Geometry of some twistor spaces of algebraic dimension one

It is shown that there exists a twistor space on the $n$-fold connected sum of complex projective planes $n\mathbb{CP}^2$, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed over $n\mathbb{CP}^2$ for any $n\ge 5$, while the latter kind of example is constructed over $5\mathbb{CP}^2$. Both of these seem to be the first such example on $n\mathbb{CP}^2$. The algebraic reduction in these examples is induced by the anti-canonical system of the twistor spaces. It is also shown that the former kind of twistor spaces contain a pair of non-normal Hopf surfaces.


Introduction
Let X be a compact complex manifold. We denoted by dim X for the complex dimension of X. A basic invariant for X is the algebraic dimension, which is usually denoted by a(X). This is defined as the transcendental degree over C of the field of meromorphic functions on X, and thus roughly measures how many meromorphic functions exist on X. For any compact complex manifold X we have 0 ≤ a(X) ≤ dim X. We have a(X) = 0 iff X has only constant meromorphic functions, and a(X) = dim X iff X is bimeromorphically equivalent to a projective algebraic manifold. For any X, there exists a projective algebraic manifold Y and a surjective meromorphic map f : X → Y which induces an isomorphism for the meromorphic function fields of Y and X, so that dim Y = a(X); the meromorphic map f : X → Y is called the algebraic reduction of X, and is known to be unique under a bimeromorphic equivalence. Moreover, fibers of f are necessarily connected.
If a(X) = dim X − 1, a general fiber of the algebraic reduction is always an elliptic curve; this is a consequence of the fact that the degree of the canonical bundle of an algebraic curve is zero only when it is an elliptic curve. In the case a(X) = dim X − 2 and dim X > 2, a possible list of a general fiber of the algebraic reduction is obtained in [26, p. 146]; basically the surfaces in the list are surfaces with non-positive Kodaira dimension. When X is 3dimensional and belongs to the class C , structure of a possible general fiber of algebraic reduction is determined by A. Fujiki [6,p. 236,Theorem]. But when X ∈ C , it is not easy to construct a (non-trivial) example of X with a(X) = 1 which has a surface in the list as a general fiber of the algebraic reduction.
As is noticed by F. Campana, C. LeBrun, Y. S. Poon, M. Ville and others, the so called the twistor spaces associated to self-dual metrics on real 4-manifolds provide examples of compact complex threefold Z which satisfies Z ∈ C and a(Z) = 1. Some interesting examples are the twistor spaces of a Ricci-flat Kähler metric on a complex torus and a K3 surface with the complex orientation reversed; the algebraic reduction of these twistor spaces is a natural holomorphic projection to CP 1 , which is differentiably a fiber bundle over CP 1 whose fibers are a complex torus or a K3 surface respectively. Also, a Hopf surface has a conformally flat structure whose twistor space enjoys a similar property. Further, in the case of Hopf surfaces, Fujiki [8] showed that if a(Z) = 1, a general fiber of its algebraic reduction is either a Hopf surface or a ruled surface over an elliptic curve. (We will call the latter as an elliptic ruled surface in the sequel.) These twistor spaces are quotient of CP 3 minus two skew lines by a free linear Z-action.
As examples of a different flavor, by Donaldson-Friedman [4], LeBrun-Poon [19,20], Poon [25] and others, if n ≥ 4, the connected sum nCP 2 of n copies of the complex projective planes admits a self-dual metric whose twistor space Z satisfies a(Z) = 1. For these examples, the algebraic reduction is induced by the natural square root K −1/2 Z of the anticanonical line bundle of Z, and from this, it may be readily seen, with a help of a useful result by Pedersen-Poon [24], that a general fiber of the algebraic reduction is a rational surface. Thus rational surfaces actually occur as a general fiber of the algebraic reduction of a twistor space on nCP 2 if n ≥ 4. Also, it is not difficult to see that, for any twistor space Z on 4CP 2 , other complex surfaces cannot be a general fiber of the algebraic reduction for Z with a(Z) = 1. However, to the best of the author's knowledge, no example is known so far of a twistor space of algebraic dimension one on nCP 2 having any other surfaces in the list as a general fiber of the algebraic reduction.
In this article, we show that there exists a twistor space Z on nCP 2 satisfying a(Z) = 1 whose general fiber of the algebraic reduction is birational to (i) a ruled surface over an elliptic curve, or (ii) a K3 surface. The former type of twistor spaces are constructed over nCP 2 for arbitrary n ≥ 5, while the latter is constructed only over 5CP 2 . The algebraic reduction for both of these twistor spaces is induced by the anti-canonical system of the twistor space, and it is a pencil, having base curves.
We briefly explain characteristic features of these twistor spaces. The twistor spaces with the pencil of elliptic ruled surfaces have a particular pair of rational curves which are base curves of the anti-canonical pencil. Moreover these are double curves of the elliptic ruled surfaces. In particular, the surfaces are non-normal. These surfaces are obtained from their normalization by identifying each of two disjoint sections of the ruling by an involution on each. (This is why the two double curves are not elliptic but rational.) These twistor spaces admit a C * -action, and each member of the pencil is C * -invariant. There exists a reducible member of the pencil, whose irreducible components consist of two surfaces which are birational to a Hopf surface. The intersection of the two components is a non-singular elliptic curve, and it is a single orbit of the C * -action. On the other hand, the twistor spaces with the pencil of K3 surfaces do not admit a C * -action, and the base locus of the pencil consists of a pair of chains of curves formed by four rational curves. From the second Betti number, these twistor spaces do not seem to have a direct generalization to the case n > 5.
At a technical level, both of these twistor spaces are characterized by the property that they contain a rational surface of special kinds whose twice is an anti-canonical divisor of the twistor space. The most difficult part for analyzing the structure is to show that under this condition on the presence of the rational surface, the anti-canonical system is necessarily a pencil. This is shown by using a variation formula for the Euler characteristic for a line bundle under a blow-up whose center is a curve. We apply the formula for a particular line bundle and the rational curves, which lead to the conclusion.

Construction of some rational surfaces
In this section, we first construct a rational elliptic surface S 0 admitting a non-trivial C * -action, which satisfies K 2 = 0. Next we blowup this surface at fixed points of the C *action, and obtain a rational surface S with C * -action which satisfies K 2 = 8 − 2n, where n > 4. Then we investigate the linear system |mK −1 S | for any m > 0. By a real structure on a complex manifold, we mean an anti-holomorphic involution on it. Throughout this section E denotes a smooth elliptic curve equipped with: • a real structure σ without fixed point, • a holomorphic involution τ which has 4 fixed points, and which commutes with σ. Also on the complex projective line CP 1 , we put: • a real structure without fixed point, which is also denoted by σ, • a holomorphic involution τ which has 2 fixed points, and which commutes with σ. From these, on the product surface E × CP 1 we obtain: • the product real structure, again denoted by σ, • the product holomorphic involution, again denoted by τ , commuting with σ. On the product E × CP 1 , the holomorphic involution τ has exactly 8 fixed points. So the quotient complex surface (E × CP 1 )/τ has 8 ordinary double points. This quotient surface also has a real structure induced by that on E × CP 1 . The projection E × CP 1 → CP 1 induces a holomorphic map (E × CP 1 )/τ → CP 1 /τ , which defines a structure of elliptic surface over CP 1 /τ ≃ CP 1 on the quotient surface. Let be the minimal resolution of the 8 double points. The real structure on (E × CP 1 )/τ naturally lifts to S 0 , and we still denote it by σ. This has no real point. We note that the surface S 0 is uniquely determined by the isomorphism class of the initial elliptic curve E. Taking the composition with the above projection (E × CP 1 )/τ → CP 1 /τ ≃ CP 1 , we obtain an elliptic fibration This has non-reduced fibers over the images of the two fixed points of τ on CP 1 . These fibers are clearly of type I * 0 in Kodaira's notation for singular fibers of elliptic fibrations. Denoting 0 and ∞ for the images of the two fixed points, we write the singular fibers as where C 0 and C 0 = σ(C 0 ) are the unique double component of each fiber respectively. Of course, all of these components are (−2)-curves. (See Figure 1 for these constructions.) The following properties on the surface S 0 are immediate to see from the construction: • The surface S 0 has an effective C * -action induced from the C * -action on E × CP 1 which is the product of the trivial action on E and the standard C * -action on CP 1 . • This C * -action on S 0 is compatible with the above elliptic fibration f 0 : S 0 → CP 1 , and induces a non-trivial but non-effective C * -action on the base space CP 1 . • All regular fibers of f 0 are isomorphic to E, and are mutually identified by the C *action. The two singular fibers of f 0 are C * -invariant, and among their components only the double components C 0 and C 0 are point-wise fixed by the C * -action. Moreover we have the following properties on the structure of the surface S 0 : Proposition 2.1. Let S 0 be as above. Then K 2 S 0 = 0, and the anti-canonical system of the surface S 0 is a pencil without a fixed point, whose associated morphism may be identified with the elliptic fibration f 0 : S 0 → CP 1 in (2.2). Moreover, S 0 is a rational surface.
