Stable Degenerations of Surfaces Isogenous to a Product II

In this note, we describe the possible singularities on a stable surface which is in the boundary of the moduli space of surfaces isogenous to a product. Then we use the $\mathbb Q$-Gorenstein deformation theory to get some connected components of the moduli space of stable surfaces.


Introduction
Stable surfaces were first introduced by Kollár and Shepherd-Barron in the paper [KoSB88]. These surfaces appear naturally as one-parameter limits of surfaces of general type: ifX → ∆ is a one-parameter Q-Gorenstein family whose general fibres X t are the canonical models of surfaces of general type and X → ∆ is its canonical model, then the central fibre X 0 is a stable surface. The singularities on stable surfaces are semi log canonical (cf. Definition 1.4) and a complete classification of them has already been given in [KoSB88]. The moduli problem for stable surfaces was solved later through the work of several authors (cf. [Ko90,A94,Hac04,HK04,Ko08,AbH09] and Theorem 1.13).
In view of the above general framework, examples are badly needed to illustrate the geometry of the moduli space of stable surfaces. One strategy is to compactify the well studied moduli spaces of smooth surfaces whose stable degenerations can be described in a satisfactory way. In this direction, van Opstall [vO05, vO06] studied the moduli space of products of curves and the stable degenerations of surfaces isogenous to a product. Recently, Alexeev and Pardini [AP09] gave an explicit compactification of moduli spaces of the more complicated Campedelli and Burniat surfaces, and Rollenske [R09] did the same for very simple Galois double Kodaira fibrations.
Here we continue the work of van Opstall ( [vO06]) on the stable degenerations of surfaces isogenous to a product and give some connected components of the moduli space of stable surfaces. A surface isogenous to a product is the quotient of a product of two smooth curves by a free group action. Van Opstall showed that a stable degeneration of such surfaces is the quotient of a product of two stable curves C × D by a not necessarily free group action (cf. Theorem 1.12). We go one step further to describe explicitly how the group acts on the product of two stable curves. To do this, we first establish some results on the smoothing of a stable curve with a group action (cf. Section 2.1). Then we can apply these results to the free smoothing of a product of two stable curves with a group action (see Sections 2.2 and 2.3). Note that X = (C × D)/G is a stable degeneration if and only if (C × D, G) admits a free smoothing (cf. the discussion after Theorem 1.12).
Therefore we get our desired description of the group action on a product of stable curves (Prop. 2.9 and 2.16). The basic tool used is Cartan's lemma which reduces the action of a stabilizer to a linear one.
Having described the singularities on a stable degeneration X (Cor. 2.11 and 2.18), we show that if the singularities are of certain type, namely (U 1a ), (U 1b ), (U 2a ) or (U 2b ), then the Q-Gorenstein deformation of X is unobstructed (Theorem 3.3) and hence the compactification of the moduli space considered by van Opstall in fact yields connected components of the moduli space of stable surfaces: . We should mention that the Q-Gorenstein deformation theory set out by Hacking ([Hac01] and [Hac04]) is indispensable for our purpose.
Acknowledgements. I would like to thank my advisor Prof. Fabrizio Catanese at Universität Bayreuth for suggesting this research and for his patience and encouragement during many discussions. The idea of classifying the singularities on stable degenerations of surfaces isogenous to a product originates from him. I want to thank Fabio Perroni and Sönke Rollenske for several discussions during the preparation of this note. I am grateful to Stephen Coughlan for improving the English presentation. Thanks also go to my domestic advisor Prof. Jinxing Cai at Beijing University for encouragement.
This work was completed at Universität Bayreuth under the financial support of China Scholarship Council "High-level university graduate program"and DFG Forschergruppe 790 "Classification of algebraic surfaces and compact complex manifolds".
1. Preliminaries 1.1. Notation. Let G be a finite group acting on a set A.
For σ ∈ G, |σ| is the order of σ. For a ∈ A, G a := {g ∈ G|g · a = a} is the stabilizer of a. If G a = {1}, we say that a is a fixed point of the action. If G a = {1} for every a ∈ A, we say that G acts freely on A.
