.

This is equivalent to αq(p − μγq) ≡ 0 mod 2a_{1}b_{1}c. Since γ|2a_{1}b_{1}c, it follows the first condition in (3.1.35) which is cδβαpq ≡ 0 mod γ.

For b-series we get the same. Thus, we have

ON CORRESPONDENCES OF K3 WITH ITSELF, II 57

Proposition 3.1.9. The condition (3.1.35) of Theorem 3.1.3 is equivalent to

αq(p − μγq) ≡ 0 mod 2a_{1}b_{1}c.  (3.1.95)

Consider (3.1.34) of Theorem 3.1.3. We consider the a-series. Then

(x, y) = β(\frac{2a_{1}b_{1}c}{γ} + αδq^{2}, αpq)  (3.1.96)

The first relation of (3.1.34) gives

m(a, b)γx + μγy ≡ 2a_{1}b_{1}βc mod 2a_{1}b_{1}γ.

Using (3.1.96), we get

m(a, b)αγδq^{2} + μαγpq ≡ 2a_{1}b_{1}c(1 − m(a, b)) mod 2a_{1}b_{1}γ.

This is equivalent to

−αδq^{2} + μαpq ≡ \frac{4a_{1}b_{1}c}{γ} mod 2a_{1}

and

αδq^{2} + μαpq ≡ 0 mod 2b_{1}.

Using (3.1.71), we can rewrite the first relation in the homogeneous form

αp(p − μγq) ≡ 0 mod 2a_{1}γ.

The second relation of (3.1.34) gives

δm(a, b)y + μγx − 2a_{1}b_{1}βμc ≡ 0 mod 2a_{1}b_{1}δ.

By (3.1.96) it is equivalent to

m(a, b)αpq + αμγδq^{2} ≡ 0 mod 2a_{1}b_{1}.

This is equivalent to

−αpq + αμγδq^{2} ≡ 0 mod 2a_{1}

and

αpq + αμγδq^{2} ≡ 0 mod 2b_{1}.

The third relation in (3.1.34) is δ (m(a, b)γx + μγy − 2βa_{1}b_{1}c) ≡ μγ (δm(a, b)y + μγx − 2μβa_{1}b_{1}c) mod (2a_{1}b_{1}c^{2}δ)(2a_{1}b_{1}).

Using (3.1.96), one can calculate that it is equivalent to

μαγpq (m(a, b) − 1) + αγq^{2} (γμ^{2} − m(a, b)δ)
≡ 2a_{1}b_{1}c (m(a, b) − 1) mod (2a_{1}b_{1}c^{2})(2a_{1}b_{1}).

This is equivalent to

4a_{1}b_{1}c ≡ αγq (2μp − (γμ^{2} + δ)q) mod (2a_{1}b_{1}c^{2})(2a_{1})

and

αγq^{2}(δ − γμ^{2}) ≡ 0 mod (2a_{1}b_{1}c^{2})(2b_{1}).

Using (3.1.71), we can rewrite the first relation in the homogeneous form

α(p − μγq)^{2} ≡ 0 mod (2a_{1}b_{1}c^{2})(2a_{1}).

Thus, we get

Proposition 3.1.10. The condition (3.1.34) of Theorem 3.1.3 is equivalent to the system of congruences: For the a-series (the sign +)

αp(p − μγq) ≡ 0 mod 2a_{1}γ,  (3.1.97)
αδq^{2} + αμpq ≡ 0 mod 2b_{1},  (3.1.98)
−αpq + αμγδq^{2} ≡ 0 mod 2a_{1},  (3.1.99)
αpq + αμγδq^{2} ≡ 0 mod 2b_{1},  (3.1.100)
α(p − μγq)^{2} ≡ 0 mod (2a_{1}b_{1}c^{2})(2a_{1}),  (3.1.101)
αγq^{2}(δ − γμ^{2}) ≡ 0 mod (2a_{1}b_{1}c^{2})(2b_{1}).  (3.1.102)

For the b-series (the sign −):

αp(p − μγq) ≡ 0 mod 2b_{1}γ,  (3.1.103)
αδq^{2} + αμpq ≡ 0 mod 2a_{1},  (3.1.104)
−αpq + αμγδq^{2} ≡ 0 mod 2b_{1},  (3.1.105)
αpq + αμγδq^{2} ≡ 0 mod 2a_{1},  (3.1.106)
α(p − μγq)^{2} ≡ 0 mod (2a_{1}b_{1}c^{2})(2b_{1}),  (3.1.107)
αγq^{2}(δ − γμ^{2}) ≡ 0 mod (2a_{1}b_{1}c^{2})(2a_{1}).  (3.1.108)

