Quotient spaces of K3 surfaces by non-symplectic involutions fixing a curve of genus 8 or more

Theorem 1.1

Let X be a K3 surface and let g be a non-symplectic involution of X such that the rank of Pic(X/〈g〉) is at most 3. We have the following.

1. Fix(g) contains a smooth curve C such that the genus of C is at least 8.

2. If the rank of Pic(X/〈g〉) is 3, then there is a closed point x ∈ 𝔽_l such thatX∉Cland X /〈g〉 ≅ Blow_x𝔽_l where l = 1, 2, or 4.

For a primitive non-symplectic automorphism g of X, the quotient space X /〈g〉 is an Enriques surface or a rational surface, and it is an Enriques surface if and only if ord(g) = 2 and Fix(g) = 0; see the classification of complex surfaces in [5]. If the quotient space X/〈g〉 is smooth, then ord(g) = 2, 3, 4 or 6 by [11, Theorem 1.5]. In [12, Theorems 4.3, 4.9, and 5.4], the author shows that if X/〈g〉 is smooth and ord(g) ≥ 3, then X/〈g〉 is reconstructed from the action of g on rational curves on X and Fix(gⁱ) for 1 ≤ i < ord(g). In this paper, for the case where ord(g) = 2 and Fix(g) contains a curve C of genus 8 or more, we show that X /〈g〉 is reconstructed from the action of g on rational curves on X and Fix(g). In the case where Fix(g) contains a curve C of genus 7 or less, the number of curves in Fix(g) may be too large to handle, so the author did not investigate it. In [11; 21] finite abelian groups are classified which act faithfully on a K3 surface and such that the quotient space is smooth. In [9; 11] the authors classify finite abelian groups which act faithfully on an Enriques surface and such that the quotient space is smooth. In general, it is not known whether the quotient space is reconstructed from the structure and action of the group. The author thinks that the answer is affirmative.

Our other main theorems are Theorem 1.2, Theorem 1.3 and Theorem 1.4.

Theorem 1.2

Let X be a K3 surface and let g be a non-symplectic involution of X such that Fix(g) contains a curve C⁽¹⁰⁾of genus 10. Then the quotient space X /〈g〉 is ℙ²or 𝔽₄. Furthermore, we have the following.

1. X/〈g〉 ≅ ℙ²if and only if Fix(g) = C⁽¹⁰⁾.

2. X /〈g〉 ≅ 𝔽₄if and only if Fix(g) = C⁽¹⁰⁾ ⊔ R where R is a smooth rational curve.

Theorem 1.3

Let X be a K3 surface and let g be a non-symplectic involution of X such that Fix(g) contains a curve C⁽⁹⁾of genus 9. Then the quotient space X /〈g〉 is either 𝔽_n with n = 0, 1, 2 or Blow_x𝔽₄and x ∉ C₄. Furthermore, we have the following.

1. X /〈g〉 ≅ 𝔽₁if and only if Fix(g) = C⁽⁹⁾and there is a smooth rational curve R on X such that g(R) = R or (g(R) ⋅ R) = 1.

2. X /〈g〉 ≅ 𝔽₂if and only if Fix(g) = C⁽⁹⁾and there is a smooth rational curve R on X such that (g(R) ⋅ R) = 0.

3. X /〈g〉 ≅ 𝔽₀if and only if Fix(g) = C⁽⁹⁾and there is no smooth rational curve R on X such that g(R) = R or (g(R) ⋅ R) = 0 or 1.

4. X /〈g〉 ≅ Blow_x𝔽₄where x ∉ C₄if and only if Fix(g) = C⁽⁹⁾ ⊔ R where R is a smooth rational curve.

Theorem 1.4

Let X be a K3 surface and let g be a non-symplectic involution of X such that Fix(g) contains a curve C⁽⁸⁾of genus 8. Then the quotient space X /〈g〉 is either Blow_x𝔽_n for n = 1, 2 with x ∉ C_n or Blow_x,y𝔽₄and x, y ∉ C₄. Furthermore, we have the following.

