Surfaces with K^{2} = 7 and p_{g} = 4

Ingrid C. Bauer - Catanese ^{∗}

Mathematisches Institut der Universität Göttingen Bunsenstr. 3-5 37075 Göttingen

January 7, 1999

0 Introduction

In this paper we will study (smooth, projective, connected,) minimal surfaces defined over the complex numbers with geometric genus p_{g} = 4 and such that the canonical divisor has selfintersection 7. The study of these surfaces has a deep historical interest. In the book Le superficie algebriche (cf. [Enr]) F. Enriques summarized the results obtained in a life long research which took place in an arc of more than fifty years. After achieving the socalled rough classification of surfaces, in chapter V III of his celebrated book he considered the classification of surfaces of general type with low invariants, and the core of this chapter dealt with the study of surfaces with p_{g} = 4, for which the analysis is based on a detailed study of the behaviour of the canonical map φ_{K} : S ⟶ Σ ⊂ ℙ^{3}. Nowadays a major clarification of Enriques’ ideas has been made by Kodaira and others, but a main question remains: with diverse (algebraic and transcendental) tools available, can we really do better than Enriques? Thus in this work we take up this challenge, and we dare say that the answer to the above question is positive. To explain in a more detailed way what we mean, let us briefly recall the status of classification for surfaces with p_{g} = 4, for which we have by Noether’s inequality K^{2} ≥ 2p_{g} − 4. Moreover, since for irregular surfaces we have K^{2} ≥ 2p_{g} (up to K^{2} = 7) we may assume q = 0. Let now S be a minimal surface of general type with p_{g}(S) = 4 and irregularity q = 0. Then there are the following results: K_{S}^{2} = 4: this case was studied by M. Noether and later E. Horikawa (cf. [Noe], [HorI-V]). Here S turns out to be the double cover of an irreducible quadric in ℙ^{3} and the surfaces with K^{2} = p_{g} = 4 form an irreducible moduli space. K_{S}^{2} = 5: this case was analized by F. Enriques and E. Horikawa (cf. [HorQ], [Enr]) giving rise to a connected, but reducible moduli space. One component corresponds to the quintic surfaces in ℙ^{3} with at most rational double points, the other irreducible component (where the canonical system of the corresponding surfaces has one simple base point) corresponds to double covers of a quadric in ℙ^{3}. K^{2} = 6: this case was also treated by F. Enriques and E. Horikawa (cf. [HorQ], [Enr]) yielding a reducible moduli space. As far as we know, up to now it is not yet known whether this moduli space is connected or not. K^{2} = 7: here Enriques proposed some special subfamilies, and some attempts to produce examples with Σ a surface of degree 7 were done by Maxwell (cf. [Max]) and Kodaira (cf. [Kod]). Ciliberto (cf. [Cil]) was the first to describe a locally complete family with Σ having ordinary singularities. Later on, two of the possible cases were analized by F. Catanese, F. Zucconi (cf. [Cat1], [Zuc]). The aim of this paper will be to complete the analysis of the (many) remaining cases. Namely, our first result consists in giving a very precise description of all the surfaces with K^{2} = 7 and p_{g} = 4. This result will allow us to show that the moduli space 𝔐_{K^{2}=7,p_{g}=4} has three irreducible components of respective dimensions 36, 36, 38. Studying the deformations of minimal surfaces with K^{2} = 7 and p_{g} = 4 very carefully will lead us to the conclusion that the the two irreducible components of dimension 36 have non empty intersection, whereas it seems rather difficult to decide whether the irreducible component of dimension 38 is in fact a connected component of 𝔐_{K^{2}=7,p_{g}=4} or not. Unfortunetaly we are up to the moment not yet able to handle this case completely, but we think that in principle it should be possible to do it in the near future. K^{2} = 8: in this case the moduli space is not connected. By results in [CiFrML] and [Ol], there are at least three connected components of 𝔐_{K^{2}=8,p_{g}=4}. A complete classification of the case K^{2} = 8 is probably rather complicated (even if theoretically possible) and the situation tends to become much worse for higher K^{2}. Our paper will be organized as follows: In the first two sections some known facts about surfaces with K^{2} = 7 and p_{g} = 4 will be recalled and some preparatory results will be presented. In particular, we start by partitioning the class of minimal surfaces with K^{2} = 7 and p_{g} = 4 in several subfamilies according to the following:

∗I would like to express my deep gratitude to Fabrizio from whom I learnt a lot about algebraic surfaces.

Proposition 0.1 Let S be a minimal (smooth, connected, algebraic) surface (over ℂ) with K_{S}^{2} = 7 and p_{g}(S) = 4. Then S belongs to exactly one of the following classes:

(F) |K_{S}| = |H| + F, where |H| is the movable part of |K_{S}|, F is the fixed part and F^{2} = −2, K_{S} · F = 0. In this case |H| has exactly one base point x ∈ S and φ_{|H|} : Ŝ(x) ⟶ Q ⊂ ℙ^{3} is a morphism of degree two from Ŝ(x) onto a quadric Q in ℙ^{3}. (F’) |K_{S}| = |H| + F, where |H| is the movable part of |K_{S}|, F · K_{S} = 1, F^{2} = −1 and H · F = 2. In this case |H| is base point free and φ_{|H|} : S ⟶ Q ⊂ ℙ^{3} is a morphism of degree two from S onto a quadric Q in ℙ^{3}.

(0) |K_{S}| doesn’t have base points and the canonical map φ_{|K_{S}|} is a birational morphism from S onto a surface of degree seven in ℙ^{3}. (I.1) |K_{S}| has exactly one (simple) base point and φ_{|K_{S}|} is a birational map from S onto a surface of degree six in ℙ^{3}. (I.2) |K_{S}| has exactly one (simple) base point and φ_{|K_{S}|} is a map of degree two from S onto a cubic surface in ℙ^{3}. (I.3) |K_{S}| has exactly one (simple) base point and φ_{|K_{S}|} is a map of degree three from S onto a quadric surface in ℙ^{3}. (III) |K_{S}| has exactly three (possibly infinitely near) base points and φ_{|K_{S}|} is a map of degree two from S onto a quadric surface in ℙ^{3}.

Since surfaces of type (0) were described by Catanese and those of type (I.3) by Zucconi, in section 3 we will study the surfaces of type (I) and the surfaces of type (III). The most intricate part is here to exclude that the three base points of the surfaces of type (III) are infinitely near. The main idea is to take the blow up Ŝ of the isolated fixed points of the involution induced by the canonical map and to show that the surface obtained is isomorphic to Horikawa’s canonical resolution S^{∗} of the singularities of the branch locus of the double cover induced by the canonical map. Therefore we have now two possibilities for calculating the number t of points on S̃ which have to be blown up in order to obtain Ŝ = S^{∗}, namely t. Using the formulas for the self intersection of the first Chern class of S^{∗} and calculating the number of isolated fixed points on S̃, we see that the case that |K_{S}| has infinitely near base points is impossible. In section four we will classify the surfaces of type (F) and (F′), where the canonical system has a non trivial fixed part. In section 5 we will count the moduli of the families of surfaces with K^{2} = 7 and p_{g} = 4 we obtained in our classification. By Kuranishi’s theorem (cf [Kur]) the dimension of the moduli space of surfaces with K^{2} = 7 and p_{g} = 4 in a generic point is at least 10χ − 2K^{2} = 36. Using this result we see that many of the families obtained in our fine classification do not give rise to an irreducible component, because their number of moduli is too small. The main result of these sections will be the following (for a more precise formulation we refer to (5.1)):

Theorem 0.2 Let S be a smooth minimal surface with K_{S}^{2} = 7 and p_{g}(S) = 4. Then S belongs to exactly one of the following families:

(0) |K_{S}| doesn’t have base points and the canonical map φ_{|K_{S}|} is a birational morphism from S onto a surface of degree seven in ℙ^{3}. These surfaces form an open unirational, irreducible set of dimension 36 in the moduli space 𝔐_{K^{2}=7,p_{g}=4} of surfaces with K_{S}^{2} = 7 and p_{g}(S) = 4. (For a more detailed description of these surfaces we refer to [Cat1]). (I.1) |K_{S}| has exactly one (simple) base point and φ_{|K_{S}|} is a birational map. In this case the canonical image Σ := φH̃(S̃) is a surface of degree six in ℙ^{3} with the following properties:

(a) the double curve Γ ⊂ Σ is a plane conic,

(b) if γ ⊂ ℙ^{3} is the plane containing Γ, then Σ has a generalized tacnode o ∈ γ\Γ with tacnodal plane α ≠ γ,

(c) φ_{|H|}(E) = α ∩ γ, where E is the exceptional curve of the blow up of the base point of |K_{S}|. Apart from rational double points Σ doesn’t have other singularities. The surfaces of type (I.1) form an irreducible unirational set of dimension 35 in 𝔐_{K^{2}=7,p_{g}=4}. (I.2) |K_{S}| has exactly one simple base point x and φ_{|K_{S}|} is a map of degree two. Let π : S̃ ⟶ S be the blow up of S in x, E the exceptional curve and |H̃| the movable part of |KS̃|. Then φ_{|H̃|} factors as the blow up of ℙ̂^{2}(x_{1}, …, x_{6}) in x_{7} ∈ E_{6} and the double cover branched along a smooth divisor D ∈ |12L − 4 ∑_{i=1}^{5} E_{i} − 2E_{6} − 4E_{7}|. These surfaces form an irreducible unirational set of dimension 35 in 𝔐_{K^{2}=7,p_{g}=4}. (I.3) |K_{S}| has exactly one base point and φ_{|K_{S}|} is a map of degree three onto a quadric in ℙ^{3}. These surfaces form an irreducible unirational set of dimension 35 in 𝔐_{K^{2}=7,p_{g}=4}. For more details we refer to [Zuc]. (III) |K_{S}| has exactly three pairwise different simple base points x_{1}, x_{2}, x_{3} ∈ S. Let π : S̃ ⟶ S be the blow up of S in x_{1}, x_{2}, x_{3}, |H̃| the movable part of |KS̃|, E_{i} := π^{−1}(x_{i}). Then we have the following possibilities: (III.α) φ_{|H̃|}(S̃) = ℙ^{1} × ℙ^{1} and φ_{|H̃|}(E_{1}), φ_{|H̃|}(E_{2}), φ_{|H̃|}(E_{3}) are pairwise disjoint lines. These surfaces form an irreducible unirational set of dimension 36 in 𝔐_{K^{2}=7,p_{g}=4}. (III.β) φ_{|H̃|}(S̃) = ℙ^{1} × ℙ^{1}, and without restriction φ_{|H̃|}(E_{1}), φ_{|H̃|}(E_{2}) ≡ Γ_{1}, φ_{|H̃|}(E_{3}) ≡ Γ_{2}. These surfaces form an irreducible unirational set of dimension 38 in 𝔐_{K^{2}=7,p_{g}=4}. (III.γ) φ_{|H̃|}(S̃) is the quadric cone. These surfaces form an irreducible unirational set of dimension 35 in 𝔐_{K^{2}=7,p_{g}=4} and they are obtained as degenerations of surfaces of type (III.α). (F) |K_{S}| has a non trivial fixed part F, with F^{2} = −2 and K_{S} · F = 0. Then, if |H| is the movable part of |K_{S}|, |H| has exactly one simple base point x. Let π : S̃ ⟶ S be the blow up of S in x and let |H̃| be the movable part of |KS̃|. Then φH̃ : S̃ ⟶ Q is a morphism of degree two from S̃ onto a quadric and φH̃(F) is a line. We have the following two cases: (F.1) φH̃(S̃) = ℙ^{1} × ℙ^{1}. These surfaces form an irreducible unirational set of dimension 35 in 𝔐_{K^{2}=7,p_{g}=4}. (F.2) φH̃(S̃) is the quadric cone. These surfaces form an irreducible unirational set of dimension 34 in 𝔐_{K^{2}=7,p_{g}=4} and they are obtained as degenerations of surfaces of type (F.1). (F’) |K_{S}| = |H| + F, where |H| is the movable part of |K_{S}|, where F · K_{S} = 1, F^{2} = −1 and H · F = 2. In this case |H| is base point free and φ_{|H|} : S ⟶ Q ⊂ ℙ^{3} is a morphism of degree two from S onto the quadric cone Q in ℙ^{3}. These surfaces form an irreducible unirational set of dimension 37 in 𝔐_{K^{2}=7,p_{g}=4} and they are obtained as degenerations of the surfaces of type (III.β).

In all the cases (III), (F) and (F′) we will give a factorization of the canonical map in a blow up of the quadric in points in special position and we give the linear system to which the branch divisor has to belong. From this considerations it follows rather quickly that the moduli space 𝔐_{K^{2}=7,p_{g}=4} of minimal surfaces with K^{2} = 7 and p_{g} = 4 has three irreducible components of dimensions 36, 36 and 38. Calculating the rank of the Kodaira Spencer map of a general surface of type (III.γ) allows us to conclude that the irreducible components of dimension 36 have non empty intersection. I.e., our main result will be the following:

Theorem 0.3 1) The moduli space 𝔐_{K^{2}=7,p_{g}=4} has three irreducible components 𝔐_{36}, 𝔐′_{36} and 𝔐_{38}, where i is the dimension of 𝔐_{i}.

2) 𝔐_{36} ∩ 𝔐′_{36} is non empty. In particular, 𝔐_{K^{2}=7,p_{g}=4} has at most two connected components.

3) 𝔐′_{36} ∩ 𝔐_{38} is empty.

Finally we conclude the paper by showing that the surfaces with K^{2} = 7 and p_{g} = 4 are simply connected, in particular all minimal surfaces with K^{2} = 7 and p_{g} = 4 are homeomorphic (cf. [F]).

Contents

0 Introduction 1

1 The canonical system 6

2 Some known results 9

3 Surfaces with K^{2} = 7, p_{g} = 4, such that the canonical system doesn’t have a fixed part 12 3.1 Surfaces of type (I.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Surfaces of type (I.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Surfaces of type (I.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Surfaces of type (III) . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.1 Q = ℙ^{1} × ℙ^{1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4.2 Q is the quadric cone in ℙ^{3} . . . . . . . . . . . . . . . . . . . . 30 3.4.3 Infinitely near base points . . . . . . . . . . . . . . . . . . . . 33

4 |K| has a (non trivial) fixed part 42 4.1 Surfaces of type (F) . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Surfaces of type (F′) . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 The moduli space 50 5.1 The ”fine” classification of surfaces with K_{S}^{2} = 7 and p_{g} = 4 (revisited) 50 5.2 Counting the number of moduli . . . . . . . . . . . . . . . . . . . . . 53 5.3 The irreducible components of the moduli space . . . . . . . . . . . . 64 5.4 The fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . 74

1 The canonical system

Let S be a minimal (smooth, connected, projective) surface over the complex numbers with K^{2} = 7 and p_{g} = 4.

Remark 1.1 1) S is of general type.

2) (cf. [Deb]) S is a regular surface, i.e. q(S) := H^{1}(S, O_{S}) = 0.

Moreover, we have the following result:

Theorem 1.2 Let S be a minimal surface with K^{2} = 7, p_{g} = 4. Then the canonical system |K_{S}| of S is not composed with a pencil.

Proof. Follows immediately from [Deb]. q.e.d First of all we treat the case, where |K_{S}| has a (non-trivial) fixed part. We will show that if |K_{S}| has a non-trivial fixed part then either F^{2} = −2, K_{S} · F = 0 or we have K_{S} · F = 1, F^{2} = −1 and H · F = 2.

Proposition 1.3 Let S be a minimal surface with K^{2} = 7 and p_{g} = 4. Moreover, let F be the fixed part of the canonical system |K_{S}| of S. Then either

(1) F = ∅, or

(2) K_{S} · F = 0 and F^{2} = −2, or

(3) K_{S} · F = 1 and F^{2} = −1. In case (2) as well as in case (3) we have H · F = 2.

Proof. Let |H| be the movable part of |K_{S}|, in particular |K_{S}| = |H| + F. Then we have:

(∗)  7 = K_{S}^{2} = H^{2} + H · F + K_{S} · F.

By the 2-connectedness of the canonical divisor of S (cf.[BPV],VII, prop.(6.1)) we have H · F ≥ 2 and therefore H^{2} + K_{S} · F ≤ 5. Using the fact that

H^{2} ≥ degφ_{|H|} · degφ_{|H|}(S) ≥ 4,

we immediately see from (∗) that K_{S} · F ≤ 1. K_{S} · F = 1 implies that H · F = 2 and F^{2} = −1. Since K_{S} is numerically effective it follows that the only other possibility is

K_{S} · F = 0

and from (∗) we see that we have the following two possibilities:

\begin{matrix}(i) & H^{2} = 5, H · F = 2 (F^{2} = −2), \\ (ii) & H^{2} = 4, H · F = 3 (F^{2} = −3), \end{matrix}

where again we use that H^{2} ≥ 4. We observe that (ii) is not possible since F^{2} = −3 implies that K_{S} · F + F^{2} = −3 is not divisible by two, contradicting the adjunction formula. Therefore the proposition is proven. q.e.d We assume now that

|K_{S}| = |H| + F,

where |H| is the movable part of the canonical system |K_{S}| of S and F is its fixed part. Moreover, we assume that H^{2} = 5, H · F = 2 and F^{2} = −2 (i.e., we consider case (2) of (1.3)).

Lemma 1.4 Under the above assumptions |H| has exactly one (simple) base point x ∈ S and φ_{|H|} : S − − ⟶ Q ⊂ ℙ^{3} is a rational map of degree two onto a quadric Q in ℙ^{3}.

Proof. First of all we remark that by [HorQ], lemma 2 |H| has at most one base point. If |H| has exactly one base point, then φ_{|H|} has to be a rational map of degree two onto a quadric Q ⊂ ℙ^{3}. We assume that |H| is without base points, then φ_{|H|} : S ⟶ W_{5} ⊂ ℙ^{3} has to be a birational morphism from S to a quintic surface W_{5} in ℙ^{3}. Let C ∈ |H| be a generic curve, which we can assume to be nonsingular. Then

2g(C) − 2 = C^{2} + C · K_{S} = 5 + C(H + F) = 12,

hence g(C) = 7. On the other hand φ_{|H|}(C) is a plane quintic curve, which has genus at most 6 and we get a contradiction. q.e.d We will now give a ”rough classification” of minimal surfaces S with K^{2} = 7 and p_{g}(S) = 4. More precisely, we will prove the following result:

Proposition 1.5 Let S be a minimal (smooth, connected, algebraic) surface (over ℂ) with K_{S}^{2} = 7 and p_{g}(S) = 4. Then S belongs to exactly one of the following classes:

(F) |K_{S}| = |H| + F, where |H| is the movable part of |K_{S}|, F^{2} = −2, K_{S} · F = 0. In this case |H| has exactly one base point x ∈ S and φ_{|H|} : Ŝ(x) ⟶ Q ⊂ ℙ^{3} is a morphism of degree two from Ŝ(x) onto a quadric Q in ℙ^{3}.

(F’) |K_{S}| = |H| + F, where |H| is the movable part of |K_{S}|, where F · K_{S} = 1, F^{2} = −1 and H · F = 2. In this case |H| is base point free and φ_{|H|} : S ⟶ Q ⊂ ℙ^{3} is a morphism of degree two from S onto a quadric Q in ℙ^{3}.

(0) |K_{S}| doesn’t have base points and the canonical map φ_{|K_{S}|} is a birational morphism from S onto a surface of degree seven in ℙ^{3}.

(I.1) |K_{S}| has exactly one (simple) base point and φ_{|K_{S}|} is a birational map from S onto a surface of degree six in ℙ^{3}.

(I.2) |K_{S}| has exactly one (simple) base point and φ_{|K_{S}|} is a map of degree two from S onto a cubic surface in ℙ^{3}.

(I.3) |K_{S}| has exactly one (simple) base point and φ_{|K_{S}|} is a map of degree three from S onto a quadric surface in ℙ^{3}.

(II I) |K_{S}| has exactly three (possibly infinitely near) base points and φ_{|K_{S}|} is a map of degree two from S onto a quadric surface in ℙ^{3}.

We remark that in the above statement the roman digits indicate the number of base points of |K_{S}|, whereas the arabic numbers in case (I) indicate the degree of the canonical map. Proof. Follows immediately from the following lemma, and (1.3), (1.4) for the case (F). q.e.d

Lemma 1.6 Let S be a minimal surface with K^{2} = 7 and p_{g}(S) = 4. We assume that |K_{S}| does not have a fixed part. Let π : S̃ ⟶ S be a minimal blow-up of S, such that the movable part |H| of |π^{∗}K_{S}| doesn’t have base points. Then we have H^{2} ∈ {4, 6, 7}. Moreover, |K_{S}| has only simple base points.

Proof. Since |K_{S}| is not composed with a pencil, by [HorQ], lemma 2, we have: 4 ≤ H^{2} ≤ 7. Therefore we only have to exclude the possibility H^{2} = 5. We assume that H^{2} = 5, i.e. |K_{S}| has exactly two (possibly infinitely near) base points x_{1}, x_{2} and setting E_{i} := π^{−1}(x_{i}) we have: |π^{∗}K_{S}| = |H|+E_{1} +E_{2}. Therefore |H| induces a birational morphism φ_{|H|} : S̃ ⟶ W_{5} ⊂ ℙ^{3} from S̃ to a quintic surface W_{5} in ℙ^{3}. Let C ∈ |H| be a generic curve, which we can assume to be smooth. By the adjunction formula it follows:

2g(C) − 2 = H^{2} + H · KS̃ = 5 + (π^{∗}K_{S} + E_{1} + E_{2}) · H = 14,

hence g(C) = 8. Again φ_{|H|}(C) is a plane quintic curve, which has genus at most 6. Hence we get a contradiction and H^{2} ∈ {4, 6, 7}. On the other hand H^{2} = K_{S}^{2} − ∑_{b∈Bs(K)} m_{b}^{2} = 7 − ∑_{b∈Bs(K)} m_{b}^{2}, where m_{b} is the multiplicity of the base point b and therefore m_{b} = 1 for each b. q.e.d It is obvious that this result implies now proposition (1.5), if we note that |K_{S}| is base point free if and only if H^{2} = 7.

2 Some known results

In this section we will recall a couple of results on surfaces with K^{2} = 7, p_{g} = 4, which have already been established in the past. We do not want to enter in the details of proof and refer to the original papers. The cases (0) and (I.3) of theorem (1.6) have already been settled by F. Catanese in [Cat1] and by F. Zucconi in [Zuc] respectively. They obtain the following results:

Theorem 2.1 (F. Catanese) Surfaces with p_{g} = 4 and K^{2} = 7 such that |K_{S}| is without base points form an irreducible, unirational open set of their moduli space 𝔐_{χ=5,K^{2}=7} of dimension 36.

Theorem 2.2 (F. Zucconi) The surfaces with K^{2} = 7 and p_{g} = 4 such that the canonical map has degree three form an irreducible, unirational set of dimension 35 in the moduli space of surfaces with K^{2} = 7 and p_{g} = 4.

In the remaining part of this section we will discuss an example of F. Enriques. In his famous book ”Superficie algebriche”, chapter 8, ([Enr]), he gives ideas and concrete examples how to classify smooth algebraic surfaces with K^{2} = 7 and p_{g} = 4. Unfortunately, there are also some mistakes. Among other things he gives the following construction:

Example 2.3 (F. Enriques) We consider two different planes α, γ of ℙ^{3}. Let C ⊂ γ ≅ ℙ^{2} be a (reduced) conic. Furthermore, let Σ ⊂ ℙ^{3} be a surface of degree six having as only singularities (apart from rational double points) C as double curve and having an (ordinary) tacnode o(∉ C) contained in the plane γ, whose tacnodal plane is α.

We recall that an ordinary tacnode is given by the following analytic equation: z^{2} = xy(x + y)(x − λy).

Before showing that Σ is the canonical image of a minimal surface S with K^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| of S has exactly one (simple) base point, i.e. S is of type (I.1) (cf. prop. (1.5)), we give another description of the hypersurfaces Σ ⊂ ℙ^{3} in Enriques’ example (2.3). First we give some definition generalizing the notion of a tacnode: A generalized tacnode is a two dimensional elliptic hypersurface singularity (X, 0), such that the fundamental cycle has self intersection (−2). In particular, (X, 0) is Gorenstein and by [Lau], theorem (1.3), (X, 0) is a double point singularity, hence the local analytic equation is given by

z^{2} = f(x, y),

where f vanishes of order four in 0. I.e., a generalized tacnode (X, 0) is the singularity obtained as the double cover branched along a curve with a quadruple point, which after blow up decomposes in at most simple triple points or double points.

Proposition 2.4 Let α, γ ⊂ ℙ^{3} be two different planes, let C ⊂ γ be a reduced conic and take a point o ∈ α ∩ γ, such that o ∉ C. Let Q ⊂ ℙ^{3} be an irreducible quadric containing C and finally let Σ ⊂ ℙ^{3} be a hypersurface having C as double curve and having a generalized tacnode in o with tacnodal plane α. Then Σ is an element of the linear system

|λα^{2}Q^{2} + γF_{5}|,

where λ ∈ ℂ, F_{5} ∈ |5H − C − 2o| and is furthermore subject to the condition that α be a tangent plane of F_{5} in o (i.e., α|F_{5}^{(2)}, where F_{5}^{(2)} is the homogeneous part of degree two of the Taylor expansion of F_{5} around o).

Let V ⊂ |5H−C−2o| be the linear subsystem given by the quintics F_{5} ∈ |5H−C−2o| such that α|F_{5}^{(2)}. Then obviously a general element G_{5} ∈ V is biplanar at o.

Remark 2.5 Let Σ_{0} be a general hypersurface of ℙ^{3} in the linear system |λα^{2}Q^{2} + γF_{5}|, where Q ⊂ ℙ^{3} is an irreducible quadric containing C, F_{5} ∈ V. Then

(i) Σ_{0} has C as double curve. (ii) Σ_{0} has an ordinary tacnode in o (with tacnodal plane α).

Proof. (i) is a routine verification. (ii) The local analytic equation of Σ_{0} in o is (note that o ∉ Q):

0 = α^{2} + γ(2αβ + G),

where β has degree 1 and G vanishes of order ≥ 3 in o. We set z := α + βγ, then the above equation becomes:

z^{2} = β^{2}γ^{2} − Gγ = γ(β^{2}γ − G).

Hence for general G, i.e. for general F_{5} ∈ V ⊂ |5H − C − 2o| biplanar at o with one tangent - plane equal to α, o is an ordinary tacnode of Σ_{0}. q.e.d

Proof (of (2.4)). Let α, γ, C, o and Σ be as in the statement of proposition (2.4). Then Σ ≡ Q^{2} · Q′mod(γ), where Q′ is a quadric such that Q′ ≡ α^{2}mod(γ). Hence we may arrange that Σ ≡ α^{2}Q^{2} + γF_{5}, where obviously F_{5} ∈ |5H − C − o|. It remains to show that F_{5} vanishes of order two in o with tangent plane α. The local analytic equation of Σ around o is:

(∗)  0 = α^{2} + γ \frac{F_{5}}{Q^{2}} .

But, since Σ has a generalized tacnode in o with tacnodal plane α, the equation of Σ (around o) has to be of the form:

0 = (α + g_{≥2})^{2} + k,

where k is a term of degree bigger or equal to four and g_{≥2} vanishes of order at least two in o. Then

(∗∗)  (α + g_{≥2})^{2} + k = α^{2} + 2αg_{≥2} + h_{≥4},

with h_{≥4} vanishing of order at least four in o. Therefore, comparing (∗) and (∗∗) we see that F_{5} has to vanish of order two in o and, if F^{(2)} denotes the homogeneous part of degree two of the Taylor series of F_{5} around o, we see that

2αg_{≥2} = γF^{(2)}.

Hence α divides F^{(2)} and we have shown the proposition. q.e.d

We will now show the following:

Proposition 2.6 Let α, γ ⊂ ℙ^{3} be two different planes, let C ⊂ γ be a (reduced) conic and let o ∈ α∩γ be a point such that o ∉ C. Moreover, let Σ ⊂ ℙ^{3} be a surface having C as double curve and having a generalized tacnode in o with tacnodal plane α. Then Σ is the canonical image of a minimal, smooth surface S with K^{2} = 7 and p_{g} = 4.

Before proving (2.6) we have to make some remarks on adjunction and subadjunction (cf. Mumford’s appendix [Mum] to [Zar]). For this purpose let X ⊂ ℙ^{3} be a surface and we consider the normalization p : Y ⟶ X of X. Moreover, let π : S ⟶ Y be a minimal desingularization of Y. We denote by Sing(Y) the singular locus of Y and i : Y \Sing(Y) ↪ Y is the natural inclusion. Then, since Y is normal, the canonical sheaf of Y is given by ω_{Y} = i_{∗}(Ω_{Y \Sing(Y)}^{2}), i.e. ω_{Y} is the sheaf of rational 2 - forms on Y having poles only on Sing(Y). Moreover, let C := Hom_{O_{X}}(O_{Y} , O_{X}) be the conductor of O_{Y} in O_{X}. We remark that C is a pure ideal of codimension one in O_{X}, supported in the singular curves of X.

Remark 2.7 (i) X is normal if and only if X has only finitely many singular points. (ii) We have:

p_{∗}ω_{Y} = Cω_{X} = CO_{X}(d − 4),

where d is the degree of X, and there is a Weil divisor Δ on Y such that

ω_{Y} = O(p^{∗}(d − 4) − Δ).