Consider the condition (iv) of Theorem 3.1.3. It means that the corresponding element h̃ is primitive. Since h̃^{2} = 2a_{1}b_{1} and the lattice N(X) is even, it is not valid only if h̃/l ∈ N(X) for some prime l such that l^{2}|a_{1}b_{1}. Thus, (3.1.36) is not valid if and only if

x ≡ y ≡ \frac{x − μy}{2a_{1}b_{1}c^{2}/γ} ≡ 0 mod l

for some prime l such that l^{2}|a_{1}b_{1}. Using (3.1.66) and (3.1.71), we get Proposition 3.1.11. The condition (iv) of Theorem 3.1.3 is equivalent to the nonexistence of a prime l such that l^{2}|a_{1}b_{1} and

x ≡ y ≡ \frac{x − μy}{2a_{1}b_{1}c^{2}/γ} ≡ 0 mod l.  (3.1.109)

Equivalently, the system of congruences

\{ \begin{matrix}αp^{2} + αγδq^{2} ≡ 0 mod 2γl \\ αpq ≡ 0 mod l \\ αp^{2} + αγδq^{2} ≡ 2αγμpq mod 4a_{1}b_{1}c^{2}l \end{matrix}  (3.1.110)

is not satisfied for any prime l such that l^{2}|a_{1}b_{1}.

Consider the condition (v) of Theorem 3.1.3. This is equivalent to γ(h̃) = γ where h̃ · N(X) = γ(h̃)ℤ. All other conditions of Theorem 3.1.3 give that γ|h̃ · N(X), and γ|γ(h̃). Since h̃^{2} = 2a_{1}b_{1}, it follows that (γ(h̃)/γ) | 2a_{1}b_{1}/γ. Equivalently, (v) is not satisfied, if and only if for some prime l|2a_{1}b_{1}/γ one has

x ≡ \frac{δy}{γ} ≡ \frac{μγx − δy}{2a_{1}b_{1}c^{2}} ≡ 0 mod l.

Using (3.1.66) and (3.1.71), we get

Proposition 3.1.12. The condition (v) of Theorem 3.1.3 is equivalent to the nonexistence of a prime l | (2a_{1}b_{1}/γ) such that

x ≡ \frac{δy}{γ} ≡ \frac{μγx − δy}{2a_{1}b_{1}c^{2}} ≡ 0 mod l.  (3.1.111)

Equivalently, the system of congruences

\{ \begin{matrix}αp^{2} + αγδq^{2} ≡ 0 mod 2γl \\ δαpq ≡ 0 mod γl \\ μαp^{2} + μαγδq^{2} ≡ 2αδpq mod 4a_{1}b_{1}c^{2}l \end{matrix}  (3.1.112)

is not satisfied for any prime l | (2a_{1}b_{1}/γ).

Now we collect analysed conditions of Theorem 3.1.3 all together. We divide them in general conditions: (G) which are valid for both a and b-series, conditions (A) which are valid for the a-series, and conditions (B) which are valid for the b-series.

60 V.V. NIKULIN

(G): General conditions

αp^{2} + αγδq^{2} ≡ 2γμαpq mod 4a_{1}b_{1}c^{2},  (3.1.113)
μαp^{2} + μαγδq^{2} ≡ 2αδpq mod 4a_{1}b_{1}c^{2},  (3.1.114)
αq(p − μγq) ≡ 0 mod 2a_{1}b_{1}c,  (3.1.115)
\{ \begin{matrix}αp^{2} + αγδq^{2} ≡ 0 mod 2γl \\ αpq ≡ 0 mod l \\ αp^{2} + αγδq^{2} ≡ 2αγμpq mod 4a_{1}b_{1}c^{2}l \end{matrix}  (3.1.116)

is not satisfied for any prime l such that l^{2}|a_{1}b_{1},

\{ \begin{matrix}αp^{2} + αγδq^{2} ≡ 0 mod 2γl \\ δαpq ≡ 0 mod γl \\ μαp^{2} + μαγδq^{2} ≡ 2αδpq mod 4a_{1}b_{1}c^{2}l \end{matrix}  (3.1.117)

is not satisfied for any prime l | (2a_{1}b_{1}/γ). (A): Conditions of the a-series