1. X/〈g〉 ≅ Blow_x𝔽₁where x ∉ C₁if and only if Fix(g) = C⁽⁸⁾and there are two smooth rational curves R₁and R₂on X such that (g(R₁)∪ R₁)∩ (g(R₂)∪ R₂) = 0, and g(R_i) = R_i or (g(R_i)⋅ R_i) = 1 for i = 1, 2.

2. X /〈g〉 ≅ Blow_x𝔽₂where x ∉ C₂if and only if Fix(g) = C⁽⁸⁾and there are two smooth rational curves R₁and R₂on X such that (g(R₁)∪ R₁)∩ (g(R₂)∪ R₂) = 0, (g(R₁)⋅ R₁) = 0, and g(R₂) = R₂or (g(R₂)⋅ R₂) = 1.

3. X /〈g〉 ≅ Blow_x,y𝔽₄where x, y ∉ C₄if and only if Fix(g) = C⁽⁸⁾ ⊔ R where R is a smooth rational curve.

Section 2 contains preliminaries: we describe the basic facts about Galois covers of algebraic varieties. We show Part (2) of Theorem 1.1 in Theorem 2.12. Part (1) of Theorem 1.1 is a consequence of Theorems 1.2, 1.3, and 1.4. In Section 3 we give examples of K3 surfaces and non-symplectic involutions such that the quotient spaces are Blow_x𝔽_l or Blow_x,y𝔽₄ where x, y ∉ C_l and l = 1, 2, or 4. Examples of the case where the quotient spaces are Hirzebruch surfaces are already known, see for example [10; 11].

In Section 4 we prove Theorems 1.2, 1.3 and 1.4 .We prove them all in a similar way. The proof becomes more complicated as the genus of curves in the fixed point set of a non-symplectic involution becomes smaller. First, we prove the case of genus 10. Quoting this result, we prove the case of genus 9. Finally, quoting the results for genus 10 and 9, we prove the case of genus 8. In Section 5, we show that if a K3 surface X has a non-symplectic involution g such that X /〈g〉 ≅ 𝔽₂ or Blow_x𝔽₂ where x ∉ C₂, then X has another non-symplectic involution h different from g.

Definition 2.1

Let f : X → M be a branched covering where M is a complex manifold and X is a normal complex space. We call f : X → M a Galois cover if there is a subgroup G of the automorphism group Aut(X) of X such that X /G ≅ M and f : X → M is isomorphic to the quotient morphism q : X → X /G ≅ M, and we call G the Galois group of f : X → M. Furthermore, if G is an abelian group, then we call f : X → M an abelian cover.

Definition 2.2

Let f : X → M be a finite branched covering where M is a complex manifold and X is a normal complex space, let Δ be the branch locus of f and let B₁, . . . , B_k be the irreducible components of Δ with codimension one. We assume that for every j, the ramification index at each irreducible component of f⁻¹(B_j) is the same, which is denoted as b_j . Then the effective divisor

is called the branch divisor of f : X → M.

The following theorem, which is called the ramification formula of the canonical divisor, will be useful.

Theorem 2.3

([13, Theorem 6.1.7]). Let X be a smooth algebraic variety, let G be a finite subgroup of Aut(X) such that X /G is smooth, and letB:=∑i=1kb_iB_i be the branch divisor of the quotient morphism q : X → X /G. Then

where K _X (respectively K _X/G) is the canonical divisor of X (respectively X /G).

Since the canonical line bundle of a K3 surface is trivial, i.e. K _X = 0, the branch divisor B must satisfy the equation KX/G+∑i=1rbi−1biBi=0in Pic_ℚ(X/G). The branch divisor B is inferred from K _X/G. Theorem 2.4 below is a slightly more convenient version of Theorem 2.3.

