On the group of automorphisms of Horikawa surfaces

Minimal algebraic surfaces of general type $X$ such that $K^2_X=2\chi(\mathcal{O}_X)-6$ are called Horikawa surfaces. In this note the group of automorphisms of Horikawa surfaces is studied. The main result states that given an admissible pair $(K^2, \chi)$ such that $K^2=2\chi-6$, every irreducible component of Gieseker's moduli space $\mathfrak{M}_{K^2,\chi}$ contains an open subset consisting of surfaces with group of automorphisms isomorphic to $\mathbb{Z}_2$.


Introduction.
We work over the field C of complex numbers.The main numerical invariants of an algebraic surface X are the self-intersection of its canonical class K 2 X and its holomorphic Euler characteristic χ(O X ).If X is minimal and of general type, the following inequalities are well known to be satisfied (cf.[2, Chapter VII]): Minimal algebraic surfaces of general type X such that K 2 X = 2χ(O X ) − 6 were already studied by Enriques [5], [6,Section VIII.11] but they are often called Horikawa surfaces because of Horikawa's contribution to their deformation theory [10].One property of Horikawa surfaces is that their canonical system is base-point-free and induces a morphism whose image is a surface.Moreover, this morphism has degree 2 (see Theorem 2).As a consequence, every Horikawa surface X has a Z 2 -action generated by the involution that sends a general point of X to the point with the same image via the canonical map of X.In particular, the group of automorphisms Aut(X) of X has a subgroup isomorphic to Z 2 and one may wonder whether this subgroup is proper or not when X is sufficiently general.
It is known that under some ampleness and generality assumptions, the group of automorphisms of a surface that can be realized as an abelian G-cover is precisely the group G (see [7] or [15]).Since Horikawa surfaces can be realized as Z 2 -covers via their canonical map, one could expect the group of automorphisms of a general Horikawa surface to be Z 2 .Nevertheless, Horikawa surfaces do not satisfy the hypothesis of [7,Theorem 4.6] nor [15, Corollary B] and, to the best of the author's knowledge, there is no reference allowing to prove this in a straightforward way.
Given an admissible pair (K 2 , χ), i.e. a pair of integers satisfying the inequalities (1), let us denote by M K 2 ,χ Gieseker's moduli space of canonical models of surfaces of general type with fixed self-intersection of the canonical class K 2 and fixed holomorphic Euler characteristic χ.The aim of this note is to prove the following: Theorem 1.Let (K 2 , χ) be an admissible pair such that K 2 = 2χ − 6.Then every irreducible component of M K 2 ,χ contains an open subset consisting of surfaces with group of automorphisms isomorphic to Z 2 .
The proof of Theorem 1 is based on the following idea.Let X be a smooth surface with an involution τ that induces a Z 2 -cover X → X/ τ with building data {L, B} (see Section 3).Then every automorphism σ of X that commutes with τ induces an automorphism σ of X/ τ such that σ(B) = B. Hence, if τ were in the center of Aut(X) and there were no non-trivial automorphism of X/ τ leaving B invariant, it would follow that Aut(X) = τ ≃ Z 2 .This idea is simple and familiar to experts on the topic but there are several technical details that have to be worked out.Furthermore, even if some of these technical details are somehow expected, as far as the author knows they have not been written down elsewhere and they are interesting on their own.The paper is structured as follows.Section 2 contains the results about Horikawa surfaces that will be needed throughout the note.In Section 3 it is explained how to construct simple cyclic covers and to obtain information about them.An elementary criterion for a surface that can be realized as a degree n simple cyclic cover to have group of automorphisms isomorphic to Z n is included.Section 4 contains a well known upper semicontinuity theorem for families of stable curves.A proof of this theorem is included for lack of a reference.Section 5 consists of a series of results about simply connected algebraic surfaces that will be needed to prove Theorem 1. Section 6 is devoted to prove Theorem 1 making use of the tools developed in the previous sections.

Horikawa surfaces on the line
This section collects the results about Horikawa surfaces that will be needed throughout the note.
Theorem 2 ([10, Lemma 1.1]).Let X be a minimal algebraic surface with K 2 X = 2χ(O X ) − 6 and χ(O X ) ≥ 4. Then the canonical system |K X | has no base point.Moreover, the canonical map ϕ KX : X → P pg(X)−1 is a morphism of degree 2 onto a surface of degree p g (X) − 2 in P pg (X)−1 .