α(γ_{a}q)^{2} ≡ 0 mod b_{1},  (3.1.118)
μαp^{2} − α(δ + μ^{2}γ)pq + μαγδq^{2} ≡ 0 mod (2a_{1}b_{1}c^{2}) (2a_{1}/(γ_{2}γ_{a})) (3.1.119)
α(δ − μ^{2}γ)pq ≡ 0 mod (2a_{1}b_{1}c^{2}) (2b_{1}/(γ_{2}γ_{b})) ,  (3.1.120)
αp(p − μγq) ≡ 0 mod 2a_{1}γ,  (3.1.121)
αδq^{2} + αμpq ≡ 0 mod 2b_{1},  (3.1.122)
−αpq + αμγδq^{2} ≡ 0 mod 2a_{1},  (3.1.123)
αpq + αμγδq^{2} ≡ 0 mod 2b_{1},  (3.1.124)
α(p − μγq)^{2} ≡ 0 mod (2a_{1}b_{1}c^{2})(2a_{1}),  (3.1.125)
αγq^{2}(δ − γμ^{2}) ≡ 0 mod (2a_{1}b_{1}c^{2})(2b_{1}).  (3.1.126)

(B): Conditions of the b-series

α(γ_{a}q)^{2} ≡ 0 mod a_{1},  (3.1.127)
μαp^{2} − α(δ + μ^{2}γ)pq + μαγδq^{2} ≡ 0 mod (2a_{1}b_{1}c^{2}) (2b_{1}/(γ_{2}γ_{b})) (3.1.128)
α(δ − μ^{2}γ)pq ≡ 0 mod (2a_{1}b_{1}c^{2}) (2a_{1}/(γ_{2}γ_{a})) ,  (3.1.129)
αp(p − μγq) ≡ 0 mod 2b_{1}γ,  (3.1.130)
αδq^{2} + αμpq ≡ 0 mod 2a_{1},  (3.1.131)

−αpq + αμγδq^{2} ≡ 0 mod 2b_{1},  (3.1.132)
αpq + αμγδq^{2} ≡ 0 mod 2a_{1},  (3.1.133)
α(p − μγq)^{2} ≡ 0 mod (2a_{1}b_{1}c^{2})(2b_{1}),  (3.1.134)
αγq^{2}(δ − γμ^{2}) ≡ 0 mod (2a_{1}b_{1}c^{2})(2a_{1}).  (3.1.135)

Now we can formulate the main results which follows from Theorem 3.1.3 and the calculations above.

Theorem 3.1.13. Let X be a K3 surface with ρ(X) = 2, and H a polarization of X of degree H^{2} = 2rs, r, s ≥ 1. Assume that the Mukai vector (r, H, s) is primitive. Let Y be the moduli space of sheaves on X with the isotropic Mukai vector v = (r, H, s) and the canonical nef element h = (−a, 0, b) mod ℤv. Let H̃ = H/d be the corresponding primitive polarization with H̃ · N(X) = γℤ. We denote by μ the invariant of the pair (N(X), H̃) and use notations of Proposition 3.1.1.

We have Y ≅ X, if

g.c.d(c, dγ) = 1

and there exists a square-free α | 2a_{1}b_{1} such that the equation

p^{2} − γδq^{2} = \frac{4a_{1}b_{1}c}{α}  (3.1.136)

has an integral solution (p, q) which satisfies the general condition (G) and one of conditions (A) ((p, q) belongs to the a-series) or (B) ((p, q) belongs to the b-series) above.

These conditions are necessary to have Y ≅ X, if X is a general K3 surface with ρ(X) = 2, i. e. the automorphism group of the transcendental periods (T (X), H^{2,0}(X)) is ±1.

Solutions (3.1.136) give all solution (x, y) of Theorem 3.1.3 as associated solutions

(x, y) = ±(\frac{2a_{1}b_{1}c}{γ} + αδq^{2}, αpq).  (3.1.137)

Now we want to interpret solutions (p, q) of Theorem 3.1.13 as elements of the Picard lattice N(X).