Theorem 2.4

Let X be a K3 surface, let G be a finite subgroup of Aut(X) such that X /G is a smooth rational surface, and let B be the branch divisor of the quotient morphism q : X → X/G. We consider a birational morphism f : X /G → S where S is a smooth surface, and we setf∗B=∑i=1lbiB˜iwhereB˜iis an irreducible curve on S for i = 1, . . . , l. Then

Proof. Let E₁, . . . , E_m be the exceptional divisors of f : X /G → S. Since X /G and S are smooth and f is a birational morphism, we get

and there are positive integers a_i for i = 1, . . . , m such that

We set B=∑i=1lbiBi+∑j=1kbj′Bj′where B_i is not an exceptional divisor of f for i = 1, . . . , l and Bj′is an exceptional divisor of f for j = 1, . . . , k. Then there are non-negative integers c_i,1, . . . , c_i,m such that

for i = 1, . . . , l. By Theorem 2.3,

Since Pic(X/G)=f∗Pic(S)⨁i=1mZEi,we have

Since S is rational, Pic(S) is torsion free. Therefore, 0=KS+∑i=1lbi−1biB˜i.2

For a set S, we write the cardinality of S as |S|. Let X be an algebraic variety, and let G be a finite subgroup of Aut(X). We denote by Y := X/G the quotient space, and by q : X → Y the quotient morphism. Then q : X → Y is the Galois cover whose Galois group is G. The branch locus Δ of q : X → Y is the subset of Y given by

It is known that Δ is an algebraic subset of codimension one if Y is smooth; see [23]. Let B₁, . . . , B_k be irreducible components of Δ whose dimension is one. For j=1,…,k let Dj,1,…,Dj,ljbe irreducible components of q⁻¹(B_j). We set

for s = 1, . . . , l_j . If X is smooth, then GDj,sis a cyclic group. Since B_j is irreducible, we get that for 1 ≤ j_s < j_t ≤ l_j there is an automorphism f ∈ G such that f(D_j,s) = D_j,t . Since f(D_j,s) = D_j,t , we have f∘GDj,s∘f−1=GDj,t.In particular, if G is an abelian group, then GDj,s=GDj,t for 1≤js\ltjt≤lj.The ramification index at D is

and B := b₁B₁ + ⋅⋅ ⋅ + b_kB_k is the branch divisor of q : X → Y.

The following theorem is important for checking the structure of G from the branch divisor.

Theorem 2.5

([11, Theorem 2.8]). Let X be a K3 surface with a finite subgroup G of Aut(X) such that the quotient space X/G is smooth. LetB:=∑i=1kbiBibe the branch divisor of the quotient morphism q : X → X/G. We putp∗Bi=∑j=1lbiCi,jwhere C_i,jis an irreducible curve for j = 1, . . . , l. LetGCi,j:=g∈G:g∣Ci,j=idCi,jand let G_i be the subgroup of G generated byGCi,1,…,GCi,l.Then the following holds.

1. Let I ⊂ {1, . . . , k} be a subset. If (X/G)\⋃_i∈IB_i is simply connected, then G is generated by {G_j}_{j∈{1,...,k}\I}.

2. The group G C i , j  is  Z / b i Z , and  G C i , j is generated by a primitive non-symplectic automorphism of order b_i.

3. If G is abelian, then there exists an automorphism g ∈ G such that⋃j=1lCi,j⊂Fix(g),and hence the C_i,jare pairwise disjoint.

4. If the self intersection number (B_i ⋅ B_i) of B_i is positive, then l = 1 and the genus ofCi,1 is |G|2bi2Bi⋅Bi+1.

Theorem 2.6

Let X be a K3 surface, let g be a non-symplectic involution of X, and letB:=∑i=1kbiBibe the branch divisor of the quotient morphism q : X → X/〈g〉. Then the following holds.

1. b ₁ = ⋅⋅ ⋅ = b_k = 2.