Z n denotes the cyclic group of order n. We work over the field C of complex numbers.
1.2. Surfaces isogenous to a product.
Definition 1.1 ([Cat00], Definition 3.1). A smooth projective surface S is isogenous to a (higher) product if it is a quotient S = (C × D)/G, where C, D are smooth curves of genus at least two, and G is a finite group acting freely on C × D.
Let S = (C ×D)/G be a surface isogenous to a product. Let G • := G∩(Aut(C)× Aut(D)); then G • acts on the two factors C, D and acts on C × D via the diagonal action. We say that S is of unmixed type if G = G • ; otherwise S is said to be of mixed type. By [Cat00,Prop. 3.13], we always assume G • acts faithfully on C, D.
By [Cat00,Prop. 3.16], surfaces S = (C × D)/G isogenous to a product and of mixed type are obtained as follows. There is a (faithful) action of a finite group G • on a curve C of genus at least 2 and a nonsplit extension , which is of order ≤ 2. Once we fix a representative ϕ of the above class, there exists an element τ ′ in G \ G • such that, setting τ = τ ′2 , we have: (I) ϕ(γ) = τ ′ γτ ′−1 , (II) G acts, under a suitable isomorphism of C and D, by the formulae: γ(P, Q) = (γP, (ϕγ)Q) for γ in G • ; whereas the lateral class of G • consists of the transformations Let Γ be the subset of G • consisting of the transformations having some fixed point. Then the condition that G acts freely amounts to: The moduli space of surfaces isogenous to a product is partly illustrated in the following abridged theorem: Cat03]). Let S be a surface isogenous to a product. Then moduli space M top S of surfaces with the same topological type as S is either irreducible and connected or it contains two connected components which are interchanged by complex conjugation.
1.3. Cartan's lemma. The following lemma is used throughout Section 2 for the (analytically) local analysis of the group actions. Lemma 1.3 (Cartan's lemma). Let z ∈ Z be an analytic singularity with Zariski tangent space T and let G be a finite group of automorphisms of z ∈ Z. Then there exists a G-equivariant embedding z ∈ Z → T ∋ 0.
1.4. Q-Gorenstein deformation theory of stable surfaces. We recall the Q-Gorenstein deformation theory of stable surfaces set out by Hacking.
(ii) X has at most double normal crossing singularities in codimension 1; (iii) the Weil divisor class K X is Q-Cartier; (iv) ifX → X is the normalization andD ⊂X is the preimage of the codimension-1 part of X sing , then the pair (X,D) is log canonical, i.e., for any resolution µ :X →X, we have with all a i ≥ −1. A stable surface is a projective slc surface with an ample dualizing sheaf.
Let F be a coherent sheaf. For n ∈ Z, we define the n-th reflexive power of F by F [n] := (F ⊗n ) * * , the double dual of the n-th tensor product. Let R be the category of Noetherian local C-algebras with residue field C. Definition 1.5. Let B a Noetherian scheme over C. A flat projective morphism f : X → B is called a Q-Gorenstein family of stable surfaces if for every closed point t ∈ B, the fibre X t is a stable surface and for every n ∈ Z, ω [n] X /B commutes with any base change.
Let R ∈ R. A Q-Gorenstein deformation of X over R is a Q-Gorenstein family f : X → Spec R together with an isomorphism X ⊗ R k(R) ∼ = X. Isomorphisms between two Q-Gorenstein deformations are defined in an obvious way.
Remark 1.6. The hypothesis that ω X /B commutes with any base change for any n is called Kollár's condition and it implies that ω For a stable surface X, we define a functor Def QG X : R → (Sets) as follows: for any R ∈ R, Def QG X (R) = {Q-Gorenstein deformations of X over R}/ ≃ . The Q-Gorenstein deformations of X are exactly those deformations which, locally at each point P ∈ X, are induced by equivariant deformations of the canonical covering of P ∈ X ([Hac01, Prop. 10.13]). If X is Gorenstein, then the Q-Gorenstein deformation theory of X coincides with the ordinary one.