As this is not difficult to see, we omit a proof. We mention that this surface S 0 is the same as the surface S ′ 0 in [13, p. 146] that appears in a construction of a twistor space on 4CP 2 admitting a C * -action, whose algebraic dimension is two.
As in Proposition 2.1, the surface S 0 satisfies K 2 = 0. Let n be any integer greater than 4. We then choose distinct (n − 4) points on the curve C 0 , any of which is different from the four points C 0 ∩ C i , i ∈ {1, 2, 3, 4}, and let be the blowup at the (n − 4) points on C 0 and the (n − 4) points on C 0 which are the real conjugations of the (n − 4) points. Evidently the real structure and the C * -action on S 0 naturally lift on the surface S. Obviously S is uniquely determined from S 0 by the choice of (n − 4) points on C 0 . For the pluri-anticanonical systems of this surface, we have Proposition 2.2. For the above rational surface S, we have: In particular κ −1 (S) = 0 for the anti-Kodaira dimension of S.
(iii) The single member of the system |2K −1 S | is concretely given by Proof. The item (i) is of course obvious. For (ii), we recall from Proposition 2.1 the relation for the anti-canonical system on S 0 . Let C 5 , C 6 , . . . , C n be the exceptional curves over the blown-up points on C 0 , and C 5 , C 6 , · · · , C n the exceptional curves over the real conjugate points over C 0 . Let be the composition, which is also an elliptic fibration. In the following, for simplicity of notation, we write C 1,2,3,4 := 1≤i≤4 C i and C 5,6,...,n := 5≤i≤n C i .
(2.8) Therefore from (2.7) we have Thus the last curve belongs to the system |2K −1 S |, and therefore h 0 (2K −1 S ) ≥ 1, implying h 0 (2mK −1 S ) ≥ 1 for any m > 0. Next we show κ −1 (S) = 0. For this it is enough to see that the divisor (2.9) itself is precisely the 'negative part' of the Zariski decomposition [27] of the divisor. For this it suffices to verify that the intersection matrix of the divisor 2C 0 + C 1,2,3,4 is negative definite. From elementary calculation, we readily see that the eigenvalues of the intersection matrix (C i , C j ) 0≤i,j≤4 are all negative. Hence the intersection matrix is actually negative definite, and thus we obtain κ −1 (S) = 0. This means that h 0 (2mK −1 S ) = 1 for any m > 0, since otherwise we have κ −1 (S) ≥ 1.
It remains to show that H 0 ((2m − 1)K −1 S ) = 0 for any m ≥ 1. The case m = 1 being already shown, suppose m > 1. Since K −1 S .C 0 = C 2 0 + 2 = (2 − n) + 2 = 4 − n which is negative as n > 4, the curve C 0 is a fixed component of |(2m − 1)K −1 S |. As K −1 S .C i = 0 for any i ∈ {1, 2, 3, 4}, and C i and C 0 intersect, this means that C i is also a fixed component for any i ∈ {1, 2, 3, 4}. By reality, the same is true for the conjugate curves C i , 0 ≤ i ≤ 4. Hence we have an isomorphism Moreover, the intersection number of the ingredient of the right-hand side with the curve C 0 is readily computed to be 2{(1 − m)n + (4m − 5)}. Since n ≥ 5 and m > 1, we have (1 − m)n + (4m − 5) ≤ 5(1 − m) + (4m − 5) = −m < 0. This implies that the right-hand side of (2.10) still has C 0 , and therefore C 0 also, as fixed components. Thus the unique member (2.9) of |2K −1 S | is a fixed component of the system Hence we obtain an isomorphism 3. Geometry of a twistor space which contains the rational surface Let n > 4 and g be a self-dual Riemannian metric on nCP 2 whose scalar curvature is positive. Let Z be the associated twistor space. In the sequel we write F for the holomorphic line bundle K −1/2 Z which makes a natural sense for a twistor space associated to a self-dual metric [1]. In this section we investigate structure of Z which has the rational surface S constructed in the last section as a real member of the linear system |F |. (The existence of such a twistor space will be shown in Section 5.) 3.1. Determination of the algebraic dimension of Z. Let n > 4 and S be any one of the rational surfaces constructed in the last section, satisfying K 2 = 8 − 2n. Suppose the twistor space Z has the surface S as a real member of the system |F |. We also suppose that the C * -action on S extends to the whole of Z in a way that it is compatible with the real structure on Z. This means that the corresponding self-dual structure on nCP 1 have a non-trivial S 1 -action, and its natural lift on the twistor space Z has the given C * -action as the complexification when restricted to the divisor S. Then we have the following properties on linear systems on Z: Proposition 3.1. Suppose n > 4, and let Z and S be a twistor space on nCP 2 and the divisor respectively as above. Then we have: is a pencil, and the associated meromorphic map gives an algebraic reduction of Z. In particular, a(Z) = 1 holds for the algebraic dimension of Z.
The meromorphic map Z → CP 1 in (ii) has a non-empty indeterminacy locus. This will be investigated in the next subsection.
The assertion (i) can be readily shown by a standard argument using Proposition 2.2, and we omit a proof. From (i), it follows a(Z) ≤ 1, since we have a(Z) ≤ 1 + κ −1 (S) in general [2], and κ −1 (S) = 0 by Proposition 2.2. The rest of this subsection is devoted to proving a(Z) > 0. We first show a transformation formula for the Euler characteristic of a line bundle over a compact complex threefold under the blowup with a 1-dimensional center in general: Proposition 3.2. Let X be a compact complex threefold and L a holomorphic line bundle over X. Let C be a smooth curve on X, µ : X 1 → X the blowup at C, and E ⊂ X 1 the exceptional divisor. Then if we write L 1 := µ * L − E, for the Euler characteristic, we have where g C is the genus of the curve C.
Proof. By the Riemann-Roch formula for a threefold, we have To compute the right-hand side, we first note that, for any line bundles F 1 and F 2 over X, we have Therefore we obtain where we used (3.4)-(3.6) for the last equality.
Next for the second term in the right-hand side of (3.2), we have where the last equality is from adjunction formula. Hence, with the aid of (3.4), we obtain from (3.8) Next, for the third term of (3.2), we first have On the other hand, for the transformation law for the second Chern class of a threefold, we have by [9, pp.609-610, Lemma] where [C] denotes the cohomology class in H 4 (X, Z) represented by the 2-cycle C. Therefore we obtain (3.9)).