2. B_i ∩ B_j = 0 for 1 ≤ i < j ≤ k.

3. (B_i ⋅ B_j) = 0 for 1 ≤ i < j ≤ k.

4. If (B_i ⋅ B_i)> 0 for some i, then (B_j ⋅ B_j)< 0 for j ≠ i.

5. q ⁻¹(B_i) is a smooth curve of genus18Bi⋅Bi+1 for 1≤i≤k.In particular, if (B_i ⋅ B_i)< 0 then (B_i ⋅ B_i) = −4 and q⁻¹(B_i) is a smooth rational curve.

Proof. Since the degree of q is two, b_i = 2 and q⁻¹(B_i) is irreducible for i = 1, . . . , k. Since q⁻¹(B_i)⊂ Fix(g) and X is smooth, q⁻¹(B_i) is smooth for i = 1, . . . , k and q⁻¹(B_i)∩ q⁻¹(B_j) = 0 for 1 ≤ i < j ≤ k. Since q⁻¹(B_i)∩ q⁻¹(B_j) = 0, we get that B_i ∩ B_j = 0, and hence (B_i ⋅ B_j) = 0 for 1 ≤ i < j ≤ k. We assume that (B_i ⋅ B_i) > 0 for some i. By the Hodge index theorem and (B_i ⋅ B_j) = 0 for 1 ≤ i < j ≤ k, we get that (B_j ⋅ B_j) < 0 for j ≠ i. By Theorem 2.5 and b_i = 2, the genus of q−1Bi is 18Bi⋅Bi+1 for i=1,…,k.We assume that (B_i ⋅ B_i) < 0. Since q^∗B_i = 2q⁻¹(B_i) and the degree of q is 2, (B_i ⋅ B_i) = 2(q⁻¹(B_i)⋅ q⁻¹(B_i)) and hence (q⁻¹(B_i)⋅ q⁻¹(B_i)) < 0. Since X is a K3 surface and q⁻¹(B_i) is irreducible, (q⁻¹(B_i)⋅ q⁻¹(B_i)) = −2 and hence (B_i ⋅ B_i) = −4. □

Let X be a K3 surface and let g be a non-symplectic automorphism of X. If Fix(g) ≠ 0, then the quotient space X /〈g〉 is rational. All non-singular rational surfaces are obtained by blow ups of ℙ² or 𝔽_n. For a Hirzebruch surface 𝔽_n where n ∈ ℤ_≥0, there is the unique fibration ϕ_n : 𝔽_n → ℙ¹ with a section C_n such that ϕl−1(s)is a smooth rational curve for any s ∈ ℙ¹. Furthermore, let Fn:=ϕn∗s for s∈P1. Then (C_n ⋅ F_n) = 1, (F_n ⋅ F_n) = 0, (C_n ⋅ C_n) = −n,

Note that for n=0,Cn=pr1∗OP1(1) and Fn=pr2∗OP1(1),and for n ≥ 1, C_n is the unique curve on 𝔽_n such that the self intersection number is negative, and F is the fibre class of ϕ_n : 𝔽_n → ℙ¹.

Lemma 2.7

([8, Chapter V Corollary 2.18]). Let 𝔽_n be a Hirzebruch surface where n ≠ 0, and let C^' ⊂ 𝔽_n be an irreducible curve. Then one of the following holds:

1. C ^' = C_n.

2. C ^' = F in Pic(𝔽_n).

3. C ^' = aC_n + bF in Pic(𝔽_n) where a ≥ 1 and b ≥ na.

Furthermore, for any integers a and b where a ≥ 1 and b ≥ na, there is a smooth curve D ⊂ 𝔽_n such that D = aC_n + bF in Pic(𝔽_n).

Note that for an irreducible curve C in 𝔽_n, if (C ⋅ C) < 0 then C = C_n for 1 ≤ n.