Remark 1.7. If a semiuniversal Q-Gorenstein deformation exists, we denote the base by Def QG X . When we refer to the ordinary deformation of X, the semiuniversal base is denoted by Def X .
, the space of first-order Q-Gorenstein deformations. Let T 1 QG,X be the sheaf associated to the presheaf defined by U → Def QG U (C[ǫ]/ǫ 2 ), for any open subset U of X. Then we have a local-to-global sequence: QG,X is considered as the tangent space to the functor Def QG X . 1.5. Stable surfaces in this paper. We are mostly interested in stable surfaces that are quotients of a product of two stable curves. We recall the definition of a stable curve first ([DM69]): Definition 1.8. Let g ≥ 2 be an integer. A stable curve of genus g is a reduced, connected, 1-dimensional scheme C over C such that: (i) C has only ordinary double points as singularities; (ii) if E is a non-singular rational component of C, then E meets the other components of C in more than 2 points; Corollary 1.11. Let C, D be stable curves. Let Z := C × D and G a group acting on Z with finitely many fixed points. Then (C × D)/G is a stable surface.
The stable degenerations of surfaces isogenous to a product are of the above form, as we will see from the following: Theorem 1.12 ( [vO06], Theorem 3.1). Suppose X * → ∆ * is a family of surfaces isogenous to a product over a punctured disk. Then, possibly after a finite change of base, totally ramified over the origin, X can be completed to a family of stable surfaces over the disk whose central fibre is a quotient of a product of stable curves.
According to the proof of the above theorem in [vO06], we give an explicit construction of the stable degenerations here. There are two cases: Unmixed case: we have, up to finite base change, G-equivariant smoothings of stable curves (cf. Section 2.1) C → ∆ and D → ∆ such that the completion X → ∆ of X * → ∆ * is (C × ∆ D)/G → ∆. In particular, the central fibre is (C 0 × D 0 )/G where G acts faithfully on C 0 , D 0 and acts diagonally on C 0 × D 0 . However, the action of G on C 0 × D 0 is not necessarily free.
Mixed case: there exists a finite group G • , a G • -equivariant smoothing C → ∆ of stable curves and a nonsplit extension yielding an automorphism ϕ of G • , such that the pairs (C t , G) with t = 0 satisfy properties (I), (II), (A), (B) in Section 1.2. On the central fibre C 0 , we still have a G-action that enjoys properties (I), (II), but not necessarily (A), (B), i.e., the action of G on C 0 × C 0 is not necessarily free. Now the completion X → ∆ is (C × ∆ C)/G → ∆, where the action of G • on the second factor is twisted by ϕ.
In both cases, the stable degeneration X 0 is of the form (C × D)/G, where C, D are stable curves and G acts on C×D with finitely many fixed points. Tautologically the pair (C × D, G) admits a free smoothing, i.e., a one-parameter family C × ∆ D → ∆ such that the following hold: (ii) The fibre C t × D t over t = 0 is smooth; (iii) G acts on C × ∆ D preserving the fibres and the action of G on the central fibre coincides with the given action of G on C × D; (iv) G acts freely on the general fibres C t × D t for t = 0.
1.6. The moduli space of stable surfaces. Let (Sch)/C be the category of Noetherian C-schemes. We define the moduli functor of stable surfaces: for any Proof. See [Kov09, Section 7.C] and [Ko08].
We shall get some connected components of this moduli space by studying the Q-Gorenstein deformations of stable degenerations of surfaces isogenous to a product in Section 3.

Stable degenerations of surfaces isogenous to a product
In this section, we will give a precise description of possible singularities on stable degenerations of surfaces isogenous to a product. This is a careful improvement of Theorem 1.12, which allows one to further study the Q-Gorenstein deformations of the stable degenerations.

2.1.
Smoothings of stable curves with group actions. Our surfaces can be constructed by taking finite quotients of products of two stable curves, so their geometry is closely related to that of stable curves. In this section, we will establish some facts about smoothings of stable curves with group actions. More precisely, in the case when the group action admits a smoothing, we will show what the stabilizers on the central fibre can be and how they act locally analytically. These facts are used in Sections 2.2 and 2.3 for the smoothing of a product of stable curves with a group action.