If X has ordinary singularities along smooth curves in ℙ^{3}, then Y is contained in a smooth blow up of ℙ^{3} and then the canonical sheaf ω_{Y} of Y is invertible. In this case, since π_{∗}ω_{S} ⊂ ω_{Y}, there is an ideal S in O_{Y} such that π_{∗}ω_{S} = Sω_{Y}.

S is called the ideal of subadjunction. Moreover, there is an ideal A ⊂ C such that

H^{0}(S, ω_{S}) = H^{0}(X, p_{∗}π_{∗}ω_{S}) = H^{0}(X, p_{∗}(Sω_{Y})) = H^{0}(X, Aω_{X}).

A is called the adjunction ideal in C.

Proof (of (2.6)). We consider as above the normalization p : Y ⟶ Σ of Σ and a minimal desingularization π : S̃ ⟶ Y of Y. Observe that for a generalized tacnode S is the maximal ideal. Then by adjunction we have:

|KS̃| = |(p ∘ π)^{∗}(2H_{ℙ^{3}} − C − o)| = (p ∘ π)^{∗}(γ + |H_{ℙ^{3}}|),

since a quadric in ℙ^{3} containing the conic C and the point o ∉ C has to contain γ. Therefore the fixed part F of |KS̃| equals to the pull-back of γ ∩ Σ under p ∘ π. Moreover, we see that |H| := |KS̃ − F | = (p ∘ π)^{∗}(|H_{ℙ^{3}}|) is base point free and p ∘ π = φ_{|H|}. Let E := φ_{|H|}^{−1}(α ∩ γ), then F = 2E, since γ ≡ α^{2} at the generic point of α ∩ γ. Moreover, H · E = 1. We have H^{2} = 6 and H · KS̃ = H · (H + 2E) = 8. The adjunction formula for E yields (since p_{a}(E) = 0):

−2 = E(KS̃ + E) = E(H + 3E) = 1 + 3E^{2},

hence E^{2} = −1. This implies, in particular, that K2 ; S̃ = H^{2} + 4H · E + 4E^{2} = 6. Moreover, E is a rational (−1) - curve and, since all exceptional divisors of the first kind are contained in the fixed part of |KS̃|, contracting E we obtain a minimal surface S with K_{S}^{2} = 7 and p_{g} = 4, having Σ as canonical image. q.e.d

3 Surfaces with K^{2} = 7, p_{g} = 4, such that the canonical system doesn’t have a fixed part

3.1 Surfaces of type (I.1)

Let S be a minimal (smooth, connected) surface with K^{2} = 7 and p_{g}(S) = 4. Assume that the canonical system |K_{S}| has one (simple) base point x ∈ S. Let π : S̃ ⟶ S be the blow up of S in x and let E := π^{−1}(x) be the exceptional curve of π. Thus we have:

|KS̃| = |π^{∗}K_{S}| + E = |H| + 2E,

where |H| is base point free. Furthermore we will assume in this paragraph that φ_{|H|} : S̃ ⟶ F_{6} is a birational morphism from S̃ onto a surface of degree six in ℙ^{3}. In the following we will give a geometric description of these surfaces and their canonical map. Essentially we will prove that the surfaces of type (I.1) are all obtained as in Enriques’ construction (cf. (2.6)). By adjunction we know that:

H^{0}(S̃, ωS̃) = H^{0}(F_{6}, AO_{F_{6}}(2)),

where A is the adjunction ideal of S̃. Hence, if A is the adjoint divisor (i.e. (φ_{|H|})_{∗}O(−A) = A), then H + 2E + A ≡ 2H. We can write A = D + E, where D_{red} = ∑_{i=1}^{k} D_{i} and D_{i} is an irreducible, reduced divisor on S̃ with D_{i} · H > 0 for each i ∈ {1, …, k} and E = ∑_{i=1}^{l} E_{i} such that for each i ∈ {1, …, l} E_{i} is an irreducible, reduced divisor on S̃ with E_{i} · H = 0. Since H ≡ A + 2E, there exists a plane γ ⊂ ℙ^{3} such that φ_{|H|}^{∗}(γ) ≡ H ≡ A + 2E and γ · F_{6} = ∑_{i=1}^{k} r_{i}(r_{i} − 1)Γ_{i} + 2L, where L := φ_{|H|}(E) ⊂ F_{6} is a line and Γ_{i} are the irreducible components (possibly infinitely near) of the singular curve Γ ⊂ F_{6}. Moreover, r_{i} is the multiplicity of the irreducible component Γ_{i}.

Remark 3.1 (i) The conductor Hom_{O_{F_{6}}}(O_{Y} , O_{F_{6}}) is the direct image p_{∗}(O_{Y} (− ∑_{i}(r_{i} − 1)Γ_{i})), where p : Y ⟶ F_{6} is the normalization of F_{6}. (ii) Let γ′ ⊂ ℙ^{3} be a generic plane. Then

γ′ · (γ ∩ F_{6}) = 2 + ∑ r_{i}(r_{i} − 1)Γ_{i} · γ′ = 6,

hence

∑ r_{i}(r_{i} − 1)Γ_{i} · γ′ = 4.

Therefore we have the following two possibilities:

1) there is exactly one i_{0} such that r_{i_{0}} = 2 and Γ_{i_{0}} · γ′ = 2, or

2) Γ_{1} · γ′ = Γ_{2} · γ′ = 1 and r_{1} = r_{2} = 2. I.e., either

1) Γ = Γ_{i_{0}} is a smooth (plane) curve of degree two, or

2) Γ = Γ_{1} + Γ_{2} is the union of two coplanar lines.

Proposition 3.2 There is at least one i ∈ {1, …, l} such that KS̃ · E_{i} > 0.

Proof. If KS̃ · E_{i} = 0 for all i ∈ {1, …, l}, then the normalization p : Y ⟶ F_{6} of F_{6} has only rational double points. Therefore:

4 = dimH^{0}(S̃, O(KS̃)) = dimH^{0}(Y, ω_{Y}) = H^{0}(F_{6}, O_{F_{6}}(2 − φ_{|H|}(D)).

Now, H^{0}(F_{6}, O_{F_{6}}(2−φ_{|H|}(D)) is the complete linear system of quadrics in ℙ^{3} cutting out a fixed conic and this vector space has dimension five. Therefore we get a contradiction. q.e.d

Remark 3.3 From E_{i} · H = 0 for all i ∈ {1, …, k} it follows that the intersection matrix (E_{i} ·E_{j})_{i,j} is negative definite. Since KS̃ = H +2E, we get KS̃ ·E_{i} = 2E ·E_{i} for all i. Therefore the E_{i}’s with E · E_{i} = 0 yield rational double points on the canonical model of S. Moreover, by (3.2), there must be at least one i ∈ {1, …, k} such that E · E_{i} > 0.

If we remark that the E_{i}’s with E · E_{i} > 0 yield singular points of Y, which are not rational double points, the following is obvious.

Lemma 3.4 Let p : Y ⟶ F_{6} be the normalization of F_{6} and let P_{1}, …, P_{r} be the singular points of Y, which are not rational double points. Then p(P_{1}), …, p(P_{r}) are contained in the line L := φ_{|H|}(E).

Proof. We recall that p(P_{1}), …, p(P_{r}) correspond exactly to the connected components E^{(i)} of E such that E · E^{(i)} > 0. Hence p(P_{i}) = φ_{|H|}(E^{(i)}) ∈ L = φ_{|H|}(E). q.e.d

Lemma 3.5 We have E ·E = 1, in particular there is only one connected component E^{(i)} of E giving rise to a singular point o (different from a rational double point) on Y.

Proof. We observe that

−D = φ^{∗}(C) = φ^{∗}(I_{Γ}),

hence

D · E = −deg(I_{D}|E) = −deg(I_{Γ}|φ(E)) = 2,

where the middle equality holds, because φ|E : E ⟶ L := φ(E) is an isomorphism. Now, we have:

−1 = E · KS̃ = (2H − D) · E − E · E = −E · E.

q.e.d

Therefore in the following we can assume: E = E^{(i)} is connected.

Proposition 3.6 With the usual notation it holds:

(i) p_{g}(Y) = 5. (ii) q(Y) = 0. (iii) E^{2} = −2 and p_{a}(E) = 1. (iv) o lies in the plane containing Γ and φ_{|H|}(E) ⊂ γ.

(v) If p : Y ⟶ F_{6} denotes again the normalization of F_{6}, then p^{−1}(o) ∉ p^{−1}(Γ).

Proof. (i) p_{g}(Y) = h^{0}(Y, ω_{Y}) = h^{0}(F_{6}, O_{F_{6}}(2 − Γ)) = 5. (ii) Since Y is normal, we have O_{Y} = π_{∗}OS̃. Therefore H^{1}(Y, O_{Y}) = H^{1}(Y, π_{∗}OS̃) = H^{1}(S̃, OS̃) = 0. (iii) We have E · E = 1 and this implies that KS̃ · E = 2. Now,

2 = KS̃ · E = (π^{∗}K_{Y} − E)E.

If Γ ⊂ F_{6} is a smooth conic, then ℙ̂^{3}(Γ) is smooth and Y ⊂ ℙ̂^{3}(Γ) is a hypersurface, hence Y is Gorenstein. In the case where Γ is the union of two coplanar lines, we will show in (3.8) that Y is Gorenstein. Therefore (in both cases) π^{∗}K_{Y} · E = 0 and it follows from the above formula that E^{2} = −2, whence p_{a}(E) = 1 by the adjunction formula. (iv) Since D · E = 2, the line φ_{|H|}(E) is contained in the plane which containes the conic Γ. But o ∈ φ_{|H|}(E) and (iv) is proved.

(v) It suffices to show that on S̃ we have D · E = 0. In fact, it holds:

2 = KS̃ · E = D · E + E^{2} + 4E · E = D · E − 2 + 4 = D · E + 2.

Hence, D · E = 0. q.e.d

Lemma 3.7 Let W be the blow up of ℂ^{3} with centre equal the union l_{x} ∪ l_{y} of two incident lines. Then:

1) W is Gorenstein;

2) if W̄ is the blow up of ℙ^{3} with centre l_{x} ∪ l_{y}, KW̄ = −4H + E, where E is the exceptional (Cartier) divisor;

3) E is the union of two copies of the Hirzebruch surface 𝔽_{1} glued along a fibre.

Proof. Let l_{x} ∪ l_{y} = {z = xy = 0}. then W ⊂ ℂ^{3} × ℙ^{1} is defined by W = {((x, y, z), (a : b)) ∈ ℂ^{3} × ℙ^{1} : za − xyb = 0}. Thus, for a = 1 we get z = xyb and W is smooth there. Whereas, for a = 0, b = 1 we obtain za = xy and W has exactly an isolated hypersurface singularity (an ordinary quadratic point). Thus 1) is shown.

2) follows right away, since KW̄ is invertible, by calculating at points of codimension 2 that KW̄ = π^{∗}K_{ℙ^{3}}.

3) Since we do not need it we leave the proof as an exercise. q.e.d.

Corollary 3.8 Assume that (with the same notation as in (3.6)) Γ is the union of two coplanar lines. Then Y is a Cartier divisor in W̄, in particular Y is Gorenstein.

Proof. The morphism φ_{H} : S̃ ⟶ F_{6} factors through the blow up π̃ of Γ ⊂ F_{6} ⊂ ℙ^{3}, since I_{Γ} has an invertible pull back in S̃. Since there are no infinitely near singular curves, it follows that the strict transform Y′ of F_{6} in W̄ has isolated singularities. Since Y′ ≡ 6H−2E, we see that Y′ is a Cartier divisor in W̄, whence Y′ is Gorenstein with ω_{Y′} ≡ O_{Y′}(2H−E). Since Y′ is Gorenstein with isolated singularities, it follows finally that Y′ is normal, i.e. Y′ is equal to the normalization Y of F_{6}. q.e.d.

Remark 3.9 We consider the spectral sequence

H^{i}(R^{j}π_{∗}OS̃) ⟹ H^{i+j}(OS̃).

Since H^{1}(OS̃) = 0, we get H^{1}(O_{Y}) = 0 and d_{2} : H^{0}(R^{1}π_{∗}OS̃) ⟶ H^{2}(O_{Y}) is injective. On the other hand p_{g}(S) = 4 and h^{2}(O_{Y}) = h^{0}(ω_{Y}) = 5, therefore h^{0}(R^{1}π_{∗}OS̃) = 1 and E gives rise to an elliptic singularity on Y respectively on F_{6}. Moreover, since o ∉ Γ, o is a hypersurface singularity, in particular o is an elliptic Gorenstein singularity with E^{2} = −2, i.e. o is a generalized tacnode.

Lemma 3.10 In the above situation we have:

(1) the tacnodal plane α of o is different from γ;

(2) φ_{|H|}(E) =: L = α ∩ γ.

Proof. First of all we recall that

div(γ)|_{F_{6}} = 2L + 2Γ,

hence F_{6} ≡ Q^{2}α̃^{2}(modγ), i.e., F_{6} ≡ Q^{2}α̃^{2} + γ · F_{5}, where α is a plane different from γ containig L. Since o ∉ Γ, Q does not vanish in o whence α̃ vanishes in o and F_{5} vanishes in o. Now, the local analytic equation of F_{6} in o is:

0 = α̃^{2} + γ · f =: f_{6}.

Since f_{6} is the equation of a generalized tacnode the homogeneous part of degree two f_{6}^{(2)} = α̃^{2} + γ · f^{(1)} has to be a square. An elementary argument shows the existence of λ ∈ ℂ such that f_{6}^{(2)} = (α̃^{2} + λγ)^{2}. Therefore we set α := α̃ + λγ. After the above linear change of coordinates we can suppose that α = α̃ is the tacnodal plane of o. Then it is obvious that α ≠ γ and φ_{|H|}(E) = α ∩ γ (note that we have already seen that φ_{|H|}(E) = L ⊂ γ). q.e.d. Summing up the above results, we get the following:

Theorem 3.11 Let S be a minimal surface with K_{S}^{2} = 7 and p_{g}(S) = 4 such that |K_{S}| has exactly one (simple) base point. Moreover, we assume that φ_{|K_{S}|} is a birational map. Then the canonical image Σ := φ_{|K_{S}|}(S) is a surface of degree six in ℙ^{3} with the following properties:

1) the double curve Γ ⊂ Σ is a (plane) conic;

2) if γ ⊂ ℙ^{3} is the plane containing Γ, then Σ has a generalized tacnode o ∈ γ\Γ with tacnodal plane α ≠ γ;

3) φ_{|H|}(E) = α ∩ γ. Apart from rational double points Σ doesn’t have other singularities.

Combining with (2.4) − (2.6) we see:

Theorem 3.12 (i) All surfaces with K^{2} = 7 and p_{g} = 4 such that |K| has one base point and φ_{|K|} is birational are obtained by taking the minimal model of a minimal resolution of singularities of a surface Σ ⊂ ℙ^{3} of degree six having a double conic Γ ⊂ γ ≅ ℙ^{2} ⊂ ℙ^{3} and a generalized tacnode o ∈ γ\Γ with tacnodal plane α ≠ γ. (ii) All surfaces with K^{2} = 7 and p_{g} = 4 such that |K| has one base point and φ_{|K|} is birational are obtained by taking the minimal model of a minimal resolution of singularities of a surface

Σ_{0} ∈ U ⊂ |λα^{2}Q^{2} + γF_{5}|,

where the above linear system is as described in (2.4) and U contains a Zariski open subset of |λα^{2}Q^{2} + γF_{5}|.

3.2 Surfaces of type (I.2)

Let S be a minimal (smooth, connected) algebraic surface with K^{2} = 7 and p_{g}(S) = 4 such that the canonical system |K_{S}| has exactly one simple base point x ∈ S. Let π : S̃ ⟶ S be the blow up of x and let E := π^{−1}(x) be the exceptional curve of π. Then we have:

|KS̃| = |π^{∗}K_{S}| + E = |H̃| + 2E,

where |H̃| is the movable part of |π^{∗}K_{S}|. Furthermore, we will assume in this section that φ_{|H̃|} : S̃ ⟶ F_{3} ⊂ ℙ^{3} is a morphism (generically) of degree two from S̃ onto a cubic surface F_{3} in ℙ^{3}.

Remark 3.13 It is wellknown that a cubic surface in ℙ^{3} is either ruled by lines (in this case it has a double line) or has only isolated singularities. In the latter case the number of singular points is at most four. Moreover, any cubic F_{3} ⊂ ℙ^{3} with only isolated singular points can be realized as F_{3} = ℙ̂^{2}(x_{1}, …, x_{6}) embedded by the linear system |H_{F_{3}}| = |3L − ∑_{i=1}^{6} E_{i}|.

Our first result is that the canonical image of a surface of type (I.2) has only isolated singularities.

Lemma 3.14 Let S be a minimal surface with K^{2} = 7 and p_{g} = 4 such that |K_{S}| has exactly one base point and φ_{|K_{S}|} : S − − ⟶ F_{3} is a map of degree two from S onto a cubic F_{3} in ℙ^{3}. Then F_{3} has only isolated singularities. In particular, F_{3} is not ruled by lines.

Proof. Let x ∈ S be the base point of |K_{S}| and let π : S̃ ⟶ S be the blow up of S in x. Moreover, let |H̃| be the movable part of |π^{∗}K_{S}|. We assume that the singular locus of F_{3} has dimension one. Let D be a hyperplane section of F_{3} and let C ∈ |H̃| = |φ_{|H̃|}^{∗}D| be the inverse image of D under φ_{|H|}. We can assume that C is irreducible and smooth (by possibly replacing D by another hyperplane section of F_{3}). Let D′ ⟶ D be the normalization of D. Then φ_{|H|}|C : C ⟶ D factors through D′. If K|C denotes the restriction of KS̃ to C, then we have dimH^{0}(C, O(K|C)) = 5, because D is a rational curve and K|C is the pull back of a divisor of degree 4 on D′ ≅ ℙ^{1}. On the other hand, we consider the exact sequence:

0 ⟶ OS̃(2E) ⟶ OS̃(KS̃) ⟶ O_{C}(KS̃) ⟶ 0.

We remark that

\begin{matrix}χ(2E) = h^{0}(2E) − h^{1}(2E) + h^{2}(2E) = 1 − h^{1}(2E) + h^{0}(H̃) = \\ = 5 − h^{1}(2E) = \frac{1}{2}(2E)(2E − KS̃) + 5 = 4, \end{matrix}

hence h^{1}(2E) = 1. Since H^{1}(S̃, OS̃) = 0, it follows from the above exact sequence that

dimH^{0}(C, O_{C}(KS̃)) =
= dimH^{0}(OS̃(KS̃)) − dimH^{0}(OS̃(2E)) + dimH^{1}(OS̃(2E)) =
= 4 − 1 + 1 = 4,

and we get a contradiction. q.e.d We want now to understand all irreducible curves C ⊂ S̃ which are contracted to a point by φ_{|H̃|}.

Lemma 3.15 Let C ⊂ S̃ be an irreducible curve on S̃ such that φ_{|H̃|}(C) is a point. Then either

1) C is a rational curve with C^{2} = −2, or

2) C · E = 1. In the second case we have the following two possibilities: 2a) p_{a}(C) = 0, C^{2} = −4, 2b) p_{a}(C) = 1, C^{2} = −2.

Proof. A curve C ⊂ S̃ is mapped to a point by φ_{|H̃|} if and only if H̃ · C = 0. Therefore, C ⊂ S̃ is contracted iff KS̃ · C = 2E · C. If C · E = 0, then by the adjunction formula we get C^{2} = 2p_{a}(C) − 2. Since C^{2} < 0 the only possibility is p_{a}(C) = 0, C^{2} = −2. Otherwise, C · E > 0. But, since H̃ · E = 1, φ_{|H|} maps E isomorphically onto a line in F_{3}. Moreover, C ∩ E is mapped to a point under φ_{|H̃|}. This implies that C · E = 1. Again by the adjunction formula we obtain:

0 > C^{2} = 2p_{a}(C) − 2 − KS̃ · C = 2p_{a}(C) − 4,

and we have the following two possibilities:

a) p_{a}(C) = 0, C^{2} = −4,

b) p_{a}(C) = 1, C^{2} = −2. q.e.d

We would now like to calculate the (singular) branch divisor of the double cover φ_{|H̃|}, but unfortunately the cubic surface in general has singularities. This problem will be avoided by making the following considerations. From now on let σ : S̃ ⟶ S̃ be the involution induced by φ_{|H̃|}. Then the fixed locus F ix(σ) equals {P_{1}, …, P_{t}}∪C_{1} ∪…∪C_{s}, where C_{1}, …, C_{s} are smooth curves contained in S̃ and P_{1}, …, P_{t} ∈ S̃ are the isolated fixed points of σ.

Proposition 3.16 σ does not have isolated fixed points, i.e. t = 0.

Proof. Let π̂ : Ŝ ⟶ S̃ be the blow up of S̃ in P_{1}, …, P_{t} and denote by E′_{1}, …, E′_{t} the exceptional curves of π̂. Furthermore, we denote by σ̂ the induced involution on Ŝ and the quotient Ŝ/σ̂ by P̂. We remark that P̂ is smooth and the branch locus Δ of the double cover φ̂ : Ŝ ⟶ P̂ is smooth and reduced. We have:

Δ ∈ |2δ|,

where

(φ̂)_{∗}OŜ ≅ OP̂(−δ) ⊕ OP̂.

Moreover, we have the following equalities:

5 = χ(S, O_{S}) = χ(S̃, OS̃) = χ(Ŝ, OŜ) =

2(p_{g}(P̂) − q(P̂) + 1) + \frac{1}{2}δ(δ + KP̂);

6 − t = K2 ; Ŝ = 2(KP̂ + δ)^{2}.

This implies that:

\begin{matrix}(∗) & δ(δ + KP̂) = 6; \\ (∗∗) & KP̂(δ + KP̂) = −3 − \frac{t}{2}. \end{matrix}

We have the following commutative diagram:

Ŝ ^{φ̂ \overline{}}P̂

[figure: π̂; φĤ]
S̃ ^{φH̃\overline{}}F_{3} ⊂ ℙ^{3}

π
S  .

Then:

\begin{matrix}(φ̂)^{∗}(Ĥ) = π̂^{∗}(H̃) = π̂^{∗}(KS̃ − 2E) = \\ KŜ − π̂^{∗}(2E) − ∑_{i=1}^{t} E′_{i} = φ̂^{∗}(KP̂ + δ) − ∑_{i=1}^{t} E′_{i} − π̂^{∗}(2E). \end{matrix}

We see that:

(φ̂^{∗}(Ĥ))^{2} = (π̂^{∗}(H̃))^{2} = H̃^{2} = 6,

hence:

Ĥ^{2} = \frac{1}{2}(φ̂^{∗}(Ĥ))^{2} = 3.

Moreover, h^{0}(P̂, OP̂(Ĥ)) = 4 and h^{2}(P̂, OP̂(Ĥ)) = h^{0}(P̂, OP̂(KP̂ − Ĥ)) = 0 (where we use that P̂ is a rational surface). Applying Riemann-Roch to Ĥ on P̂ we obtain:

\begin{matrix}h^{1}(P̂, OP̂(Ĥ)) = 4 − χ(P̂, OP̂(Ĥ)) = 4 − (1 + \frac{1}{2}(Ĥ^{2} − Ĥ · KP̂)) = \\ 3 − \frac{3}{2} + \frac{1}{2} Ĥ · KP̂ = \frac{3}{2} + \frac{1}{2} Ĥ · KP̂. \end{matrix}

It remains now to calculate Ĥ · KP̂. In fact,

\begin{matrix}Ĥ · KP̂ = \frac{1}{2}(φ̂^{∗}Ĥ · φ̂^{∗}KP̂) = \\ \frac{1}{2}(φ̂^{∗}(KP̂ + δ) · φ̂^{∗}(KP̂) − 2E · φ̂^{∗}KP̂ − ∑_{i=1}^{t} E′_{i} · φ̂^{∗}KP̂). \end{matrix}

We remark that σ̂|E′_{1} = id_{E′_{1}}, …, σ̂|E′_{t} = id_{E′_{t}} and σ̂|E = id_{E}. Hence the images A_{1}, …, A_{t}, A of these curves under φ̂ are smooth rational curves with selfintersection −2 on P̂. This implies that

A_{1} · KP̂ = … = A_{t} · KP̂ = A · KP̂ = 0

and therefore:

2E · φ̂^{∗}KP̂ = φ̂^{∗}A · φ̂^{∗}KP̂ = 0,
E′_{1} · φ̂^{∗}KP̂ = \frac{1}{2}(φ̂^{∗}A_{1} · φ̂^{∗}KP̂) = 0,
⋮
E′_{t} · φ̂^{∗}KP̂ = \frac{1}{2}(φ̂^{∗}A_{t} · φ̂^{∗}KP̂) = 0.

Together with the above equation we get:

Ĥ · KP̂ = \frac{1}{2}(φ̂^{∗}(KP̂ + δ) · φ̂^{∗}(KP̂)) = KP̂(δ + KP̂) = −3 − \frac{t}{2}.

Therefore:

h^{1}(P̂, OP̂(Ĥ)) = \frac{3}{2} − \frac{3}{2} − \frac{t}{4} = \frac{−t}{4}

and we see that t has to be equal to zero. q.e.d

Remark 3.17 1) Moreover, we see from the above proof that H^{1}(P̂, Ĥ) = 0 and KP̂ · Ĥ = −3, KP̂ · (KP̂ + δ) = −3.

2) Since σ : S̃ ⟶ S̃ doesn’t have isolated fixed points, Y := S̃/σ is smooth and the branch divisor Δ on Y is smooth and reduced.

We consider the commutative diagram:

S̃  ^{φH̃  \overline{}}F_{3} ⊂ ℙ^{3}

[figure: φ; φ_{H_{Y}}]
[figure: π; S̃/σ =: Y; ψ]

S  [figure: η]
P,

where η : Y ⟶ P is a birational morphism onto a smooth surface P and ψ : P ⟶ F_{3} ⊂ ℙ^{3} does not contract any exceptional curve of the first kind of P.

Remark 3.18 P is a minimal desingularization of F_{3}. Hence P ≅ ℙ̂^{2}(x_{1}, …, x_{6}) and ψ = φ_{|H_{P} |} with |H_{P} | = |3L − ∑_{i=1}^{6} E_{i}|, where E_{1}, …,E_{6} are the exceptional divisors of the blow up π′ : ℙ̂^{2}(x_{1}, …, x_{6}) ⟶ ℙ^{2}. Moreover, we can choose the points x_{1}, …,x_{6} such that φH̃(E) = E_{6}.

We consider now the morphism φ̃ : S̃ ⟶ P, φ̃ = η ∘ φ, and we will now calculate the branch divisor of this (generic) double cover on P. Let W be a smooth surface and let B be a divisor on W which does not have multiple components and such that there is a line bundle L = O(δ) on W with 2L = O(B). Let f : S′ ⟶ W be the double cover of W branched along B. Then S′ is non singular if and only if B is. Moreover by [HorQ], lemma 5, S′ has at most rational double points if and only if B has no singular point of multiplicity greater than 3 and any triple point s of B decomposes into singularities of multiplicity smaller or equal to two on the proper transform of B after one blow up with center s. A singularity s of B is called negligible (or non essential) if and only if it gives rise to at most a rational double point on S′, otherwise s is called an essential singularity. Let b ∈ B be a singular point of B and let q : W_{1} ⟶ W be the blow up of W in b. If m is the multiplicity of b, we set B_{1} := q^{∗}B−2[\frac{m}{2}]E and L_{1} := q^{∗}L−[\frac{m}{2}]O(E), where E := q^{−1}(b). Since O(B_{1}) = 2L_{1} we can consider the double cover f_{1} : S′_{1} ⟶ W_{1} branched along B_{1}. If B_{1} has a singular point we proceed in the same way and after a finite number of steps we obtain a non singular model S^{∗} od S′, which is called the canonical resolution of S′. We remark that in general S^{∗} is different from the minimal resolution of S′, but if B has only negligible singularities then the canonical resolution of S′ equals the minimal resolution of S′ (cf. [HorQ], lemma 5). In the paper quoted above Horikawa gives formulas for the selfintersection of the canonical divisor of S^{∗} and for the geometric genus. We are going to state them and apply them right away. We recall that two minimal smooth surfaces of general type are deformations of each other, if and only if their canonical models are (cf. e.g. [Cat3], paragraph 19). Hence in the future we will not care about negligible singularities of the branch locus of the double cover or equivalently about the rational double points they will cause on S′ (since from the point of view of deformation theory there is no difference whether we consider a surface of general type with at most rational double points or a minimal resolution of singularities of it). Anyhow, as we will show in section 5, if |Δ| denotes the linear system of branch divisors of the families of double covers in our situations, then a general element D ∈ |Δ| is smooth.

Using the same notation as Horikawa we have by lemma 6 in [HorQ]:

(∗) 5 = χ(S^{∗}, O_{S^{∗}}) = χ(OS̃) = \frac{1}{2}δ(K_{P} + δ) + 2 − \frac{1}{2} ∑[\frac{m_{i}}{2} ]([\frac{m_{i}}{2} ] − 1),

and (if we denote the ”half multiplicities” [\frac{m_{i}}{2} ] by x_{i}):

(∗∗) c_{1}(S̃)^{2} = 6 ≥ c_{1}(S^{∗})^{2} = 2(K_{P} + δ)^{2} − 2 ∑(x_{i} − 1)^{2}.