Lemma 2.8

Let 𝔽_l be the Hirzebruch surface of degree l and let C_l ⊂ 𝔽_l be the smooth rational curve with (C_l ⋅ C_l) = −l for 0 ≤ l. Take a closed point x ∈ C_l and let Blow_x𝔽_l be the blow up of 𝔽_l at x. Then there is a closed point y ∈ 𝔽_l+1such that y ∉ C_l+1and Blow_x𝔽_l ≅ Blow_y𝔽_l+1.

Proof. Let ϕ_l : 𝔽_l → ℙ¹ be the fibration, let s ∈ ℙ¹ be the closed point such that x∈Fs:=ϕl−1(s),and let E be the exceptional divisor of the blow up π : Blow_x𝔽_l → 𝔽_l .We set Fs′:=π∗−1Fs and Cl′:=π∗−1Cl.Since Cl∩Fs={x},Cl′∩Fs′=∅.Since C_l is smooth at x,π∗Cl=Cl′+E and Cl′⋅Cl′=−l−1. Since F_s is smooth at x and F_s ≅ ℙ¹, we get that Fs′≅P1 and π∗Fs=Fs′+E.Since (F_s ⋅ F_s) = 0 and (E ⋅ E) = −1, we have Fs′⋅Fs′=−1.Then there is a birational morphism f : Blow_x𝔽_l → Z such that Z is a smooth surface and the exceptional locus of f is Fs′.Since Blow_x𝔽_l is rational, Z is rational. Since the rank of the Picard group of Blow_x𝔽_l is 3, that of Z is 2. Then Z is a Hirzebruch surface. Since Cl′∩Fs′=∅,fCl′⊂Zis a smooth rational curve such that fFs′∉fCl′and fCl′⋅fCl′=−l−1.Therefore Z is 𝔽_l+1 and fCl′=Cl+1.2

Let X be a K3 surface, let G be a finite subgroup of Aut(X) such that X/G is smooth, and let B:=∑i=1lbiBibe the branch divisor of the quotient morphism q : X → X /G. Then 0=KX/G+∑i=1kbi−1biB. We assume that X/G ≅ ℙ²; since the degree of KP2=−3,

Now we assume that X/G ≅ 𝔽_l with l ≥ 0. Let∑i=1kbiαiC+βiFbe the numerical class of B where B_i = α_iC+β_iF in Pic(𝔽_l) for i = 1, . . . , k. We have KFl=−2Cl−(l+2)Fn in PicFl, hence

Theorem 2.9

([11, Lemma 4.2 and Proposition 4.3]). Let X be a K3 surface, let G ⊂ Aut(X) be a finite subgroup such that X/G is a smooth rational surface, and letB=∑i=1kbiBibe the branch divisor of the quotient morphism q : X → X/G. For a birational morphism f : X/G → S where S is a smooth surface, let E_i be the exceptional divisor of f for i = 1, . . . , m. Then for i = 1, . . . , m we have that f(E_i)∈ Supp(f_∗B). In addition, ifBi˜is smooth for 1 ≤ i ≤ k, then for 1 ≤ j ≤ m there exist integers s, t with 1 ≤ s < t ≤ k such that f(e_j)∈ f_∗B_s ∩ f_∗B_t.

Let X be a K3 surface and let g be a non-symplectic involution of X. We assume that X/〈g〉 is rational. If X/〈g〉 is not ℙ² or 𝔽₀, then X/〈g〉 has an irreducible curve with negative self intersection number. The following Proposition 2.10 establishes which negative integers can occur as self intersection numbers of irreducible curves on X/〈g〉.

Proposition 2.10

Let X be a K3 surface, let g be a non-symplectic involution of X, and let q : X → X/〈g〉 be the quotient morphism. If X/〈g〉 has an irreducible curve C such that (C ⋅ C) < 0, then (C ⋅ C) = −1, −2, or −4. Furthermore, we have the following.