Definition 2.1. Let G be a finite group acting faithfully on a stable curve C. A smoothing of the pair (C, G) is a (flat) family of stable curves C → ∆ over the unit disk such that The fibre C t of the family over t = 0 is smooth; (iii) G acts on C preserving the fibres, and the action on C 0 coincides with the given one on C under the isomorphism of (i).
Remark 2.2. We also call C → ∆ a G-equivariant smoothing of C.
Now let (C, G) be as in the definition and assume C → ∆ is a smoothing of (C, G).
Lemma 2.3. There are only finitely many points on C having non-trivial stabilizers, or, equivalently, there are only finitely many fixed points for the G-action.
Proof. Otherwise there is a τ = 1 ∈ G acting as identity on some irreducible component D of C. Pick a smooth point P of D. Then C is smooth around P and we can take local coordinate z of D and local coordinate t of ∆ such that (z, t) form local coordinates of C around P and τ acts as (z, t) → (z, t). So τ = 1 ∈ G is the identity, a contradiction.
Lemma 2.5. If P ∈ C is a smooth point, and 1 = τ ∈ G P , then τ also fixes points on C t , for t = 0.
Proof. As in the proof of Lemma 2.3, we can find local coordinates (z, t) for C around P such that τ acts as (z, t) → (ξz, t), where ξ ∈ C * is a primitive |τ |-th root of unity. So τ fixes (0, t) ∈ C t , for t = 0.
Lemma 2.6. If P ∈ C is a node, then G P is either cyclic or dihedral.
Proof. The germ of C around P can be seen as a deformation of a node. We can find an embedding of the germ into (C 3 , 0) such that the equation of the germ is xy − t k = 0, k ≥ 1. In fact, let be the semiuniversal family of a node. Then locally around P , C → ∆ is just the pull-back by where k ≥ 1.
By Cartan's lemma, we can assume the action of G P is given by for any τ ∈ G P . Since G P acts on the central fibre C 0 : xy = 0, we have where ξ is a primitive |τ |-th root of unity and η is some non-zero number. Let π : G P → Z 2 be the determinant homomorphism: for any τ ∈ G P , π(τ ) := det(τ ) = 1, if τ does not interchange the branches at P, −1, if τ interchanges the branches at P.
Then the kernel H of π consists of τ ∈ G P whose action is given by and it is easy to see in this case that Lemma 2.7. Let P ∈ C be a node. Suppose 1 = τ ∈ G P : then τ fixes points on C t for t = 0 if and only if π(τ ) = −1, where π : G P → Z 2 is as in the proof of the previous lemma.
On the other hand, if π(τ ) = −1, then τ (x, y, t) = (ηy, η −1 x, t) with η ∈ C * . So τ (x, y, t) = (x, y, t) if and only if ηy = x. Taking the equation xy = t k into consideration, τ fixes 2 points: (η t k η , t) and (−η t k η , t) on C t , for t = 0. Now we can state our main theorem in this section: Theorem 2.8. A pair (C,G) admits a smoothing if and only if for any node P ∈ C, we can find local (analytic) embedding of C : (xy = 0) ⊂ C 2 such that, for any τ ∈ G P , the action of τ is given by either Proof. The "only if"part is shown in the proof of Lemma 2.6. For the "if "part, we divide the proof into two steps.
Step 1: The germ P ∈ C has a local G-equivariant smoothing. More precisely, let U ⊂ C be a neighborhood around P defined by xy = 0 ⊂ C 2 as in the hypothesis, we will show that the pair (U, G P ) is G P -smoothable. In fact we can consider the family U : (xy − s = 0) ⊂ C 2 × ∆ → ∆ with s ∈ ∆ as the parameter. For any τ ∈ G P , τ (x, y) = (ξx, ξ −1 y) or (ηy, η −1 x), and it is easily seen that the action of G P on U extends to the family U → ∆.