Let (p, q) be a solution of (3.1.136). Then p^{2} − γδq^{2} = 4a_{1}b_{1}c/α and

γp^{2} − δ(γq)^{2} = \frac{4a_{1}b_{1}cγ}{α} .  (3.1.138)

Assume that (p, q) satisfies the condition (A) (i.e. it belongs to the a-series). By (3.1.125), α(p − μγq)^{2} ≡ 0 mod 4a_{1}^{2}c^{2}. Since α is square-free, it follows that

p − μγq ≡ 0 mod 2a_{1}c.  (3.1.139)

Let us put

t = b_{1}c.  (3.1.140)

Then

tp − tμq ≡ 0 mod 2a_{1}b_{1}c^{2}.  (3.1.141)

It follows, tp − tμq ≡ 0 mod 2a_{1}b_{1}c^{2}/γ and

h̃_{1} = t\frac{pH̃ + γqf(H̃)}{2a_{1}b_{1}c^{2}/γ} ∈ N(X).  (3.1.142)

We have

h̃_{1}^{2} = \frac{t^{2}(γp^{2} − δ(γq)^{2})}{2a_{1}b_{1}c^{2}/γ} = \frac{t^{2}4a_{1}b_{1}cγ}{2αa_{1}b_{1}c^{2}/γ} = \frac{2t^{2}γ^{2}}{αc} = \frac{2γ^{2}b_{1}^{2}c}{α}.  (3.1.143)

We have

H̃ · h̃_{1} = b_{1}cp ≡ 0 mod b_{1}c and p = \frac{H̃ · h̃_{1}}{b_{1}c} .  (3.1.144)

Also

−f(H̃) · h̃_{1} = b_{1}cγδq ≡ 0 mod b_{1}cγδ and q = −\frac{f(H̃) · h̃_{1}}{b_{1}cγδ} .  (3.1.145)

Here

−γδ = det N(X)  (3.1.146)

and we can rewrite (3.1.145) as follows:

f(H̃) · h̃_{1} = b_{1}c det N(X)q ≡ 0 mod b_{1}c det N(X) and q = \frac{f(H̃) · h̃_{1}}{b_{1}c det N(X)}.

(3.1.147)

Another calculation of p and q is as follows. We have

h̃_{1} = \frac{uH̃ + vf(H̃)}{2a_{1}b_{1}c^{2}/γ}  (3.1.148)

where

u ≡ 0 mod b_{1}c, p = \frac{u}{b_{1}c},

and

v ≡ 0 mod b_{1}cγ, q = \frac{v}{b_{1}cγ} .  (3.1.149)

We remind that here ℤf(H̃) is the orthogonal complement to H̃ in N(X). Both these calculations of p and q are equivalent.

Changing the letters a and b places we get the same results for the b-series. We get Theorem 3.1.14. Let X be a K3 surface with ρ(X) = 2, and H a polarization of X of degree H^{2} = 2rs, r, s ≥ 1. Assume that the Mukai vector (r, H, s) is primitive. Let Y be the moduli space of sheaves on X with the isotropic Mukai vector v = (r, H, s) and the canonical nef element h = (−a, 0, b) mod ℤv. Let H̃ = H/d be the corresponding primitive polarization with H̃ · N(X) = γℤ. We denote by μ the invariant of the pair (N(X), H̃) and use notations of Proposition 3.1.1.

We have Y ≅ X, if

g.c.d(c, dγ) = 1,

and at least for one square-free α | 2a_{1}b_{1} there exists h̃_{1} ∈ N(X) which belongs to the a-series or to the b-series described below.

The h̃_{1} belongs to the a-series, if

(h̃_{1})^{2} = \frac{2γ^{2}b_{1}^{2}c}{α}, H̃ · h̃_{1} ≡ 0 mod b_{1}c, f(H̃) · h̃_{1} ≡ 0 mod b_{1}c det N(X),

(3.1.150)

and

p = \frac{H̃ · h̃_{1}}{b_{1}c} , q = \frac{f(H̃) · h̃_{1}}{b_{1}c det N(X)}

satisfy conditions (G) and (A) above (see (3.1.148) and (3.1.149) about an alternative definition of p and q).

The h̃_{1} belongs to the b-series, if

(h̃_{1})^{2} = \frac{2γ^{2}a_{1}^{2}c}{α}, H̃ · h̃_{1} ≡ 0 mod a_{1}c, f(H̃) · h̃_{1} ≡ 0 mod a_{1}c det N(X),

(3.1.151)

and

p = \frac{H̃ · h̃_{1}}{a_{1}c} , q = \frac{f(H̃) · h̃_{1}}{a_{1}c det N(X)}  (3.1.152)

satisfy the conditions (G) and (B) above.

These conditions are necessary to have Y ≅ X, if X is a general K3 surface with ρ(X) = 2, i. e. the automorphism group of the transcendental periods (T (X), H^{2,0}(X)) is ±1.