1. If (C ⋅ C) = −1, then there is a smooth rational curve R on X such that R ∉ Fix(g), q(R) = C, and g(R) = R or (g(R) ⋅ R) = 1.

2. If (C⋅C) = −2, then there is a smooth rational curve C^'on X such that R ⊄ Fix(g), q(R) = C, and g(R)∩ R = 0, i.e. (g(R) ⋅ R) = 0.

3. If (C ⋅ C) = −4, then there is a smooth rational curve C^'on X such that R ⊂ Fix(g) and q(R) = C.

Proof. Since the degree of q is 2, we obtain that q^∗C is 2R, R, or R + R^' where R and R^' are distinct irreducible curves on X.

We assume that q^∗C = 2R. Since the degree of q is 2, we have R ⊂ Fix(g) and (C⋅C) = 2(R⋅R). Since (C⋅C) < 0, (R⋅R) < 0. Since X is a K3 surface, (R⋅R) = −2 and R is a smooth rational curve. Since (C⋅C) = 2(R⋅R), (C⋅C) = −4.

We assume that q^∗C = R. Then R ⊄ Fix(g) and g(R) = R. Since the degree of q is two, 2(C ⋅ C) = (R ⋅ R). As before, R is a smooth rational curve and (C ⋅ C) = −1.

We assume that q^∗C = R + R^'. Since the degree of q is two, we obtain that g(R) = R^' and 2(C ⋅ C) = (R ⋅ R) + 2(R ⋅ R^') + (R^' ⋅ R^'). Since g(R) = R^', we have (R ⋅ R) = (R^' ⋅ R^') and hence 2(C ⋅ C) = 2(R ⋅ R) + 2(R ⋅ R^'). Since R ≠ R^', (R ⋅ R^')≥ 0. Since (C ⋅ C) < 0, we have (R ⋅ R^')≥ 0 and (C ⋅ C) = (R ⋅ R) + (R ⋅ R^'), hence (R ⋅ R) < 0. Then (R ⋅ R) = (R^' ⋅ R^') = −2 and R and R^' are smooth rational curves. Since (C ⋅ C) = (R ⋅ R) + (R ⋅ R^'), we obtain (C ⋅ C) = −2 + (R ⋅ R^'). Since (C ⋅ C) < 0 and (R ⋅ R^')≥ 0, we get that (C ⋅ C) = −1 or −2, and we have (C ⋅ C) = −1 (respectively −2) if and only if (R ⋅ R^') = 1 (respectively 0). □

Let X be a K3 surface and let g be a non-symplectic involution of X. If X/〈g〉 ≅ 𝔽_n with n ≥ 0, then by Proposition 2.10 we have n = 0, 1, 2, or 4. A similar result is obtained even if X/〈g〉 is not a Hirzebruch surface.

Proposition 2.11

Let X be a K3 surface and let g be a non-symplectic involution of X such that there is a birational morphism f : X/〈g〉 → 𝔽_n where n ≥ 0. Then 0 ≤ n ≤ 4.

Proof. Let C_n ⊂ 𝔽_n be the unique smooth rational curve with (C_n ⋅ C_n) = −n, and let C′:=f∗−1Cn⊂Xbe an irreducible curve. Since X/〈g〉 and 𝔽_n are smooth, f : X/〈g〉 → 𝔽_n is given by blow ups. As a result, (C^' ⋅ C^') ≤ (C_n ⋅ C_n) = −n. By Proposition 2.10 we have −4 ≤ (C^' ⋅ C^'). Therefore 0 ≤ n ≤ 4. □

Theorem 2.12

Let X be a K3 surface and let g be a non-symplectic involution of X. If the rank of Pic(X/〈g〉) is 3, then there is a closed point x ∈ 𝔽_n such that X/〈g〉 ≅ Blow_x𝔽_n where x ∉ C_n and n = 1, 2, or 4.