For proving the assertion (ii) in Proposition 3.1, we next recall that, on the divisor S, there are (n−4) numbers of (−1)-curves C 5 , C 6 , . . . , C n . (See the construction in the last section.) Since the restriction of the twistor projection π : Z → nCP 2 to the divisor S is necessarily diffeomorphic around the curves C i and C i for any i ∈ {5, 6, . . . , n} in the orientationreversing way, the image π(C i ) = π(C i ) is a sphere in nCP 2 whose self-intersection number is 1. Moreover, these spheres are disjoint since π is degree two on S. Let ξ i ∈ H 2 (nCP 2 , Z), i ∈ {5, 6, . . . , n}, be the cohomology class represented by the sphere π(C i ). These form a part of orthonormal basis of , and we write O Z (α i ) when we regard the element α i ∈ H 2 (Z, Z) as an element in Pic Z. We then have Then we have the following key vanishing property: Proof. In this proof we write α 5,6,...,n and C 5,6,...,n for the sum 5≤i≤n α i and 5≤i≤n C i respectively, for simplicity. From the inclusion C 0 ⊂ S ⊂ Z we have an exact sequence where the two ideal sheaves are taken in O Z . As S ∈ |F | we have I S ≃ −F . By taking a tensor product with the sheaf O Z (−α 5,6,...,n ), noting ,...,n + C 5,6,...,n ) by (3.13) from the choice of the class α i , we obtain an exact sequence ,...,n + C 5,6,...,n ) → 0. (3.14) Now from the Hitchin's vanishing theorem [11], noting that the line bundle O Z (±α 5,6,...,n ) are real and of degree 0, we have Therefore from the cohomology exact sequence of (3.14), we obtain an isomorphism ,...,n + C 5,6,...,n )). (3.15) In order to compute the right-hand side, we consider an obvious exact sequence Here we have used the fact that the curves C 5 , C 6 , . . . , C n are disjoint from the union C 0 ∪ C 5 ∪ C 6 ∪ · · · ∪ C n . Then noting that C 0 ∪ C 5 ∪ C 6 ∪ · · · ∪ C n is simply connected from their configuration, we have H 1 (O C 0 ∪C 5 ∪C 6 ∪···∪Cn ) = 0. Moreover, we clearly have H 0 (O S (−C 0 − C 5,6,...,n + C 5,6,...,n )) = 0. Therefore from the connectedness of C 0 ∪ C 5 ∪ C 6 ∪ · · · ∪ C n , the cohomology exact sequence of (3.16) gives an isomorphism Moreover, since C 5 , C 6 , . . . , C n are (−1)-curves and the surface S is rational, we readily obtain H 1 (O S (C 5,6,...,n )) = 0. Hence from (3.15) we obtain the required vanishing result.
For the purpose of proving the assertion (ii) of Proposition 3.1, let be the blowup of the twistor space Z at the curves C 0 ⊔ C 0 , and let E 0 and E 0 be the exceptional divisors over C 0 and C 0 respectively. From the inclusion C 0 ⊂ S ⊂ Z, recalling that (C 0 ) 2 S = 2 − n, for the normal bundle of C 0 in Z, we readily have Then by utilizing the transformation formula in Proposition 3.2, we can prove Lemma 3.4. Let µ 1 : Z 1 → Z be as above. Then we have for the Euler characteristic of the invertible sheaf on Z 1 .
Proof. We decompose the blowup µ 1 : where ν 1 is the blowup at C 0 and ν 2 is the blowup at C 0 . (We use this strange notation only in this proof.) Then we have, with the same notation as in the proof of Lemma 3.3, Moreover, since ν 2 is birational, we have, for any coherent sheaf L on Z 1/2 , Hence we have Now we are going to compute the right-hand side by using Proposition 3.2 on putting L = O Z − α 5,6,...,n , C = C 0 , and µ = ν 1 . For this, we first compute as Therefore by the formula in Proposition 3.2, we obtain, recalling that C 0 is rational, The first term of right-hand side may be computed by Riemann-Roch formula as follows: where in the second equality we used that α i . α j . α k = 0 for any i, j, k (since α i -s are lifts from the 4-manifold), and c 2 (Z) are lifts from the 4-manifold [12]), c 1 (Z). c 2 (Z) = 12(χ − τ ) = 24 (Hitchin [12, p.135]).
We are still preparing for the proof of Proposition 3.1 (ii). We next show the following: where we are identifying the curve C i in S with the strict transform into S 1 . Moreover, we have an isomorphism Proof. Since the center C 0 ⊔ C 0 for the blowup µ 1 is contained in the divisor S, the intersections S 1 ∩ E 0 and S 1 ∩ E 0 are naturally isomorphic to C 0 and C 0 under µ 1 respectively. Hence for the first restriction, we calculate, under the identification S 1 ≃ S given by µ 1 , where in the third isomorphism we have used the isomorphism ..,n − C 5,6,...,n , which is obtained in (2.7). This means the first isomorphism (3.26) in the lemma. On the other hand, for the latter restriction, we have Hence we get the second isomorphism (3.27) in the lemma. Now we are able to prove a key proposition, which readily implies a(Z) > 0: Proposition 3.6. Let Z and S be as in Proposition 3. 1

. Then we have a non-vanishing
Therefore, the system F − α 5 − · · · − α n consists of a single member.
Proof. We consider the restriction homomorphism of the line bundle µ * 1 F − α 5,6,...,n − 2E 0 to the divisor S 1 ∪ E 0 . The kernel of this homomorphism is computed as, noting µ * For the section of the last non-trivial term, we have, by (3.26) and (3.27), recalling that On the other hand, by the Leray spectral sequence and Lemma 3.3, we obtain and this vanishes by Lemma 3.3. Moreover, H 0 (µ * 1 O Z (−α 5,6,...,n ) − E 0 ) = 0 clearly. Hence from the cohomology exact sequence of (3.30), we obtain the desired non-vanishing result.
In this and the next subsections we denote by X for the unique member of the system |F − α 5 − · · · − α n |. Then X is the unique member of the system |F + α 5 + · · · + α n |.
Proof for (ii) of Proposition 3.1. We have h 0 (2F ) ≥ 2, because 2S and X + X are linearly independent divisors belonging to |2F |. For the reverse inequality, the cohomology exact sequence of the exact sequence 0 . But the right-hand side is 1 + 1 = 2 by Propositions 2.2 and 3.1. Thus we obtain h 0 (2F ) = 2. Hence a(Z) ≥ 1. Since we have a(Z) ≤ 1 as explained right after Proposition 3.1, we obtain a(Z) = 1.
Let Φ : Z → CP 1 be the rational map induced by the pencil |2F |, and let Z ′ → Z be an elimination of the indeterminacy locus of Φ which preserves the real structure. (We will an explicit elimination later in the next subsection, but here we do not need it.) If Φ is not an algebraic reduction, we get a factorization We emphasize that the pencil |2F | is generated by the non-reduced member 2S and a reducible member X + X as in the above proof.

3.2.
Structure of members of the anti-canonical system of Z. Let Z → nCP 2 and S ∈ |F | be as in the previous subsection. In this subsection we investigate structure of the pencil |2F |. In particular we determine structure of a general fiber of the algebraic reduction of Z. We first investigate the unique member X of the system |F − 5≤i≤n α i | found in the last subsection.
Proposition 3.7. Let X be the divisor as above.
(i) As a divisor on the surface S ∈ |F |, we have Further it is easy to see from intersection numbers that the system |2C 0 + C 1,2,3,4 | has no member other than 2C 0 + C 1,2,3,4 itself. Thus we obtain (i).