Since

(i) |K_{P} | = | − 3L + ∑_{i=1}^{6} E_{i}| = | − H_{P} |,
(ii) φ̃^{∗}(| − K_{P} | + E_{6}) = |H| + 2E = |KS̃| = φ^{∗}(K_{Y} + δ_{Y}) =
= φ̃^{∗}(η^{∗}K_{P} + ∑_{i=1}^{m} E_{i} + δ_{Y}),

we obtain that

δ_{Y} ≡ −2η^{∗}K_{P} + η^{∗}E_{6} − ∑_{i=1}^{m} E_{i}.

Therefore for the branch divisor δ on P we have:

δ ≡ −2K_{P} + E_{6}.

Using now (∗) and (∗∗) we obtain:

\begin{matrix}5 = \frac{1}{2}(6L − ∑_{i=1}^{5} 2E_{i} − E_{6})(3L − ∑_{i=1}^{5} E_{i}) + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1) \\ = \frac{8}{2} + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1) \end{matrix}

and this implies:

(∗′) ∑ x_{i}(x_{i} − 1) = 2.

\begin{matrix}c_{1}(S̃)^{2} = 6 ≥ c_{1}(S^{∗})^{2} = 2(3L − ∑_{i=1}^{5} E_{i})^{2} − 2 ∑(x_{i} − 1)^{2} \\ = 2 · 4 − 2 ∑(x_{i} − 1)^{2} \end{matrix}

and therefore

(∗∗′) ∑(x_{i} − 1)^{2} ≥ 1.

Putting (∗′), (∗∗′) together we see that the only possibility is x_{1} = 2, i.e. Δ = 2δ has exactly one essential singularity. In this case S^{∗} = S̃. Moreover, since S^{∗} = S̃, we see that Δ = 2δ has to have a quadruple point as only singularity (for a (3, 3)

- point would imply that S^{∗} = ˆ S̃(y)). Furthermore, since φ(E)^{2} = −2 on Y, but φ̃(E)^{2} = E_{6}^{2} = −1 on P, we conclude that the quadruple point of Δ has to lie on E_{6}.

Concluding we see that Y = P̂(x_{7}) = ℙ̂^{2}(x_{1}, …, x_{6}, x_{7}), where x_{7} is infinitely near to x_{6}. Moreover the branch locus on Y equals:

\begin{matrix}|Δ_{Y} | = | − 4η^{∗}K_{P} + 2η^{∗}E_{6} − 4E_{7}| = |12L − ∑_{i=1}^{5} 4E_{i} − 2E′_{6} − 6E_{7}| \\ = | − 4K_{Y} + 2E′_{6} + 2E_{7}| = E′_{6} + |12L − ∑_{i=1}^{5} 4E_{i} − 3E′_{6} − 6E_{7}|. \end{matrix}

Vice versa, let x_{1}, …, x_{6} be points in ℙ^{2} such that P := ℙ̂^{2}(x_{1}, …, x_{6}) is mapped to a normal cubic surface in ℙ^{3} by the linear system |H_{P} | = |3L − ∑_{i=1}^{6} x_{i}|. Moreover, let x_{7} ∈ E_{6} and let Y := P̂(x_{7}) be the blow up of P := ℙ̂^{2}(x_{1}, …, x_{6}) in x_{7}, E_{i} denotes the inverse image of x_{i} in Y. We write again E_{6} = E′_{6} + E_{7}. Let be |Δ| = |12L − ∑_{i=1}^{5} 4E_{i} − 2E′_{6} − 6E_{7}| and let D = D′ + E′_{6} ∈ |Δ| be smooth (resp. with only negligible singularities)^{1}. We take the double cover φ : Z ⟶ Y branched along Δ. Then Z is a smooth surface (resp. a surface with at most rational double points) with

K_{Z}^{2} = (φ^{∗}(K_{Y} + δ)) =
(φ^{∗}(−3L + ∑_{i=1}^{5} E_{i} + E′_{6} + 2E_{7} + 6L − ∑_{i=1}^{5} 2E_{i} − E′_{6} − 3E_{7}))^{2} =
(φ^{∗}(3L − ∑_{i=1}^{5} E_{i} − E_{7}))^{2} = 2 · 3 = 6.

Moreover,

|K_{Z}| = |φ^{∗}(3L − ∑_{i=1}^{5} E_{i} − E_{7})| = φ^{∗}(E′_{6}) + φ^{∗}(|3L − ∑_{i=1}^{6} E_{i}|),

hence h^{0}(O_{Z}(K_{Z})) = h^{0}(O_{P} (3L − ∑_{i=1}^{6} E_{i})) = 4. Furthermore, if E := φ^{−1}(E′_{6}), then φ^{∗}E′_{6} = 2E and E^{2} = −1. Therefore blowing down E we get a minimal surface S with K_{S}^{2} = 7, p_{g}(S) = 4 and such that |K_{S}| has one simple base point. Summing up all these results we have proven the following theorem:

Theorem 3.19 Let S be a minimal surface with K_{S}^{2} = 7 and p_{g}(S) = 4 such that |K_{S}| has exactly one simple base point x ∈ S and φ_{|K_{S}|} : S − − ⟶ F_{3} ⊂ ℙ^{3} is a map of degree two. Furthermore, let π : S̃ ⟶ S be the blow up of S in x and let |H̃| = |KS̃ − 2E| be the movable part of |π^{∗}K_{S}|. Then φ_{|H̃|} : S̃ ⟶ F_{3} factors as φ_{|H̃|} = η ∘ φ, where η : Y ⟶ ℙ̂^{2}(x_{1}, …, x_{6}) is the blow up of ℙ̂^{2}(x_{1}, …, x_{6}) in a point x_{7} ∈ E_{6} (- we assume here w.r. that φH̃(E) = E_{6} -) and φ : S̃ ⟶ Y is the double cover branched along a divisor D′ + E′_{6} ∈ |12L − 4 ∑_{i=1}^{5} E_{i} − 2E′_{6} − 6E_{7}| having at most negligible singularities.

1The existence of a smooth divisor D ∈ |12L − ∑_{i=1}^{5} 4E_{i} − 2E′_{6} − 6E_{7}| will be shown in section 5.2. In fact, this result will come out as a byproduct, when we calculate the number of moduli of these surfaces.

Corollary 3.20 All surfaces of type (I.2) are birationally equivalent to a double cover of ℙ̂^{2}(x_{1}, …, x_{6}, x_{7}), where x_{7} is infinitely near to x_{6} (and x_{1}, …, x_{6} ∈ ℙ^{2} are such that |3L− ∑_{i=1}^{6} x_{i}| gives a morphism from ℙ̂^{2}(x_{1}, …, x_{6}) to a normal cubic surface in ℙ^{3}), branched along D′ +E′_{6} ∈ |Δ| = |12L−4 ∑_{i=1}^{5} E_{i} −2E′_{6} −6E_{7}|, where D′ has at most negligible singularities.

3.3 Surfaces of type (I.3)

In this section we will only very briefly state the result of F. Zucconi (cf. [Zuc]). We consider the 3 - fold ℙ = ℙ(O_{ℙ^{1}}(1) ⊕ O_{ℙ^{1}}(2) ⊕ O_{ℙ^{1}}(4)). Let T be a tautological divisor on ℙ, Π a fibre of the natural projection π : ℙ ⟶ ℙ^{1} and let X_{0} ∈ H^{0}(ℙ, O_{ℙ}(T − Π)), X_{1} ∈ H^{0}(ℙ, O_{ℙ}(T − 2Π)), X_{2} ∈ H^{0}(ℙ, O_{ℙ}(T − 4Π)) be sections which give a projective coordinate system on Π. Moreover, let {t_{0}, t_{1}} be a basis of H^{0}(ℙ, O_{ℙ}(Π)) and let y be a fibre coordinate of the line bundle O(2T − 6Π) on ℙ. Then we have:

Theorem 3.21 (F. Zucconi) S is a minimal surface with K_{S}^{2} = 7, p_{g}(S) = 4 and degφ_{|K_{S}|} = 3 if and only if there exists a linear subsystem |F | of |K_{S}| which is a rational pencil of non - hyperelliptic curves of genus 3 with a simple base point P′ and such that the relative canonical model of the fibration induced on the blow up S′ of S in P′ is the complete intersection in the total space of |2T −6Π| of the following two hypersurfaces:

\begin{matrix}t_{0}y = X_{0}X_{2}, \\ αy^{2} + Qy + c_{1}X_{1}^{4} + X_{2}P = 0; \end{matrix}

where α ∈ H^{0}(ℙ, O_{ℙ}(4Π)), α|{t_{0} = 0} ≠ 0, c_{1} ∈ ℂ\{0}. Moreover, Q ∈ |2T − 2Π| and Q = c_{0}X_{0}^{2} + α_{1}X_{0}X_{1} + α_{2}X_{1}^{2} + α_{4}X_{1}X_{2} + α_{6}X_{2}^{2}, where c_{0} ∈ ℂ\{0}, α_{i} ∈ H^{0}(ℙ, O_{ℙ}(iΠ)), P ∈ |3T − 4Π|, P = β_{1}X_{1}^{3} + β_{2}X_{1}^{2}X_{2} + β_{3}X_{1}X_{2}^{2} + β_{4}X_{2}^{3}, where β_{i} ∈ H^{0}(ℙ, O_{ℙ}(2iΠ)).

Corollary 3.22 The moduli space of surfaces of type (I.3) is irreducible and unirational of dimension 35.

For more details and for the proofs of the above results we refer to the original paper [Zuc].

3.4 Surfaces of type (III)

In this chapter we will study the surfaces of type (III), i.e. those which have a fixed part free canonical system with three base points. First of all we will consider the case where these three base points are pairwise different. Afterwards we will concentrate on the case of infinitely near base points of the canonical system and will indeed show that these surfaces do not exist. Let S be a minimal surface with K_{S}^{2} = 7 and p_{g}(S) = 4 such that the canonical system |K_{S}| has three base points. First we shall assume that the three base points x_{1}, x_{2}, x_{3} ∈ S are pairwise distinct, i.e. there are no infinitely near base points of |K_{S}|. Let π : S̃ ⟶ S be the blow up of S in x_{1}, x_{2}, x_{3} and let E_{i} := π^{−1}(x_{i}), 1 ≤ i ≤ 3, be the exceptional curves of π. Moreover, let |H̃| be the movable part of |π^{∗}K_{S}|, in particular KS̃ ≡ H̃ + 2E_{1} + 2E_{2} + 2E_{3}. Then φ_{|H̃|} : S̃ ⟶ Q ⊂ ℙ^{3} is a morphism (generically) of degree two from S̃ onto a quadric in ℙ^{3}.

Remark 3.23 1) The canonical map φ_{|KS̃|} induces a biregular involution σ on S. It is obvious that the set of base points of the canonical system of S is left invariant under σ. In principle, if there are three pairwise different base points of |K_{S}|, it could be possible that two of the three base points are interchanged by the involution σ and the third is left fixed. It is not difficult to exclude this case and we will postpone this argument until the end of section (3.4.3). Therefore we will assume in the following that σ(x_{i}) = x_{i} for all 1 ≤ i ≤ 3.

2) Let Q ⊂ ℙ^{3} be an irreducible quadric. Then either Q = ℙ^{1} × ℙ^{1} embedded in ℙ^{3} by the linear system |H_{Q}| = |Γ_{1} + Γ_{2}|, where Γ_{1}^{2} = Γ_{2}^{2} = 0 and Γ_{1} · Γ_{2} = 1 or Q is the quadric cone in ℙ^{3}, i.e. Q is the image of 𝔽_{2} = ℙ(O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(−2)) under the morphism given by the linear system |H_{𝔽_{2}}| = |2Γ + Γ_{∞}|, where Γ_{∞}^{2} = −2, Γ^{2} = 0 and Γ · Γ_{∞} = 1.

First we treat the case

3.4.1 Q = ℙ^{1} × ℙ^{1}

Since |H_{Q}| = |Γ_{1} + Γ_{2}| with Γ_{1}^{2} = Γ_{2}^{2} = 0, Γ_{1} · Γ_{2} = 1 and since |H̃| = φ_{|H̃|}^{∗}(|H_{Q}|) we can write |H̃| = |D_{1} + D_{2}| with D_{1}^{2} = D_{2}^{2} = 0 and D_{1} · D_{2} = 2.

Remark 3.24 Since H̃ · E_{i} = 1, for i = 1, 2, 3, we see that φ_{|H̃|} maps E_{i} isomorphically onto a line L_{i} on Q, i.e. φ_{|H̃|}|E_{i} : E_{i} ⟶ L_{i} is an isomorphism.

We have for j ∈ {1, 2}:

D_{j} · KS̃ = D_{j}(H + 2 ∑_{i=1}^{3} E_{i}) = 2 + 2 ∑_{i=1}^{3}(D_{j} · E_{i}).

Since H ·E_{i} = (D_{1} +D_{2})·E_{i} = 1 and D_{j} ·E_{i} ≥ 0, it follows that for each i ∈ {1, 2, 3}

D_{1} · E_{i} = 1 and D_{2} · E_{i} = 0, or
D_{1} · E_{i} = 0 and D_{2} · E_{i} = 1.

Therefore we have the following result:

Lemma 3.25 In the above situation we have (up to a permutation of D_{1} and D_{2}) the following two possibilities:

\begin{matrix}I) & D_{1} · KS̃ = 8, & D_{2} · KS̃ = 2, \\ II) & D_{1} · KS̃ = 6, & D_{2} · KS̃ = 4. \end{matrix}

Using this information we can calculate the bidegree of the branch locus Δ ⊂ ℙ^{1}×ℙ^{1} of φ_{|H̃|}. In fact, since KS̃ = φ∗ ; H̃(K_{Q} + δ), we get in case I):

2(δ · Γ_{1}) = φ∗ ; H̃δ · D_{1} = KS̃ · D_{1} − φ∗ ; H̃K_{Q} · D_{1} = 8 − 2 · (−2) = 12,

and

2(δ · Γ_{2}) = φ∗ ; H̃δ · D_{2} = KS̃ · D_{2} − φ∗ ; H̃K_{Q} · D_{2} = 2 + 4 = 6.

Therefore δ has bidegree (3, 6) on ℙ^{1} × ℙ^{1}, hence

Δ ≡ 2δ ∈ |6Γ_{1} + 12Γ_{2}|.

In the same way, we see in case II)

Δ ≡ 2δ ∈ |8Γ_{1} + 10Γ_{2}|.

Remark 3.26 In case I) the exceptional curves E_{1}, E_{2}, E_{3} are mapped to three pairwise disjoint lines L_{1}, L_{2}, L_{3} in Q, i.e. L_{1}, L_{2}, L_{3} ∈ |Γ_{2}|, while in case II) we have (up to permutation of x_{1}, x_{2}, x_{3}) φH̃(E_{1}) = L_{1} ∈ |Γ_{2}|, φH̃(E_{2}) = L_{2} ∈ |Γ_{2}| and φH̃(E_{3}) = L_{3} ∈ |Γ_{1}|.

CASE I). We will now determine the singular points of the branch curve Δ on ℙ^{1} × ℙ^{1}. By [HorQ], lemma 6, we have the following numerical conditions:

\begin{matrix}5 = χ(S^{∗}, O_{S^{∗}}) = χ(OS̃) = \frac{1}{2}δ(K_{Q} + δ) + 2 − \frac{1}{2} ∑[\frac{m_{i}}{2} ]([\frac{m_{i}}{2} ] − 1) = \\ \frac{1}{2} · (3, 6) · (1, 4) + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1) \end{matrix}

and this implies, if we again denote [\frac{m_{i}}{2} ] by x_{i}:

(∗) ∑ x_{i}(x_{i} − 1) = 12.

On the other hand:

\begin{matrix}c_{1}(S̃)^{2} = 4 ≥ c_{1}(S^{∗})^{2} = 2(K_{Q} + δ)^{2} − 2 ∑(x_{i} − 1)^{2} = \\ 2(1, 4)^{2} − 2 ∑(x_{i} − 1)^{2}, \end{matrix}

Hence:

(∗∗) ∑(x_{i} − 1)^{2} ≥ 6.

Therefore we obtain the following numerical possibilities for the x_{i}’s from (∗) and (∗∗):

\begin{matrix}(I.1) & x_{1} = x_{2} = … = x_{6} = 2. \\ (I.2) & x_{1} = 3, x_{2} = x_{3} = x_{4} = 2. \\ (I.3) & x_{1} = 3, x_{2} = 3. \\ (I.4) & x_{1} = 4. \end{matrix}

Remark 3.27 Let σ̃ : S̃ ⟶ S̃ be the involution, induced by φH̃. Then σ̃|E_{i} : E_{i} ⟶ E_{i} is the identity (since the base points of |K_{S}| are fixed points of the involution σ on S (cf. (3.23))). Therefore the image of E_{i} in Y := S̃/σ̃ is a line of self - intersection (−2). Hence there must be exactly two (since φH̃(E_{i})^{2} = 0 on Q) essential singularities of the branch curve |Δ| on each of the L_{1}, L_{2}, L_{3}. This implies that we can exclude the cases (I.2) - (I.4).

Therefore it remains to treat the case (I.1). We first remark that in this situation S^{∗} = S̃, hence there cannot be (3, 3) - points, since then S^{∗} would be different from S̃. Therefore it follows that the branch curve Δ has to have six (simple) quadruple points. Moreover, on each L_{i} there have to be exactly two quadruple points. Therefore we have shown the following result:

Theorem 3.28 Let S be a minimal surface with K_{S}^{2} = 7, p_{g}(S) = 4 such that |K_{S}| has exactly three pairwise different base points x_{1}, x_{2}, x_{3}. Moreover, we suppose that

(i) φH̃(S̃) = ℙ^{1} × ℙ^{1} =: Q,
(ii) φH̃(E_{1}), φH̃(E_{2}), φH̃(E_{3}) are pairwise disjoint lines.

Then φH̃ = p ∘ φ, where p : Q̂(y_{1}, …, y_{6}) ⟶ Q is the blow up of Q = ℙ^{1} × ℙ^{1} in y_{1}, …, y_{6} such that y_{1}, y_{2} are contained in a fibre ∈ |Γ_{2}|, as well as y_{3}, y_{4}, and y_{5}, y_{6} in different fibres (∈ |Γ_{2}|). Finally φ : S̃ ⟶ Q̂(y_{1}, …, y_{6}) is the double cover of Q̂(y_{1}, …, y_{6}) branched along

D = D̃ + L_{1} + L_{2} + L_{3},

where L_{1} ≡ Γ_{2} − E_{y_{1}} − E_{y_{2}}, L_{2} ≡ Γ_{2} − E_{y_{3}} − E_{y_{4}}, L_{3} ≡ Γ_{2} − E_{y_{5}} − E_{y_{6}} and D̃ ∈ |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{y_{i}}| has at most non essential singularities ^{1}.

1The existence of these surfaces will be shown in section 5.2.

Vice versa, let L_{1}, L_{2}, L_{3} ≡ Γ_{2} be three pairwise disjoint lines in Q = ℙ^{1} × ℙ^{1} and let y_{1}, y_{2} ∈ L_{1}, y_{3}, y_{4} ∈ L_{2} and y_{5}, y_{6} ∈ L_{3} be (pairwise distinct) points. On Q̂(y_{1}, …, y_{6}) we consider the linear system

|Δ| = |6Γ_{1} + 12Γ_{2} − 4 ∑_{i=1}^{6} E_{y_{i}}| = L_{1} + L_{2} + L_{3} + |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{y_{i}}|.

Let D̃ ∈ |6Γ_{1} +9Γ_{2} −3 ∑_{i=1}^{6} E_{y_{i}}| be smooth (resp. with only negligible singularities). We take the double cover φ : Z ⟶ Q̂(y_{1}, …, y_{6}) branched along D̃ + L_{1} + L_{2} + L_{3}. Then Z is a smooth surface (resp. a surface with at most rational double points) with

K_{Z}^{2} = φ^{∗}(K_{Q̂(y_{1},... ,y_{6})} + δ)^{2} = φ^{∗}(Γ_{1} + 4Γ_{2} − ∑_{i=1}^{6} E_{y_{i}})^{2} = 4

and

|K_{Z}| = φ^{∗}(|Γ_{1} + Γ_{2}|) + φ^{∗}(Γ_{2} − E_{y_{1}} − E_{y_{2}}) + φ^{∗}(Γ_{2} − E_{y_{3}} − E_{y_{4}}) + φ^{∗}(Γ_{2} − E_{y_{5}} − E_{y_{6}}).

Therefore p_{g}(Z) = h^{0}(O_{ℙ^{1}×ℙ^{1}}(Γ_{1} + Γ_{2})) = 4 and Z has three pairwise disjoint exceptional curves of the first kind. Blowing them down we obtain a minimal surface S with K_{S}^{2} = 7, p_{g}(S) = 4 such that the canonical system |K_{S}| has three pairwise distinct base points.

Corollary 3.29 All minimal surfaces of type (III) having three pairwise distinct base points of |K_{S}| such that

(i) φH̃(S̃) = ℙ^{1} × ℙ^{1},
(ii) φH̃(E_{1}), φH̃(E_{2}), φH̃(E_{3})are pairwise disjoint lines.

are obtained by blowing down the three exceptional curves of the double cover of Q̂(y_{1}, …, y_{6}) branched along D ∈ |Δ| = |6Γ_{1} + 12Γ_{2} − 4 ∑_{i=1}^{6} E_{y_{i}}|, where y_{1}, …, y_{6} are six points such that y_{1}, y_{2} ∈ L_{1}, and y_{3}, y_{4} ∈ L_{2} and y_{5}, y_{6} ∈ L_{3} and L_{1}, L_{2}, L_{3} ≡ Γ_{2} are three pairwise different fibres of ℙ^{1} × ℙ^{1}.

CASE II). As in the previous case we have the following numerical conditions:

\begin{matrix}5 = χ(S^{∗}, O_{S^{∗}}) = χ(OS̃) = \frac{1}{2}δ(K_{Q} + δ) + 2 − \frac{1}{2} ∑[\frac{m_{i}}{2} ]([\frac{m_{i}}{2} ] − 1) = \\ \frac{1}{2} · (4, 5) · (2, 3) + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1) \end{matrix}

and this implies, if we again denote [\frac{m_{i}}{2} ] by x_{i}:

(∗) ∑ x_{i}(x_{i} − 1) = 16.

On the other hand:

\begin{matrix}c_{1}(S̃)^{2} = 4 ≥ c_{1}(S^{∗})^{2} = 2(K_{Q} + δ)^{2} − 2 ∑(x_{i} − 1)^{2} = \\ 2(2, 3)^{2} − 2 ∑(x_{i} − 1)^{2}, \end{matrix}

Hence:

(∗∗) ∑(x_{i} − 1)^{2} ≥ 10.

Therefore we obtain the following numerical possibilities for the x_{i}’s from (∗) and (∗∗):

\begin{matrix}(II.1) & x_{1} = x_{2} = 3, x_{3} = x_{4} = 2. \\ (II.2) & x_{1} = 4, x_{2} = x_{3} = 2. \end{matrix}

Remark 3.30 As in the previous case we conclude that on each of the images of E_{1}, E_{2}, E_{3} on Q there have to be exactly two essential singularities of Δ. But since (up to permutation of the indices) we have φH̃(E_{1}) = L_{1} ∈ |Γ_{2}|, φH̃(E_{2}) = L_{2} ∈ |Γ_{2}| and φH̃(E_{3}) = L_{3} ∈ |Γ_{1}|, we see that Δ has to have at least four essential singularities and we can exclude case (II.2). More precisely, since in case (II.1) there are exactly four essential singularities, the singular points y_{1}, y_{2}, y_{3}, y_{4} of Δ have to fulfill the following conditions:

{y_{1}} = φH̃(E_{1}) ∩ φH̃(E_{3}),
{y_{2}} = φH̃(E_{2}) ∩ φH̃(E_{3}),
y_{3} ∈ φH̃(E_{1}), y_{4} ∈ φH̃(E_{2}).

Since in case (II.1) we have S^{∗} = S̃, we see that y_{1}, y_{2} are simple sextuple points and y_{3}, y_{4} are simple quadruple points. Summing up we see that we have obtained the following result:

Theorem 3.31 Let S be a minimal surface with K_{S}^{2} = 7, p_{g}(S) = 4, such that the canonical system |K_{S}| of S has exactly three pairwise different (simple) base points x_{1}, x_{2}, x_{3}. Moreover, we assume that

\begin{matrix}(i) & φH̃(S̃) = ℙ^{1} × ℙ^{1} =: Q, \\ (ii) & φH̃(E_{1}), φH̃(E_{2}) ≡ Γ_{2}, φH̃(E_{3}) ≡ Γ_{1}. \end{matrix}

Then φH̃ = p ∘ φ, where p : Q̂(y_{1}, …, y_{4}) ⟶ Q is the blow up of Q = ℙ^{1} × ℙ^{1} in y_{1}, …, y_{4} such that {y_{1}} = φH̃(E_{1})∩φH̃(E_{3}), {y_{2}} = φH̃(E_{2})∩φH̃(E_{3}), y_{3} ∈ φH̃(E_{1}) and y_{4} ∈ φH̃(E_{2}). Finally φ : S̃ ⟶ Q̂(y_{1}, …, y_{4}), where is the double cover branched along

D = D̃ + L_{1} + L_{2} + L_{3},

where L_{1} ≡ Γ_{2} − E_{y_{1}} − E_{y_{3}}, L_{2} ≡ Γ_{2} − E_{y_{2}} − E_{y_{4}}, L_{3} ≡ Γ_{1} − E_{y_{1}} − E_{y_{2}} and D̃ ∈ |7Γ_{1} + 8Γ_{2} − 4E_{y_{1}} − 4E_{y_{2}} − 3E_{y_{3}} − 3E_{y_{4}}| has at most negligible singularities.

Vice versa, let L_{1}, L_{2} ∈ |Γ_{2}| be two different fibres and suppose that L_{3} ∈ |Γ_{1}|. We consider on Q̂(y_{1}, …, y_{4}) the linear system

|Δ| = |8Γ_{1} + 10Γ_{2} − 6E_{y_{1}} − 6E_{y_{2}} − 4E_{y_{3}} − 4E_{y_{4}}|
= |7Γ_{1} + 8Γ_{2} − 4E_{y_{1}} − 4E_{y_{2}} − 3E_{y_{3}} − 3E_{y_{4}}|
+(Γ_{1} − E_{y_{1}} − E_{y_{2}}) + (Γ_{2} − E_{y_{1}} − E_{y_{3}}) + (Γ_{2} − E_{y_{2}} − E_{y_{4}}).

Let D̃ ∈ |7Γ_{1} + 8Γ_{2} − 4E_{y_{1}} − 4E_{y_{2}} − 3E_{y_{3}} − 3E_{y_{4}}| be smooth (up to negligible singularities) and let φ : Z ⟶ Q̂(y_{1}, …, y_{4}) be the double cover of Q̂(y_{1}, …, y_{4}) branched along

D̃ + (Γ_{2} − E_{y_{1}} − E_{y_{3}}) + (Γ_{2} − E_{y_{2}} − E_{y_{4}}) + (Γ_{1} − E_{y_{1}} − E_{y_{2}}).

Then Z is a smooth surface (up to rational double points) with

\begin{matrix}|K_{Z}| = φ^{∗}(|K_{Q̂(y_{1},... ,y_{6})} + δ|) = φ^{∗}(|2Γ_{1} + 3Γ_{2} − 2E_{y_{1}} − 2E_{y_{2}} − E_{y_{3}} − E_{y_{3}}|) = \\ φ^{∗}(|Γ_{1} + Γ_{2}|) + φ^{∗}(Γ_{2} − E_{y_{1}} − E_{y_{3}}) + φ^{∗}(Γ_{2} − E_{y_{2}} − E_{y_{4}}) + φ^{∗}(Γ_{1} − E_{y_{1}} − E_{y_{2}}). \end{matrix}

In particular, we see that K_{Z}^{2} = 2(2Γ_{1} + 3Γ_{2} − 2E_{y_{1}} − 2E_{y_{2}} − E_{y_{3}} − E_{y_{4}})^{2} = 4 and p_{g}(Z) = 4. Moreover Z has three pairwise disjoint exceptional curves E_{1}, E_{2}, E_{3} of the first kind. Blowing them down we obtain a minimal surface S with K_{S}^{2} = 7, p_{g}(S) = 4 such that the canonical system |K_{S}| has three pairwise distinct base points fulfilling the following two conditions:

\begin{matrix}(i) & φH̃(S̃) = ℙ^{1} × ℙ^{1}, \\ (ii) & φH̃(E_{1}), φH̃(E_{2}) ≡ Γ_{2}, φH̃(E_{3}) ≡ Γ_{1}. \end{matrix}

We finally remark that it is easy to see that the points x_{1}, …, x_{4} have to be pairwise different.