Proof. Since the rank of Pic(X/〈g〉) is 3, X/〈g〉 is not ℙ² and 𝔽_n where n ≥ 0. Then there is a birational morphism f : X → X/〈g〉 → 𝔽_n where n ≥ 0. By Proposition 2.11 we have 0 ≤ n ≤ 4. Since f : X → X/〈g〉 → 𝔽_n is given by blow ups, the rank of Pic(𝔽_n) is 2, and that of Pic(X/〈g〉) is 3, we get that X/〈g〉 ≅ Blow_x𝔽_n. By Proposition 2.11 and Lemma 2.8 we have 1 ≤ n ≤ 4 and x ∉ C_n. We assume that n = 3 and set C′:=f∗−1C3⊂X/〈g〉.Since x ∉ C₃, (C^' ⋅ C^') = (C₃ ⋅ C₃) = −3. By Proposition 2.10, this is a contradiction. Therefore, n = 1, 2, or 4. □

Theorem 3.1

([5, Lemma 17.1]). Let M be a smooth projective variety, and let D be a smooth effective divisor on M. If the class D /n ∈ Pic(M), then there is a Galois cover f : X → M whose branch divisor is nD and the Galois group is isomorphic to ℤ/nℤ as a group.

For n ≥ 0, it is known that the Hirzebruch surface 𝔽_n is isomorphic to a variety F_n in ℙ¹ × ℙ², see for example [6, Section 6], namely to

In this section, we assume that 𝔽_n = F_n with n ≥ 1. The first projection gives the fibre space structure ϕ_n : 𝔽_n → ℙ¹ such that the numerical class of the fibre of ϕ_n is F_n, and

is the unique irreducible curve on 𝔽_n such that the self intersection number is negative. Let a and b be positive integers such that b ≥ na. We put

where t_i,j,k ∈ ℂ, and

We assume that t_i,0,0 ≠ 0 for some i with 0 ≤ i ≤ b − na. Then

is a homogeneous polynomial of degree b − na with X₀ and X₁ as variables, and

For [s : t]∈ ℙ¹,

is a homogeneous polynomial of degree a with Y₀, Y₁ and Y₂ as variables, and

Therefore, Cn⋅BFa,b=b−na and Fn⋅BFa,b=a, and hence BFa,b=aCn+bFn in PicFn. 

Example 3.2

Let B^' be an irreducible curve on 𝔽₁, defined by the equation

Then B^' = 4C₁ + 6F in Pic(𝔽₁), and the point x := ([1 : 0 : 0], [1 : 0]) ∈ 𝔽₁ is the unique singular point of B^'. Let π : Blow_x𝔽₁ → 𝔽₁ be the blow up at x, and E be the exceptional divisor of π. We set B1:=π∗−1B′⊂ Blow XF1.Then B₁ is a smooth curve such that π^∗B^' = B₁+ 2E and B₁ = 4π^∗C₁+6π^∗F−2E in Pic(Blow_x𝔽₁). By Theorem 3.1 and since B₁ = 2(2π^∗C₁ + 3π^∗F − E) in Pic(Blow_x𝔽₁), there is the Galois cover q : X → Blow_x𝔽₁ such that the branch divisor is 2B₁ and the Galois group is ℤ/2ℤ as a group. By Theorem 2.3, the canonical divisor of X is numerically trivial. Then X is a bi-elliptic surface, an abelian surface, an Enriques surface or a K3 surface. By [22, Theorem 3], X is not a bi-elliptic surface. Since Blow_x𝔽₀ has a smooth curve whose self intersection number is negative, so does X. As a result, X is not an abelian surface. If X is an Enriques surface, then there is a Galois cover q^' : X^' → Blow_x𝔽₀ of degree 4 such that X^' is a K3 surface, and the branch divisor is 2B₁. By Theorem 2.5, the Galois group of q^' : X^' → Blow_x𝔽₀ is ℤ/2ℤ as a group. This contradicts the fact that the degree of q^' is 4. Therefore, X is a K3 surface. Let g be a generator of G. Then g is a non-symplectic involution of X such that X/〈g〉 = Blow_x𝔽₁.