Simple cyclic abelian covers.
Let Y be a smooth surface.Suppose there exist a line bundle L and an effective divisor B on Y such that B ∈ |nL|.We denote by V (L) the total space of the bundle L, by π : V (L) → Y the bundle projection and by t ∈ H 0 (V (L), π * L) the tautological section.If s ∈ H 0 (Y, nL) is a section vanishing exactly along B, then the zero divisor X of the section t n − π * s ∈ H 0 (V (L), π * (nL)) defines a surface in V (L).Moreover, if we denote by µ a primitive n-root of unity, the map τ : t → µ • t induces a Z n -action on X such that f := π| X can be realized as the quotient map X → X/Z n ≃ Y .The morphism f : X → Y is said to be a simple cyclic cover of degree n with branch locus B (see [2, Section I.17]).The set {L, B} is known as the building data of the simple cyclic cover.
Simple cyclic covers are a special type of abelian cover.Let G be a finite abelian group.A G-cover of a surface Y is a finite map f : X → Y together with a faithful action of G on X such that f exhibits Y as X/G.Abelian covers in general were first studied by Pardini [18] but simple cyclic covers were already considered by Comessatti [4].Other references where particular types of covers were studied are [3], [17], [19] or [20].
In this note we are mainly interested in Z 2 -covers.Since every Z 2 -cover is simple cyclic and some of the results needed to prove Theorem 1 can be easily generalized to simple cyclic covers, we will also deal with this type of covers.
Remark 1.Let f : X → Y be a simple cyclic cover of degree n with branch locus B. Note that if the Picard group of Y has no n-torsion then the line bundle L can be deduced from the divisor B. In this note we are only going to consider covers of simply connected surfaces, for which the Picard group has no torsion (cf.[16,Remark 3.10]).[18,Proposition 3.1]) and in this case we will say that f : X → Y is a smooth simple cyclic cover.

Proposition 1 ([18, Proposition 4.2]
).Let Y be a smooth surface and f : X → Y a smooth degree n simple cyclic cover with building data {L, B}.Then: Remark 3. Let Y be a smooth surface and let us consider a smooth simple cyclic cover f : X → Y of degree n with building data {L, B}.Let us assume that If we denote by i : Y P N −1 the (possibly rational) map defined by the complete linear system |K Y + (n − 1)L|, it follows that i • f is the map induced by the complete linear system |K X |, i.e. it is the canonical map of X.In particular i(Y ) is the canonical image of X.The following result is clearly inspired by [14,Lemma 5.3] and [15].Although it is probably well known, a proof is included for lack of a reference.

Theorem 4.
Let Y be a smooth surface and f : X → Y a degree n ≥ 2 simple cyclic cover with building data {L, B} such that the branch locus B is a general member of the linear system |nL|.Suppose that the map defined by the complete linear system |K Y + (n − 1)L| is birational and that If we denote by τ an order n automorphism of X such that f can be realized as the quotient of X by the action of τ ≃ Z n , then: Proof.First of all, it follows from Remark 3 that the canonical map Φ of X is the composition of f with the map Ψ induced by the complete linear system |K Y + (n − 1)L|.In particular, Φ has degree deg(Φ) = n because Ψ is birational.
Let us consider the Galois group G = {g ∈ Aut(X) : Φ • g = Φ} of Φ.On the one hand, τ ≃ Z n is contained in G. Given that the order of G is at most deg(Φ) = n, we infer that G = τ ≃ Z n .On the other hand, G coincides with {g ∈ Aut(X) : g * C = C for every C ∈ |K X |} and this is clearly a normal subgroup of Aut(X).Hence, we conclude that τ ≃ Z n is a normal subgroup of Aut(X).If n = 2, then τ ≃ Z 2 is in the center of Aut(X) because order 2 normal subgroups are always central.
That being said, we deduce from the fact that τ ≃ Z n is normal in Aut(X) that h induces an automorphism h of X/Z n ≃ Y for every h ∈ Aut(X).Moreover, denoting by R = (f * B) red the ramification of f , I claim that h(R) = R and therefore h(B) = B. Indeed, let T = fix(τ ) be the set of points fixed by τ .Then the set fix(hτ h −1 ) of points fixed by hτ h −1 is equal to h(T ).Now, since G is normal, there exists an integer k coprime to n such that hτ h −1 = τ k and therefore h(T ) = fix(hτ