For the assertion (ii), since the pencil |2F | is generated by 2S and X + X as above, by (3.31), we have C 0 ∪ C 0 ⊂ Bs |2F |. Then while it is tempting to conclude that the divisor X has double points along the curve C 0 , we need to exclude the possibility that X is tangent to S along C 0 . For this purpose, as in the previous subsection, let µ 1 : Z 1 → Z be blowup at C 0 ⊔ C 0 , and E 0 ⊔ E 0 the exceptional divisor. Then we put . We have an identification |2F | ≃ |L ′ |. We now show that the system |L ′ | has the divisor E 0 + E 0 as fixed components.
As in (3.19), When the former is the case, writing O(0, 1) for the fiber class of the projection E 0 → C 0 , we have Hence we obtain This means that, if n > 5, the exceptional divisor E 0 , and therefore E 0 also, are fixed components of |L ′ |. (The case n = 5 will be considered later.) When , if we denote A for the (−2)-section of the projection E 0 ≃ F 2 → C 0 , and f denotes its fiber class, we have, in a similar way for obtaining (3.33), If E 0 ≃ F 0 and n = 5, we can show E 0 ⊂ Bs |L ′ | in the following way. If this did not hold, by (3.33), there would exist a divisor Y ′ ∈ |L ′ | which satisfies Y ′ | E 0 ∈ |O(1, 0)|. On the other hand, if we still denote C i and C i (i ∈ {1, 2, 3, 4}) for the strict transforms of the curves in Z into Z 1 , the curve C i (and also C i ) is clearly a base curve of |L ′ | by (3.31). In particular, Y ′ has to include the four curves C 1 , . . . , C 4 , and therefore the point C i ∩ E 0 has to be on a (1, 0)-curve for any i ∈ {1, 2, 3, 4}. On the other hand, if S 1 again denotes the strict transform of the divisor S into Z 1 , we have by (3.32) Moreover, since µ 1 blows up the curve C 0 (and C 0 ) in the divisor S, the intersection S 1 ∩ E 0 is isomorphic to the center C 0 , and therefore irreducible. Furthermore, S 1 clearly contains the curves C 1 , . . . , C 4 . Therefore all the intersection points E 0 ∩ C i (i ∈ {1, 2, 3, 4}) have to pass an irreducible (1, 1)-curve on E 0 . This contradicts the above conclusion that the 4 points C i ∩ E 0 are on a (1, 0)-curve, which is derived from the assumption E 0 ⊂ Bs |L ′ |. Therefore E 0 ⊂ Bs |L ′ |, as claimed.
Still we need to show that E 0 ⊂ Bs |L ′ | holds also in the case E 0 ≃ F 2 and n ∈ {5, 6}. But this can be shown in a similar manner to the above argument, so we omit the detail; we just mention that, instead of (3.35), we have Now since the pencil |µ * 1 (2F )| has the divisor 2E 0 +2E 0 as fixed components, any member of the pencil |2F | has multiplicity two along the curve C 0 . Hence so is the member X + X. But from (3.31) we have X ∩ C 0 = ∅. Therefore the divisor X itself has double points along the curve C 0 at least. But from (3.31), X can have at most double points along C 0 . Therefore the multiplicity of X along C 0 is two. Finally, since X ∈ |F − α 5,6,...,n |, if X were reducible, we have X = D + D ′ , for some degree-one divisors D and D ′ . Then D + D is a member of |F |, and D + D = S as S is irreducible. This contradicts h 0 (F ) = 1 in Proposition 3.1. Hence X is irreducible. Now the next proposition is obvious from (3.31) if we recall that the pencil |2F | is generated by the divisors 2S and X + X: Proposition 3.8. For the base locus of the anti-canonical system |2F |, we have Therefore if µ 1 : Z 1 → Z is the blowup at C 0 ∪ C 0 and E 0 ∪ E 0 is the exceptional divisor as before, and if we put this time, then by Proposition 3.7 (ii), we have an isomorphism |2F | ≃ |L 1 |. In order to determine structure of a general member of the original pencil |2F |, we look at the behavior of the pencil |L 1 | when restricted to E 1 : Lemma 3.9.
(i) We have an isomorphism Therefore the restriction of the pencil |L 1 | to E 0 remains to be a pencil. (iii) The last pencil on E 1 is generated by the twice of S 1 | E 0 ∈ |K −1/2 E 0 | which is smooth, and an anti-canonical curve which intersects the curve S 1 | E 0 transversally at four points. (iv) A general member of the last pencil is a non-singular elliptic curve, which is 2 : 1 over C 0 under the projection µ 1 | E 0 : E 0 → C 0 .
Proof. Let S 1 ⊂ Z 1 be the strict transform of S into Z 1 as before. Then since Then from (3.35) (in the case E 0 ≃ F 0 ) and (3.36) (in the case E 0 ≃ F 2 ), we obtain the desired isomorphism (3.38).
Next, to show the injectivity of the restriction map, let X 1 be the strict transform of the divisor X into Z 1 . Then since the pencil |L 1 | is generated by 2S 1 and X 1 + X 1 , it suffices to show the two curves 2S 1 | E 0 and (X 1 + X 1 )| E 0 are different. By (3.28), we have For (iii), since the pencil |L 1 | is generated by 2S 1 and X 1 + X 1 , again writing C 0 for the intersection S 1 ∩ E 0 , it suffices to show that the restriction X 1 | E 0 , which is an anti-canonical by (3.38), is a curve which intersects C 0 transversally at the four points p i : On the other hand we have . This means (X ∩ E 0 ) ∩ C 0 = {p 1 , p 2 , p 3 , p 4 } and each of the intersections is transverse. Thus we obtain (iii). This means that the base locus of the pencil is {p 1 , p 2 , p 3 , p 4 }. Hence by Bertini's theorem, a general member of the pencil can have singularities only at the 4 points. But from the last transversality, this cannot happen. Thus a general member of the pencil is a non-singular, and it has to be an elliptic curve which is 2 : 1 over C 0 , since it is an anti-canonical curve. Thus we obtain (iv).
Later in the proof of Theorem 3.11 we will see that the intersection X 1 | E 0 is a smooth elliptic curve.
In order to reveal interesting geometric properties of the twistor space, we determine the set of C * -fixed points on the twistor space. For this, we recall that each of the curves C i , 1 ≤ i ≤ n, is C * -invariant and there are exactly two fixed points on each of these curves, one of which is the intersection with the curve C 0 . Let l i be the twistor line that passes the other fixed point on C i . These n twistor lines are C * -invariant. Then we have: Proof. The divisor S ∈ |F | is C * -invariant, and from the construction of S, the set of C *fixed points on S consists of the curve C 0 ∪ C 0 and one of the two fixed points on the curves C i and C i for 1 ≤ i ≤ n. From computations using local coordinates, it is easy to see that if i ∈ {1, 2, 3, 4}, the twistor line l i has exactly two fixed points l i ∩ (C i ∪ C i ), while if i ∈ {5, 6, . . . , n}, the twistor line l i is point-wise fixed by the C * -action. Then by Lefschetz fixed-point formula, noting χ(Z) = 2n + 4, it follows readily from this that these are all the fixed points of the C * -action on Z.
We call these (n − 4) twistor lines l i , i ∈ {5, 6, . . . , n}, as C * -fixed twistor lines. Now we can prove the main result of this subsection, which in particular implies that the twistor spaces we have discussed satisfy the property that a general fiber of its algebraic reduction is birational to an elliptic ruled surface: Theorem 3.11. Let n > 4 and Z be a twistor space on nCP 2 with C * -action, which has the rational surface S constructed in Section 2 as a real (single) member of |F | as in Proposition 3.1. Then we have: (i) A general member of the pencil |2F | is birational to a ruled surface over an elliptic curve. Moreover, the two rational curves C 0 and C 0 are ordinary double curves of these surfaces. (ii) The unique member X of the system |F −α 5 −· · ·−α n | in Proposition 3.7 is birational to a Hopf surface. Moreover, the curve C 0 is an ordinary double curve of X.