3.4.2 Q is the quadric cone in ℙ^{3}

In this section we consider the following situation: Let S be a minimal surface with K_{S}^{2} = 7 and p_{g}(S) = 4 such that the canonical system |K_{S}| has three pairwise distinct base points. Let π : S̃ ⟶ S be the blow up of S in x_{1}, x_{2}, x_{3} and let E_{i} := π^{−1}(x_{i}), 1 ≤ i ≤ 3, be the exceptional curves of π. Moreover, let |H̃| be the movable part of |π^{∗}K_{S}|, in particular KS̃ ≡ H̃ + 2E_{1} + 2E_{2} + 2E_{3}. We assume furthermore that φ_{|H̃|} : S̃ ⟶ Q ⊂ ℙ^{3} is a morphism (generically) of degree two from S̃ onto the quadric cone Q in ℙ^{3}. We will first show that in this case φ_{|H̃|} factors through the Hirzebruch surface 𝔽_{2}.

Proposition 3.32 Let |H̃| be the movable part of |π^{∗}K_{S}| and assume that φ_{|H̃|} : S̃ ⟶ Q ⊂ ℙ^{3} is a morphism of degree two from S̃ onto the quadric cone Q in ℙ^{3}. Then φ_{|H̃|} factors through 𝔽_{2}, i.e. there is a morphism φ̃ : S̃ ⟶ 𝔽_{2} such that the diagram

S̃  ^{^{φH̃}  \overline{}}Q ⊂ ℙ^{3}

[figure: φ̃; |2Γ+Γ_{∞}|]
𝔽_{2}

is commutative.

Remark 3.33 In the above proposition we do not need the hypothesis that the base points x_{1}, x_{2}, x_{3} of the canonical system of S are pairwise distinct, i.e. the above claim holds also if the base points of K_{S} are infinitely near.

Proof. We have KS̃ = H̃ + 2E, where E is the exceptional divisor of the blow up π : S̃ ⟶ S of the base points of |K_{S}|. Since φ_{|H̃|} : S̃ ⟶ Q, where Q is the quadric cone in ℙ^{3}, we can write:

H̃ ≡ 2D + G, where H̃ · G = 0.

This implies that G^{2} < 0 or G = 0 (i.e., if v is the vertex of the quadric cone Q, then φ−1 ; H̃(v) is a point). On the other hand:

4 = H̃^{2} = H̃(2D + G) = 2D · H̃,

whence D · H̃ = 2. But D · H̃ = 2D^{2} + D · G, and D^{2} ≥ 0, D · G ≥ 0, therefore we have the following two possibilities:

\begin{matrix}(i) & D^{2} = 1, D · G = 0, or \\ (ii) & D^{2} = 0, D · G = 2. \end{matrix}

But D · KS̃ = D · H̃ + 2E · D = 2 + 2E · D ≡ 0(mod2), hence by the adjunction formula also D^{2} ≡ 0(mod2) and this excludes case (i). Therefore G has to be a non trivial divisor and we have proven the proposition. q.e.d

We have D · KS̃ = D · H̃ + 2 ∑_{i=1}^{3} D · E_{i} = 2 + 2 ∑_{i=1}^{3} D · E_{i}. But (2D + G) · E_{i} = H̃ · E_{i} = 1 and therefore D · E_{i} = 0, G · E_{i} = 1, which implies that D · KS̃ = 2. Moreover, G·KS̃ = (H̃ +2 ∑_{i=1}^{3} E_{i})·G = 6. Then φ̃^{∗}δ ·D = KS̃ ·D−φ̃^{∗}K_{𝔽_{2}} ·D = 2− φ^{∗}(−4Γ−2Γ_{∞})·D = 2+2·2 = 6 and φ̃^{∗}δ·G = KS̃·G−φ̃^{∗}K_{𝔽_{2}}·G = 6−2·(−4+4) = 6. Therefore δ · Γ = 3 and δ · Γ_{∞} = 3 and we conclude that δ ≡ 9Γ + 3Γ_{∞}, whence Δ ≡ 18Γ + 6Γ_{∞}. (We recall that Γ, Γ_{∞} ∈ Div(𝔽_{2}) with Γ^{2} = 0, Γ · Γ_{∞} = 1 and Γ_{∞}^{2} = −2.) We have now the following numerical conditions:

5 = χ(S^{∗}, O_{S^{∗}}) = χ(OS̃) = \frac{1}{2}δ(K_{𝔽_{2}} + δ) + 2 − \frac{1}{2} ∑[\frac{m_{i}}{2} ]([\frac{m_{i}}{2} ] − 1) =
\frac{1}{2}(9Γ + 3Γ_{∞})(5Γ + Γ_{∞}) + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1) =
\frac{1}{2}(9 + 15 − 6) + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1)

and this implies, if we again denote [\frac{m_{i}}{2} ] by x_{i}:

(∗) ∑ x_{i}(x_{i} − 1) = 12.

On the other hand:

\begin{matrix}c_{1}(S̃)^{2} = 4 ≥ c_{1}(S^{∗})^{2} = 2(K_{𝔽_{2}} + δ)^{2} − 2 ∑(x_{i} − 1)^{2} = \\ 2(5Γ + Γ_{∞})^{2} − 2 ∑(x_{i} − 1)^{2}. \end{matrix}

Hence:

(∗∗) ∑(x_{i} − 1)^{2} ≥ 6.

From this again we get the following numerical possibilities for the x_{i}’s:

\begin{matrix}(1) & x_{1} = x_{2} = … = x_{6} = 2. \\ (2) & x_{1} = 3, x_{2} = x_{3} = x_{4} = 2. \\ (3) & x_{1} = 3, x_{2} = 3. \\ (4) & x_{1} = 4. \end{matrix}

Since E_{1}, E_{2}, E_{3} are mapped to three pairwise disjoint lines on 𝔽_{2} we can exclude by the same argument as in (3.4.1) the cases (2) − (4). For case (1) we remark that (since c_{1}(S^{∗})^{2} = 4 = c_{1}(S̃)^{2}) we have S^{∗} = S̃, hence there can only be quadruple points as essential singularities. Again there have to be exactly two quadruple points on each φ̃(E_{i}). Therefore we have shown the following result:

Theorem 3.34 Let S be a minimal surface with K_{S}^{2} = 7 and p_{g}(S) = 4 such that the canonical system |K_{S}| has exactly three pairwise distinct base points. Furthermore, we assume that φH̃(S̃) is the quadric cone in ℙ^{3}. Then (by (3.30)) φH̃ factors through 𝔽_{2} and φ̃ = p ∘ φ, where p : 𝔽̂_{2}(y_{1}, …, y_{6}) ⟶ 𝔽_{2} is the blow up of 𝔽_{2} in y_{1}, …, y_{6}, where y_{1}, y_{2} ∈ φ̃(E_{1}), y_{3}, y_{4} ∈ φ̃(E_{2}) and y_{5}, y_{6} ∈ φ̃(E_{3}). Moreover, φ : S̃ ⟶ 𝔽̂_{2}(y_{1}, …, y_{6}) is the double cover of 𝔽̂_{2}(y_{1}, …, y_{6}) branched along

D = D̃ + L_{1} + L_{2} + L_{3},

where L_{1} ≡ Γ − E_{y_{1}} − E_{y_{2}}, L_{2} ≡ Γ − E_{y_{3}} − E_{y_{4}}, L_{3} ≡ Γ − E_{y_{5}} − E_{y_{6}} and D̃ ∈ |6Γ_{∞} + 15Γ − 3 ∑_{i=1}^{6} E_{y_{i}}| has at most negligible singularities.

Vice versa, let L_{1}, L_{2}, L_{3} ≡ Γ be three pairwise different lines in 𝔽_{2} and let y_{1}, y_{2} ∈ L_{1}, y_{3}, y_{4} ∈ L_{2} and y_{5}, y_{6} ∈ L_{3} be (generic) points. On 𝔽̂_{2}(y_{1}, …, y_{6}) we consider the linear system

|Δ| = |6Γ_{∞} + 18Γ − 4 ∑_{i=1}^{6} E_{y_{i}}| = L_{1} + L_{2} + L_{3} + |6Γ_{∞} + 15Γ − 3 ∑_{i=1}^{6} E_{y_{i}}|.

Let D̃ ∈ |6Γ_{∞} +15Γ−3 ∑_{i=1}^{6} E_{y_{i}}| be smooth and we take the double cover φ : Z ⟶ 𝔽̂_{2}(y_{1}, …, y_{6}) branched along D̃ + L_{1} + L_{2} + L_{3}. Then Z is a smooth surface with

K_{Z}^{2} = φ^{∗}(K_{𝔽̂_{2}(y_{1},... ,y_{6})} + δ)^{2} = φ^{∗}(Γ_{∞} + 5Γ − ∑_{i=1}^{6} E_{y_{i}})^{2} = 4

and

|K_{Z}| = φ^{∗}(|Γ_{∞} + 2Γ|) + φ^{∗}(Γ − E_{y_{1}} − E_{y_{2}}) + φ^{∗}(Γ − E_{y_{3}} − E_{y_{4}}) + φ^{∗}(Γ − E_{y_{5}} − E_{y_{6}}).

Therefore p_{g}(Z) = h^{0}(O_{𝔽_{2}}(Γ_{∞}+Γ)) = 4 and Z has three pairwise disjoint exceptional curves of the first kind. Blowing them down we obtain a minimal surface S with K_{S}^{2} = 7, p_{g}(S) = 4 such that the canonical system |K_{S}| has three pairwise distinct base points and φH̃(S̃) is the quadric cone in ℙ^{3}.

Corollary 3.35 All minimal surfaces of type (III) having three pairwise distinct base points of |K_{S}| such that φH̃(S̃) is the quadric cone in ℙ^{3}, are obtained by blowing down the three exceptional curves of the double cover of 𝔽̂_{2}(y_{1}, …, y_{6}) branched along D ∈ |Δ| = |6Γ_{∞} + 15Γ − 4 ∑_{i=1}^{6} E_{y_{i}}|, where y_{1}, …, y_{6} are six points such that y_{1}, y_{2} ∈ L_{1}, y_{3}, y_{4} ∈ L_{2} and y_{5}, y_{6} ∈ L_{3}, where L_{1}, L_{2}, L_{3} ≡ Γ are three pairwise different fibres of 𝔽_{2}^{1}.

An easy calculation shows that never more than one of the points y_{1}, …, y_{6} lie on the curve Γ_{∞}.

3.4.3 Infinitely near base points

In the rest of the paragraph we will concentrate on the case, where the three base points of |K_{S}| are infinitely near. In fact we will show that there do not exist minimal surfaces S with K_{S}^{2} = 7, p_{g}(S) = 4 such that the canonical system has infinitely near base points.

Remark 3.36 Let π : S̃ ⟶ S be the blow up of the base points x_{1}, x_{2}, x_{3} of |K_{S}|. If x_{1}, x_{2}, x_{3} are not pairwise distinct then we have the following two possibilities for the exceptional divisor E of π: A) E = E_{1} + E_{2} + E_{3}, where E_{1} and E_{3} are irreducible rational (−1) - curves, whereas E_{2} = E′_{2} + E_{3}. E′_{2} is an irreducible rational curve of self - intersection (−2) and E′_{2} · E_{3} = 1. B) E = E_{1} + E_{2} + E_{3}, where E_{3} is an irreducible rational (−1) - curve, whereas E_{1} = E′_{1} + E′_{2} + E_{3} and E_{2} = E′_{2} + E_{3}. E′_{1}, E′_{2} are irreducible rational curves of self - intersection (−2) and E′_{1} · E′_{2} = E′_{2} · E_{3} = 1 and E′_{1} · E_{3} = 0. All the other configurations of exceptional curves giving rise to three base points, which are not pairwise distinct, are excluded by the fact that |K_{S}| has only simple base points.

1The existence of these surfaces will be shown in section 5.2.

CASE A). We remark that the canonical map induces a biregular involution σ on S, since S is minimal. Since there is a base point infinitely near to x_{2}, but not to x_{1}, we conclude that σ(x_{1}) = x_{1} and σ(x_{2}) = x_{2}. Therefore, if σ̃ is the involution induced on the blow up S̃ of S in x_{1}, x_{2}, x_{3}, we see that σ̃|E′_{2} = id_{E′_{2}}, but σ̃|E_{3} ≠ id_{E_{3}}. But this contradicts the fact that H̃ · E_{3} = 1 (where |H̃| is the movable part of |KS̃|), hence φH̃ maps E_{3} isomorphically to a line in Q. Therefore CASE A) is not possible and we are reduced to the analysis of CASE B). CASE B) We suppose now that S is a minimal surface with K_{S}^{2} = 7, p_{g}(S) = 4, such that the base points of |K_{S}| are as in case B) of the above remark. Let σ : S̃ ⟶ S̃ be the (biregular) involution on S̃ induced by φH̃. Then the fixed locus F ix(σ) of σ equals {P_{1}, …, P_{t}} ∪ C_{1} ∪ … ∪ C_{s}, where C_{1}, …, C_{s} are smooth curves contained in S̃ and P_{1}, …, P_{t} ∈ S̃ are the isolated fixed points of σ. Let π̂ : Ŝ ⟶ S̃ be the blow up of S̃ in P_{1}, …, P_{t} and let E′′_{1}, …, E′′_{t} be the exceptional curves of π̂. Furthermore, we denote by σ̂ the induced involution on Ŝ and the quotient Ŝ/σ̂ by P̂. We remark that P̂ is smooth and the branch locus Δ of the double cover φ̂ : Ŝ ⟶ P̂ is smooth and reduced. We have:

Δ ∈ |2δ|,

where

(φ̂)_{∗}OŜ ≅ OP̂(−δ) ⊕ OP̂.

Moreover, we have the following equalities:

5 = χ(S, O_{S}) = χ(S̃, OS̃) = χ(Ŝ, OŜ) =

2(p_{g}(P̂) − q(P̂) + 1) + \frac{1}{2}δ(δ + KP̂);

4 − t = K2 ; Ŝ = 2(KP̂ + δ)^{2}.

This implies that:

\begin{matrix}(∗) & δ(δ + KP̂) = 6; \\ (∗∗) & KP̂(δ + KP̂) = −4 − \frac{t}{2}. \end{matrix}

We have the following commutative diagram:

Ŝ ^{φ̂ \overline{}}P̂

[figure: π̂; φĤ]
S̃ ^{φH̃\overline{}}Q ⊂ ℙ^{3}

π
S  .

Then:

\begin{matrix}(φ̂)^{∗}(Ĥ) = π̂^{∗}(H̃) = π̂^{∗}(KS̃ − 2E) = \\ KŜ − π̂^{∗}(2E) − ∑_{i=1}^{t} E′′_{i} = φ̂^{∗}(KP̂ + δ) − ∑_{i=1}^{t} E′′_{i} − π̂^{∗}(2E). \end{matrix}

We see that:

(φ̂^{∗}(Ĥ))^{2} = (π̂^{∗}(H̃))^{2} = H̃^{2} = 4,

hence:

Ĥ^{2} = \frac{1}{2}(φ̂^{∗}(Ĥ))^{2} = 2.

Since φĤ : P̂ ⟶ Q is the blow up of a quadric in ℙ^{3} any curve C ∈ |Ĥ| is rational, hence

−2 = 2p_{a}(C) − 2 = Ĥ^{2} + Ĥ · KP̂,

and therefore

Ĥ · KP̂ = −4.

On the other hand

\begin{matrix}Ĥ · KP̂ = \frac{1}{2} · φ̂^{∗}Ĥ · φ̂^{∗}KP̂ = \\ \frac{1}{2}(φ̂^{∗}(KP̂ + δ) · (φ̂^{∗}KP̂) − ∑_{i=1}^{t} E′′_{i} · φ̂^{∗}KP̂ − 2π̂^{∗}E · φ̂^{∗}KP̂). \end{matrix}

Exactly by the same argument as in proposition (3.14) we conclude that E′′_{i}·φ̂^{∗}KP̂ = 0 for all i ∈ {1, …, t}. Therefore

Ĥ · KP̂ = −4 − \frac{t}{2} − π̂^{∗}E · φ̂^{∗}KP̂

and it remains to calculate π̂^{∗}E · φ̂^{∗}KP̂. We recall that E = E_{1} + E_{2} + E_{3} = E′_{1} + 2E′_{2} + 3E_{3}. Moreover, necessarily σ̂|E′_{1} = id_{E′_{1}} and σ̂|E_{3} = id_{E_{3}}, whereas σ̂|E′_{2} ≠ id_{E′_{2}}. This implies that φ̂(E′_{1}) has self - intersection −4 and so φ̂(E′_{1})·KP̂ = 2. Analogously we see that E_{3} · φ̂^{∗}KP̂ = 0. We get:

\begin{matrix}Ĥ · KP̂ = −4 − \frac{t}{2} − (E′_{1} + 2E′_{2}) · φ̂^{∗}KP̂ = \\ −4 − \frac{t}{2} − \frac{1}{2}φ^{∗}φ̂(E′_{1}) · φ̂^{∗}KP̂ − 2E′_{2} · φ̂^{∗}KP̂ = −4 − \frac{t}{2} − 2 − 2E′_{2} · φ̂^{∗}KP̂. \end{matrix}

From this formula we see that, if there are isolated fixed points on E′_{2}, then Ĥ ·KP̂ ≤ −6 − \frac{t}{2} and this is a contradiction to Ĥ · KP̂ = −4. If there are no isolated fixed points on E′_{2}, then we obtain:

Ĥ · KP̂ = −6 − \frac{t}{2} − 4(φ̂(E′_{2}) · KP̂) = −2 − \frac{t}{2},

hence t has to be equal to four. Summing up the above considerations we have shown the following:

Proposition 3.37 Let S is a minimal surface with K_{S}^{2} = 7, p_{g}(S) = 4 such that the canonical system has infinitely near base points. Then the configuration of the base points is as in case B) and the involution induced by the canonical map on the blow up S̃ of S in the base points of |K_{S}| has exactly four isolated fixed points, i.e. with the above notation we have t = 4.

In the sequel we will compare Ŝ to the canonical resolution S^{∗} of the double cover branched along a singular divisor (after Horikawa). In fact we will show that in case that the image of the morphism of degree two is smooth the two surfaces Ŝ and S^{∗} coincide. Using Horikawa’s method we have therefore another way of calculating t and it will turn out that this allows us to exclude easily the existence of surfaces with K_{S}^{2} = 7, p_{g}(S) = 4 such that the canonical system has infinitely near base points. We consider the following situation: Let φ : X ⟶ Y be a morphism between smooth algebraic surfaces, which is generically of degree two. We consider the minimal resolution of singularities ψ : Ŷ ⟶ Y of the branch divisor of φ (cf. [HorQ]) and take then the associated double cover φ_{can} : X^{∗} ⟶ Ŷ. Then X^{∗} is a smooth algebraic surface, which does not need to coincide with X. In general, we have a birational morphism π′ : X^{∗} ⟶ X. We will now prove the following result:

Theorem 3.38 Let φ : X ⟶ Y be a morphism, which is generically of degree two, between smooth algebraic surfaces and we assume that the involution σ induced by φ on X is biregular. Then the blow up X̂ of X in the isolated fixed points of σ equals the canonical resolution X^{∗} after Horikawa.

Let S be a minimal surface such that the canonical map is generically of degree two onto a smooth surface. Then |K_{S}| induces a birational involution σ on S, which is biregular, since S is minimal. Let π : S̃ ⟶ S be the blow up of the base points of |K_{S}|. Then (since σ(Z) = Z, where Z is the base scheme of |K_{S}|) σ induces a biregular involution on S̃ and we get as a consequence of the above theorem:

Theorem 3.39 Let S be a minimal surface such that the canonical map is generically of degree two onto a smooth surface. Let π : S̃ ⟶ S be the blow up of the base points of |K_{S}| and let |H̃| be the movable part of |π^{∗}K_{S}|. Then S^{∗} = Ŝ, where S^{∗} is obtained applying Horikawa’s construction to φH̃, i.e. taking minimal resolution of the singularities of the branch divisor of φH̃ and then taking the associated double cover, and Ŝ is the blow up of the isolated fixed points of the involution σ : S̃ ⟶ S̃, induced by φH̃.

Proof (of (3.38)). We have the following diagram:

X̂  ^{φ̂ \overline{}}P̂ = X̂/σ̂

[figure: β; α]

[figure: π̂; X^{∗}; φ_{can}; η; Ŷ]

[figure: π′; ψ]
X  ^{φ \overline{}}Y

Here, π̂, π, π′, η, ψ are birational morphisms, whereas φ̂ and φ_{can} are finite morphisms of degree two. Moreover, α : P̂ − − ⟶ Ŷ, β : X^{∗} − − ⟶ X̂ are the obvious birational maps such that ψ ∘ α = η and π̂ ∘ β = π′. We will prove now that α and β are indeed morphisms. Let

g : Z = Z_{n} ^{g_{n} \overline{}}Z_{n−1} ^{\overline{}g_{n−1}} … ^{g_{1} \overline{}}Z_{0} = P̂

be a minimal sequence of blow ups such that there is a (birational) morphism f : Z ⟶ Ŷ with α ∘ g = f. Moreover, let E ⊂ Z be the exceptional curve of g_{n}. By the minimality of g, we have that f(E) =: C is a curve in Ŷ and C^{2} ≥ E^{2} = −1. Since C is contracted to a point by ψ it follows that C^{2} = −1. Let

Ŷ = Y_{m} ^{\overline{}ψ_{m}} Y_{m−1} ^{\overline{}ψ_{m−1}} … ^{ψ_{1} \overline{}}Y_{0} = Y

be a minimal resolution of the singularities of the branch divisor of Y and let i_{0} ∈ {0, …, m} be maximal such that (ψ_{m} ∘ ψ_{m−1} ∘ … ∘ ψ_{i_{0}})(C) is a point. Since −1 = C^{2} ≥ ψ_{m}(C)^{2} ≥ … ≥ (ψ_{m} ∘ ψ_{m−1} ∘ … ∘ ψ_{i_{0}+1})(C)^{2} < 0, we have that ψ_{m}(C)^{2} = … = (ψ_{m} ∘ ψ_{m−1} ∘ … ∘ ψ_{i_{0}+1})(C)^{2} = −1. Therefore Z, Y_{m}, …, Y_{i_{0}} are isomorphic in a neighbourhood of E, C, …, (ψ_{m} ∘ ψ_{m−1} ∘ … ∘ ψ_{i_{0}+1})(C) respectively. Moreover, (ψ_{m} ∘ ψ_{m−1} ∘ … ∘ ψ_{i_{0}})(C) is a singular point of the branch locus of Y_{i_{0}−1}, hence the branch locus of Z_{n−1} is singular (namely g_{n}(E) is a singular point of the branch locus of Z_{n−1}). But this contradicts the fact that the branch locus of P̂ is smooth and we have shown that α is a morphism.

On the other hand, if there is an element x ∈ X^{∗} such that β : X^{∗} − − ⟶ X̂ is not defined, i.e. there is a curve E′ ⊂ X̂ which is contracted to the point x ∈ X^{∗} by β^{−1}. This means that x is an isolated fixed point of the involution σ_{X^{∗}} on X^{∗}, contradicting the smoothness of Ŷ. Therefore, since every subdiagram of the above diagramis commutative, we obtain the following commutative diagram:

X̂  ^{φ̂ \overline{}}P̂

[figure: β; α]
X^{∗}  ^{\overline{}φ_{can}} Ŷ

and this implies immediately that β is an isomorphism, since if there was a curve C ⊂ X^{∗} contracted by β to a point x ∈ X̂, then (α ∘ φ̂∘ β)(C) = (φ̂∘ β)(x), whereas φ_{can}(C) is a curve. Analogously we see that α is an isomorphism. q.e.d

We will now apply this result in order to exclude that |K_{S}| has infinitely near base points. Let S be a smooth, minimal surface with K^{2} = 7 and p_{g}(S) = 4. We assume that |K_{S}| has three base points x_{1}, x_{2}, x_{3}, where x_{2} is infinitely near to x_{1} and x_{3} is infinitely near to x_{2}, i.e., the configuration of case B) in (3.36). Let π : S̃ ⟶ S be the blow up of S in x_{1}, x_{2}, x_{3} and let E be the exceptional divisor of π. Then E = E_{1} + E_{2} + E_{3}, where E_{3} is an irreducible rational (−1) - curve, whereas E_{1} = E′_{1} + E′_{2} + E_{3} and E_{2} = E′_{2} + E_{3}; E′_{1}, E′_{2} are irreducible rational curves of self - intersection (−2) and E′_{1} · E′_{2} = E′_{2} · E_{3} = 1 and E′_{1} · E_{3} = 0. Let |H̃| be the movable part of |KS̃|, then |KS̃| = |H̃| + 2E. We have to distinguish two cases:

(i) φH̃(S̃) = ℙ^{1} × ℙ^{1},
(ii) φH̃(S̃) is the quadric cone.

(i) φH̃(S̃) = ℙ^{1} × ℙ^{1} Again we can write |H̃| = |D_{1} + D_{2}| with D_{1}^{2} = D_{2}^{2} = 0 and D_{1} · D_{2} = 2.

Remark 3.40 Since H̃ · E_{3} = 1, φH̃ maps E_{3} isomorphically onto a line in Q = ℙ^{1} × ℙ^{1}. H̃ · E′_{1} = H̃ · E′_{2} = 0, hence E′_{1} and E′_{2} are contracted to points by φH̃.

We have the following conditions:

D_{1} · E_{3} = 1 or 0,
(D_{1} + D_{2}) · E_{3} = 1,
D_{1} · E′_{1} = D_{2} · E′_{1} = D_{1} · E′_{2} = D_{2} · E′_{2} = 0.

Then D_{i} · KS̃ = D_{i}(H̃ + 2 ∑_{j=1}^{3} E_{j}) = 2 + 6D_{i} · E_{3}, and we get the following result:

Proposition 3.41 In the above situation we have (up to a permutation of D_{1} and D_{2}):

D_{1} · KS̃ = 8, D_{2} · KS̃ = 2.

From this we can calculate the bidegree of the branch locus Δ ⊂ ℙ^{1} × ℙ^{1}, namely we see that we have:

Δ ∈ |6Γ_{1} + 12Γ_{2}|.

Using again Horikawa’s formulas we get:

\begin{matrix}5 = χ(S^{∗}, O_{S^{∗}}) = χ(OS̃) = \frac{1}{2}δ(K_{Q} + δ) + 2 − \frac{1}{2} ∑[\frac{m_{i}}{2} ]([\frac{m_{i}}{2} ] − 1) = \\ \frac{1}{2} · (3, 6) · (1, 4) + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1) \end{matrix}

and this implies, if we again denote [\frac{m_{i}}{2} ] by x_{i}:

(∗) ∑ x_{i}(x_{i} − 1) = 12.

On the other hand:

\begin{matrix}c_{1}(S̃)^{2} = 4 ≥ c_{1}(S^{∗})^{2} = 2(K_{Q} + δ)^{2} − 2 ∑(x_{i} − 1)^{2} = \\ 2(1, 4)^{2} − 2 ∑(x_{i} − 1)^{2}, \end{matrix}

whence:

(∗∗) ∑(x_{i} − 1)^{2} ≥ 6.

Therefore we obtain the following numerical possibilities for the x_{i}’s from (∗) and (∗∗):

\begin{matrix}(1) & x_{1} = x_{2} = … = x_{6} = 2. \\ (2) & x_{1} = 3, x_{2} = x_{3} = x_{4} = 2. \\ (3) & x_{1} = 3, x_{2} = 3. \\ (4) & x_{1} = 4. \end{matrix}

From this we see that c_{1}(S^{∗})^{2} ∈ {4, 2, 0, −2}. Therefore by (3.37), (3.39) only (3) is possible. I.e., the only essential singular points of the branch divisor have to be two points p_{1}, p_{2} such that x_{1} = x_{2} = 3. It is easy to see that this case is impossible, e.g. S^{∗} is S̃ blown up in 4 points, which is impossible if there are only two essential singularities of the branch divisor. Hence we get the following:

Proposition 3.42 Minimal surfaces with K^{2} = 7 and p_{g} = 4 such that the canonical system has three base points x_{1}, x_{2}, x_{3}, such that x_{2} is infinitely near to x_{1}, x_{3} is infinitely near to x_{2} and such that φH̃(S̃) = ℙ^{1} × ℙ^{1} do not exist.

We will now analize the second case.

(ii) φH̃(S̃) is the quadric cone. We have seen that in this case there is a morphism φ̃ : S̃ ⟶ 𝔽_{2} such that the following diagram

S̃  ^{^{φH̃}  \overline{}}Q ⊂ ℙ^{3}

[figure: φ̃; |2Γ+Γ_{∞}|]
𝔽_{2}

is commutative (cf. (3.32)). We write H̃ = 2D + G, where G · H̃ = 0, D^{2} = 0 and D · G = 2. Then

D · HS̃ = D · H̃ + 2 ∑_{i=1}^{3} E_{i} · D = 2 + 2 ∑_{i=1}^{3} E_{i} · D.

But H̃ · E_{i} = 1 for all i ∈ {1, 2, 3} and therefore D · E_{i} = 0 and G · E_{i} = 1. This implies that

D · KS̃ = 2

and

G · KS̃ = G · H̃ + 2 ∑_{i=1}^{3} E_{i} · G = 6.