Automorphisms of a family of stable curves.
Let f : X → B be a fibration, i.e. a proper and surjective morphism with connected fibers from a surface X to a smooth connected curve B. As in [7] we will denote by Aut X/B the B-scheme of automorphisms of the fibers of f .In particular, the fiber of Aut X/B → B over b ∈ B is isomorphic to the group of automorphisms of the fiber of f over b.The aim of this section is to prove the following: Theorem 5. Let X → ∆ be a genus g ≥ 2 fibration over the unit disc ∆ ⊂ C such that X is smooth and X t is a stable curve for every t ∈ ∆.Then Aut X /∆ → ∆ is proper and the map sending t ∈ ∆ to the order of the group of automorphisms of X t is an upper semicontinuous function.
This result is well known but we include a proof, that was pointed out to the author by Rita Pardini, for lack of a reference.
Proof of Theorem 5. Denoting ∆ * = ∆\{0} and X * = X \X 0 , the properness of Aut X /∆ → ∆ will follow if we show that every section of Aut X * /∆ * → ∆ * can be extended uniquely to a section of Aut X /∆ → ∆ by the valuative criterion of properness.Let σ be a section of Aut X * /∆ * → ∆ * and denote by σ : X X the birational map induced by σ.Suppose that σ cannot be extended to a morphism and consider a minimal sequence of blow-ups ε : where each E i , i ∈ {1, . . ., r} is an f -exceptional curve.By construction the last irreducible (−1)-curve Γ arising from ε is not contracted by f and so Γ r i=1 E i ≥ 0 because Γ is not a component of r i=1 E i .In addition, since X t is a stable curve for every t ∈ ∆ the divisor K X is relatively nef and Γf * K X ≥ 0. We conclude that: which is a contradiction.Hence we can assume σ to be a morphism because it can be extended to X .Moreover, this extension is unique because X * is an open and dense subset of X .Since Aut X /∆ → ∆ has finite fibers because X t is a stable curve for every t ∈ ∆, there exists an integer m > 0 such that (σ| X * ) m is the trivial automorphism and therefore σm is the trivial automorphism on X .In particular, σ is an automorphism and σ can be extended uniquely to a section of Aut X /∆ → ∆.It follows that Aut X /∆ → ∆ is proper.
Write Aut X /∆ = Y ⊔Z where Y is the union of the 1-dimensional components of Aut X /∆ and Z consists of isolated points.Then the restriction Y → ∆ of Aut X /∆ → ∆ to Y is flat by [9,Proposition III.9.7].Moreover, Y → ∆ is étale.Indeed, by [9,Exercise III.10.3] it suffices to show that Y → ∆ is unramified, but this is clear since Aut(X t ) is reduced (because it is a complex group scheme) and finite (because X t is a stable curve) for every t ∈ ∆.In particular, the cardinality #Y t of the fibers of Y → ∆ is constant for every t ∈ ∆ (cf.[8, Corollary to Proposition 3.13]).The semicontinuity of the map sending t ∈ ∆ to the order |Aut(X t )| of the group of automorphisms of X t follows taking into account that:

Simply connected algebraic surfaces.
In this section we gather some results about simply connected algebraic surfaces that will be needed to prove Theorem 1.They will come up as corollaries of the following: Theorem 6.Let S be a smooth and simply connected algebraic surface and Λ a very ample linear system on S such that the general curve in Λ has genus ≥ 3 and is non-hyperelliptic.Then there exists a dense open subset of Λ consisting of curves without non-trivial automorphisms.
Proof.By [11, 10.6.18]all but at most finitely many of the curves in a general 1-dimensional linear subspace of Λ have no non-trivial automorphisms.Therefore a general element of Λ has no non-trivial automorphism.
As a consequence we obtain the following (see Remark 4): Corollary 1.Let F e be the Hirzebruch surface with negative section ∆ 0 of self-intersection (−e) and fiber F .Then a general member D of the linear system |a∆ 0 + bF | with a > 2, b > max{ae, (a − 1)e + 2} satisfies Aut(F e , D) = {1}.
Proof.First of all, the linear system |a∆ 0 + bF | is very ample by [9, Corollary V.2.18].On the other hand, by the Adjunction Formula is very ample again by [9, Corollary V.2.18], we have that K D is very ample and therefore D is a non-hyperelliptic curve of genus greater than 2. Therefore we can apply Theorem 6 to the smooth and simply connected surface e and the linear system |a∆ 0 + bF | to conclude that D has no non-trivial automorphisms.
Let us consider h ∈ Aut(F e , D).We are going to show that it is necessarily the trivial automorphism of F e .Firstly, h| D is an automorphism of D and therefore it is trivial.Hence D belongs to the fixed locus of h and a general fiber F of F e has at least F D = a points fixed by h.Then F and h(F ) are irreducible curves such that  ii) S is the Hirzebruch surface F 2k+2 with k ≥ 2 and D is a general member of the linear system |O F 2k+2 (6∆ 0 + 10(k + 1)F )|.
iii) S = P 2 and D is a general member of the linear system |O P 2 (d)| with even d ≥ 8.
Then Aut(X) ≃ Z 2 where X → S is a Z 2 -cover of S branched along D.
Proof.We know that Aut(S, D) = {1} by Corollary 1, Corollary 2 or Corollary 3 depending on whether we are in case i), ii) or iii) respectively.The result follows now from Theorem 4.iii) taking into account that h 0 (K S ) = 0 and K S + 1 2 D is very ample.
6 Group of automorphisms of Horikawa surfaces.
The aim of this section is to prove Theorem 1.We will need the following: Lemma 1.Let (K 2 , χ) be an admissible pair such that K 2 = 2χ − 6. Suppose there exists a smooth surface X ∈ M K 2 ,χ whose group of automorphisms is isomorphic to Z 2 and denote by M the irreducible component of M K 2 ,χ containing X. Then the group of automorphisms of a general surface in M is isomorphic to Z 2 .We are ready to prove Theorem 1.