Proof. The assertions about double curves in (i) and (ii) are now obvious from Lemma 3.9 (iv). For proving the remaining assertion in (i), we first see that each member of the pencil |2F | is C * -invariant. For this we notice that each point of the image of C 0 under the twistor fibration Z → nCP 2 is fixed by the S 1 -action on nCP 2 corresponding to the C * -action on Z. This means that under holomorphic coordinates (x, y, z) in a neighborhood of a point of C 0 for which C 0 = {x = y = 0}, the S 1 -action is concretely given by (x, y, z) → (tx, ty, z), t ∈ S 1 = U (1). Therefore the C * -action induced on the exceptional divisor E 0 is trivial. Since the restriction map H 0 (Z 1 , L 1 ) → H 0 (E 0 , L 1 ) is injective as in Lemma 3.9 and is C * -equivariant from C * -invariance of C 0 , this means that each member of the pencil |2F | is C * -invariant. Take a general member Y of the pencil |2F |, and let Y 1 ∈ |L 1 | be the strict transform of Y into the blowup Z 1 . Then Y 1 is also C * -invariant, and the two intersections Y 1 ∩ E 0 and Y 1 ∩ E 0 are smooth elliptic curves on E 0 by Lemma 3.9 (iv), whose points are C * -fixed. Therefore, ifỸ 1 denotes an equivariant resolution of Y 1 (including the caseỸ 1 = Y 1 if Y 1 is non-singular), the strict transforms of the intersections Y 1 ∩ E 0 and Y 1 ∩ E 0 intoỸ 1 are also C * -fixed elliptic curves. In particular, a non-singular minimal model of the surfaceỸ 1 has a C * -fixed point. Therefore by the classification of compact complex surfaces admitting a non-trivial C * -action which has a fixed point obtained in [23] and [10], Y 1 is birational to one of the following surfaces: a rational surface, a ruled surface over a curve of genus ≥ 1, or a VII-surface. But a non-trivial C * -action on a rational surface cannot have an elliptic curve which is point-wise fixed. Also, a ruled surface over a curve of genus > 1 does not have an elliptic curve. Therefore Y 1 is birational to an elliptic ruled surface or a VII-surface. But by the classification of a non-trivial C * -action on a VII-surface with a fixed point obtained in [10], such a C * -action on a VII-surface cannot have two elliptic curves which are point-wise fixed. Hence Y 1 is birational to an elliptic ruled surface, and we obtain (i) of the theorem.
For the assertion (ii), let X 1 be the strict transform of X into Z 1 . Then the intersection X 1 ∩ E 0 is a curve whose points are fixed by the induced C * -action. Let ν :X 1 → X 1 be a desingularization which preserves the C * -action. (ν 1 might be an identity map.) We first show thatX 1 does not contain an irreducible curve D which is point-wise fixed by the C * -action and which is different from any component of the curve ν −1 (X 1 ∩ E 0 ). If the image µ 1 (ν(D)) is a curve, then it must be one of the fixed twistor line l i -s from Lemma 3.10. But if this were actually the case, we would have l i ⊂ X for some fixed twistor line l i (5 ≤ i ≤ n). This contradicts X| S = 2C 0 + C 1,2,3,4 (3.40) obtained in Proposition 3.7, since l i is disjoint from the right-hand side of (3.40) if i ∈ {5, 6, . . . , n}. Thus we have µ 1 (ν(D)) = l i for any i ∈ {5, 6, . . . , n}. Hence µ 1 (ν(D)) has to be a point (if D would exist). But if µ 1 (ν(D)) is a point, then the closure of a general orbit of the C * -action on X passes the point µ 1 (ν(D)). This cannot happen as is readily seen from the fact that the point on nCP 2 over which the point µ 1 (ν(D)) belongs must be an isolated fixed point of the S 1 -action corresponding to the C * -action on Z. Thus we have shown that the desingularizationX 1 does not contain an irreducible curve D which is point-wise fixed by the C * -action and which is different from any component of ν −1 (X 1 ∩ E 0 ).
Next by using this we show thatX 1 is a VII-surface. From Lemma 3.9 (iii), the intersection X 1 ∩ E 0 is an anti-canonical curve on E 0 . Moreover, regardless of whether E 0 ≃ F 0 or F 2 , it can be readily seen that the curve X 1 | E 0 does not have a non-reduced component. This curve X 1 | E 0 is point-wise fixed by the C * -action. Hence if X 1 is birational to a ruled surface, from the absence of the curve D above, there has to be a point on X 1 such that the closure of a general orbit on X 1 of the C * -action intersects. This cannot happen as before, and therefore X 1 cannot be a ruled surface. Hence by the above cited result of [10],X 1 is birational to a VII-surface. Then a finer classification result [10,Klassifikationssatz] means that if a VII-surface admits a non-trivial C * -action which has a fixed point, then the surface is a Hopf surface or a parabolic Inoue surface [16,3]. Both of these surfaces with C * -action has exactly two connected curves, one of which is a non-singular elliptic curve which is point-wise fixed by the C * -action. The two possibilities are distinguished by another curve; if the surface is Hopf, then the curve is a non-singular elliptic curve which is an orbit of the C * -action, and if the surface is parabolic Inoue, then the curve is a cycle of rational curves.
We show that the surface cannot be a parabolic Inoue surface. For this, we first note that the intersection X 1 | E 0 has to be a smooth elliptic curve from the above description of the C *action on VII-surfaces. This in particular means that X 1 is smooth at points on X 1 | E 0 . So the self-intersection number in X 0 makes sense. Recalling that X 1 ∈ |µ * 1 (F −α 5,6,...,n )−2E 0 |, it can be computed as 6,...,n − C 5,6,...,n , C 0 ) S = −(n − 4). (3.43) Further, in the case E 0 ≃ F 0 , we have (3.44) and the same conclusion for the case E 0 ≃ F 2 . Substituting (3.42)-(3.44) into (3.41), we obtain Therefore, if X min denotes the minimal model of X 0 , since the self-intersection number of the elliptic curve in X min corresponding to E 0 | X 1 is zero, we can obtain the minimal model X min from the desingularizationX 1 by blowing-up (−1)-curves that intersect the elliptic curve X 1 | E 0 2(n − 2) times repeatedly. In particular,X 1 has at least 2(n − 2) isolated fixed points of the C * -action.
We next show that the curves C 1 , C 2 , C 3 and C 4 (the strict transforms into Z 1 ) are (−1)curves on X 1 . For this, from the inclusions C i ⊂ S 1 ⊂ X 1 , we obtain an exact sequence Hence from the last exact sequence, we obtain . From this, we obtain . Hence by using the inclusion C i ⊂ X 1 ⊂ Z 1 , we can derive N C i /X 1 ≃ O(−1). Thus each C i is a (−1)-curve on S 1 . Hence we can contract these four curves on S 1 , and consequently the self-intersection number of the curve E 0 | X 1 becomes 2(2 − n) + 4 = 2(4 − n) on the blowing-down.
By Lemma 3.10, as X ≃ F − α 5,6,...,n , we have X. l i = 2, and the intersection X ∩ l i consists of two points for any i ∈ {5, 6, . . . , n} in a generic situation, and each of the two intersection points is joined with the elliptic curve E 0 | X 1 by a (−1)-curve. Moreover if we blow-down all these 2(n − 4)-curves, the self-intersection number of the curve E 0 | X 1 becomes precisely zero. If X ∩ l i happened to consist of one point (for some i), this point is an ordinary double point, and by resolving it we obtain a (−2)-curve. This curve has exactly two fixed points of the C * -action, and one of the two points intersects (−1)-curve which joins the (−2)-curve with the elliptic curve E 0 | X 1 . Therefore again by contracting these (−1)-curve and (−2)-curve subsequently, we again obtain that the contribution from the fixed twistor line l i for increasing the self-intersection number of the elliptic curve E 0 | X 1 is again 2.