Hence Δ ≡ 6Γ_{∞} + 18Γ on 𝔽_{2}. (We recall that Γ_{∞}^{2} = −2, Γ^{2} = 0 and Γ · Γ_{∞} = 1). Setting Γ_{0} := Γ+Γ_{∞} we have Γ_{0}^{2} = 0, Γ·Γ_{0} = 1, Δ ≡ 6Γ_{0}+12Γ and K_{𝔽_{2}} = −2Γ_{0}−2Γ. Using again Horikawa’s (in-)equalities we get:

\begin{matrix}5 = χ(S^{∗}, O_{S^{∗}}) = χ(OS̃) = \frac{1}{2}δ(K_{Q} + δ) + 2 − \frac{1}{2} ∑[\frac{m_{i}}{2} ]([\frac{m_{i}}{2} ] − 1) = \\ \frac{1}{2} · (3, 6) · (1, 4) + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1) \end{matrix}

and this implies, if we again denote [\frac{m_{i}}{2} ] by x_{i}:

(∗) ∑ x_{i}(x_{i} − 1) = 12.

On the other hand:

\begin{matrix}c_{1}(S̃)^{2} = 4 ≥ c_{1}(S^{∗})^{2} = 2(K_{Q} + δ)^{2} − 2 ∑(x_{i} − 1)^{2} = \\ 2(1, 4)^{2} − 2 ∑(x_{i} − 1)^{2}, \end{matrix}

whence:

(∗∗) ∑(x_{i} − 1)^{2} ≥ 6.

Therefore we obtain the following numerical possibilities for the x_{i}’s from (∗) and (∗∗):

\begin{matrix}(1) & x_{1} = x_{2} = … = x_{6} = 2. \\ (2) & x_{1} = 3, x_{2} = x_{3} = x_{4} = 2. \\ (3) & x_{1} = 3, x_{2} = 3. \\ (4) & x_{1} = 4. \end{matrix}

Using again (3.37), (3.39) we see that only (3) is possible and this is excluded as in the case of the smooth quadric before.

Proposition 3.43 Minimal surfaces with K^{2} = 7 and p_{g} = 4 such that the canonical system has three base points x_{1}, x_{2}, x_{3}, such that x_{2} is infinitely near to x_{1}, x_{3} is infinitely near to x_{2} and such that φH̃(S̃) is the quadric cone do not exist.

Summing up the preceeding results we finally obtain:

Theorem 3.44 Let S be a minimal surface with K^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has three base points x_{1}, x_{2}, x_{3}. Then the base points cannot be infinitely near, i.e. x_{1}, x_{2}, x_{3} have to be pairwise distinct points on S.

Remark 3.45 1) We want to end this section by sketching how this methods also exclude that, if we have three pairwise distinct base points x_{1}, x_{2}, x_{3} of the canonical system, the involution σ interchanges e.g. x_{1} and x_{2} and leaves fixed x_{3}. It is easy to see that the number of isolated fixed points of the involution on S̃ has to be equal to 4 using the fact that φ_{|H̃|}|(E_{1} + E_{2}) is an unramified covering of a rational (−1) - curve on Q. We observe that all the numerical calculations in Horikawa’s formulas remain the same, hence there is only one numerically possible case, namely two singular points of the branch curve with ”half multiplicity” x_{1} = x_{2} = 3 and φH̃(E_{1} + E_{2}) and φH̃(E_{3}) are two disjoint lines on ℙ^{1} × ℙ^{1} resp. on 𝔽_{2}. But we have necessarily to have at least four essential singularities (note that S^{∗} is equal to S̃ in four points). Therefore this is not possible and we have that σ(x_{i}) = x_{i} for all 1 ≤ i ≤ 3.

2) We remark that in the case where |K_{S}| has exactly three different base points, we can see immediately (by the same calculations as in the beginning of section 3.4.3) that the number t of isolated fixed points of the involution σ̃ : S̃ ⟶ S̃ induced by the canonical map is equal to zero. Hence it follows immediately that the only numerical possibilities for the essential singularities of the branch divisor are in this situation (I.1) and (II.1) (cf. p. 27 and p. 29).

3) The importance of theorem (3.38) lies in the fact that the number t of isolated fixed points of the involution on S̃ induced by the canonical map can be calculated also by Horikawa’s method as the number of blow ups between S^{∗} and S̃. This (as we saw before) excludes a lot of numerical possibilities for the essential singularities of the considered double cover and simplifies the proofs, which is important to make achievable a classification for surfaces with p_{g} = 4 and K^{2} ≥ 8.

4 |K| has a (non trivial) fixed part

4.1 Surfaces of type (F)

Let S be a minimal surface with K^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has a non trivial fixed part F, such that K_{S} · F = 0. Let |H| be the movable part of |K_{S}| then we have seen in section 1 that F^{2} = −2, H^{2} = 5, K_{S} · F = 0 and |H| has one simple base point x ∈ S. Let π : S̃ ⟶ S be the blow up of S in x and let E be the exceptional divisor of π. Then |KS̃| = |H̃| + F + 2E, where H̃ ≡ H − E and φH̃ : S̃ ⟶ Q is a morphism (generically) of degree two from S̃ onto a quadric Q ⊂ ℙ^{3}. We firstly treat the case that Q is the quadric cone.

Proposition 4.1 We assume in the above situation that φH̃(S̃) is the quadric cone. Then there is a morphism φ̃ : S̃ ⟶ 𝔽_{2} such that the diagram

S̃  ^{^{φH̃}  \overline{}}Q ⊂ ℙ^{3}

[figure: φ̃; |2Γ+Γ_{∞}|]
𝔽_{2}

is commutative, i.e. φH̃ factors through 𝔽_{2}.

Proof. We have |KS̃| = |H̃| + F + 2E, and H̃^{2} = 4. We write again |H̃| = |2D + G| with H̃ · G = 0. This implies that G = 0 or G^{2} < 0. But

4 = H̃^{2} = H̃(2D + G) = 2H̃ · D,

hence H̃ · D = 2. On the other hand H̃ · D = 2D^{2} + D · G, which implies that either D^{2} = 0 and D · G = 2 or D^{2} = 1 and D · G = 0 (which means G = 0, since H̃ is connected). Assume that D^{2} = 1, i.e. G = 0, then

1 = H̃ · E = 2D · E + G · E = 2D · E

and we get a contradiction. Hence φH̃ has to factor through 𝔽_{2}. q.e.d

We have:

2 = H̃ · F = (2D + G) · F = 2D · F + G · F,

hence there are two possibilities (noting that on S we have K_{S}·G = 2p_{a}(G)−2+G^{2} ≥ 1, hence G is not a component of F and therefore G · F ≥ 0):

\begin{matrix}(i) & D · F = 0, G · F = 2, \\ (ii) & D · F = 1, G · F = 0. \end{matrix}

But D^{2} = 0 and KS̃ · D = (H̃ + F + 2E) · D = 2 + F · D + 2E · D ≡ 0(mod2), hence F · D ≡ 0(mod2). Therefore (ii) is not possible. This implies that:

D · KS̃ = (H̃ + F + 2E) · D = 2 + F · D + 2E · D = 2

and

G · KS̃ = F · G + 2E · G = 4.

From this we conclude that for the branch divisor Δ on 𝔽_{2} it holds: Δ ≡ 6Γ_{∞}+16Γ ≡ 6Γ_{0} + 10Γ. Using again Horikawa’s (in-) equalities we obtain:

5 = χ(S^{∗}, O_{S^{∗}}) = χ(OS̃) = \frac{1}{2}δ(K_{𝔽_{2}} + δ) + 2 − \frac{1}{2} ∑[\frac{m_{i}}{2} ]([\frac{m_{i}}{2} ] − 1) =
\frac{1}{2}(5Γ + 3Γ_{0})(3Γ + Γ_{0}) + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1) =
9 − \frac{1}{2} ∑ x_{i}(x_{i} − 1)

and this implies, if we again denote [\frac{m_{i}}{2} ] by x_{i}:

(∗) ∑ x_{i}(x_{i} − 1) = 8.

On the other hand:

\begin{matrix}c_{1}(S̃)^{2} = 6 ≥ c_{1}(S^{∗})^{2} = 2(K_{𝔽_{2}} + δ)^{2} − 2 ∑(x_{i} − 1)^{2} = \\ 2(3Γ + Γ_{0})^{2} − 2 ∑(x_{i} − 1)^{2}. \end{matrix}

Hence:

(∗∗) ∑(x_{i} − 1)^{2} ≥ 3.

From this again we get the following numerical possibilities for the x_{i}’s:

\begin{matrix}(1) & x_{1} = x_{2} = … = x_{4} = 2. \\ (2) & x_{1} = 3, x_{2} = 2. \end{matrix}

Remark 4.2 In case (1) S^{∗} is the blow up of S̃ in two points and in case (2) S^{∗} is S̃ blown up in four points.

Remark 4.3 Since D · F = 0, we see immediately that F has to be mapped two to one onto a line L ∈ |Γ| in 𝔽_{2} under φH̃. In particular, φH̃(F) is not contained in the branch divisor. On the other hand φH̃ maps E isomorphically onto a line L′ ⊂ 𝔽_{2}. In particular, L · L′ = 0. Since L′ is contained in the branch divisor of φH̃ there must necessarily be two essential singularities on L′ having x_{i} = 2 both. Hence case (2) cannot occur.

If we consider case (1) we see that there have to be two of the essential singularities on L′, namely two simple quadruple points. Since t = 2, the other two singular points have to be (3, 3) - points lying on L. Therefore we obtain that φH̃ is given by the double cover of 𝔽_{2} blown up in x_{1}, …, x_{6} branched along

L′ + E′_{3} + E′_{5} + B,

where L′ ∈ |Γ − E_{1} − E_{2}|, B ∈ |6Γ_{0} + 9Γ − 3 ∑_{i=1}^{6} E_{i}| has at most negligible singularities, and where x_{1}, x_{2} are two points on L′, x_{3}, x_{5} are two points on L and x_{4} is infinitely near to x_{3}, x_{6} is infinitely near to x_{5}. Let φ : S^{∗} ⟶ 𝔽̂_{2}(x_{1}, …, x_{6}) be the double cover of 𝔽̂_{2}(x_{1}, …, x_{6}) branched along L′ + E′_{3} + E′_{5} + B, then

K_{S^{∗}} ≡ φ^{∗}(Γ_{0} + 3Γ − E_{1} − E_{2} − E_{4} − E_{6}).

Hence K_{S^{∗}}^{2} = 2(6 − 4) = 4. Moreover, if 2C_{1} := φ^{∗}E′_{3}, 2C_{2} := φ^{∗}E′_{5}, and 2E = φ^{∗}(L′), then

|K_{S^{∗}}| = |φ^{∗}(Γ_{0} + Γ)| + 2C_{1} + 2C_{2} + 2E.

In particular, h^{0}(O_{K_{S^{∗}}}) = 4. We remark that C_{1}, C_{2} are rational (−1) - curves and after contracting them, we obtain a surface S̃ with K2 ; S̃ = 6. Contracting E we obtain a minimal surface S with K_{S}^{2} = 7 and p_{g} = 4 of type (F). Concluding we have shown the following theorem:

Theorem 4.4 Let S be a minimal surface with K_{S}^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has a non trivial fixed part F with K_{S} ·F = 0. Then, if |H| is the movable part of |K_{S}|, |H| has exactly one simple base point x. Let π : S̃ ⟶ S be the blow up of S in x and let |H̃| be the movable part of |KS̃|. Then φH̃ : S̃ ⟶ Q is a morphism of degree two from S̃ onto a quadric. If we assume that Q is the quadric cone, then φH̃ factors through 𝔽_{2} and if σ is the involution on S̃ induced by φH̃, then σ has two isolated fixed points. If π̂ : Ŝ(= S^{∗}) ⟶ S̃ is the blow up of S̃ in these two points, then

φ̃ ∘ π̂ = p ∘ φ,

where p : 𝔽̂_{2}(x_{1}, …, x_{6}) ⟶ 𝔽_{2} is the blow up of 𝔽_{2} in x_{1}, …, x_{6}, such that x_{1}, x_{2} are two points on φ̃(E), x_{3}, x_{5} ∈ φ̃(F) and x_{4} is infinitely near to x_{3}, x_{6} is infinitely near to x_{5}. Moreover, φ : S^{∗} ⟶ 𝔽̂_{2}(x_{1}, …, x_{6}) is the double cover branched along

L′ + E′_{3} + E′_{5} + B,

where L′ ∈ |Γ − E_{1} − E_{2}|, B ∈ |6Γ_{0} + 9Γ − 3 ∑_{i=1}^{6} E_{i}| smooth (up to negligible singularities).

Finally, we will treat the case (F), where φH̃(S̃) = ℙ^{1} × ℙ^{1}. We write again |H̃| = |D_{1} + D_{2}| with D_{1}^{2} = D_{2}^{2} = 0, D_{1} · D_{2} = 2. We remark that H̃ · E = (D_{1} + D_{2}) · E = 1, hence without restriction D_{1} · E = 1, D_{2} · E = 0. Then

KS̃ · D_{1} = D_{1} · H̃ + D_{1} · F + 2E · D_{1} = 2 + 2 + D_{1} · F,

and

KS̃ · D_{2} = 2 + D_{2} · F.

By the adjunction formula we see that D_{i} · KS̃ ≡ 0(mod2), hence either D_{1} · F = 2, D_{2} · F = 0 or D_{1} · F = 0, D_{2} · F = 2. In particular we see that F is not contained in the ramification divisor, hence φH̃ maps F two to one onto a line in ℙ^{1} × ℙ^{1}. Therefore we have two possibilities:

\begin{matrix}(i) & D_{1} · KS̃ = 6, D_{2} · KS̃ = 2, \\ (ii) & D_{1} · KS̃ = 4, D_{2} · KS̃ = 4. \end{matrix}

This implies that the branch divisor Δ satisfies:

(1) Δ ≡ |6Γ_{1} + 10Γ_{2}|,

or

(2) Δ ≡ |8Γ_{1} + 8Γ_{2}|.

In case (1) we have φH̃(F)·φH̃(E) = 0, whereas in case (2) it holds: φH̃(F)·φH̃(E) = 1.

Using Horikawa’s (in-) equalities we obtain:

\begin{matrix}5 = χ(S^{∗}, O_{S^{∗}}) = χ(OS̃) = \frac{1}{2}δ(K_{Q} + δ) + 2 − \frac{1}{2} ∑[\frac{m_{i}}{2} ]([\frac{m_{i}}{2} ] − 1) = \\ \frac{1}{2} · (3, 5) · (1, 3) + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1) \end{matrix}

and this implies, if we again denote [\frac{m_{i}}{2} ] by x_{i}:

(∗) ∑ x_{i}(x_{i} − 1) = 8.

On the other hand:

\begin{matrix}c_{1}(S̃)^{2} = 6 ≥ c_{1}(S^{∗})^{2} = 2(K_{Q} + δ)^{2} − 2 ∑(x_{i} − 1)^{2} = \\ 2(1, 3)^{2} − 2 ∑(x_{i} − 1)^{2}, \end{matrix}

Hence:

(∗∗) ∑(x_{i} − 1)^{2} ≥ 3.

Therefore we obtain the following numerical possibilities for the x_{i}’s from (∗) and (∗∗):

\begin{matrix}(1.1) & x_{1} = x_{2} = … = x_{4} = 2. \\ (1.2) & x_{1} = 3, x_{2} = 2. \end{matrix}

We remark that in case (1.1) t = 2 and in case (1.2) t = 4.

In case (2) we have:

\begin{matrix}5 = χ(S^{∗}, O_{S^{∗}}) = χ(OS̃) = \frac{1}{2}δ(K_{Q} + δ) + 2 − \frac{1}{2} ∑[\frac{m_{i}}{2} ]([\frac{m_{i}}{2} ] − 1) = \\ \frac{1}{2} · (4, 4) · (2, 2) + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1) \end{matrix}

and this implies, if we again denote [\frac{m_{i}}{2} ] by x_{i}:

(∗) ∑ x_{i}(x_{i} − 1) = 10.

On the other hand:

\begin{matrix}c_{1}(S̃)^{2} = 6 ≥ c_{1}(S^{∗})^{2} = 2(K_{Q} + δ)^{2} − 2 ∑(x_{i} − 1)^{2} = \\ 2(2, 2)^{2} − 2 ∑(x_{i} − 1)^{2}, \end{matrix}

Hence:

(∗∗) ∑(x_{i} − 1)^{2} ≥ 5.

Therefore we obtain the following numerical possibilities for the x_{i}’s from (∗) and (∗∗):

\begin{matrix}(2.1) & x_{1} = x_{2} = … = x_{5} = 2. \\ (2.2) & x_{1} = 3, x_{2} = x_{3} = 2. \end{matrix}

In case (2.1) we have t = 0 and in case (2.2) we have t = 2. As in the case of the quadric cone before we can immediately exclude the possibility (1.2). In case (1.1) we see as before where the canonical image was the quadric cone that we have the following result:

Theorem 4.5 Let S be a minimal surface with K_{S}^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has a non trivial fixed part F with K_{S} · F = 0. Moreover, we assume that φH̃(S̃) = ℙ^{1} × ℙ^{1} =: Q and that φH̃(F) · φH̃(E) = 0. Then t = 2 and if π̂ : Ŝ(= S^{∗}) ⟶ S̃ is the blow up of S̃ in these two isolated fixed points, then

φ̃ ∘ π̂ = p ∘ φ,

where p : Q̂(x_{1}, …, x_{6}) ⟶ Q is the blow up of Q = ℙ^{1} × ℙ^{1} in x_{1}, …, x_{6} such that x_{1}, x_{2} are two points on φ̃(E), x_{3}, x_{5} ∈ φ̃(F) and x_{4} is infinitely near to x_{3}, x_{6} is infinitely near to x_{5}. Moreover, φ : S^{∗}(= Ŝ) ⟶ Q̂(x_{1}, …, x_{6}) is the double cover branched along

L′ + E′_{3} + E′_{5} + B,

where L′ ∈ |Γ_{2} − E_{1} − E_{2}|, B ∈ |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}| has at most negligible singularities.

Now it remains to consider the case (2). We remark that in case (2.1) we have t = 0, hence all the five essential singularities have to be quadruple points (i.e. there cannot be (3, 3) - points since then S^{∗} would be different from S̃). Since φH̃(F) is not in the branch locus and S^{∗} = S̃, after resolving the singularities of the branch curve the selfintersection of φH̃(F) has to be (−1). But this means that there is exactly one essential singularity on φH̃(F), which is not sufficient for getting a fixed part of the canonical system of S̃. In case (2.2) we have t = 2. But this is only possible if we have two (3, 3) - points. These have to be contained in φH̃(F) and then it remains only one singular point for L′ = φH̃(E). Therefore one of the (3, 3) - points has to be on φH̃(F) ∩ φH̃(E), which is easily seen to be impossible.

Remark 4.6 Using the same arguments as in 3.4.3 we can immediately see that the number of isolated fixed points of the involution on S̃ induced by the canonical map is at least two, and equal to two iff there are two isolated fixed points on F. By (3.38) this excludes immediately the numerical cases (1.1) and (2.1).

Proposition 4.7 Minimal surfaces with K_{S}^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has a non trivial fixed part F with K_{S} · F = 0 and such that

1) φH̃(S̃) = ℙ^{1} × ℙ^{1} and

2) φH̃(F) · φH̃(E) = 1 do not exist.

4.2 Surfaces of type (F′)

Let S be a minimal surface with K_{S}^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has a non trivial fixed part F with K_{S} · F = 1. Let |H| be the movable part of |K_{S}|. Then we have seen in section 1 that φ_{|H|} : S ⟶ Q ⊂ ℙ^{3} is a morphism of degree two from S onto a quadric in ℙ^{3}. We will first treat the case Q = ℙ^{1} × ℙ^{1}. In this case |H| = |D_{1} + D_{2}| with D_{i}^{2} = 0 and D_{1} · D_{2} = 2. It is immediate that (up to exchanging D_{1} and D_{2}) we have

K_{S} · D_{1} = 2, K_{S} · D_{2} = 4.

Therefore we see that the bidegree of the branch divisor Δ is equal to (6, 8). Using Horikawa’s formulas we see immediately that we have two singular points p_{1}, p_{2} of the branch divisor with ”half multiplicities” x_{i} equal to two. On the other hand, S^{∗} is the blow up of S in three points, which is impossible.

Remark 4.8 The case that φ_{|H|}(S) is the quadric cone and φ_{|H|} factors through the Hirzebruch surface 𝔽_{2} is excluded in the same way, if we note that in this case |H| = φ^{∗}(|Γ + Γ_{0}|), hence H ≡ D_{1} + D_{2}, with D_{i}^{2} = 0, D_{1} · D_{2} = 2 as above. (In fact, D_{1} = φ^{∗}(Γ_{0}) and D_{2} = φ^{∗}(Γ)).

Therefore we are reduced to the case that the canonical image of S is the quadric cone and φ_{H} does not factor through 𝔽_{2}. Hence the inverse image of the vertex of Q under φ_{H} is a point x ∈ S. Let π : S̃ ⟶ S be the blow up of S in x. Then

KS̃ ≡ π^{∗}H + π^{∗}F + E.

We observe that x is an isolated fixed point of the involution σ induced by K_{S}.

Remark 4.9 If F is contained in the ramification divisor of φ_{H}, then F is isomorphic to a conic in 𝔽_{2}, because H · F = 2. But this is impossible, since F has genus 1, whereas a conic in 𝔽_{2} has genus zero. Therefore there is at least one isolated fixed point of the involution on S induced by the canonical map on F and it is easy to see that x ∈ F.

Therefore

KS̃ ≡ π^{∗}H + F̃ + 2E,

where F̃ = π^{∗}F −E. Now on S̃, |H| = |2D +G| = φ_{H}^{∗}(Γ_{∞} +2Γ) and φ_{H}^{∗}(Γ_{∞}) = 2E. It is straightforward that for the branch divisor Δ on 𝔽_{2} we have now:

Δ ≡ 6Γ + 8Γ_{0} ≡ 14Γ + 8Γ_{∞}.

By Horikawa’s formulas we obtain that

\begin{matrix}5 = χ(S^{∗}, O_{S^{∗}}) = χ(OS̃) = \frac{1}{2}δ(K_{𝔽_{2}} + δ) + 2 − \frac{1}{2} ∑[\frac{m_{i}}{2} ]([\frac{m_{i}}{2} ] − 1) = \\ \frac{1}{2} · (3, 4) · (1, 2) + 2 − \frac{1}{2} ∑ x_{i}(x_{i} − 1) \end{matrix}

and this implies, if we again denote [\frac{m_{i}}{2} ] by x_{i}:

(∗) ∑ x_{i}(x_{i} − 1) = 4.

On the other hand:

\begin{matrix}c_{1}(S̃)^{2} = 6 ≥ c_{1}(S^{∗})^{2} = 2(K_{𝔽_{2}} + δ)^{2} − 2 ∑(x_{i} − 1)^{2} = \\ 2(1, 2)^{2} − 2 ∑(x_{i} − 1)^{2}, \end{matrix}

Hence:

(∗∗) ∑(x_{i} − 1)^{2} ≥ 1.

Therefore the only possibility for the x_{i}’s is:

x_{1} = x_{2} = 2.

Moreover, S^{∗} is the blow up of S in two points. Therefore there have to be two (3, 3) - points on the branch curve. It is easy to see that they have to be contained in the image of F.

We also remark that Γ_{∞} is a component of the branch divisor Δ. Therefore we obtain that φ_{H} is given by the double cover of 𝔽_{2} blown up in x_{1}, …, x_{4} branched along

Γ_{∞} + E′_{1} + E′_{3} + B,

where B ∈ |7Γ_{0}+7Γ−3 ∑_{i=1}^{6} E_{i}| is smooth (up to negligible singularities), and where x_{1}, x_{2} are two points on L′ = φ_{H}(F), x_{2} is infinitely near to x_{1}, x_{4} is infinitely near to x_{3}. Let φ : S^{∗} ⟶ 𝔽̂_{2}(x_{1}, …, x_{4}) be the double cover of 𝔽̂_{2}(x_{1}, …, x_{4}) branched along Γ_{∞} + E′_{1} + E′_{3} + B, then

K_{S^{∗}} ≡ φ^{∗}(2Γ_{0} + Γ − E_{2} − E_{4}).

Hence K_{S^{∗}}^{2} = 2(6 − 4) = 4. Moreover, if 2C_{1} := φ^{∗}E′_{1}, 2C_{2} := φ^{∗}E′_{3}, and 2E = φ^{∗}(Γ_{∞}), then

|K_{S^{∗}}| = |φ^{∗}(Γ_{0} + Γ)| + 2C_{1} + 2C_{2} + 2E.

In particular, h^{0}(O_{K_{S^{∗}}}) = 4. We remark that C_{1}, C_{2} are rational (−1) - curves and after contracting them, we obtain a surface S̃ with K2 ; S̃ = 6. Contracting E we obtain a minimal surface S with K_{S}^{2} = 7 and p_{g} = 4 of type (F′).

Theorem 4.10 Let S be a minimal surface with K_{S}^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has a non trivial fixed part F with K_{S} · F = 1. Let |H| be the movable part of |K_{S}|.

1) φ_{|H|}(S) is the quadric cone and φ_{|H|} does not factor through 𝔽_{2}.

2) Let x be the inverse image of the vertex of the quadric cone and let π : S̃ ⟶ S be the blow up of S in x. If σ is the involution induced by φ_{|H|} on S̃, then σ has two isolated fixed points. If π̂ : Ŝ(= S^{∗}) ⟶ S̃ is the blow up of S̃ in these two points, then

φ̃ ∘ π̂ = p ∘ φ,

where p : 𝔽̂_{2}(x_{1}, …, x_{4}) ⟶ 𝔽_{2} is the blow up of 𝔽_{2} in x_{1}, …, x_{4} such that x_{1}, x_{3} are two points on φ̃(F), x_{2} is infinitely near to x_{1} and x_{4} is infinitely near to x_{3}. Moreover, φ : S^{∗} ⟶ 𝔽̂_{2}(x_{1}, …, x_{4}) is the double cover branched along

Γ_{∞} + E′_{1} + E′_{3} + B,

where B ∈ |7Γ_{0} + 7Γ − 3 ∑_{i=1}^{6} E_{i}| has at most negligible singularities.

Remark 4.11 As in the other cases obviously also the converse of the theorem is true, i.e. if we take a configuration as above on 𝔽_{2} and take the minimal model of the associated double cover, then we obtain a surface of type (F′).

5 The moduli space

5.1 The ”fine” classification of surfaces with K_{S}^{2} = 7 and p_{g} = 4 (revisited)

For the sake of clarity we will restate again the main results we have obtained in the previous sections.

Theorem 5.1 Let S be a smooth minimal surface with K_{S}^{2} = 7 and p_{g}(S) = 4. Then S belongs to exactly one of the following families:

(0) |K_{S}| doesn’t have base points and the canonical map φ_{|K_{S}|} is a birational morphism from S onto a surface of degree seven in ℙ^{3}. (For a more detailed description of these surfaces we refer to [Cat1]). (I.1) |K_{S}| has exactly one (simple) base point and φ_{|K_{S}|} is a birational map. In this case the canonical image Σ := φH̃(S̃) is a surface of degree six in ℙ^{3} with the following properties:

(a) the double curve Γ ⊂ Σ is a plane conic,

(b) if γ ⊂ ℙ^{3} is the plane containing Γ, then Σ has a generalized tacnode o ∈ γ\Γ with tacnodal plane α ≠ γ,

(c) φ_{H}(E) = α ∩ γ, where E is the exceptional curve of the blow up of the base point of |K_{S}|. Apart from rational double points Σ doesn’t have other singularities. (I.2) |K_{S}| has exactly one simple base point x and φ_{|K_{S}|} is a map of degree two. Let π : S̃ ⟶ S be the blow up of S in x, E the exceptional curve and |H̃| the movable part of |KS̃|. Then we have a commutative diagram:

S̃ ^{φH̃\overline{}}F_{3} ⊂ ℙ^{3}

[figure: ∥; φ̃; |3L−∑^{6} E_{i}|]
S̃  ℙ̂^{2}(x_{1}, …, x_{6})

[figure: φ; p]
ℙ̂^{2}(x_{1}, …, x_{7})

where p is the blow up of ℙ̂^{2}(x_{1}, …, x_{6}) in x_{7} ∈ E_{6} (we assume here without restriction that φ(E) = E_{6}) and φ is the double cover branched along

D′ + E′_{6},

where D′ ∈ |12L − 4 ∑_{i=1}^{5} E_{i} − 3E_{6} − 3E_{7}| has at most negligible singularities. (I.3) |K_{S}| has exactly one base point and φ_{|K_{S}|} is a map of degree three onto a quadric in ℙ^{3}. For more details we refer to [Zuc]. (III) |K_{S}| has exactly three pairwise different simple base points x_{1}, x_{2}, x_{3} ∈ S.