Note that U → ∆ is the semiuniversal deformation of the node P ∈ U and the tangent space of the base space at 0 is Ext 1 Step 2: We will use the local-to-global exact sequence ) → 0 to prove that local smoothings of nodes with stabilizers lift to a smoothing of (C, G). To do this, first note that where, for any coherent sheaf F on C, F P denotes the stalk of F at P . For any τ ∈ G, τ acts on H 0 (C, Ext 1 OC (Ω C , O C )) and maps the Ext 1 OC,P (Ω C,P , O C,P ) summand isomorphically to the Ext 1 O C,τ (P ) (Ω C,τ (P ) , O C,τ (P ) ) summand. Let n(P ) := |G/G P | and τ 1 , . . . , τ n(P ) ∈ G representatives of elements of G/G P . Then τ 1 (P ), . . . , τ n(P ) (P ) is the orbit of P under the action of G. And G acts on the vector space The invariant subspace V G P is 1-dimensional, spanned by (τ 1 (σ), . . . , τ n(P ) (σ)), where σ is an element spanning Ext 1 OC,P (Ω C,P , O C,P ) ∼ = C. In view of (2.1), the dimension of H 0 (C, Ext 1 OC (Ω 1 C , O C )) is exactly the number of node orbits under the action of G. Taking the G-invariants of the local-to-global sequence, we get In particular, there exists λ ∈ Ext 1 OC (Ω C , O C ) G such that the π(λ)'s P -summand is nonzero for any node P ∈ C. Then λ gives a smoothing of (C, G).

2.2.
Singularities of degenerations of surfaces isogenous to a product of unmixed type. We study smoothings of products of two stable curves with a group action in a similar way as G-equivariant smoothings of curves in Section 2.1.
We treat the unmixed case first. Proof. For the " ⇒ " direction, suppose Z = C × ∆ D → ∆ is a free smoothing of (C × D, G). Let (P, Q) be any point on C × D. Note that G (P,Q) = G P ∩ G Q . We divide our further discussion into 3 cases: Case (U 0 ): P, Q are both smooth points on C, D respectively. We will show in this case that G (P,Q) = {1}. Suppose 1 = τ ∈ G (P,Q) : then P, Q are both fixed points of τ . Since P, Q are both smooth points on C, D respectively, τ fixes points on C t , D t for t = 0 by Lemma 2.5. So τ fixes points on C t × D t for t = 0, which contradicts the assumption that G acts freely on C t × D t for t = 0.
Case (U 1 ): One of P, Q, say P , is a node and the other is a smooth point. Suppose G (P,Q) = {1}. By Lemma 2.4, G Q is cyclic and hence its subgroup G (P,Q) is also cyclic. Let G (P,Q) = τ , τ = 1. By Lemma 2.5, τ fixes points of D t for t = 0. Since G acts freely on C t × D t for t = 0, τ does not fix any point of C t , t = 0. By Lemma 2.7, τ does not interchange the branches of C around P . Hence there are a local embedding C : (xy = 0) ⊂ C 2 around P and a local coordinate z of D around Q such that the action of τ on C × D is (x, y, z) → (ξx, ξ −1 y, ξ q z) where ξ is a primitive root of unity of order |τ | and (q, |τ |) = 1.
Case (U 2 ): P, Q are both nodes on C, D.
Suppose G (P,Q) = {1}. By Lemma 2.6, we have that G P and G Q are either cyclic or dihedral. This implies that G (P,Q) = G P ∩ G Q is either cyclic or dihedral. Suppose G (P,Q) is dihedral. Then G P and G Q are both dihedral and, by the proof of Lemma 2.6, there is τ 1 ∈ G (P,Q) (resp. τ 2 ∈ G (P,Q) ) such that τ 1 (resp. τ 2 ) interchanges the branches of C at P (resp. the branches of D at Q). By Lemma 2.7, τ 1 (resp. τ 2 ) fixes points of C t (resp. D t ) for t = 0. Since neither τ 1 nor τ 2 fixes points on C t × D t , τ 1 (resp. τ 2 ) does not fix points on D t (resp. C t ). Again by Lemma 2.7, τ 1 (resp. τ 2 ) does not interchange the branches of D at Q (resp. the branches of C at P ). Now set τ := τ 1 τ 2 , then τ interchanges the branches of C as well as those of D. This implies that τ fixes points on C t × D t for t = 0, a contradiction.