Thus all the intersections X ∩l i are joined by exceptional curves for obtaining the minimal model X min . On the other hand, since a general member of the pencil |2F | is irreducible, the special member X + X is connected, and therefore X ∩ X has to be a curve, which is of course C * -invariant. The classification result [10,Klassifikationssatz] means that this curve is either a smooth elliptic curve or a cycle of rational curve, depending on whether X 1 is birational to a Hopf surface or a parabolic Inoue surface respectively. But the latter cannot happen since X ∩ X cannot have any fixed point of the C * -action anymore. Hence X 1 , and so X, is birational to a Hopf surface, as asserted.
The intersection X ∩ X has the following characteristic properties: Proposition 3.12. The intersection X ∩ X is a smooth elliptic curve, which is a single orbit of the C * -action on Z. Moreover, X ∩ X is homologous to zero.

3.3.
Elimination of the base locus of the pencil |2F |. As in Proposition 3.8, the anticanonical system |2F | of the present twistor spaces has base curves. In this subsection, we give a sequence of blowups which completely eliminates the base locus. This gives an explicit holomorphic model for the algebraic reduction of Z. We begin with the following easy Proposition 3.13. Denoting C 1 , . . . , C 4 for the strict transforms of the original curves in Z into the blowup Z 1 , for the line bundle Moreover, a general member of the pencil |L 1 | is non-singular.
Thus the double curves of a general member of the pencil |2F | are completely resolved by the blowup µ 1 . For the resolved divisors, we have the following Proposition 3.14. Let Y ∈ |2F | be a general anti-canonical divisor of Z, and Y 1 its strict transform into Z 1 . Then we have . Moreover, Y 1 contains the curves C i and C i (i ∈ {1, 2, 3, 4}) as (−1)-curves.
Since this can be shown easily by using adjunction formula and the standard exact sequence associated to the inclusions C i ⊂ S 1 ⊂ Z 1 and C i ⊂ Y 1 ⊂ Z 1 , we omit a proof.
Next we investigate the way how general members of the pencil |2F | intersect along the curve C i for i ∈ {1, 2, 3, 4}: Proposition 3.15. A general member of the pencil |2F | is tangent to the divisor X along the curve C i ∪ C i for any i ∈ {1, 2, 3, 4} to the second order.
Proof. Let x be a defining section of the divisor X ∈ |F − α 5,6,...,n |. Similarly, let s ∈ H 0 (F ) be a non-zero element, so that (s) = S. Then since S is non-singular, and since X is nonsingular at least on the curve C i (i ∈ {1, 2, 3, 4}) except possibly at the intersection point C i ∩ C 0 , from (3.31), we can take s and x as a part of local coordinates of Z around the point. Then since the pencil |2F | is generated by 2S and X + X, a defining section of a general member of the pencil is of the form xx = cs 2 , c ∈ C * . (3.48) Moreover, from (3.31), the divisor X does not intersect C 1,2,3,4 , and therefore x = 0 for any point of C 1,2,3,4 . From the equation (3.48), this means that a general member of |2F | is tangent to X along C 1,2,3,4 to the second order.
Note however that the intersection of the divisors S and X is transverse along C 1,2,3,4 . Proposition 3.15 means that, even after we blow up Z 1 at the base curves (3.47), the strict transform of the pencil still has base curves (which is identified with the original base curve (3.47)). Because the tangential order of general members of the pencil is two, it is not difficult to show that if we blow up the base curves (3.47), then the resulting pencil has no base point. In summary, we need the next 3 steps for the elimination of the base locus of |2F | on Z: 1. The blowup µ 1 : Z 1 → Z at the curves C 0 ∪ C 0 . This resolves the double curves of general members of the pencil |2F |. 2. The blowup Z 2 → Z 1 at the curves C 1 ∪ C 2 ∪ C 3 ∪ C 4 and the conjugate curves.
This transforms the touching situation of general members along these eight curves into transversal intersections. 3. The blowup Z 3 → Z 2 at the transversal intersections of general members of the pencil. This blowup separates the intersections completely.

4.
Existence of a twistor space on 5CP 2 with a pencil of K3 surfaces In this section, we show that, on 5CP 2 , there exists a twistor space Z satisfying a(Z) = 1 whose general fiber of the algebraic reduction is a K3 surface. A proof proceeds in a similar way to the case of elliptic ruled surfaces employed in Sections 2 and 3, but the proof is pretty easier in that we do not need complicated calculations for cohomology groups like we did in Section 3.1.
First we construct a rational surface S with c 2 1 (S) = −2 which will be included in a twistor space on 5CP 2 as a real member of the linear system |F |. For this, we start from the product surface CP 1 × CP 1 . On it we put a real structure that is the product of the complex conjugation and the anti-podal map. We mean by a (k, l)-curve on CP 1 × CP 1 a curve of bidegree (k, l). We first take distinct two non-real (1, 0)-curves. We suppose that these are not mutually conjugate to each other. Next we take distinct two (0, 1)-curves, which are not mutually conjugate to each other. (Each one cannot be real from the choice of the real structure on CP 1 × CP 1 .) Let D be the sum of all these four curves. This is a reduced, non-real (2, 2)-curve. In Figure 3, the components of D are displayed as solid lines. Then the conjugate curve D is also a reduced, non-real (2, 2)-curve, whose components are displayed as dashed lines in  Proof. Let C be the strict transform of the curve D into S, and C 5 := ǫ −1 (p) the exceptional curve over p. Then a double point p of D is separated by ǫ, and C is a chain of four rational curves. The self-intersection numbers of these four rational curves are readily seen to be −3, −2, −2, −3 (4.1) respectively. Of course the same is true for the other chain C. Note that C ∩ C = ∅. Moreover, noting C ≃ ǫ * D − 2C 5 , we easily obtain linear equivalences From this we obtain 2K −1 S ≃ C + C. Thus the sum C + C belongs to |2K −1 S |. Since the intersection matrices for C and C may be readily seen to be negative definite from (4.1), by a property of the Zariski decomposition, we obtain that H 0 (2mK −1 S ) ≃ C for any m > 0. The other assertion H 0 ((2m − 1)K −1 S ) = 0 can be shown in a similar way to the same assertion in Proposition 2.2 (ii). So we omit the detail.
Similarly to what we did in Section 3.1, let α 5 ∈ H 2 (Z, Z) be the cohomology class determined from the curve C 5 on S. So we have α 5 | S = C 5 − C 5 in H 2 (S, Z). The next proposition amounts to Propositions 3.1 and 3.6 in the case for twistor spaces studied in Section 3, but the proof is much easier basically because n = 5. (iii) The anti-canonical system |2F | of Z is a pencil, and is generated by a non-reduced member 2S and the reducible member X + X. (iv) a(Z) = 1, and the algebraic reduction of Z is induced by the meromorphic map associated to the pencil |2F |.
Proof. The item (i) immediately follows from |K −1 S | = ∅ in Proposition 4.1. For (ii), we consider the standard exact sequence Then from the fact that (C 5 ) 2 S = −1 on S, exactly in the same way to the final part of the proof of [14,Proposition 3.3], we obtain, by Riemann-Roch formula and Hitchin's vanishing theorem, that H 1 (O Z (−α 5 )) = 0. Hence from the cohomology exact sequence of (4.3) we obtain an isomorphism H 0 (F −α 5 ) ≃ H 0 ((F − α 5 )| S ). Moreover, from an isomorphism in (4.2), we have ). Irreducibility of the unique member X of |F − α 5 | again follows from |F | = {S} as in the proof of Proposition 3.7 (ii). This proves the assertion (ii) of the proposition.