Let π : S̃ ⟶ S be the blow up of S in x_{1}, x_{2}, x_{3}, |H̃| the movable part of |KS̃|, E_{i} := π^{−1}(x_{i}). Then we have the following possibilities: (III.α) φ_{|H̃|}(S̃) = Q = ℙ^{1} × ℙ^{1} and φH̃(E_{1}), φH̃(E_{2}), φH̃(E_{3}) are pairwise disjoint lines. Then φH̃ = p ∘ φ, where p : Q̂(y_{1}, …, y_{6}) ⟶ Q is the blow up of ℙ^{1} × ℙ^{1} in y_{1}, …, y_{6} such that y_{1}, y_{2} are contained in a fibre ∈ |Γ_{2}|, as well as y_{3}, y_{4}, and y_{5}, y_{6} in different fibres (∈ |Γ_{2}|)). Finally, φ : S̃ ⟶ { \atop ℙ^{1} × ℙ^{1}}(y_{1}, …, y_{6}) is the double cover branched along

D = D̃ + L_{1} + L_{2} + L_{3},

where L_{1} ≡ Γ_{2} − E_{y_{1}} − E_{y_{2}}, L_{2} ≡ Γ_{2} − E_{y_{3}} − E_{y_{4}}, L_{3} ≡ Γ_{2} − E_{y_{5}} − E_{y_{6}} and D̃ ∈ |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{y_{i}}| has at most negligible singularities. (III.β) φH̃(S̃) = Q = ℙ^{1} × ℙ^{1}, and without restriction φH̃(E_{1}), φH̃(E_{2}) ≡ Γ_{1}, φH̃(E_{3}) ≡ Γ_{2}. In this case φH̃ = p ∘ φ, where p : Q̂(y_{1}, …, y_{4}) ⟶ Q is the blow up of Q = ℙ^{1} × ℙ^{1} in y_{1}, …, y_{4} such that {y_{1}} = φH̃(E_{1}) ∩ φH̃(E_{3}), {y_{2}} = φH̃(E_{2}) ∩ φH̃(E_{3}), y_{3} ∈ φH̃(E_{1}) and y_{4} ∈ φH̃(E_{2}). Finally, φ : S̃ ⟶ Q̂(y_{1}, …, y_{4}) is the double cover branched along

D = D̃ + L_{1} + L_{2} + L_{3},

where L_{1} ≡ Γ_{2} − E_{y_{1}} − E_{y_{3}}, L_{2} ≡ Γ_{2} − E_{y_{2}} − E_{y_{4}}, L_{3} ≡ Γ_{1} − E_{y_{1}} − E_{y_{2}} and D̃ ∈ |7Γ_{1} + 8Γ_{2} − 4E_{y_{1}} − 4E_{y_{2}} − 3E_{y_{3}} − 3E_{y_{4}}| has at most negligible singularities. (III.γ) In this case we have a commutative diagram:

S̃ ^{^{φH̃} \overline{}}Q ⊂ ℙ^{3}

[figure: ∥; φ̃; |Γ_{0}+Γ|]
S̃  𝔽_{2}

[figure: φ; p]
𝔽̂_{2}(y_{1}, …, y_{6})

where p : 𝔽̂_{2}(y_{1}, …, y_{6}) ⟶ 𝔽_{2} is the blow up of 𝔽_{2} in y_{1}, …, y_{6} such that y_{1}, y_{2} ∈ φ̃(E_{1}), y_{3}, y_{4} ∈ φ̃(E_{2}) and y_{5}, y_{6} ∈ φ̃(E_{3}). Moreover, φ : S̃ ⟶ 𝔽̂_{2}(y_{1}, …, y_{6}) is the double cover of 𝔽̂_{2}(y_{1}, …, y_{6}) branched along

D = D̃ + L_{1} + L_{2} + L_{3},

where L_{1} ≡ Γ − E_{y_{1}} − E_{y_{2}}, L_{2} ≡ Γ − E_{y_{3}} − E_{y_{4}}, L_{3} ≡ Γ − E_{y_{5}} − E_{y_{6}} and D̃ ∈ |6Γ_{∞} + 15Γ − 3 ∑_{i=1}^{6} E_{y_{i}}| smooth (up to negligible singularities). (F) |K_{S}| has a non trivial fixed part F with K_{S} · F = 0. Then, if |H| is the movable part of |K_{S}|, |H| has exactly one simple base point x. Let π : S̃ ⟶ S be the blow up of S in x and let |H̃| be the movable part of |KS̃|. Then φH̃ : S̃ ⟶ Q is a morphism of degree two from S̃ onto a quadric and φH̃(F) is a line. We have the following two cases: (F.1) φH̃(S̃) = Q := ℙ^{1} × ℙ^{1}, then φ̃ ∘ π̂ = p ∘ φ, where p : Q̂(x_{1}, …, x_{6}) ⟶ Q is the blow up of Q = ℙ^{1} × ℙ^{1} in x_{1}, …, x_{6}, such that x_{1}, x_{2} are two points on φ̃(E), x_{3}, x_{5} ∈ φ̃(F) and x_{4} is infinitely near to x_{3}, x_{6} is infinitely near to x_{5}. Moreover, φ : S^{∗}(= Ŝ) ⟶ Q̂(x_{1}, …, x_{6}) is the double cover branched along

L′ + E′_{3} + E′_{5} + B,

where L′ ∈ |Γ_{2} − E_{1} − E_{2}|, B ∈ |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}| has at most negligible singularities. (F.2) φH̃(S̃) is the quadric cone. Then φH̃ factors through 𝔽_{2} and if σ is the involution on S̃ induced by φH̃, then σ has two isolated fixed points. If π̂ : Ŝ(= S^{∗}) ⟶ S̃ is the blow up of S̃ in these two points, then

φ̃ ∘ π̂ = p ∘ φ,

where p : 𝔽̂_{2}(x_{1}, …, x_{6}) ⟶ 𝔽_{2} is the blow up of 𝔽_{2} in x_{1}, …, x_{6} such that x_{1}, x_{2} are two points on φ̃(E), x_{3}, x_{5} ∈ φ̃(F) and x_{4} is infinitely near to x_{3}, x_{6} is infinitely near to x_{5}. Moreover, φ : S^{∗} ⟶ 𝔽̂_{2}(x_{1}, …, x_{6}) is the double cover branched along

L′ + E′_{3} + E′_{5} + B,

where L′ ∈ |Γ − E_{1} − E_{2}|, B ∈ |6Γ_{0} + 9Γ − 3 ∑_{i=1}^{6} E_{i}| has at most negligible singularities. In particular, we have the following commutative diagram:

S̃ ^{^{φH̃} \overline{}}Q ⊂ ℙ^{3}

[figure: π̂; φ̃; |Γ+Γ_{0}|]
Ŝ  𝔽_{2}

[figure: φ; p]
𝔽̂_{2}(y_{1}, …, y_{6}).

(F’) |K_{S}| has a non trivial fixed part F with K_{S} · F = 1. Then, if |H| is the movable part of |K_{S}|, φ_{|H|}(S) is the quadric cone and φ_{|H|} does not factor through 𝔽_{2}. Let x be the inverse image of the vertex of the quadric cone and let π : S̃ ⟶ S be the blow up of S in x. If σ is the involution induced by φ_{|H|} on S̃, then σ has two isolated fixed points. If π̂ : Ŝ(= S^{∗}) ⟶ S̃ is the blow up of S̃ in these two points, then

φ̃ ∘ π̂ = p ∘ φ,

where p : 𝔽̂_{2}(x_{1}, …, x_{4}) ⟶ 𝔽_{2} is the blow up of 𝔽_{2} in x_{1}, …, x_{4} such that x_{1}, x_{3} are two points on φ̃(F), x_{2} is infinitely near to x_{1} and x_{4} is infinitely near to x_{3}. Moreover, φ : S^{∗} ⟶ 𝔽̂_{2}(x_{1}, …, x_{4}) is the double cover branched along

Γ_{∞} + E′_{1} + E′_{3} + B,

where B ∈ |7Γ_{0} + 7Γ − 3 ∑_{i=1}^{4} E_{i}| has at most negligible singularities.

Remark 5.2 As we saw in the preceeding sections, the constructions work also the other way round, i.e. if we take in one of the above cases e.g. a smooth (or without essential singularities) divisor in the given linear system, take the resolution of the singularities and then the associated double cover we obtain (after blowing down the exceptional (−1) - curves) a minimal surface with K^{2} = 7 and p_{g} = 4 with the desired properties.

In the next paragraph we will now proceed in counting the number of moduli of the above families.

5.2 Counting the number of moduli

In this section we will determine the dimension of the families of minimal surfaces with K^{2} = 7 and p_{g} = 4 we classified in the previous part of the paper. First of all we want to make some remarks on the deformation theory of surfaces of general type. It is nowadays wellknown that there exists a quasiprojective coarse moduli space 𝔐 for surfaces of general type modulo birational equivalence (cf. [Gies]). By the results of E. Bombieri (cf. [Bom]) the surfaces of general type with given numerical invariants χ and K^{2} belong to a finite number of families. We introduce the number m = m(S) which is called the number of moduli of S and it is by definition the dimension of 𝔐 at the point [S] representing the isomorphism class of S in 𝔐. By the well known results of Kuranishi (cf. [Kur]) locally around the point [S] (corresponding to the isomorphism class of S) 𝔐 is analytically isomorphic to the quotient of the base B of the Kuranishi family by the finite group Aut(S). Hence, if T_{S} is the tangent sheaf of S,

dimH^{1}(S, T_{S}) ≥ m(S) ≥ dimH^{1}(S, T_{S}) − dimH^{2}(S, T_{S}) = 10χ − 2K^{2}.

In particular, it holds the following very useful:

Theorem 5.3 Let [S] be a generic point of the moduli space 𝔐. Then m(S) ≥ 10χ(S) − 2K_{S}^{2}.

We will heavily rely on this result, since several of our families of surfaces with K^{2} = 7 and p_{g} = 4 will have dimension smaller or equal to 35, which is strictly smaller than 10χ(S)−2K_{S}^{2} = 36 and therefore they do not give rise to an irreducible component of 𝔐. We will now proceed in calculating m(S) for each family in theorem (5.1). First we recall again the result of F. Catanese for the case (0).

Theorem 5.4 Surfaces with K^{2} = 7, p_{g} = 4 such that the canonical system |K| is free from base points form an irreducible unirational open set of dimension 36 of their moduli space 𝔐_{K^{2}=7,p_{g}=4}.

For a proof of this result we refer to the original article [Cat1]. Now we consider the case of surfaces with K^{2} = 7 and p_{g} = 4 such that the canonical system has exactly one base point. We will show that in these cases the number of moduli is equal to 35, hence there doesn’t arise a new component of the moduli space.

Theorem 5.5 Let S be a minimal surface with K^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has exactly one simple base point and φ_{|K|} is a birational map. Then m(S) = 35. More precisely, the surfaces with K^{2} = 7 and p_{g} = 4 such that the canonical system has exactly one base point and φ_{|K|} is birational form an irreducible set of dimension 35 in their moduli space.

Proof. Let S be a minimal surface with K^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has exactly one simple base point and φ_{|K|} is a birational map. Then S is the minimal desingularization of a surface Σ ⊂ ℙ^{3} of degree six with the following properties:

(a) the double curve C ⊂ Σ is a plane conic,

(b) if γ ⊂ ℙ^{3} is the plane containing C, then Σ has a generalized tacnode o ∈ γ\C with tacnodal plane α ≠ γ,

(c) φ_{H}(E) = α ∩ γ. By (3.12) we know that for a general surface of type (I.1) the canonical image Σ is an element of the linear system

|λα^{2}Q^{2} + γF_{5}|,

where F_{5} is an element of the linear subsystem Δ ⊂ |5H − C − 2o| of quintics with tangent plane α in o and Q ⊂ ℙ^{3} is an irreducible quadric containing C. Therefore (using that Σ is of general type, hence there can be at most a finite number of projectivities φ ∈ P GL(3) such that φ(Σ) = Σ):

m(S) = dim|λα^{2}Q^{2}+γF_{5}|−dim{φ ∈ P GL(3) : φ ”preserves the whole configuration”}.

It is easy to verify that

dim|λα^{2}Q^{2} + γF_{5}| = dimΔ + 1.

We see immediately that dimΔ = 38, since dimH^{0}(O_{ℙ^{3}}(5)) = 56, the property that a quintic contains the conic C gives 11 linear conditions, having a double point in o give 4 conditions and finally having tangent plane α in o are 3 conditions. Moreover,

dim{φ ∈ P GL(3) : φ preserves the whole configuration} =

= dim{φ ∈ P GL(3) : φ(α) = α, φ(γ) = γ, φ(L) = L, φ(C) = C} =
= dim{φ ∈ P GL(3) : φ(α) = α, φ|γ = id_{γ}} = 3,

which implies that m(S) = 35. q.e.d Proceeding now with case (I.2) we first observe the following:

Remark 5.6 As we see directly from the construction, the surfaces with K^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has exactly one simple base point and φ_{|K|} has degree two form an irreducible set in their moduli space.

Let S be a minimal surface with K^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has exactly one simple base point x and φ_{|K_{S}|} is a map of degree two. Then we have seen that S is birationally equivalent to a double cover of ℙ̂^{2}(x_{1}, …, x_{6}, x_{7}), where x_{7} is infinitely near to x_{6} (and x_{1}, …, x_{6} ∈ ℙ^{2} are such that |3L − ∑_{i=1}^{6} x_{i}| gives a morphism from ℙ̂^{2}(x_{1}, …, x_{6}) to a normal cubic surface in ℙ^{3}), branched along D′ + E′_{6} ∈ |Δ| = |12L − 4 ∑_{i=1}^{5} E_{i} − 2E′_{6} − 6E_{7}|.

Proposition 5.7 1) dim|12L − 4 ∑_{i=1}^{5} E_{i} − 2E′_{6} − 6E_{7}| = 28.

2) |12L − 4 ∑_{i=1}^{5} E_{i} − 3E′_{6} − 6E_{7}| does not have base points.

3) A general element D ∈ 12L − 4 ∑_{i=1}^{5} E_{i} − 3E′_{6} − 6E_{7}| is smooth. In particular, the surfaces of type (I.2) exist.

Proof. 1) Let C be an element of the linear system |12L−4 ∑_{i=1}^{5} E_{i}−3E_{6}−3E_{7}|, then by Riemann - Roch and using the fact that h^{2}(O(C)) = h^{0}(O(K − C) ≤ h^{0}(K) = 0 we have:

χ(O(C)) = h^{0}(O(C)) − h^{1}(O(C)) + h^{2}(O(C))
= h^{0}(O(C)) − h^{1}(O(C)) = \frac{1}{2}C(C − K) + 1
= \frac{1}{2}(12L − 4 ∑_{i=1}^{5} E_{i} − 3E_{6} − 3E_{7}) · (15L − 5 ∑_{i=1}^{5} E_{i} − 4E_{6} − 4E_{7}) + 1
= \frac{1}{2}(180 − 100 − 24) + 1 = 29.

But H^{1}(O(C)) = H^{1}(O(K − C))^{∨} and on ℙ̂^{2}(x_{1}, …, x_{7}) we have:

K − C ≡ −(5H − 3E_{7} + E′_{6}),

where H ≡ 3L − ∑_{i=1}^{6} E_{i}. Let be D in |C − K|, then it holds:

(i) D^{2} > 0, (ii) there exists D′ ∈ |C − K|, D′ is reduced and connected, in particular, D′ is 1 - connected. By Ramanujam’s’s vanishing theorem this implies that H^{1}(O(K −C)) = H^{1}(O(C))^{∨} = 0 and we obtain: dim|12L − 4 ∑_{i=1}^{5} E_{i} − 2E′_{6} − 6E_{7}| = 28.

2) is straightforward.

3) follows from 2) using the theorem of Bertini. q.e.d From this we conclude:

Theorem 5.8 Minimal surfaces with K^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has exactly one simple base point x and φ_{|K_{S}|} is a map of degree two form an irreducible, unirational set of dimension 33 in their moduli space.

Proof. Let S be a minimal surface with K^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has exactly one simple base point x and φ_{|K_{S}|} is a map of degree two. Then we have seen that S is birationally equivalent to a double cover of ℙ̂^{2}(x_{1}, …, x_{6}, x_{7}), where x_{7} is infinitely near to x_{6} (and x_{1}, …, x_{6} ∈ ℙ^{2} are such that |3L − ∑_{i=1}^{6} x_{i}| gives a morphism from ℙ̂^{2}(x_{1}, …, x_{6}) to a normal cubic surface in ℙ^{3}), branched along D′ + E′_{6} ∈ |Δ| = |12L − 4 ∑_{i=1}^{5} E_{i} − 2E′_{6} − 6E_{7}|. Therefore m(S) = dim|Δ| + n − dimP GL(2), where n is the number of moduli for the points x_{1}, …, x_{7}. Since x_{7} is infinitely near to x_{6} we see immediately that n = 6·2+1 = 13. Hence m(S) = 28 + 13 − 8 = 33. q.e.d In the case (I.3) we quote the result of F. Zucconi (cf [Zuc]).

Theorem 5.9 Surfaces with K^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has exactly one simple base point x and φ_{|K_{S}|} is a map of degree three form an irreducible, unirational set of dimension 35 in their moduli space.

Now we will calculate the number of moduli of the surfaces K^{2} = 7 and p_{g} = 4 such that the canonical system |K_{S}| has exactly three (pairwise distinct) base points. It will turn out that in this case there will show up new components of the moduli space (coming from the cases (III.α) and (III.β)).

We consider on Q := ℙ^{1}×ℙ^{1} three pairwise different lines L_{1}, L_{2}, L_{3} ∈ Γ_{2}. Moreover, let x_{1}, …, x_{6} be six points in ℙ^{1} × ℙ^{1} such that x_{1}, x_{2} ∈ L_{1}, x_{3}, x_{4} ∈ L_{2} and x_{5}, x_{6} ∈ L_{3}. We remark that x_{2} can be infinitely near to x_{1} (as well as x_{4} to x_{3} and x_{6} to x_{5}), but in the direction of L_{1} (resp. L_{3}, L_{5}). We consider the blow up p : Q̂(x_{1}, …, x_{6}) ⟶ Q of Q = ℙ^{1} × ℙ^{1} in x_{1}, …, x_{6} and denote by E_{i} the inverse image of x_{i} under p.

Proposition 5.10 H^{1}(Q̂(x_{1}, …, x_{6}), 6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}) = 0.

Proof. We have on ℙ^{1} × ℙ^{1} the following exact sequence of sheaves:

0 ⟶ ∏_{i=1}^{6} m_{x_{i}}^{3}O(6Γ_{1} + 9Γ_{2}) ⟶ O(6Γ_{1} + 9Γ_{2}) ^{ρ \overline{}}⨁_{i=1}^{6} O_{x_{i}}/m_{x_{i}}^{3} ⟶ 0.

Let i ∈ {1, …, 6} be arbitrary. Then we claim that there is a divisor Σ_{i} ∈ |3Γ_{1}+6Γ_{2}| such that x_{i} ∉ Σ_{i} and such that Σ_{i} has triple points in x_{1}, …, x̂_{i}, …, x_{6}. In fact, e.g. for i = 1 we take Σ_{1} = 3L_{2} + 3L_{3} + 3L^{2}, where L^{2} is the unique line in |Γ_{1}| containing x_{2}. Now, for each i we consider the subspace W_{i} := σ_{i} ·H^{0}(O(3Γ_{1} +3Γ_{2})) of H^{0}(O(6Γ_{1} +9Γ_{2})), where σ_{i} is defined by Σ_{i} := {σ_{i} = 0} and we see that W_{i} maps onto V_{i} := H^{0}(O_{x_{i}}/m_{x_{i}}^{3}) under H^{0}(ρ), in particular H^{0}(6Γ_{1}+9Γ_{2}) ⟶ ⨁_{i=1}^{6} O_{x_{i}}/m_{x_{i}}^{3} is surjective. In the case that e.g. x_{2} is infinitely near to x_{1}, we consider in the above proof (instead of W_{1}) W′_{1} := s_{1} · H^{0}(O(6Γ_{1} + 3Γ_{2})) where div(s_{1}) = 3L_{2} + 3L_{3}. We observe that W′_{1} maps onto O/(p_{∗}O(−3E_{1}−3E_{2})) = ℂ[x, y]/(y^{3}, y^{2}x^{2}, yx^{4}, x^{6}), since H^{0}(O(6Γ_{1} + 3Γ_{2})) provides us with all monomials x^{a}y^{b} with a ≤ 6 and b ≤ 3. q.e.d.

Corollary 5.11 We assume that none of the x_{1}, …, x_{6} is an infinitely near point. Then we have:

1) |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}| is without base points.

2) Let D ∈ |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}| be generic, then D is smooth. In particular, surfaces of type (III.α) exist.

Proof. 1) Let K_{i} be the kernel of W_{i} ⟶ V_{i} := H^{0}(O_{x_{i}}/m_{x_{i}}^{3}). Then for the base locus of |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} x_{i}| we have:

{3x_{1}, …, 3x_{6}} ⊂ Bs(|6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} x_{i}|) ⊂ ⋂_{i=1}^{6} Bs(K_{i}) ⊂ {3x_{1}, …, 3x_{6}},

where the last inclusion holds, because H^{0}(O(3Γ_{1} + 3Γ_{2})) ⟶ O_{x_{i}}/𝔪_{x_{i}}^{4} is surjective whence K_{i} ⟶ 𝔪_{x_{i}}^{3}/𝔪_{x_{i}}^{4} is surjective. Therefore |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}| is without base points.

2) follows from Bertini’s theorem. q.e.d.

Now we are able to calculate the number of moduli of the surfaces of type (III.α).

Theorem 5.12 For a surface S of type (III.α) we have: m(S) = 36.

Proof. We recall that all surfaces of type (III.α) are obtained by blowing down the three exceptional curves of the double cover of Q̂(y_{1}, …, y_{6}) branched along D ∈ |Δ| = |6Γ_{1} +12Γ_{2} −4 ∑_{i=1}^{6} E_{y_{i}}|, where y_{1}, …, y_{6} are six points such that y_{1}, y_{2} ∈ L_{1}, and y_{3}, y_{4} ∈ L_{2} and y_{5}, y_{6} ∈ L_{3} and L_{1}, L_{2}, L_{3} ≡ Γ_{2} are three pairwise different fibres of Q = ℙ^{1} × ℙ^{1}. By Riemann - Roch we have for C ∈ |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{y_{i}}|:

χ(O(C)) = h^{0}(O(C)) − h^{1}(O(C)) = \frac{1}{2}C(C − K) + 1
= \frac{1}{2}(6Γ_{1} + 9Γ_{2} − 4 ∑_{i=1}^{6} E_{y_{i}})(8Γ_{1} + 11Γ_{2} − 4 ∑_{i=1}^{6} E_{y_{i}}) + 1
= \frac{1}{2}(66 + 72 − 6 · 12) + 1 = 34.

Moreover, by (5.10), we see that h^{1}(O(C)) = 0, hence dim|6Γ_{1}+12Γ_{2}−4 ∑_{i=1}^{6} E_{y_{i}}| = 33. Therefore, for a surface S of type (III.α) we have:

m(S) = 33 + n − dim(Aut(ℙ^{1} × ℙ^{1}),

where n is the number of moduli of the x_{1}, …, x_{6}. Since the x_{i}’s lie pairwise on a line we see that n = 3 · 2 + 3 = 9. Moreover, dim(Aut(ℙ^{1} × ℙ^{1}) = 6 and we have proven that m(S) = 36. q.e.d Remark 5.13 We see immediately from our description of the surfaces of type (III.α) that they form an irreducible unirational subset of the moduli space.

Remark 5.14 The above theorem together with (5.4) proves that the moduli space of surfaces with K^{2} = 7 and p_{g} = 4 is not irreducible.

In the case of the quadric cone instead of ℙ^{1} × ℙ^{1} we prove the following results:

Proposition 5.15 Let x_{1}, …, x_{6} be six points on 𝔽_{2} such that they lie pairwise on three different fibres of 𝔽_{2}. We assume that at most one of the six points lies on the section at infinity Γ_{∞}. Then on the blow up of 𝔽_{2} in x_{1}, …, x_{6} we have the following:

1) H^{1}(O(6Γ_{∞} + 15Γ − 3 ∑_{i=1}^{6} E_{i})) = 0;

2) if none of the points x_{1}, …, x_{6} is an infinitely near and none lies on Γ_{∞}, then |6Γ_{∞} +15Γ−3 ∑_{i=1}^{6} E_{i}| is base point free. In particular, a general element of |6Γ_{∞} + 15Γ − 3 ∑_{i=1}^{6} E_{i}| is smooth.

Proof. 1) We consider the linear system |B| := |8Γ_{∞} +19Γ−4 ∑_{i=1}^{6} E_{i}|. We will first treat the case that none of the points x_{i} lie on Γ_{∞} and none of the x_{i}’s is an infinitely near point. Let Δ̃_{1} be the strict transform of an irreducible curve Δ_{1} ∈ |Γ_{∞} + 2Γ| containing x_{1}, x_{3} and x_{5}; and let Δ̃_{2} be the strict transform of an irreducible curve Δ_{2} ∈ |Γ_{∞} + 2Γ| containing x_{2}, x_{4} and x_{6}. Moreover we denote by L̃_{1} (resp. L̃_{2}, L̃_{3}) the strict transform of the fibre of 𝔽_{2} containing x_{1}, x_{2} (resp. x_{3}, x_{4}; and x_{5}, x_{6}). Then |B| contains as linear subsystems:

(i) 4Δ̃_{1} + 4Δ̃_{2} + |3Γ| and (ii) 4L̃_{1} + 4L̃_{2} + 4L̃_{3} + 5Γ_{∞} + |3Γ_{∞} + 7Γ|. Since |3Γ| is base point free any base point p of |B| has to belong to Δ̃_{1} ∪ Δ̃_{2}. Using the fact that

Δ̃_{i} · L̃_{j} = Δ̃_{i} · Γ_{∞} = 0,

and Δ̃_{i}, (L̃_{j} and Γ_{∞}) are irreducible, we conclude that

(4Δ̃_{1} + 4Δ̃_{2}) ∩ (4L̃_{1} + 4L̃_{2} + 4L̃_{3} + 5Γ_{∞}) = ∅.

Moreover, there exists a curve D ∈ |3Γ_{∞} + 7Γ| such that p ∉ D and we see that |B| is base point free. In particular a general element of |B| is smooth. Therefore by the vanishing theorem of Ramanujam we obtain that

H^{1}(𝔽̂_{2}, O(−B)) = 0.

By Serre duality this is equivalent to the vanishing of H^{1}(O(6Γ_{∞}+15Γ−3 ∑_{i=1}^{6} E_{i})). Now we are reduced to the analysis of the case where (without restriction) x_{1} ∈ Γ_{∞} and x_{2}, …, x_{6} ∉ Γ_{∞}. It is easy to see that in this case we have:

|B| = (Γ_{∞} − E_{1}) + (Γ − E_{3} − E_{4}) + (Γ − E_{5} − E_{6}) + |D|,

where |D| = |7Γ_{∞} + 17Γ − 3 ∑_{i=1}^{6} E_{i} − E_{2}|. We claim that (if none of the points x_{1}, …, x_{6} is infinitely near) |D| does not have base points. Assuming this for the moment, we conclude that this implies that there is a 1 - connected divisor D ∈ |B| and by the vanishing theorem by Ramanujam we are done as in the previous case. In order to show that |D| is base point free let Δ̃_{1} be the strict transform of an irreducible curve in |Γ_{∞} + 2Γ| containing x_{2}, x_{4} and x_{6}. Furthermore let Δ̃_{2} be the strict transform of an irreducible curve in |Γ_{∞} + 2Γ| containing x_{2}, x_{3} and x_{5}. Then |D| contains the following two linear subsystems:

(i) 3Δ̃_{1} + Δ̃_{2} + |3Γ_{∞} + 9Γ − 3E_{1} − 2E_{3} − 2E_{5}| and (ii) 3L̃_{1} + 3L̃_{2} + 3L̃_{3} + (Γ − E_{2}) + 4Γ_{∞} + |3Γ_{∞} + 7Γ|.

It is easy to see that |3Γ_{∞} + 9Γ − 3E_{1} − 2E_{3} − 2E_{5}| is without base points and using the fact that Δ̃_{1} and Δ̃_{2} are irreducible and have intersection zero with L̃_{i} as well as with Γ − E_{2} and 4Γ_{∞}, we conclude that |D| does not have base points. We observe that in the case where the two points lying on one fibre are possibly infinitely near we can show the vanishing of the first cohomology group by showing by a brute force calculation (simply by calculating all the possible decompositions of a divisor in 3L̃_{1} + 3L̃_{2} + 3L̃_{3} + (Γ − E_{2}) + 4Γ_{∞} + |3Γ_{∞} + 7Γ| into two effective divisors and showing that their intersection product is positive) that there is a numerical connected divisor in |B| (note that if two of the points are infinitely near, they are not contained both in Γ_{∞}).

2) is shown exactly in the same way as the base point freeness of |B|. q.e.d.

Theorem 5.16 Let S be a minimal surface of type (III.γ). Then m(S) = 35

Proof. Let S be a minimal surface of type (III.γ). It is easy to see that at most one of the essential singularities of the branch divisor of the canonical map is on the section at infinity of 𝔽_{2}. By the above proposition and Riemann - Roch we have dim|Δ| = 33, where Δ is the branch divisor of the double cover induced by the canonical map. Hence we have:

m(S) = 33 + n − dim(Aut(𝔽_{2})) = 33 + 9 − 7 = 35.

q.e.d

Corollary 5.17 The surfaces of type (III.γ) form an irreducible, unirational set of dimension 35 in their moduli space.

Remark 5.18 In particular this implies that the surfaces of type (III.γ) do not give a new component of the moduli space of surfaces with K^{2} = 7 and p_{g} = 4. On the other hand it is obvious that a general surface of type (III.γ) deforms to a surface of type (III.α). In fact, the deformation is simply given by the degeneration of ℙ^{1} × ℙ^{1} to the quadric cone.