So G (P,Q) is cyclic and we can assume that G (P,Q) = τ . Since τ does not fix any point on C t × D t for t = 0, τ interchanges the branches of at most one of C and D. If τ does interchange the branches of one of C and D, say C, then the order of τ is 2 (Lemma 2.6) and we can choose local embeddings C : (xy = 0) ⊂ C 2 and D : (zw = 0) ⊂ C 2 such that τ acts as (x, y, z, w) → (y, x, −z, −w).
For the " ⇐ " direction, note that C and D admit G-equivariant smoothings C → ∆, D → ∆ by Theorem 2.8. In each of the cases (U 0 ), (U 1 ), (U 2 ), any nontrivial element τ k ∈ G (P,Q) = τ interchanges at most the local branches of one of the factors. This guarantees that τ k acts locally freely on at least one of the factors of C t × D t for t = 0 (Lemma 2.7). So Z := C × ∆ D → ∆ is a required free smoothing.
Remark 2.10. In the unmixed case, G (P,Q) is always cyclic.
According to Theorem 1.12 and the discussion thereafter, a surface X is a stable degeneration of surfaces isogenous to a product of unmixed type if and only if X = (C × D)/G, where C, D are stable curves and G is a finite group acting on C, D and acting diagonally on C × D such that (C × D, G) admits a free smoothing. So we have Corollary 2.11. The possible singularities of a stable degeneration X of surfaces isogenous to a product of unmixed type are as follows: (U 1a ) Double normal crossing singularities: (xy = 0) ⊂ C 3 . These are the general singularities of X. (U 1b ) Quotients of the above singularity under the group action: where ξ is a primitive n-th root of unity, (q, n) = 1. In this case, the index of the singularity is n and the canonical covering is the singularity (U 1a ).
where ξ is a primitive n-th root of unity, (q, n) = 1. In this case, the singularity is still a (Gorenstein) degenerate cusp.
We give some examples of the singularities in Cor. 2.11.
Example 2.12. Let G = σ ∼ = Z 2 . Let C, D ′ be two hyperelliptic curves. Suppose σ acts on C and D ′ as the respective hyperelliptic involutions. Let {Q ′ 1 , . . . , Q ′ 2k } be the fixed points of σ on D ′ . We obtain a stable curve D from D ′ by identifying Q ′ 2i−1 and Q ′ 2i for any 1 ≤ i ≤ k. Note that σ also acts on D. Let G act on C × D diagonally. Then the quotient (C × D)/G has singularities of type (U 1a ) or (U 1b ).
Example 2.13. Let C, D be two stable curves. Let G be a finite group acting freely on C × D. Then (C × D)/G has singularities of type (U 1a ) or (U 2a ).
Example 2.14. Let G = σ ∼ = Z 2 . Let C ′ and D ′ be two smooth curves of genus ≥ 1 such that G acts (faithfully) on both. Assume σ fixes 2k points P ′ 1 , P ′ 2 , . . . , P ′ 2k on C ′ . Let C be the stable curve obtained by identifying P ′ 2i−1 and P ′ 2i for 1 ≤ i ≤ k. Assume σ acts freely on D ′ . Pick a point Q ′ ∈ D ′ . Let D be the stable curve obtained by identifying Q ′ and σ(Q ′ ). Note that G acts on C and D. Let G act on C × D diagonally. Then (C × D)/G only has singularities of type (U 1a ) or (U 2b ).
Example 2.15. Let G = σ ∼ = Z 2 . Let C ′ and D ′ be two smooth curves of genus ≥ 1 such that G acts (faithfully) on both. Assume σ fixes 2k points P ′ 1 , P ′ 2 , . . . , P ′ 2k on C ′ . We obtain a stable curve C from C ′ by identifying P ′ 2i−1 and P ′ 2i for 1 ≤ i ≤ k. Similarly, we can obtain a stable curve D from D ′ . Note that σ also acts on C and D. Let G acts on C × D diagonally. Then the quotient (C × D)/G has singularities of type (U 1a ) or (U 2c ).