Proof
Therefore we need to construct another surface for this case.Let us consider a Z 2 -cover f : X → P 2 whose branch locus B is a general element of the linear system |O P 2 (8)|.The formulas for simple cyclic covers (see Proposition 1) yield K 2 X = 2 and χ(O X ) = 4.Moreover, K X is ample because it is the pullback via f of the ample line bundle O P 2 (1).Hence, X ∈ M 2,4 and Aut(X) = Z 2 by Corollary 4.
Remark 5.There are many examples of Horikawa surfaces whose group of automorphisms is not isomorphic to Z 2 .Indeed, let (K 2 , χ) be an admissible pair such that K 2 = 2χ − 6. Then: -Every irreducible component of M K 2 ,χ contains surfaces whose group of automorphisms has a subgroup isomorphic to Z

Remark 4 .
Let D be an effective divisor on Y .In what follows we will denote by Aut(Y, D) the subset of Aut(Y ) consisting of automorphisms h of Y such that h(D) = D.

Corollary 2 .
only irreducible self-intersection 0 curves of F e are the elements of |F |.If e = 0 the only irreducible self-intersection 0 curves of F e are the elements of the pencils |F | and |∆ 0 |.In both cases the fact that F contains a > 2 fixed points implies h(F ) = F .It follows that h| F is an automorphism of F ≃ P 1 with at least a > 2 fixed points.We conclude that h| F is the trivial automorphism of F .As a consequence h is trivial in an open and dense subset of F e and it must be the trivial automorphism.Let F 2k+2 be the Hirzebruch surface with negative section ∆ 0 of self-intersection −(2k + 2) and fiber F .If k ≥ 1 then a general member D of the linear system |O F 2k+2 (5∆ 0 + 10(k + 1)F )| satisfies Aut(F 2k+2 , D) = {1}.In particular Aut(F 2k+2 , ∆ 0 + D) = {1} Proof.Let us denote by |Aut(C)| the order of the group of automorphisms of a member C of the linear system |5∆ 0 + 10(k + 1)F |.By [7, Corollary 4.5] the map C → |Aut(C)| is an upper semicontinuous function on the subset of |5∆ 0 + 10(k + 1)F | consisting of smooth divisors.Hence, there will be an open and dense subset of curves in |5∆ 0 + 10(k + 1)F | without non-trivial automorphisms as long as the set of smooth curves with this property is not empty.Moreover, arguing like we did in the proof of Corollary 1 we can show that a divisor C ∈ |5∆ 0 +10(k+1)F | without non-trivial automorphisms satisfies Aut(F 2k+2 , C) = {1}.Therefore the result will follow if we find a smooth divisor C ∈ |5∆ 0 + 10(k + 1)F | without non-trivial automorphisms.Now, by Theorem 6 a general curve B ′ ∈ |4∆ 0 +10(k+1)F | has no non-trivial automorphisms.We can assume B ′ to be smooth and irreducible and to intersect ∆ 0 transversally in 2(k + 1) different points.Then it is clear that B = ∆ 0 + B ′ ∈ |5∆ 0 + 10(k + 1)F | has trivial group of automorphisms.Denote by ∆ ⊂ C the unit disc.We can consider a family p : X → ∆ of curves of |5∆ 0 + 10(k + 1)F | such that B is the central fiber, the rest of fibers are smooth and X is smooth.By Theorem 5 the cardinality of the fiber of Aut X /∆ → ∆ is an upper semicontinuous function.Hence, the general fiber of p has trivial group of automorphisms because B does.Corollary 3. A general member D of the linear system |O P 2 (d)| with d ≥ 4 satisfies Aut(P 2 , D) = {1}.Proof.The proof of this result is analogous to the proof of Corollary 1.

Corollary 4 .
Let (S, D) be one of the following pairs: i) S is the Hirzebruch surface F e with e ≥ 0 and D is a general member of the linear system |a∆ 0 + bF | with even a ≥ 6 and even b > max{ae, (a − 1)e + 2, (a − 2)e + 4}.

.
Let us denote by |Aut(Y )| the order of the group of automorphisms of a surface Y ∈ M K 2 ,χ .By [7, Corollary 4.5] the map Y → |Aut(Y )| is an upper semicontinuous function on the subset of M consisting of smooth surfaces.The result follows taking into account that the group of automorphisms of every Horikawa surface has a subgroup isomorphic to Z 2 (see Section 1).