Example 3.3

Let B^' be an irreducible curve on 𝔽₂, defined by the equation

Then B^' = 4C₂ + 8F in Pic(𝔽₂), and the point x := ([1 : 0 : 0], [1 : 0]) ∈ 𝔽₂ is the unique singular point of B^'. Let π : Blow_x𝔽₂ → 𝔽₂ be the blow up at x, and let E be the exceptional divisor of π. We set B1:=π∗−1B′⊂ Blow XF2.Then B₁ is a smooth curve such that π^∗B^' = B₁ + 2E and B = 4π^∗C₂ + 8π^∗F −2E in Pic(Blow_x𝔽₂). By Theorem 3.1 and since B₁ = 2(2π^∗C₂ + 4π^∗F − E) in Pic(Blow_x𝔽₂), there is a Galois cover q : X → Blow_x𝔽₂ such that the branch divisor is 2B₁ and the Galois group is ℤ/2ℤ as a group. In the same way as for Example 3.2, we get that X is a K3 surface and G is generated by a non-symplectic involution g such that X /〈g〉 = Blow_x𝔽₂.

Example 3.4

Let B^' be an irreducible curve on 𝔽₄, defined by the equation

Then B^' = 3C₄ + 12F in Pic(𝔽₄), and the point x := ([1 : 0 : 0], [1 : 0]) ∈ 𝔽₄ is the unique singular point of B^'. Let π : Blow_x𝔽₄ → 𝔽₄ be the blow up at x, and let E be the exceptional divisor of it. We set B:=π∗−1B′⊂ Blow XF4.Then B is a smooth curve such that π^∗B^' = B + 2E and B = 3π^∗C₄ + 12π^∗F − 2E in Pic(Blow_x𝔽₄). By Theorem 3.1 and since we have π⁻¹(C₄)∩ B = 0 and π^∗C₄ + B = 2(2π^∗C₄ + 6π^∗F − E) in Pic(Blow_x𝔽₄), there is a Galois cover q : X → Blow_x𝔽₄ such that the branch divisor is 2π^∗C₄ + 2B and the Galois group is ℤ/2ℤ as a group. In the same way as for Example 3.2, we get that X is a K3 surface and G is generated by a non-symplectic involution g such that X /〈g〉 = Blow_x𝔽₄.

Example 3.5

Let B^' be an irreducible curve on 𝔽₄, defined by the equation

Then B^' = 3C₄ + 12F in Pic(𝔽₄), and the singular locus of B^' is {([1 : 0 : 0], [1 : 0]), ([0 : 1 : 0], [0 : 1])}. We set x := ([1 : 0 : 0], [1 : 0]) and y := ([0 : 1 : 0], [0 : 1]). Let π : Blow_x,y𝔽₄ → 𝔽₄ be the blow up at x and y, and let E_x and E_y be the exceptional divisors of π such that E_x = π⁻¹(x) and E_y = π⁻¹(y). We set B:=π∗−1B′⊂ Blow x,yF4.Then B is a smooth curve such that π^∗B^' = B + 2E_x + 2E_y and B = 3π^∗C₄ + 12π^∗F − 2E in Pic(Blow_x,y𝔽₄). By Theorem 3.1 and since we have π⁻¹(C₄)∩ B = 0 and π^∗C₄ + B = 2(2π^∗C₄ + 6π^∗F − E) in Pic(Blow_x,y𝔽₄), there is a Galois cover q : X → Blow_x,y𝔽₄ such that the branch divisor is 2π^∗C₄ + 2B and the Galois group is ℤ/2ℤ as a group. In the same way as for Example 3.2, we get that X is a K3 surface and G is generated by a non-symplectic involution g such that X /〈g〉 = Blow_x,y𝔽₄.