Next we will determine structure of a general member of the pencil |2F |. For this we first see Proof. Since the pencil |2F | is generated by the two divisors 2S and X +X as in Proposition 4.2, we have Bs |2F | = S ∩ (X ∪ X). This is the chains C ∪ C by Proposition 4.2 (ii).
Thus the chains C ∪ C play a similar role to the curves (2.5) in the case of the twistor spaces discussed in Section 3. Main difference is that in the present case, the restriction X| S is a reduced divisor on S, while it is non-reduced for the other type of twistor spaces as in (3.31). This yields the following difference on the structure of a general member of the pencil |2F |: Theorem 4.4. Let Z be a twistor space on 5CP 2 which includes the surface S as a real member of |F | as in Proposition 4.2. Then a general member of the pencil |2F | is birational to a K3 surface. In particular, a general fiber of the algebraic reduction of Z is birational to a K3 surface.
Proof. Recalling that Bs |2F | = C ∪ C, let µ 1 : Z 1 → Z be the blowup of Z at the two chains C ∪ C, and E 1 ∪ E 1 the exceptional divisor over C ∪ C. The restrictions µ 1 | E 1 : E 1 → C and µ 1 | E 1 : E 1 → C are CP 1 -bundle maps, and Z 1 has one ordinary double point on each of the fibers over a double point of C and C. Then as X| S = C as in Proposition 4.2, the divisors X and S are already completely separated in Z 1 . Define Then from Proposition 4.3 we have an isomorphism |L 1 | ≃ |2F |. Since the pencil |2F | is generated by 2S and X + X, the pencil |L 1 | is generated by, this time, where S 1 , X 1 and X 1 are strict transforms of S, X and X into Z 1 respectively. Since X and S are already separated in Z 1 as above, we have Both X 1 ∩ E 1 and X 1 ∩ E 1 are chains of curves which are isomorphic to the original chains C and C respectively by µ 1 .
As above, Z 1 has one ordinary double point over each of the double points of the chains C and C. Let q ∈ C be any one of the three double points of the chain C, and suppose that the divisor X (in Z) is singular at q. Then the strict transform X 1 passes the singular point of Z 1 over the point q iff X has a singularity at q, as one can readily be seen from computations using local coordinates. Regardless of whether q ∈ Sing X or not, let be the blowup at the base curve (4.4) of the pencil |L 1 |. Let E 2 and E 2 be the exceptional divisors over the chains X 1 ∩ E 1 and X 1 ∩ E 1 respectively. While Z 2 has singularities, the blowup µ 2 already eliminates the base locus of the pencil |2F | and |L 1 |. Namely, if we put Type of the singularities of Z 2 depends on whether q ∈ Sing X. If q ∈ Sing X, over the point q, the variety Z 2 has exactly two ordinary double points, one of which appears from the first blowup µ 1 : Z 1 → Z, and the other of which appears from the second blowup µ 2 : Z 2 → Z 1 . So both of the singularities admit a small resolution. On the other hand, if q ∈ Sing X, the ordinary double point of Z 1 splits by the blowup µ 2 : Z 2 → Z 1 into two singularities, both of which can be locally written by an equation xy = zw 2 in C 4 (x, y, z, w) in local coordinates. This singularity also admits a small resolution, which may be concretely obtained by blowing-up the plane {x = w = 0} in the last expression. Taking this small resolution for each of the two singularities, both are transformed into an ordinary double point. So these again admit a small resolution. Let be any one of the small resolutions obtained this way, which preserves the real structure. We put L 3 := µ * 2 L 2 . Then since Bs |L 2 | = ∅ as above, we have Bs |L 3 | = ∅. Let Φ 3 : Z 3 −→ CP 1 be the morphism induced by the pencil Bs |L 3 |. From the construction, there is a natural identification between the two pencils |2F | on Z and |L 3 | on Z 3 , and members of |2F | are naturally identified with fibers of the morphism Φ 3 . Hence in order to show the theorem, it is enough to see that a general fiber of Φ 3 is a K3 surface.
For this, we write µ for the composition µ 1 • µ 2 • µ 3 , and in the following E 1 and E 2 mean the strict transforms of the original exceptional divisors into Z 3 . Then from the usual transformation formula for the canonical bundle under a blowup, outside the 6 singularities of the chains C ∪ C in Z, we have Since this is valid outside codimension two locus, by Hartogs' theorem, (4.5) is valid on the whole of Z 3 . But the right-hand side of (4.5) is exactly the line bundle L 3 . Hence a general fiber of Φ 3 is an anti-canonical divisor on Z 3 . Hence for a general fiber Y 3 of Φ 3 , we have To complete a proof, it suffices to show H 1 (Y 3 , O) = 0. For this we consider the standard exact sequence . The latter is of course isomorphic to H 2 (O Z ) from birational invariance. Moreover, we have H 2 (O Z ) = 0 since this is again dual to H 1 (K Z ), which vanishes by Hitchin's vanishing theorem. Thus Y 3 is a K3 surface, as asserted.
Of course, the morphism Φ 3 : Z 3 → CP 1 obtained above may be regarded as a holomorphic family of K3 surfaces. The two fibers corresponding to the divisors 2S and X + X are degenerate fibers of this fibration. Thus the twistor spaces in Theorem 4.4 provide degeneration of K3 surfaces under a non-Kähler setting. In this regard, it might be interesting to see if how one can make these degenerations to be semi-stable, and after that, which type of degenerations occur [5,22]. We note that the morphism Φ 3 is induced by the anti-canonical system of Z 3 , and therefore the canonical bundle of Z 3 is trivial around the singular fibers.

5.
Existence of the twistor spaces and dimension of the moduli spaces 5.1. Existence. In this subsection we show that the twistor spaces investigated in Sections 3 and 4 actually exist. Recall that both types of the twistor spaces are characterized by the complex structure of the (unique) member S of |F |. Thus it is enough to see that for such a prescribed surface S, there exists a twistor space Z having S as a member of |F |. We show this by deforming known example of a twistor space Z 0 on nCP 2 having a divisor belonging to |F Z 0 |, in such a way that the divisor is preserved and deformed into the prescribed surface S.
Let n ≥ 5 be any integer. First we show the existence of a twistor space Z on nCP 2 enjoying the assumptions in Theorem 3.11. For this, we consider the twistor space of a self-dual metric on nCP 2 constructed by Joyce [17]. These metrics admit a T 2 -action, and its natural lift to the twistor space generates a holomorphic (C * × C * )-action. By the result of Fujiki [7], the closure of a generic orbit of the (C * × C * )-action is a non-singular toric surface, and the toric surface is uniquely determined from (the isomorphism class of) the T 2 -action on nCP 2 . Conversely, the the toric surface uniquely determines the T 2 -action on nCP 2 of a Joyce metric. Thus in order to specify the T 2 -action of the Joyce metric, it is enough to specify the toric surface. For specifying the toric surface, we consider the product CP 1 × CP 1 equipped with the real structure used in the beginning of Section 4. We again take a reducible (2, 2)-curve consisting two (1, 0)-curves and two (0, 1)-curves, but this time we take them in a way that the (2, 2)-curve becomes real. We make an iterated blowup at each of the four double points in the way as displayed in Figure 4. Here, the arrows mean to blowup the exceptional curve at the point which corresponds to the direction indicated by the arrow. Let S J be the toric surface obtained by this blowup. Since we blowup CP 1 × CP 1 precisely 2n times, we have K 2 = 8 − 2n for S J . Also S J admits a natural real structure as a lift from CP 1 × CP 1 . Then there exists a Joyce metric on nCP 2 whose twistor space has the toric surface S J as a T 2 -invariant member of the system |F |.
We consider a deformation of S J preserving the real structure in a way displayed in Figure 4. Namely we shift the pair of a point and one arrow on lower left in the vertical direction, and at the same time, displace all vertical (n − 4) arrows on upper left in the vertical direction to split them into distinct (n − 4) points. We move the points on the right side as determined by the real structure from the left side.