We will now turn to the surfaces of type (III.β). Surprisingly they form a family of dimension 38. These surfaces were already considered by C. Ciliberto, P. Francia and M. Mendes Lopes (cf. [CiFrML]). In fact they show up in the list of surfaces where the bicanonical map is not birational and the surfaces do not have a genus 2 pencil. We consider on Q = ℙ^{1} × ℙ^{1} three pairwise different lines L_{1}, L_{2} ∈ Γ_{2}, L_{3} ∈ Γ_{1}. Moreover, let x_{1}, …, x_{4} be four points in ℙ^{1}×ℙ^{1} such that x_{1} ∈ L_{3}∩L_{1}, x_{2} ∈ L_{3}∩L_{2}, x_{3} ∈ L_{1}, x_{4} ∈ L_{2}. We consider the blow up p : Q̂(x_{1}, …, x_{4}) ⟶ Q of Q = ℙ^{1} × ℙ^{1} in x_{1}, …, x_{4} and denote by E_{i} the inverse image of x_{i} under p.

Proposition 5.19 Under the above conditions we have: H^{1}(Q̂(x_{1}, …, x_{4}), O(7Γ_{1}+ 8Γ_{2} − k_{i} ∑_{i=1}^{4} E_{i})) = 0, where k_{1} = k_{2} = 4 and k_{3} = k_{4} = 3.

Proof. We have on ℙ^{1} × ℙ^{1} the following exact sequence of sheaves:

0 ⟶ ∏_{i=1}^{4} m_{x_{i}}^{k_{i}}O(7Γ_{1} + 8Γ_{2}) ⟶ O(7Γ_{1} + 8Γ_{2}) ⟶ ⨁_{i=1}^{4} O_{x_{i}}/m_{x_{i}}^{k_{i}} ⟶ 0,

where k_{1} = k_{2} = 4 and k_{3} = k_{4} = 3. Let i ∈ {1, 2} be arbitrary. Then we claim that there is a divisor Σ_{i} ∈ |3Γ_{1} + 4Γ_{2}| such that x_{i} ∉ Σ_{i} and such that Σ_{i} has (at least) triple points in x_{3}, x_{4} and a quadruple point in x_{j}, where j ≠ i, and j ∈ {1, 2}. In fact, e.g. for i = 1 we take Σ_{1} = 4L_{2} + 3L^{3}, where L^{3} is the unique line in |Γ_{1}| containing x_{3}. In the same way for i = 3, 4 there is a divisor Σ_{i} ∈ |4Γ_{1} + 3Γ_{2}| such that x_{i} ∉ Σ_{i} and such that Σ_{i} has quadruple points in x_{1}, x_{2} and a triple point in x_{j}, where j ≠ i, and j ∈ {3, 4}. In fact e.g. for i = 3 we take Σ_{1} = 3L_{1} + 4L_{3}. Now for each i we consider the subspace W_{i} := σ_{i} · H^{0}(O(l_{i}Γ_{1} + 4Γ_{2})) of H^{0}(O(7Γ_{1} + 8Γ_{2})), where σ_{i} is defined by Σ_{i} := {σ_{i} = 0}, and l_{1} = l_{2} = 4, l_{3} = l_{4} = 3 and we see that the canonical morphism W_{i} ⟶ V_{i} := H^{0}(O_{x_{i}}/m_{x_{i}}^{k_{i}}) is surjective. In particular, H^{0}(7Γ_{1} + 8Γ_{2}) ⟶ ⨁_{i=1}^{4} O_{x_{i}}/m_{x_{i}}^{k_{i}} is surjective. Also here the proof has to be modified in the same way as in (5.10), if x_{3} is infinitely near to x_{1} (resp. x_{4} is infinitely near to x_{2}). q.e.d

Corollary 5.20 We assume furthermore that none of the x_{1}, …, x_{4} is infinitely near. Then we have:

1) |7Γ_{1} + 8Γ_{2} − 4E_{1} − 4E_{2} − 3E_{3} − 3E_{4}| is base point free.

2) Let D ∈ |7Γ_{1} + 8Γ_{2} − k_{i} ∑_{i=1}^{4} E_{i}| be generic, then D is smooth. In particular, surfaces of type (III.β) exist.

Proof. 1) Let K_{i} be the kernel of W_{i} ⟶ V_{i} := H^{0}(O_{x_{i}}/m_{x_{i}}^{k_{i}}). Then for the base locus of |7Γ_{1} + 8Γ_{2} − k_{i} ∑_{i=1}^{4} x_{i}| we have:

{4x_{1}, 4x_{2}, 3x_{3}, 3x_{4}} ⊂ Bs(|7Γ_{1} +8Γ_{2} −k_{i} ∑_{i=1}^{4} x_{i}|) ⊂ ⋂_{i=1}^{4} Bs(K_{i}) ⊂ {4x_{1}, 4x_{2}, 3x_{3}, 3x_{4}},

where the last inclusion holds because H^{0}(O(l_{i}Γ_{1} + 4Γ_{2})) ⟶ O_{x_{i}}/𝔪_{x_{i}}^{l_{i}} is surjective, whence K_{i} ⟶ 𝔪_{x_{i}}/𝔪_{x_{i}}^{l_{i}} is surjective.

2) follows again from Bertini’s theorem. q.e.d.

Now we are ready to count the moduli of the surfaces of type (III.β).

Theorem 5.21 The surfaces of type (III.β) form an unirational, irreducible set of dimension 38 in their moduli space.

Proof. The surfaces of type (III.β) are obtained by blowing down the three exceptional curves of the double cover of Q̂(y_{1}, …, y_{4}) branched along D ∈ |Δ| = |8Γ_{1} + 10Γ_{2} − 6E_{y_{1}} − 6E_{y_{2}} − 4E_{y_{3}} − 4E_{y_{4}}|, where y_{1}, …, y_{4} are four points such that y_{1} ∈ L_{3} ∩ L_{1}, y_{2} ∈ L_{3} ∩ L_{2}, y_{3} ∈ L_{1}, y_{4} ∈ L_{2} and L_{1}, L_{2} ≡ Γ_{2}, L_{3} ≡ Γ_{1} are three pairwise different fibres of ℙ^{1} × ℙ^{1}. By Riemann - Roch we have for C ∈ |7Γ_{1} + 8Γ_{2} − 4E_{y_{1}} − 4E_{y_{2}} − 3E_{y_{3}} − 3E_{y_{4}}|:

χ(O(C)) = h^{0}(O(C)) − h^{1}(O(C)) = \frac{1}{2}C(C − K) + 1
= \frac{1}{2}(7Γ_{1} + 8Γ_{2} − k_{i} ∑_{i=1}^{4} E_{y_{i}})(9Γ_{1} + 10Γ_{2} − (k_{i} + 1) ∑_{i=1}^{4} E_{y_{i}}) + 1
= \frac{1}{2}(70 + 72 − 64) + 1 = 40.

Moreover, by (5.19), we see that h^{1}(O(C)) = 0, hence dim|8Γ_{1} + 10Γ_{2} − 6E_{y_{1}} − 6E_{y_{2}} − 4E_{y_{3}} − 4E_{y_{4}}| = 39. Therefore, for a surface S of type (III.β) we have:

m(S) = 39 + n − dim(Aut(ℙ^{1} × ℙ^{1}),

where n is the number of moduli of the x_{1}, …, x_{4}. We see immediately that n = 5. Moreover, dim(Aut(ℙ^{1} × ℙ^{1}) = 6 and we have proven that m(S) = 38. q.e.d

In the remaining part of this section we will now concentrate on those families of surfaces where the canonical system has a fixed part, i.e. surfaces of type (F) and (F′). Let L_{1}, L ∈ |Γ_{2}| be two different lines in ℙ^{1} × ℙ^{1} and let x_{1}, x_{2} ∈ L, x_{5}, x_{6} ∈ L_{1} be points. Moreover, let be x_{2} infinitely near to x_{1} and x_{4} infinitely near to x_{3}. Then we have the following:

Proposition 5.22 1) H^{1}(6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}) = 0.

2) If furthermore x_{5} and x_{6} (as well as x_{1} and x_{3}) are not infinitely near and neither x_{2} nor x_{3} are contained in a fibre Γ_{1}, the linear system |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}| is without base points, in particular a generic element D ∈ |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}| is smooth.

3) dim|6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}| = 33.

Proof. 1) We consider the linear system |A| = |8Γ_{1} + 11Γ_{2} − 4 ∑_{i=1}^{6} E_{i}|, where we assume that for the moment that the two points x_{5} and x_{6} as well as (x_{1} and x_{2}) are not infinitely near. Moreover we assume that neither x_{2} nor x_{4} are contained in Γ_{1}. Let Δ̃_{1} be the strict transform of an irreducible curve in |Γ_{1} + Γ_{2}| passing through x_{1} in direction of x_{2} and containing x_{5}. (We remark that if x_{5} is an element of the fibre Γ_{1} containing x_{1}, then we replace x_{5} by x_{6} and note that then x_{5} and x_{3} cannot be on the same fibre in |Γ_{1}|.) Furthermore let Δ̃_{2} be the strict transform of an irreducible curve in |Γ_{1} +Γ_{2}| passing through x_{3} in direction of x_{4} and containing x_{6}. Finally let L̃_{1} (resp. L̃) be the strict transform of L_{1} (resp. L). Then |A| contains the following subsystem:

4Δ̃_{1} + 4Δ̃_{2} + |3Γ_{2}|.

It is a straightforward verification that 4Δ̃_{1} + 4Δ̃_{2} + 3Γ_{2} is 1 - connected, hence by Ramanujam’s vanishing theorem we are done. We remark that in the remaining special cases the proof that the linear system contains a 1 - connected divisor becomes somewhat more tedious (since we do not have anymore the existence of irreducible curves Δ̃_{i}), but it is still possible. We omit this, since it is again just the verification that all effective decompositions of a divisor have positive intersection number.

2) We consider the linear system |B| = |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}|, where we assume that the two points x_{5} and x_{6} as well as (x_{1} and x_{2}) are not infinitely near. Let Δ̃_{1} be the strict transform of an irreducible curve in |Γ_{1} + Γ_{2}| passing through x_{1} in direction of x_{2} and containing x_{5}. (We remark that if x_{5} is an element of the fibre Γ_{1} containing x_{1}, then we replace x_{5} by x_{6} and note that then x_{5} and x_{3} cannot be on the same fibre in |Γ_{1}|.) Furthermore let Δ̃_{2} be the strict transform of an irreducible curve in |Γ_{1} + Γ_{2}| passing through x_{3} in direction of x_{4} and containing x_{6}. Finally let L̃_{1} (resp. L̃) be the strict transform of L_{1} (resp. L). Then |B| contains the following subsystems:

(i) 3Δ̃_{1} + 3Δ̃_{2} + |3Γ_{2}|, (ii) 6L̃_{1} + 3L̃ + 3E′_{1} + 3E′_{3} + |6Γ_{1}|,

Analogously as in the previous cases we conclude that |B| does not have base points. q.e.d.

Remark 5.23 Analogously we see that on the blow up of 𝔽_{2} it holds:

dim|6Γ_{0} + 9Γ − 3 ∑_{i=1}^{6} E_{i}| = 33.

From this we conclude:

Theorem 5.24 1) The surfaces of type (F.1) form an irreducible unirational set of dimension 35 in their moduli space.

2) The surfaces of type (F.2) form an irreducible unirational set of dimension 34 in their moduli space. Moreover, the surfaces of type (F.2) are degenerations of surfaces of type (F.2) simply by the degeneration of the smooth quadric to the quadric cone in ℙ^{3}.

In particular we see that these surfaces do not give rise to a new component of the moduli space of surfaces with K^{2} = 7 and p_{g} = 4. Proof (of the theorem). 1) For a surface of type (F.1) we have:

m(S) = dim|6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}| + n − dim(Aut(ℙ^{1} × ℙ^{1}) = 33 + n − 6.

Since n = 2 · 2 + 2 + 2 = 8, the claim follows.

2) follows in the same way only using the fact that dim(Aut(𝔽_{2})) = 7. q.e.d

As in the previous cases we show for the surfaces of type (F′):

Proposition 5.25 Let x_{1}, …, x_{4} be four points on 𝔽_{2} such that x_{1}, x_{3} lie on the same fibre L ∈ |Γ|, and x_{2} is infinitely near to x_{1}, x_{4} is infinitely near to x_{3}. Moreover we assume that none of the points lie on Γ_{∞} Then:

1) H^{1}(O(7Γ_{0} + 7Γ − 3 ∑_{i=1}^{4} E_{i})) = 0,

2) if x_{1} and x_{3} are not infinitely near, then |7Γ_{0} + 7Γ − 3 ∑_{i=1}^{4} E_{i}| is without base points,

3) a general element D ∈ |7Γ_{0} + 7Γ − 3 ∑_{i=1}^{4} E_{i}| is smooth.

Proof. 1) We assume for the moment that x_{1} and x_{3} are not infinitely near and we consider on 𝔽̂_{2}(x_{1}, …, x_{4}) the linear system |Δ| = |9Γ_{0} + 9Γ − 4 ∑_{i=1}^{4} E_{i}|. Let C_{i} ∈ |Γ_{∞} + 2Γ| be an irreducible curve on 𝔽_{2} such that x_{i} ∈ C_{i}, (i = 1, 3), in direction of x_{i+1}. We denote the strict transform of C_{i} on 𝔽̂_{2}(x_{1}, …, x_{4}) by C̃_{i}. Let furthermore Γ̃ be the strict transform of the fibre Γ containing x_{1} and x_{3}. Then |Δ| contains as subsystems:

(i) 4C̃_{1} + 4C̃_{3} + |Γ_{∞} + 2Γ|, and (ii) 8Γ̃ + 4E′_{1} + 4E′_{3} + 9Γ_{∞} + |10Γ|. Since |Γ_{∞} +2Γ| does not have base points, a base point p of |Δ| has to be an element of 4C̃_{1} + 4C̃_{3}. Using the fact that

4C̃_{i} · Γ̃ = 4C̃_{i} · E′_{1} = 4C̃_{i} · E′_{3} = 4C̃_{i} · Γ_{∞} = 0,

and Γ̃, C̃_{1}, C̃_{3}, E′_{1}, E′_{3} and Γ_{∞} are irreducible (we note here that x_{1} and x_{3} are not in Γ_{∞}), we conclude that

(4C̃_{1} + 4C̃_{3}) ∩ (8Γ̃ + 4E′_{1} + 4E′_{3} + 9Γ_{∞}) = ∅.

Moreover, there exists a curve D ∈ |8Γ| such that p ∉ D and we see that |Δ| is base point free. In particular a general element of |Δ| is smooth. Therefore by the vanishing theorem of Ramanujam we obtain that

H^{1}(𝔽̂_{2}, O(−Δ)) = 0.

By Serre duality this is equivalent to the vanishing of H^{1}(O(7Γ_{0} + 7Γ − 3 ∑_{i=1}^{4} E_{i})). Again in the previous cases it is possible to show that in the case that x_{1} and x_{3} are infinitely near there is a 1 - connected divisor in |Δ|.

2) is shown exactly in the same way as the base point freeness of |Δ|.

3) follows from Bertini’s theorem. q.e.d.

Now we are able to calculate the number of moduli of surfaces of type (F′) and we conclude this section with the following result:

Theorem 5.26 The surfaces of type (F′) form an irreducible unirational set of dimension 37 in their moduli space.

Proof. Using Riemann - Roch it follows that

χ(O(C)) = h^{0}(O(C)) − h^{1}(O(C)) = 40,

for C ∈ |7Γ_{0} +7Γ−3 ∑_{i=1}^{4} E_{i}|. Hence by the above proposition (note that it is easily seen that none of the four essential singularities lie on Γ_{∞})

dim|7Γ_{0} + 7Γ − 3 ∑_{i=1}^{4} E_{i}| = 39.

Now,

m(S) = dim|7Γ_{0} + 7Γ − 3 ∑_{i=1}^{4} E_{i}| + n − dim(Aut(𝔽_{2}) = 39 + n − 7.

Since n = 2 + 3 = 5, the claim follows. q.e.d.

Remark 5.27 The surfaces of type (III.β) were already constructed in [CiFrML], but they are described in a slightly different way. Namely as double covers of 𝔽_{2} branched along Γ_{∞} + B, where B ∈ |14Γ + 7Γ_{∞}| not intersecting Γ_{∞} and having two (3, 3) - points. From this description it is obvious that the surfaces of type (F′) (which are obtained as double cover of 𝔽_{2} branched along Γ_{∞} + B, where B ∈ |14Γ + 7Γ_{∞}| not intersecting Γ_{∞} and having two (3, 3) - points lying on one fibre Γ) are degenerations of surfaces of type (III.β), hence the surfaces of type (F′) do not give rise to a new irreducible component of the moduli space 𝔐_{K^{2}=7,p_{g}=4}.

5.3 The irreducible components of the moduli space

The aim of this section is to determine the irreducible components of the moduli space of surfaces with K^{2} = 7 and p_{g} = 4. By theorem (5.3) we know that for each generic point [S] ∈ 𝔐_{K^{2}=7,p_{g}=4} we have m(S) ≥ 10χ(S) − 2K_{S}^{2} = 36. Therefore by the results of the previous section we immediately get:

Proposition 5.28 The moduli space 𝔐_{K^{2}=7,p_{g}=4} of surfaces with K^{2} = 7 and p_{g} = 4 has at most three irreducible components.

Proof. Follows immediately from the fact that there are only four families of surfaces with K^{2} = 7 and p_{g} = 4 having dimension greater or equal to 36, namely the surfaces of type (0), (III.α), (F′) and (III.β) The result follows now from (5.27). q.e.d Since the set 𝔐_{(0)} of surfaces of type (0) is open in 𝔐_{K^{2}=7,p_{g}=4} (cf. (5.4)), the surfaces of type (0) give rise to an irreducible component of the moduli space. Therefore there are at least two irreducible components of 𝔐_{K^{2}=7,p_{g}=4}. In order to prove that generic surfaces of type (III.α) do not deformat to surfaces of type (III.β) it suffices to show that the dimension of the Zariski tangent space of 𝔐_{K^{2}=7,p_{g}=4} in a general point [S] ∈ 𝔐_{(III.α)} is 36 and this will be done in the sequel.

Let L_{1}, L_{2}, L_{3} ∈ |Γ_{2}| be three pairwise distinct lines on Q := ℙ^{1} × ℙ^{1} and let x_{1}, …, x_{6} be six pairwise distinct points in ℙ^{1} × ℙ^{1} such that x_{1}, x_{2} ∈ L_{1}, x_{3}, x_{4} ∈ L_{2} and x_{5}, x_{6} ∈ L_{3}. Moreover, we assume that never two of the points x_{i} lie on one fibre in |Γ_{1}|. We consider the blow up π : Q̂(x_{1}, …, x_{6}) =: Y ⟶ Q of Q = ℙ^{1} × ℙ^{1} in x_{1}, …, x_{6} and denote by E_{i} the inverse image of x_{i} under π. We consider the following linear system on Y:

\begin{matrix}|B| = |6Γ_{1} + 12Γ_{2} − 4 ∑_{i=1}^{6} E_{i}| = \\ (Γ_{2} − E_{1} − E_{2}) + (Γ_{2} − E_{3} − E_{4}) + (Γ_{2} − E_{5} − E_{6}) + |B′|, \end{matrix}

where |B′| = |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}|.

Remark 5.29 We have: h^{0}(Y, T_{Y}) = h^{2}(Y, T_{Y}) = 0 and h^{1}(Y, T_{Y}) = 6.

Proof. This follows immediately from the fact that h^{2}(Y, T_{Y}) = h^{2}(ℙ^{1}×ℙ^{1}, T_{ℙ^{1}×ℙ^{1}}) = 0 and h^{0}(ℙ^{1} × ℙ^{1}, T_{ℙ^{1}×ℙ^{1}}) = 6). q.e.d

Let B′ ∈ |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{i}| be smooth. Then B = (Γ_{2} − E_{1} − E_{2}) + (Γ_{2} − E_{3} − E_{4}) + (Γ_{2} − E_{5} − E_{6}) + B′ is smooth, since B is the disjoint union of B′ with the strict transforms of L_{1}, L_{2} and L_{3}. In fact, since on ℙ^{1} × ℙ^{1} L_{i} · π_{∗}(B′) = 6, the branch curve on ℙ^{1} × ℙ^{1} can only have two simple triple points on each of the L_{i}’s. Therefore L_{i} and B′ are disjoint curves on Y for each i. We consider the double cover φ : X := S̃ ⟶ Y of Y branched along B ≡ 2L. By Serre duality it holds:

h^{i}(X, T_{X}) = h^{2−i}(X, Ω_{X}^{1} ⊗ Ω_{X}^{2}),

for all i ∈ {1, 2, 3}. We have the following:

Theorem 5.30 In the above situation we have:

φ_{∗}(Ω_{X}^{1} ⊗ Ω_{X}^{2}) ≅ (Ω_{Y}^{1} (logB) ⊗ Ω_{Y}^{2} ) ⊕ (Ω_{Y}^{1} ⊗ Ω_{Y}^{2} (L)).

For a proof of this result we refer to [Cat2]. Before stating our main result we observe the following:

Remark 5.31 Since φ is a finite morphism we obtain (using the above theorem):

H^{1}(X, Ω_{X}^{1} ⊗ Ω_{X}^{2}) = H^{1}(Y, φ_{∗}(Ω_{X}^{1} ⊗ Ω_{X}^{2}))
= H^{1}(Y, Ω_{Y}^{1} (logB) ⊗ Ω_{Y}^{2} ) ⊕ H^{1}(Y, Ω_{Y}^{1} ⊗ Ω_{Y}^{2} (L)).

Theorem 5.32 Let X be as above, then

\begin{matrix}1) & dimH^{1}(Y, Ω_{Y}^{1} (logB) ⊗ Ω_{Y}^{2} ) = 36, \\ 2) & dimH^{1}(Y, Ω_{Y}^{1} ⊗ Ω_{Y}^{2} (L)) = 6. \end{matrix}

This result together with the above remark implies immediately the following:

Corollary 5.33 Let X be as above, then dimH^{1}(X, T_{X}) = 42.

Proof of the theorem. 1) We consider the following two exact sequences of O_{Y} - modules:

(1) 0 ⟶ Ω_{Y}^{1} (K_{Y}) = Ω_{Y}^{1} ⊗ Ω_{Y}^{2} ⟶ Ω_{Y}^{1} (logB)(K_{Y}) ⟶ O_{B}(K_{Y}) ⟶ 0,

and

(2)  0 ⟶ O_{Y} (K_{Y} − B) ⟶ O_{Y} (K_{Y}) ⟶ O_{B}(K_{Y}) ⟶ 0.

Taking the long exact cohomology sequence associated to (2) we obtain:

(3) 0 ⟶ H^{0}(O_{Y} (K_{Y} − B)) ⟶ H^{0}(O_{Y} (K_{Y})) ⟶ H^{0}(O_{B}(K_{Y})) ⟶
⟶ H^{1}(O_{Y} (K_{Y} − B)) ⟶ H^{1}(O_{Y} (K_{Y})) ⟶ H^{1}(O_{B}(K_{Y})) ⟶
⟶ H^{2}(O_{Y} (K_{Y} − B)) ⟶ H^{2}(O_{Y} (K_{Y})) ⟶ 0.

We remark that H^{0}(Y, O_{Y} (K_{Y})) = 0, H^{1}(Y, O_{Y} (K_{Y})) = H^{1}(O_{Y})^{∨} = 0, since Y is regular. Moreover, dimH^{2}(Y, O_{Y} (K_{Y})) = dimH^{0}(O_{Y}) = 1.

We have

h^{2}(O_{Y} (K_{Y} − B)) = h^{0}(O_{Y} (B)) = h^{0}(O_{Y} (B′)) = 34.

Hence,

h^{1}(O_{B}(K_{Y})) = h^{2}(O_{Y} (K_{Y} − B)) − h^{2}(O_{Y} (K_{Y})) = 34 − 1 = 33.

On the other hand, dimH^{1}(Y, O_{Y} (K_{Y} −B)) = dimH^{0}(O_{B}(K_{Y})) and B is the disjoint union of B′ with the strict transforms of L_{1}, L_{2} and L_{3}. Therefore it follows:

\begin{matrix}H^{1}(O_{Y} (B))^{∨} ≅ H^{1}(O_{Y} (K_{Y} − B)) \\ ≅ H^{0}(O_{B′}(K_{Y})) ⊕ H^{0}(O_{L_{1}}(K_{Y})) ⊕ H^{0}(O_{L_{2}}(K_{Y})) ⊕ H^{0}(O_{L_{3}}(K_{Y})) ≅ ℂ^{3}. \end{matrix}

We apply this to the long exact sequence associated to (1):

(4) 0 ⟶ H^{0}(Ω_{Y}^{1} (K_{Y})) ⟶ H^{0}(Ω_{Y}^{1} (logB)(K_{Y})) ⟶ H^{0}(O_{B}(K_{Y})) ⟶
⟶ H^{1}(Ω_{Y}^{1} (K_{Y})) ⟶ H^{1}(Ω_{Y}^{1} (logB)(K_{Y})) ⟶ H^{1}(O_{B}(K_{Y})) ⟶
⟶ H^{2}(Ω_{Y}^{1} (K_{Y})) ⟶ H^{2}(Ω_{Y}^{1} (logB)(K_{Y})),

noting that h^{2}(Ω_{Y}^{1} (K_{Y})) = h^{0}(T_{Y}) = 0, h^{0}(Ω_{Y}^{1} (K_{Y})) = h^{2}(T_{Y}) = 0. We claim that the map H^{0}(O_{B}(K_{Y})) ⟶ H^{1}(Ω_{Y}^{1} (K_{Y})) is injective, in particular, h^{0}(Ω_{Y}^{1} (logB)(K_{Y})) = 0. In fact, since (−K_{Y}) ≡ |2Γ_{1} + 2Γ_{2} − ∑_{i=1}^{6} E_{i}| is not empty and doesn’t have a fixed part there is a morphism

O_{Y} (K_{Y}) ⟶ O_{Y} ,

which is not identically zero on L_{i}. We have now the following commutative diagram:

0 ⟶ Ω_{Y}^{1}  ⟶ Ω_{Y}^{1} (logL) ⟶  ⨁_{i=1}^{3} O_{L_{i}}

[figure]
0 ⟶ Ω_{Y}^{1} (K_{Y}) ⟶ Ω_{Y}^{1} (logL)(K_{Y}) ⟶ (⨁_{i=1}^{3} O_{L_{i}})(K_{Y}),

where the two rows are exact and the vertical arrows are induced by O_{Y} (K_{Y}) ⟶ O_{Y}. From this we obtain the following commutative diagram:

⨁_{i=1}^{3} H^{0}(O_{L_{i}}) ⟶  H^{1}(Ω_{Y}^{1} )

[figure: ≅]
⨁_{i=1}^{3} H^{0}(O_{L_{i}}(K_{Y})) ⟶ H^{1}(Ω_{Y}^{1} (K_{Y})).

By [Cat2], lemma (3.7) we see that the image of γ : ⨁_{i=1}^{3} H^{0}(O_{L_{i}}) ⟶ H^{1}(Ω_{Y}^{1} ) is generated by the first Chern classes of the L_{i}’s, which are obviously linearly independent in our situation, hence γ is injective and by the above diagram also ⨁_{i=1}^{3} H^{0}(O_{L_{i}}(K_{Y})) ⟶ H^{1}(Ω_{Y}^{1} (K_{Y})) is injective and our claim is proven. Hence we obtain:

h^{1}(Ω_{Y}^{1} (logB)(K_{Y})) = 6 − 3 + 33 = 36.

Therefore the first part of the theorem is proven.

2) We consider the exact sequence

0 ⟶ T_{Y} (−L) ⟶ π^{∗}T_{ℙ^{1}×ℙ^{1}}(−L) ⟶ (⨁_{i=1}^{6} N_{E_{i}/Y}^{∨})(−L) ⟶ 0,

and the associated long cohomology sequence:

(5) 0 ⟶ H^{0}(T_{Y} (−L)) ⟶ H^{0}(π^{∗}T_{ℙ^{1}×ℙ^{1}}(−L)) ⟶ H^{0}((⨁_{i=1}^{6} N_{E_{i}/Y}^{∨})(−L)) ⟶

⟶ H^{1}(T_{Y} (−L)) ⟶ H^{1}(π^{∗}T_{ℙ^{1}×ℙ^{1}}(−L)) ⟶ H^{1}((⨁_{i=1}^{6} N_{E_{i}/Y}^{∨})(−L)) ⟶

⟶ H^{2}(T_{Y} (−L)) ⟶ H^{2}(π^{∗}T_{ℙ^{1}×ℙ^{1}}(−L)) ⟶ 0.

We remark that, since N_{E_{i}/Y}^{∨} (−L) ≅ O_{ℙ^{1}}(−1) it follows that

H^{0}((⨁_{i=1}^{6} N_{E_{i}/Y}^{∨})(−L)) = H^{1}((⨁_{i=1}^{6} N_{E_{i}/Y}^{∨})(−L)) = 0.