Important Remark 3.1.15. Applying Theorem 3.1.3 and the formula (3.1.137) for the associated solution, we get that

±h_{1} = \{ \begin{matrix}\frac{−H}{c} + \frac{α(H̃·h̃_{1})h̃_{1}}{b_{1}^{2}c^{2}γ} , & if h̃_{1} is from a-series, \\ \frac{−H}{c} + \frac{α(H̃·h̃_{1})h̃_{1}}{a_{1}^{2}c^{2}γ} , & if h̃_{1} is from b-series \end{matrix}  (3.1.153)

belongs to N(X) and

(Y, h̃) ≅ (X, ±w(h_{1})) for some w ∈ W^{(−2)}(N(X)).  (3.1.154)

Applying additionally Theorem 2.3.3, we get the following sufficient condition to have Y ≅ X which is valid for X with any ρ(X). This is one of the main results of the paper.

In Theorem 3.1.16 below, for H̃ ∈ N we apply the same definitions and notations: f(H̃), δ, μ, as for H̃ ∈ N = N(X) of Proposition 3.1.1.

Theorem 3.1.16. Let X be a K3 surface and H a polarization of X of degree H^{2} = 2rs, r, s ≥ 1. Assume that the Mukai vector (r, H, s) is primitive. Let Y be the moduli space of sheaves on X with the isotropic Mukai vector v = (r, H, s) and the canonical nef element h = (−a, 0, b) mod ℤv. Let H̃ = H/d be the corresponding primitive polarization with H̃ · N(X) = γℤ. We assume that

g.c.d(c, dγ) = 1.

Then Y ≅ X, if at least for one square-free α | 2a_{1}b_{1} there exists h̃_{1} ∈ N(X) which belongs to the a-series or to the b-series described below (moreover, one has (3.1.153) and (3.1.154)).

The h̃_{1} belongs to the a-series, if there exists a primitive sublattice N ⊂ N(X) of the rank two containing H̃ and h̃_{1} such that H̃ · N = γℤ and

(h̃_{1})^{2} = \frac{2γ^{2}b_{1}^{2}c}{α}, H̃ · h̃_{1} ≡ 0 mod b_{1}c, f(H̃) · h̃_{1} ≡ 0 mod b_{1}c det N, (3.1.155)

and

p = \frac{H̃ · h̃_{1}}{b_{1}c} , q = \frac{f(H̃) · h̃_{1}}{b_{1}c det N}  (3.1.156)

satisfy conditions (G) and (A) above.

The h̃_{1} belongs to the b-series, if there exists a primitive sublattice N ⊂ N(X) of the rank two containing H̃ and h̃_{1} such that H̃ · N = γℤ and

(h̃_{1})^{2} = \frac{2γ^{2}a_{1}^{2}c}{α}, H̃ · h̃_{1} ≡ 0 mod a_{1}c, f(H̃) · h̃_{1} ≡ 0 mod a_{1}c det N, (3.1.157)

and

p = \frac{H̃ · h̃_{1}}{a_{1}c} , q = \frac{f(H̃) · h̃_{1}}{a_{1}c det N}  (3.1.158)

satisfy the conditions (G) and (B) above.

These conditions are necessary to have Y ≅ X, if ρ(X) ≤ 2 and X is a general X K3 surface with its Picard lattice, i. e. the automorphism group of the transcendental periods (T (X), H^{2,0}(X)) is ±1.

These paper treats in general problems considered in [MN1], [MN2] and [N4] where the additional condition γ = d = 1 had been imposed. It makes results of this paper much more complicated and less efficient.

4. Divisorial conditions on moduli of (X, H) which imply Y ≅ X and γ(H̃) = γ

Using notations above, for \overline{μ} = \{μ, −μ\} ⊂ (ℤ/(2a_{1}b_{1}c^{2}/γ))^{∗}, α |2a_{1}b_{1} and g.c.d(c, dγ) = 1 we denote by

D(r, s, d, γ; A)\overline{μ} ; α  (4.1)

the set of all δ ∈ ℕ such that δ ≡ γμ^{2} mod 4a_{1}b_{1}c^{2}/γ and the equation p^{2} − γδq^{2} = 4a_{1}b_{1}c/α has an integral solution (p, q) satisfying (G) and (A) (the a-series). Similarly, we define

D(r, s, d, γ; B)\overline{μ} ; α  (4.2)

(the b-series).