Let S be the surface obtained from CP 1 × CP 1 by the above blowing up. Note that if n = 4, S is nothing but the relatively minimal elliptic surface S 0 constructed in Section 2. We also note that this deformation (of S J into S) preserves a C * -action; we just need to consider the product of a standard C * -action on horizontal direction and the trivial C *action on the vertical direction. In summary the surface S in Theorem 3.11 is obtained from the toric surface S J by a small deformation preserving the real structure and a C * -action.
Once this is obtained, it is almost automatic to show the existence of a twistor space Z on nCP 2 having S as a real member of the system |F |. For this, letting Z 0 be the twistor space of a Joyce metric having the divisor S J as above, we consider deformation of the pair (Z 0 , S J ) preserving the C * -action and the real structure. Then because H 2 (Z 0 , Θ Z 0 (−S J )) = 0 from [2, Lemma 1.9], the divisor S J is co-stable in Z J [15,Theorem 8.3]. This proves the existence of a twistor space on nCP 2 satisfying the assumptions in Theorem 3.11. Remark 5.1. Because the twistor spaces of Joyce metrics do not contain a non-algebraic surface, one may wonder from where a VII surface in the present twistor space comes from. An answer is that on the twistor space of a Joyce metric, there exist two degree-one divisors D and D ′ such that the sum D + D ′ belongs to the same cohomology class as the divisor X, and under the small deformation the sum becomes the (irreducible) VII surface X.
The existence of a twistor space satisfying the assumptions in Theorem 4.4 can be shown in a similar way, with slightly more effort. We again start from a toric surface, for which we denote S ′ J this time, which is contained in the twistor space of a Joyce metric as a T 2invariant member of |F |. This toric surface S ′ J is obtained from CP 1 ×CP 1 by blowing up 10 times as displayed in Figure 5 (a) and (b). Namely we first blowup CP 1 ×CP 1 at the 4 double points (Figure 5 (a)) of the reducible (2, 2)-curve, and then blowup the resulting surface 6 times at the points and arrows in Figure 5 (b). Let S ′ J be the resulting toric surface. We have K 2 = −2 for this toric surface. Next we move each arrow to a point in that direction ( as indicated in Figure 5 (c)). By blowing up these 6 points we obtain a non-toric surface as a small deformation of S ′ J . This surface still admits a C * -action. In terms of the initial surface CP 1 × CP 1 , the way of blowing up for obtaining this surface with C * -action can be indicated as in Figure 5 (d). Next we deform this configuration in a way that the three arrows from the point on lower left move along lines in each direction, in a way as displayed in Figure 5 (e). Here, 'line' means a line in the usual sense, viewing CP 1 × CP 1 as a compactification of C × C = C 2 . Let S be the surface obtained from CP 1 × CP 1 by blowing up the 10 points in Figure 5 (e). Then this is exactly the surface S constructed at the beginning of Section 4. Hence we have seen that the surface S is obtained from the toric surface S ′ J by first deforming it preserving a C * -action, and then further deforming the resulting surface by a small deformation. Since the cohomology vanishing property is preserved under small deformation, by the same reasoning to the above case, we again obtain that there actually exists a twistor space Z on 5CP 2 having the surface S as a member of |F |. Thus we have shown the existence of a twistor space satisfying the assumption of Theorem 4.4.

5.2.
Dimension of the moduli spaces. In this subsection we calculate dimension of the moduli spaces of the twistor spaces discussed in Sections 3 and 4. First let n ≥ 5 and Z be the twistor space on nCP 2 studied in Section 3, and S ∈ |F | the divisor which was supposed to exist. As in Proposition 3.1, the divisor S is the unique member of the system |F |. Hence if (Z ′ , S ′ ) is another pair of a twistor space and the divisor on it enjoying the assumption in Theorem 3.11, and if S ≃ S ′ , we have Z ≃ Z ′ . Moreover, as long as the twistor space Z is obtained as a small deformation of the twistor space of a Joyce metric as in the last subsection, we have H 2 (Θ Z (−S)) = 0 by upper semi-continuity, and therefore by [15,Theorem 8.3], any small deformation of S may be realized by a deformation of the pair (Z, S). Hence in order to compute the dimension of the moduli space, it is enough to compute the dimension of the moduli space for which the complex structure of the divisor S is fixed.
For this, we consider an exact sequence 0 −→ Θ Z (−S) −→ Θ Z,S −→ Θ S −→ 0. (5.1) As we have been assuming that the C * -action on S extends to the twistor space Z, together with H 2 (Θ Z (−S)) = 0 as above, we get an exact sequence Then as χ(Θ Z ) = 15 − 7n, h 0 (Θ Z ) = 1, and H 2 (Θ Z ) = 0 by upper semi-continuity, we have h 1 (Θ Z ) = 7n − 14. On the other hand, we readily have h 1 (K −1 S ) = 2n − 9. Therefore from the exact sequence (5.3) we obtain h 1 (Θ Z,S ) = 5n − 5. Moreover we have χ(Θ S ) = 6 − 4n (as S is a blow-up of CP 1 × CP 1 at 2n points). Hence since h 0 (Θ S ) = 1 and H 2 (Θ S ) = 0, we get h 1 (Θ S ) = 4n − 5. Therefore from the cohomology exact sequence of (5.1), under an assumption that the C * -action on S always extends to that on the twistor space Z, we obtain h 1 (Θ Z (−S)) = n. Therefore, we obtain that the moduli space of our twistor spaces with the complex structure of the divisor S fixed is n-dimensional. On the other hand, as in Section 2, our surface S is obtained from the relatively minimal elliptic surface S 0 by blowing up 2(n − 4) points on the rational curve C 0 . Moreover the moduli space for S 0 is identified with the moduli space of elliptic curves with a real structure, and therefore real 1-dimensional. The contribution from the choice of the 2(n − 4) points is real 2(n − 4)-dimensional, so the moduli space for our surface S is real (2n − 7)-dimensional. Thus under the above assumption on the extension of the C * -action from S to Z, the moduli space of our twistor space is (3n − 7)dimensional. Note that the assumption on the extension of C * -action might look somewhat strong, but as we argued in Section 3, we can derive the conclusion a(Z) = 1 without using the presence of C * -action, and just under the presence of the special divisor S.
We also remark that the natural C * -action on the cohomology group H 1 (K −1 S ) ≃ C 2n−9 has exactly 1-dimensional subspace on which C * -act trivially. This seems to mean that the divisor S disappears under a generic small deformation of Z which preserves C * -action. It might be interesting to ask what is the algebraic dimension of a twistor space obtained as these C * -equivariant deformation.
Next we compute the dimension of the moduli space of the twistor spaces investigated in Section 4. First by the same reason to the case of twistor spaces in Section 3 discussed above, the complex structure of the twistor space Z deforms if that of the divisor S deforms. But in the present case, the complex structure of S cannot be deformed, as is readily seen from the construction of the surface S. Further we have H 0 (K −1 S ) = 0, and so the exact sequence (5.3) is again valid, as long as the twistor space is obtained as a small deformation of the twistor space of a Joyce metric as in the last subsection. Moreover, for these twistor spaces we have h 1 (Θ Z ) = −χ(Θ Z ) = 7 · 5 − 15 = 20 and h 1 (K −1 S ) = 2 · 5 − 9 = 1. Hence from the exact sequence (5.3) we obtain h 1 (Θ Z,S ) = 19. Noting that the exact sequence (5.2) is also valid and noticing h 1 (Θ S ) = −χ(Θ S ) = 4 · 5 − 6 = 14, we obtain h 1 (Θ Z (−S)) = 19 − 14 = 5. Thus the moduli space of the twistor spaces in Theorem 4.4 is 5-dimensional.