Therefore we see from the exact sequence (5) that H^{k}(T_{Y} (−L))(= H^{2−k}(Ω_{Y}^{1} ⊗ Ω_{Y}^{2} (L))^{∨}) = H^{k}(π^{∗}T_{ℙ^{1}×ℙ^{1}}(−L)). But

H^{k}(Y, π^{∗}T_{ℙ^{1}×ℙ^{1}}(−L)) =
= H^{k}(ℙ^{1} × ℙ^{1}, π^{∗}O_{ℙ^{1}×ℙ^{1}}(2, 0)(−L)) ⊕ H^{k}(π^{∗}O_{ℙ^{1}×ℙ^{1}}(0, 2)(−L)) =
= H^{k}(O(−Γ_{1} − 6Γ_{2} + ∑_{i=1}^{6} 2E_{i})) ⊕ H^{k}(O(−3Γ_{1} − 4Γ_{2} + ∑_{i=1}^{6} 2E_{i})).

We see immediately that H^{k}(O(−Γ_{1} − 6Γ_{2} + ∑_{i=1}^{6} 2E_{i})) = H^{2−k}(O(−Γ_{1} + 4Γ_{2} − ∑_{i=1}^{6} E_{i})) = 0 for i = 0, 2. Using Riemann - Roch we obtain that

\begin{matrix}χ(O(−Γ_{1} + 4Γ_{2} − ∑_{i=1}^{6} E_{i})) = \\ = \frac{1}{2}(−Γ_{1} + 4Γ_{2} − ∑_{i=1}^{6} E_{i})(Γ_{1} + 6Γ_{2} − ∑_{i=1}^{6} 2E_{i}) + 1 = −6 \end{matrix}

and we see that h^{1}(−Γ_{1} − 6Γ_{2} + ∑_{i=1}^{6} 2E_{i}) = 6. Furthermore we see that H^{k}(O(−3Γ_{1} − 4Γ_{2} + ∑_{i=1}^{6} 2E_{i})) = H^{2−k}(O(Γ_{1} + 2Γ_{2} − ∑_{i=1}^{6} E_{i}))^{∨} = 0 for i = 0, 2 and using again Riemann - Roch we conclude that H^{k}(O(−3Γ_{1} − 4Γ_{2} + ∑_{i=1}^{6} 2E_{i})) = 0 and we have proven (2). q.e.d

Lemma 5.34 Let S be a surface of general type and let f : S̃ ⟶ S be the blow up of S in k pairwise different points. Then h^{1}(S, T_{S}) = h^{1}(S̃, TS̃) − 2k.

The proof follows by induction on k from [Man], lemma (2.2). Putting together the above results we finally obtain:

Theorem 5.35 Let S be general surface of type (III.α). Then h^{1}(S, T_{S}) = 36.

Proof. Let S be a generic surface of type (III.α) and let π : S̃ ⟶ S be the blow up of the three (pairwise different) base points of the canonical system of S. Then the configuration of singular points of the branch curve of the double cover induced by the canonical map on ℙ^{1} × ℙ^{1} fulfills the conditions we posed at the beginning of the section. Then by (5.33) we have h^{1}(S̃, TS̃) = 42 and by lemma (5.34) we see that h^{1}(S, T_{S}) = 36. q.e.d Corollary 5.36 1) The surfaces of type (III.α) give rise to an irreducible component of the moduli space 𝔐_{K^{=}7,p_{g}=4} of surfaces with K^{2} = 7 and p_{g} = 4.

2) Let 𝔐̄_{(III.α)} be the Zariski closure of the set of surfaces of type (III.α) in 𝔐_{K^{=}7,p_{g}=4}. Then 𝔐̄_{(III.α)} is generically smooth.

Proof. Follows immediately from the fact that h^{1}(S, T_{S}) is the dimension of the Zariski tangent space of 𝔐̄_{(III.α)} and dim𝔐̄_{(III.α)} = 36.

Definition 5.37 𝔐̄_{(0)} is the Zariski closure of the set of surfaces of type (0) in 𝔐_{K^{=}7,p_{g}=4}, and analogously 𝔐̄_{(III.β)} is the Zariski closure of the set of surfaces of type (III.β) in 𝔐_{K^{=}7,p_{g}=4}.

Concluding we can formulate the following result:

Theorem 5.38 The decomposition of the moduli space 𝔐_{K^{=}7,p_{g}=4} of surfaces with K^{2} = 7, p_{g} = 4 in irreducible components writes as follows:

𝔐_{K^{=}7,p_{g}=4} = 𝔐̄_{(0)} ∪ 𝔐̄_{(III.α)} ∪ 𝔐̄_{(III.β)},

where the dimensions of these irreducible components is 36, 36, 38 respectively.

We will now conclude the section with the following result:

Theorem 5.39 The surfaces of type (III.γ) are contained in 𝔐̄_{(0)} ∩ 𝔐̄_{(III.α)}. In particular the moduli space 𝔐_{K^{=}7,p_{g}=4} of minimal surfaces with K^{2} = 7 and p_{g} = 4 has at most two connected components.

In order to prove (5.39) we have to show that a general surface of type (III.γ) is contained in the intersection of 𝔐̄_{(III.α)} with 𝔐̄_{(0)}. First of all we will calculate dimH^{1}(S, T_{S}) and dimH^{2}(S, T_{S}) for a general surface S of type (III.γ), namely this dimensions will turn out to be equal to 37 resp. 1.

Theorem 5.40 Let S be a general surface of type (III.γ). Then we have:

1) dimH^{1}(S, T_{S}) = 37,

2) dimH^{2}(S, T_{S}) = 1

Proof. We consider on 𝔽_{2} three pairwise distinct lines L_{1}, L_{2}, L_{3} ∈ |Γ|. Let x_{1}, …, x_{6} be six pairwise distinct points in 𝔽_{2} such that x_{1}, x_{2} ∈ L_{1}, x_{3}, x_{4} ∈ L_{2} and x_{5}, x_{6} ∈ L_{3}. Moreover, we assume that none of the points x_{i} lie on Γ_{∞}. We consider the blow up π : 𝔽̂_{2}(x_{1}, …, x_{6}) =: Y ⟶ 𝔽_{2} of 𝔽_{2} in x_{1}, …, x_{6} and denote by E_{i} the inverse image of x_{i} under π. We consider the following linear system on Y:

\begin{matrix}|B| = |6Γ_{0} + 12Γ − 4 ∑_{i=1}^{6} E_{i}| = \\ (Γ − E_{1} − E_{2}) + (Γ − E_{3} − E_{4}) + (Γ − E_{5} − E_{6}) + |B′|, \end{matrix}

where |B′| = |6Γ_{0} + 9Γ − 3 ∑_{i=1}^{6} E_{i}|.

We have: h^{0}(Y, T_{Y}) = h^{2}(Y, T_{Y}) = 0 and h^{1}(Y, T_{Y}) = 6. Let B ∈ |6Γ_{0} + 12Γ − 4 ∑_{i=1}^{6} E_{i}| be smooth. We consider the double cover φ : X := S̃ ⟶ Y of Y branched along B ≡ 2L. Then by blowing down the three exceptional curves in S̃ we obtain a general surface of type (III.γ). As in the proof of (5.32), 1) we obtain:

dimH^{1}(Y, Ω_{Y}^{1} (logB) ⊗ Ω_{Y}^{2} ) = 36

and

dimH^{0}(Y, Ω_{Y}^{1} (logB) ⊗ Ω_{Y}^{2} ) = 0.

Claim. i) dimH^{1}(Y, Ω_{Y}^{1} ⊗ Ω_{Y}^{2} (L)) = dimH^{1}(Y, T_{Y} (−L)) = 7. ii) dimH^{2}(Y, T_{Y} (−L)) = 1. The claim together with lemma (5.34) imply immediately 1). For 2) we only have to remark that dimH^{2}(Y, T_{Y} (−L)) = dimH^{0}(Y, Ω_{Y}^{1} ⊗ Ω_{Y}^{2} (L)) is a birational invariant of smooth surfaces and our theorem is reduced to proving the claim. Proof of the claim. We consider the exact sequence

0 ⟶ T_{Y} (−L) ⟶ π^{∗}T_{𝔽_{2}}(−L) ⟶ (⨁_{i=1}^{6} N_{E_{i}/Y}^{∨})(−L) ⟶ 0,

and the associated long cohomology sequence:

(5) 0 ⟶ H^{0}(T_{Y} (−L)) ⟶ H^{0}(π^{∗}T_{𝔽_{2}}(−L)) ⟶ H^{0}((⨁_{i=1}^{6} N_{E_{i}/Y}^{∨})(−L)) ⟶

⟶ H^{1}(T_{Y} (−L)) ⟶ H^{1}(π^{∗}T_{𝔽_{2}}(−L)) ⟶ H^{1}((⨁_{i=1}^{6} N_{E_{i}/Y}^{∨})(−L)) ⟶

⟶ H^{2}(T_{Y} (−L)) ⟶ H^{2}(π^{∗}T_{𝔽_{2}}(−L)) ⟶ 0.

We remark that, since N_{E_{i}/Y}^{∨} (−L) ≅ O_{ℙ^{1}}(−1) it follows that

H^{0}((⨁_{i=1}^{6} N_{E_{i}/Y}^{∨})(−L)) = H^{1}((⨁_{i=1}^{6} N_{E_{i}/Y}^{∨})(−L)) = 0.

Therefore we see from the exact sequence (5) that H^{k}(T_{Y} (−L))(= H^{2−k}(Ω_{Y}^{1} ⊗ Ω_{Y}^{2} (L))^{∨}) = H^{k}(π^{∗}T_{𝔽_{2}}(−L)). We have the exact sequence

0 ⟶ O_{𝔽_{2}}(2Γ_{0}) ⟶ T_{𝔽_{2}} ⟶ O_{𝔽_{2}}(2Γ) ⟶ 0

and we get on Y the exact sequence

0 ⟶ π^{∗}O_{𝔽_{2}}(2Γ_{0})(−L) ⟶ π^{∗}T_{𝔽_{2}}(−L) ⟶ π^{∗}O_{𝔽_{2}}(2Γ)(−L) ⟶ 0.

Considering the associated long exact cohomology sequence:

(∗) 0 ⟶ H^{0}(π^{∗}O_{𝔽_{2}}(2Γ_{0})(−L)) ⟶ H^{0}(π^{∗}T_{𝔽_{2}}(−L)) ⟶ H^{0}(π^{∗}O_{𝔽_{2}}(2Γ)(−L)) ⟶

⟶ H^{1}(π^{∗}O_{𝔽_{2}}(2Γ_{0})(−L)) ⟶ H^{1}(π^{∗}T_{𝔽_{2}}(−L)) ⟶ H^{1}(π^{∗}O_{𝔽_{2}}(2Γ)(−L)) ⟶

⟶ H^{2}(π^{∗}O_{𝔽_{2}}(2Γ_{0})(−L)) ⟶ H^{2}(π^{∗}T_{𝔽_{2}}(−L)) ⟶ H^{2}(π^{∗}O_{𝔽_{2}}(2Γ)(−L)) ⟶ 0.

Now, we see immediately that H^{k}(π^{∗}O_{𝔽_{2}}(2Γ_{0})(−L)) = H^{k}(O(−Γ_{0} − 6Γ + ∑_{i=1}^{6} 2E_{i})) = 0 for i = 0, 2. Using Riemann - Roch we obtain that

\begin{matrix}χ(O(−Γ_{0} + 4Γ − ∑_{i=1}^{6} E_{i})) = \\ = \frac{1}{2}(−Γ_{0} + 4Γ − ∑_{i=1}^{6} E_{i})(Γ_{0} + 6Γ − ∑_{i=1}^{6} 2E_{i}) + 1 = −6 \end{matrix}

and therefore h^{1}(−Γ_{1} − 6Γ_{2} + ∑_{i=1}^{6} 2E_{i}) = 6. On the other hand H^{k}(π^{∗}O_{𝔽_{2}}(2Γ)(−L)) = H^{k}(O(−3Γ_{0} − 4Γ + ∑_{i=1}^{6} 2E_{i})) = H^{2−k}(O(Γ_{0} + 2Γ − ∑_{i=1}^{6} E_{i}))^{∨} = H^{2−k}(O(Γ_{∞} + 3Γ − ∑_{i=1}^{6} E_{i}))^{∨} and this is equal to ℂ for i = 2 and equal to 0 for i = 0. Using again Riemann - Roch we conclude that dimH^{1}(O(−3Γ_{0} − 4Γ + ∑_{i=1}^{6} 2E_{i})) = 1 and the claim follows from the exact sequence (∗). q.e.d

We remark now that if S is a general surface of type (III.γ) and ρ : T_{B,[S]} ⟶ H^{1}(S, T_{S}), where B is the Kuranishi family in [S], is the Kodaira - Spencer map, then the image of ρ has dimension at most 36, since the surfaces of type (III.γ) are contained in 𝔐̄_{(III.α)}, which has dimension 36. In fact we will show now that the rank of ρ is exactly 36.

Theorem 5.41 The Kodaira - Spencer map ρ : T_{B,[S]} ⟶ H^{1}(S, T_{S}) of a general surface S of type (III.γ) has rank 36.

Proof. Let F ⟶ T be the Kuranishi family of deformations of 𝔽_{2}. It is a ℙ^{1} - bundle over ℙ^{1} × T, and T is one dimensional. We consider the family P′ of 6-tuples of distinct points on F_{t} with the property that p_{2i+1} and p_{2(i+1)} (for 0 ≤ i ≤ 2) belong to the same fibre. Defining F^{(2)} = F ×_{T ×ℙ^{1}} F, we obtain that P′ is an open set in F^{(2)} ×_{T} F^{(2)} ×_{T} F^{(2)}, and that P′ ⟶ T is a fibre bundle with a rational fibre of dimension 9. Then the fibre product P′ ×_{T} F posseses six canonical sections and blowing them up we obtain a family of rational surfaces f : F̃ ⟶ P′. Moreover, let D be the relative Cartier divisor on F̃ inducing |6Γ_{1} + 12Γ_{2} − 4 ∑_{i=1}^{6} E_{i}| (resp. |6Γ_{0}+12Γ−4 ∑_{i=1}^{6} E_{i}|) on F̃_{t}. Finally we define B′ := ℙ(f_{∗}(D)); in view of (5.10) and (5.15) the open set B of B′ parametrizing the surfaces S̃ of type (III.α) respectively (III.γ) is contained in a ℙ^{33} - bundle over an open set P ⊂ P′. We summarize the above in the following diagram:

S̃ ⟶ F̃ ⟶ F
↓  ↓  ↓
B ⟶ P ⟶ T.

We consider now the tangent spaces of the bases in a general (but fixed) point giving rise to a surface of type (III.γ) and denote them by T_{B}, T_{P} and T_{T}. By the consruction of this family we have an exact sequence

(∗) 0 ⟶ H^{0}(O_{D}(D)) ⟶ T_{B} ⟶ T_{P} ⟶ 0,

where D is the branch curve of the double cover φ : S ⟶ 𝔽̂_{2}. Moreover we have the exact sequence:

0 ⟶ H^{0}(TS̃) ⟶ H^{0}(φ^{∗}T𝔽̂_{2}) ⟶ H^{0}(O_{D}(D)) ⟶ H^{1}(TS̃) ⟶ H^{1}(φ^{∗}T𝔽̂_{2}).

We remark that H^{0}(φ^{∗}T𝔽̂_{2}) = 0, hence δ : H^{0}(O_{D}(D)) ⟶ H^{0}(TS̃) is injective. Since H^{1}(O𝔽̂_{2}) = 0, we have dimH^{0}(O_{D}(D)) = dimH^{0}(O𝔽̂_{2}(D)) − 1 = 33. Moreover, we remark that imδ ⊂ H^{1}(TS̃)^{+}, where H^{1}(TS̃)^{+} is the invariant part of H^{1}(TS̃) under the involution σ̃ on S̃ given by φ. By [Cat2], p.497, δ : H^{0}(O_{D}(D)) ⟶ H^{0}(TS̃) is the restriction of the Kodaira - Spencer map ρ : T_{B,[S]} ⟶ H^{1}(S, T_{S}) to H^{0}(O_{D}(D)) via the inclusion in (∗). We have the following commutative diagram:

0 ⟶ H^{0}(O_{D}(D)) ⟶ T_{B} ⟶  T_{P}  ⟶  0
∥  ↓
0 ⟶ H^{0}(O_{D}(D)) ⟶ H^{1}(TS̃) ⟶ H^{1}(φ^{∗}T𝔽̂_{2}) ⟶ H^{1}(O_{D}(D)),

where the vertical arrow is exactly ρ. Therefore there is a map α : T_{P} ⟶ H^{1}(φ^{∗}T𝔽̂_{2}) such that α commutes with the above diagram. Claim. The Kodaira - Spencer map ρ : T_{B,[S]} ⟶ H^{1}(S̃, TS̃)^{+} is surjective. Proof. We recall that dimH^{1}(S̃, TS̃)^{+} = dimH^{1}(Y, Ω_{Y}^{1} (logB) ⊗ Ω_{Y}^{1} ) = 36. By the above arguments it is enough to show that the image of

\begin{matrix}α : T_{P} ⟶ H^{1}(φ^{∗}T𝔽̂_{2}) = H^{1}(φ^{∗}T𝔽̂_{2})^{+} ⊕ H^{1}(φ^{∗}T𝔽̂_{2})^{−} \\ = H^{1}(T𝔽̂_{2}) ⊕ H^{1}(T𝔽̂_{2}(−L)) ⟶ H^{1}(T𝔽̂_{2}) \end{matrix}

has dimension three. Since P ⟶ T is a fibre - bundle with a rational fibre of dimension 9 we get an exact sequence

0 ⟶ V ⟶ T_{P} ⟶ T_{T} = ℂ ⟶ 0,

where V has dimension equal to 9. Since the Kodaira - Spencer class of F ⟶ T is non - trivial we have an isomorphism T_{T} = ℂ ⟶ H^{1}(T_{𝔽_{2}}). Now, we have the following diagram with exact rows:

0 ⟶ H^{0}(T_{𝔽_{2}}) ⟶ ⊕_{i=1}^{6}T_{𝔽_{2}}/m_{y_{i}} ⟶ H^{1}(T𝔽̂_{2}) ⟶ H^{1}(T_{𝔽_{2}}) ⟶ 0
↑  ↑  ∥
0  ⟶  V  ⟶ T_{P}  ⟶  ℂ  ⟶ 0.

V is a subspace of ⊕_{i=1}^{6}T_{𝔽_{2}}/m_{y_{i}} and γ : V ⊂ ⊕_{i=1}^{6}T_{𝔽_{2}}/m_{y_{i}} ⟶ H^{1}(T𝔽̂_{2}) is the restriction of the Kodaira - Spencer map to V. Since dimH^{0}(T_{𝔽_{2}}) = 7, and dimV = 9 we see immediately that the dimension of the image of γ has to be at least two, hence the claim is proved. q.e.d. for the claim.

Continuing with the proof of (5.41), we remark that the base points x_{i} of |K_{S}| are isolated fixed points of the involution σ (induced by φ_{K}), hence σ_{∗} : T_{x_{i}} ⟶ T_{x_{i}} equals (−id) and this implies that the invariant part of ⊕_{i=1}^{3}T_{S}/m_{x_{i}} is trivial. Let π : S̃ ⟶ S be the blow up of S in x_{1}, x_{2}, x_{3}. Then we consider again the exact sequence:

⟶ H^{0}(T_{S}) = 0 ⟶ ⊕_{i=1}^{3}T_{S}/m_{x_{i}} ⟶ H^{1}(TS̃) ⟶ H^{1}(T_{S}) ⟶ 0.

Therefore we get that the canonical map H^{1}(TS̃)^{+} ⟶ H^{1}(T_{S})^{+} is an isomorphism and we have shown that the Kodaira - Spencer map has rank 36. q.e.d.

Proof of (5.39). By (5.40) we know that the Kuranishi family for a general surface of type (III.γ) is a hypersurface in ℂ^{37}; let {f = 0} be a local defining equation. By (5.41) there exist coordinates y, t_{2}, …, t_{37} in ℂ^{37} such that the image of the above constructed family B in the moduli space is locally defined by y = 0. Hence f = y · φ(y, t_{2}, …, t_{37}). First of all we remark that φ cannot be a unit, since this would contradict that the dimension of H^{1}(T_{S}) is equal to 37. On the other hand, if we assume that φ equals the product of y^{k}, k ≥ 1, with a unit, then this contradicts the generic smoothness of the component 𝔐̄_{(III.α)}. Here we use the fact that H^{0}(T_{S}) = 0, hence the Kuranishi family (B_{S}, 0) of S is universal and by a result of J. Wavrik (cf. [Wav]) B_{S} is the Kuranishi family of [S′], where [S′] is sufficiently near to [S]. Therefore 𝔐̄_{(III.α)} has to intersect another component of 𝔐_{K^{2}=7,p_{g}=4} in general surfaces of type (III.γ). We remark that if 𝔐̄_{(III.α)} intersected 𝔐̄_{(III.β)} in a general surface of type (III.γ), then the dimension of the Zariski tangent space of 𝔐_{K^{2}=7,p_{g}=4} in a general surface of type (III.γ) would be at least 39 contradicting (5.40), hence theorem (5.39) is proven. q.e.d.

Theorem 5.42 𝔐̄_{III.α} ∩ 𝔐̄_{III.β} = ∅.

Proof. First of all we remark that, if S/T is a one parameter family of minimal surfaces such that S_{t} is of type (III.β) for t ≠ 0, S_{0} is either of type (III.β) or of type (F′). Since surfaces of type (F′) are the only ones having a fixed part of the canonical system, which gives rise to a fixed part of the canonical system on the canonical model, it follows that: if S/T is a one parameter family of minimal surfaces such that S_{t} is of type (F′) for t ≠ 0, S_{0} is of type (F′) as well. In particular, 𝔐_{III.β} ∪𝔐_{F′} is a closed subset of 𝔐_{K^{2}=7,p_{g}=4}. On the other hand, if S/T is a one parameter family of minimal surfaces such that S_{t} admits a genus 2 pencil for t ≠ 0 (e.g. of type (III.α), (III.γ), (F.1) or (F.2)), then S_{0} admits a genus 2 pencil. In particular S_{0} is not of type (III.β) or (F′). q.e.d.

Unfortunately up to now we are not able to decide whether 𝔐̄_{III.β} is a connected component of the moduli space, so we raise the following question which we hope to answer in the near future.

Question. Is 𝔐̄_{III.β} a connected component of the moduli space of surfaces with K^{2} = 7 and p_{g} = 4 or does 𝔐̄_{III.β} intersect 𝔐̄_{0}?

5.4 The fundamental group

In this section we will calculate the fundamental group of the surfaces with p_{g} = 4 and K^{2} = 7. More precisely, we will prove the following result.

Theorem 5.43 The surfaces with p_{g} = 4 and K^{2} = 7 are simply connected.

Remark 5.44 By the results of the previous section it suffices to show (5.43) for minimnal surfaces S of type (III.α) and of type (III.β).

Proof (of (5.43). 1) Let S be a general surface of type (III.α), then S is obtained by blowing down the three exceptional curves of the double cover of Q̂(y_{1}, …, y_{6}) =: Q̃,where y_{1}, …, y_{6} are six points such that y_{1}, y_{2} ∈ L_{1}, and y_{3}, y_{4} ∈ L_{2} and y_{5}, y_{6} ∈ L_{3} and L_{1}, L_{2}, L_{3} ≡ Γ_{2} are three pairwise different fibres of ℙ^{1} × ℙ^{1}, branched along

D = D̃ + L_{1} + L_{2} + L_{3},

where L_{1} ≡ Γ_{2} − E_{y_{1}} − E_{y_{2}}, L_{2} ≡ Γ_{2} − E_{y_{3}} − E_{y_{4}}, L_{3} ≡ Γ_{2} − E_{y_{5}} − E_{y_{6}} and D̃ ∈ |6Γ_{1} + 9Γ_{2} − 3 ∑_{i=1}^{6} E_{y_{i}}| is smooth. Since D^{2} > 0, by Nori’s theorem, the fundamental group π_{1}(Q̃−D) is abelian. Then

π_{1}(Q̃ − D − L_{i}) ⟶ π_{1}(Q̃ − D)

is surjective and we denote the kernel of this map by K. Now, K = π_{1}(T_{i}−L_{i}) ≅ ℤ/2, where T_{i} is a tubular neighbourhood of L_{i}, hence K is central and therefore π_{1}(Q̃ − D−L_{i}) is abelian. Repeating this argument, we obtain that π_{1}(Q̃−D−L_{1}−L_{2}−L_{3}) is abelian. By [Cat2], (1.8) it follows that π_{1}(Q̃ − D − L_{1} − L_{2} − L_{3}) = coker(r), where r : H^{2}(Q̃, ℤ) ≅ ℤ^{8} ⟶ H^{2}(D ∪ L_{1} ∪ L_{2} ∪ L_{3}, ℤ) ≅ ℤ^{4} is the canonical map. The matrix of r in the basis π^{∗}Γ_{1}, π^{∗}Γ_{2}, E_{1}, …, E_{6} is given by

( \begin{matrix}9 & 6 & 3 & 3 & 3 & 3 & 3 & 3 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} ) ,

hence rmod2 is surjective onto {(x_{1}, …, x_{4}) ∈ (ℤ/2)^{4} : ∑ x_{i} = 0} = kerρ, where ρ : ⨁^{4} ℤ/2 ⟶ ℤ/2, ρ(1) = 1. Again by [Cat2], (1.8) it follows that π_{1}(S) = π_{1}(S̃) = ker(ρ)/im(r(mod2)) = {1}.

2) If S is a general surface of type (III.β), the result follows exactly by the same argument as above. q.e.d.

Corollary 5.45 All minimal surfaces with K^{2} = 7 and p_{g} = 4 are homeomorphic.

Proof. Follows from the results of M. Freedman (cf. [F]). q.e.d.

References

[BPV] W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces. Ergeb. Math. Grenzgeb. (3) vol. 4, Springer Verlag (1984).

[Bom] E. Bombieri, Canonical models of surfaces of general type. Publ. Scient. I.H.E.S. 42 (1973), 447-495.

[Cat1] F. Catanese, Commutative algebra methods and equations of regular surfaces. ”Alg. Geom. - Bucharest 1982”, Springer LNM 1056 (1984), 68-111.

[Cat2] F. Catanese, On the moduli space of surfaces of general type. J. Diff. Geom. 19 (1984), 483-515.

[Cat3] F. Catanese, Moduli of algebraic surfaces. Theory of Moduli, Montecatini Terme, 1985, Springer LNM 1337 (1988), 1-83.

[Cil] C. Ciliberto,Canonical surfaces with p_{g} = p_{a} = 4 and K^{2} = 5, …, 10. Duke Math. Journal 48,1, 121-157 (1981).

[CiFrML] C. Ciliberto, P. Francia, M. Mendes Lopes, Remarks on the bicanonical map for surfaces of general type. Math. Z. 224 (1997), 137-166.

[Deb] O. Debarre, Inegalitees numeriques pour les surfaces de type general. Bull. Soc. Math. France 110 (1982), 319-346.

[Enr] F. Enriques, Le Superficie Algebriche. Zanichelli, Bologna (1949).

[F] M. H. Freedman, The topology of fourdimensional manifolds. J. Diff. Geometry 17, 357 - 453 (1982).

[Gies] D. Gieseker, Global moduli for surfaces of general type. Inv. Math. 43 (1977), 233-282.

[HorQ] E. Horikawa, On deformations of quintic surfaces. Inv. Math. 31 (1975), 43-85.

[HorI-V] E. Horikawa, Algebraic surfaces of general type with small c_{1}^{2}. I, Ann. of Math. (2) 104 (1976), 357-387; II, Inv. Math. 37 (1976), 121-155; III, Inv. Math. 47 (1978), 209-248; IV, Inv. Math. 50 (1979), 103-128; V, J. Fac. Sci. Univ. Tokyo, Sect. A. Math. 283 (1981),745-755.

[Kod] K. Kodaira, On characteristic systems of families of surfaces with ordinary singularities in a projective space. Amer. J. of Math. 87, (1965), 227-256.

[Kon] K. Konno, Algebraic surfaces of general type with c_{1}^{2} = 3p_{g} − 6. Math. Ann. 290 (1991), 71-107.

[Kur] M. Kuranishi, Deformations of compact complex manifolds. Presses de l’Universite de Montreal (1969).

[Lau] H. B. Laufer, On minimally elliptic singularities. Am. J. of Math. 99 (1977), 1257-1295.

[Man] M. Manetti, Degeneration of Algebraic Surfaces and Applications to Moduli Problems. Tesi di Perfezionamento. Scuola Normale Suoperiore (1996).

[Max] E. A. Maxwell, Regular canonical surfaces of genus three and four. Proceedings of the Cambridge Philosophical Society 23 (1937), 306-310.

[Mum] D. Mumford, Appendix to [Zar].

[Noe] M. Noether, Zur Theorie des eindeutigen Entsprechens algebraischer Gebilde. Math. Ann. 2 (1870), 293-316; 8 (1875), 495-533.

[Ol] P. Oliverio, Spin surfaces of general type with K^{2} = 8, p_{g} = 4, q = 0. Preprint, 24.04.98.

[Wav] J. J. Wavrik, Obstructions to the existence of a space of moduli. Global Analysis, Princeton Math. Series 29 (1969), 403-414.

[Zar] O. Zariski, Algebraic Surfaces. Second edition. Erg. Math. 61, Springer Verlag, Heidelberg (1971).

[Zuc] F. Zucconi, Surfaces with canonical map of degree three and K^{2} = 3p_{g} −5. Preprint, Feb 1997.