2.3.
Singularities of degenerations of surfaces isogenous to a product of mixed type. Now we consider the mixed case (cf. Section 1.2).
Proposition 2.16 (Criterion for free smoothings in the mixed case). Let C be a stable curve and G • < Aut(C) a finite group. Let Proof. "⇒ "Let Z → ∆ be a free smoothing of (C × C, G). It is necessarily of the form C 1 × ∆ C 2 → ∆, where C 1 → ∆ is a smoothing of (C, G • ) and C 2 → ∆ is a smoothing of C yet with a different G • -action given by G • ϕ − → G • < Aut(C). Note also that Z → ∆ is automatically a free smoothing of (C × C, G • ). Hence (C × C, G • ) satisfies (U 0 ), (U 1 ), (U 2 ) in Prop. 2.9.
Let Γ be the subset of G • consisting of the transformations having fixed points on C 1t for t = 0. Since C 1 × ∆ C 2 → ∆ is a free smoothing of C × C, we have In particular, ϕ(γ)τ γ = 1. The above two conditions simply say that G acts freely on C 1t × C 2t for t = 0.
For (iii), we discuss the possible stabilizer of a point (P, Q) ∈ C × C in the following 2 cases.
Case (M 0 ): P, Q are both smooth points of C. We will show G (P,Q) = {1} in this case. Since G • acts freely on C 1t ×C 2t for t = 0. Note that G • ∩ G (P,Q) = {1} by the claim for the unmixed (U 0 ) case. Suppose on the contrary that there is a 1 = τ 1 ∈ G (P,Q) , then This contradicts (B) above.
Case (M 1 ): One of P, Q is a node, while the other is a smooth point. We will show that G (P,Q) ⊂ G • in this case. Otherwise (P, Q) is fixed by τ ′ γ ∈ G \ G • for some γ ∈ G • : (P, Q) = τ ′ γ(P, Q) = ((ϕγ)Q, τ γP ), so P = (ϕγ)Q and Q = τ γP . In particular, either P, Q are both nodes or they are both smooth points of C, a contradiction. Hence (iii) follows.
"⇐ "By Prop. 2.9, condition (i) implies that (C, G • ) has a smoothing C 1 → ∆ and there is another smoothing C 2 → ∆ of C with a different G • -action induced by ϕ. These two smoothings of C are in fact isomorphic via ϕ. Set Z := C 1 × ∆ C 2 → ∆. We can introduce an action of G on Z → ∆ by the formulae in (II): for any It remains to check that G acts freely on Z t for t = 0. Note that G • acts freely by hypothesis (i) and Prop. 2.9. Now let τ ′ γ ∈ G \ G • for some γ ∈ G • . If τ ′ γ does not fix points on C × C, then obviously τ ′ γ does not fix points on C t × C t for t = 0. If τ ′ γ ∈ G (P,Q) for some (P, Q) ∈ C × C, then both P, Q must be nodes by (iii) and we can find local embeddings of the first factor: xy = t n (resp. of the second factor: zw = t m ) such that the action of τ ′ γ around (P, Q) is given by: (x, y, z, w, t) → (az, bw, x, y, t), where a, b ∈ C * are nonzero numbers. Hence (τ ′ γ) 2 (x, y, z, w, t) = (ax, by, az, bw, t). If t = 0, then xy = t n , zw = t m implies that xyzw = 0. Suppose τ ′ γ fixes some point (x, y, z, w, t) ∈ C 1t × C 2t for t = 0, then az = x, bw = y, x = z, y = w.
According to Theorem 1.12, a surface X is a stable degeneration of surfaces isogenous to a product of mixed type if and only if X = (C × C)/G where C is a stable curve and G is a finite group acting in the way described in Prop. 2.16. where ξ is a primitive n-th root of unity, (q, n) = 1 and ab ∈ ξ \ ξ q+1 . In this case, the index of the singularity is 2 and the canonical covering is a singularity of type (U 2c ).