We denote

D(r, s, d, γ)\overline{μ} = ( ⋃_{α|2a_{1}b_{1}} D(r, s, d, γ; A)\overline{μ} ; α) ⋃ ( ⋃_{α|2a_{1}b_{1}} D(r, s, d, γ; B)\overline{μ} ; α) . (4.3)

By Theorem 3.1.13, the set D(r, s, d, γ)\overline{μ} describes all possible pairs H ∈ N(X) of general polarized K3 surfaces (X, H) with rk N(X) = 2, the primitive polarization H̃ = H/d ∈ N(X), the invariant γ(H̃) = γ (i. e. H̃ ·N(X) = γℤ) and the invariant ±μ for H̃ ∈ N(X)), such that Y ≅ X. By general results of [N1] and [N2] the pair H̃ ∈ N(X) defines the irreducible 18-dimensional moduli of such pairs (X, H̃), i. e. a (irreducible) divisorial condition on 19-dimensional moduli of polarized K3 surfaces (X, H) which implies that γ(H̃) = γ and Y ≅ X. Thus, we can interpret our results as follows.

Theorem 4.1. The set

D(r, s, d, γ) = \{(\overline{μ}, δ) | \{μ, −μ\} ⊂ (ℤ/(2a_{1}b_{1}/γ))^{∗} , δ ∈ D(r, s, d, γ)\overline{μ}\} (4.4)

describes all (irreducible) divisorial conditions on moduli of polarized K3 surfaces (X, H) with H^{2} = 2rs and the primitive polarization H̃ = H/d (here d^{2}|rs), which imply that H̃ · N(X) = γℤ (i.e. γ(H̃) = γ)) and Y ≅ X.

We have (see (4.3))

D(r, s, d, γ)\overline{μ} = ( ⋃_{α|2a_{1}b_{1}} D(r, s, d, γ; A)\overline{μ} ; α) ⋃ ( ⋃_{α|2a_{1}b_{1}} D(r, s, d, γ; B)\overline{μ} ; α) . (4.5)

where each set D(r, s, d, γ; A)\overline{μ} ; α and D(r, s, d, γ; B)\overline{μ} ; α is infinite, if it is not empty.

Proof. We need to prove the last statement only. Assume that D(r, s, d, γ; A)\overline{μ} ; α is not empty. Thus, there exist integral (p_{0}, q_{0}) such that

\{ \begin{matrix}\frac{p_{0}^{2}−4a_{1}b_{1}c/α}{γq_{0}^{2}} ≡ γμ^{2} mod \frac{4a_{1}b_{1}c^{2}}{γ} \\ p_{0}^{2} − 4a_{1}b_{1}c/α > 0 \\ (p_{0}, q_{0}) satisfies (G) and (A) \end{matrix} .  (4.6)

Then

δ_{0} = \frac{p_{0}^{2} − 4a_{1}b_{1}c/α}{γq_{0}^{2}} ∈ D(r, s, d, γ; A)\overline{μ} ; α.  (4.7)

The (4.6) is equivalent to

\{ \begin{matrix}p_{0}^{2} − 4a_{1}b_{1}c/α ≡ γ^{2}μ^{2}q_{0}^{2} mod 4a_{1}b_{1}c^{2}q_{0}^{2} \\ p_{0}^{2} − 4a_{1}b_{1}c/α > 0 \\ (p_{0}, q_{0}) satisfies (G) and (A) \end{matrix} .  (4.8)

Clearly, (p, q_{0}) where

p ≡ p_{0} mod 4a_{1}b_{1}c^{2}q_{0}^{2}a_{1}b_{1}, and p^{2} − 4a_{1}b_{1}c/α > 0  (4.9)

also satisfies (4.8) and defines

δ = \frac{p^{2} − 4a_{1}b_{1}c/α}{γq_{0}^{2}} ∈ D(r, s, d, γ; A)\overline{μ} ; α.  (4.10)

Obviously, their number is infinite. This proves the statement.

The key question is:

Problem 4.2. When D(r, s, d, γ; A)\overline{μ} ; α and D(r, s, d, γ; B)\overline{μ} ; α are non-empty?

We hope to consider this question in further publications on the subject. About d = 1 and γ = 1 see [N4] and also [MN1], [MN2].

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Deptm. of Pure Mathem. The University of Liverpool, Liverpool L69 3BX, UK; Steklov Mathematical Institute, ul. Gubkina 8, Moscow 117966, GSP-1, Russia

E-mail address: vnikulin@liv.ac.uk  vvnikulin@list.ru