Proof. Let (C × C, G) be as in Prop. 2.16 such that X = (C × C)/G and let π : C × C → X be the quotient map. If (P, Q) ∈ C × C is such that G (P,Q) ⊂ G • , then the singularity π(P, Q) ∈ X is of type (U 1a ), (U 1b ), (U 2a ), (U 2b ) or (U 2c ). If (P, Q) ∈ C × C is such that G (P,Q) G • , then P, Q are both nodes of C by Prop. 2.9. We want to know the action of G (P,Q) around (P, Q). Note that (C×C, G • ) is of unmixed type and G • ∩G (P,Q) is just the stabilizer of (P, Q) ∈ C×C under the action of G • . By the analysis done for the unmixed type, G • ∩ G (P,Q) = τ 1 for some τ 1 ∈ G • and τ 1 interchanges at most the branches of one factors of C × C. We will show that τ 1 does not interchange any branches at P or Q.
We give an example of singularity of type (M ).
Example 2.19. Let G = σ ∼ = Z 4 . Then τ 1 := σ 2 has order 2. Let C ′ be a smooth curve of genus ≥ 2. Suppose τ 1 acts on C ′ so that there are exactly two fixed points P ′ 1 and P ′ 2 . We obtain a stable curve C from C ′ by identifying P ′ 1 and P ′ 2 . Let P ∈ C denote the image of P ′ 1 and P ′ 2 . Then τ 1 also acts on C and has exactly one fixed point P .

Connected components of the moduli space
In this section we will study the Q-Gorenstein deformations of the stable degenerations of surfaces isogenous to a product. As a result, we get some connected components of the moduli space of stable surfaces M st a,b defined in Section 1.6. Let Z = C×D be a product of two stable curves and let G be a finite group acting on Z with finitely many fixed points. Let π : Z → X be the quotient map. For any G-equivariant coherent sheaf F on Z, we define an O X -module π G * F := (π * F ) G . The following lemma is well-known.
Lemma 3.1. Let F be a G-equivariant coherent sheaf on Z. Then for any p ≥ 0, Lemma 3.2. Suppose all the (possible) singularities on X are of type (U 1a ), (U 1b ), (U 2a ) or (U 2b ). Then π G * T Z = T X and π G * T 1 Z = T 1 QG,X .
Proof. First observe that both π G * T Z and T X are S 2 -sheaves of O X -modules ([AbH09, Lemma 5.1.1]). Since π : Z → X isétale off a finite subset, π G * T Z and T X coincide off the finite subset. Then the S 2 -property guarantees that π G * T Z and T X are isomorphic on the whole of X.
For π G * T 1 Z = T 1 QG,X , we view π G * T 1 Z (resp. T 1 QG,X ) as the sheaf of first-order, G-equivariant, local deformations of Z (resp. first-order, Q-Gorenstein, local deformations of X). Let P be any point on X and let π −1 (P ) = {Q j } j be inverse image of P . Since the possible singularities on X are of type (U 1a ), (U 1b ), (U 2a ) or (U 2b ), every germ Q j ∈ Z is a canonical covering of P ∈ X and they are permuted under the action of G. Since Q-Gorenstein deformations of the germ P ∈ X are precisely those deformations which lift to deformations of the canonical covering, we have a natural identification π G * T 1 Z = T 1 QG,X sending a first-order, G-equivariant, local deformations of Z to its quotient under G.
Theorem 3.3. If all the (possible) singularities on X are of type (U 1a ), (U 1b ), (U 2a ) or (U 2b ), then a semiuniversal Q-Gorenstein deformation of X exists and hence the base Def QG X is defined. Moreover, Def QG X is smooth.
Proof. Since Z = C × D is Gorenstein, Def QG Z exists and is just Def Z . Let f : Z → Def Z be a semiuniversal deformation of Z. Then the action of G on Z induces actions of G on Z and Def Z such that f becomes a G-equivariant morphism. Taking the G-invariant part Def G Z of Def Z and the G-quotient of f −1 ((Def Z ) G ), we get a deformation of X = Z/G: Note that Def Z = Def C × Def D is smooth ([vO05, Cor. 2.3] and [DM69]), so (Def Z ) G is also smooth by Cartan's lemma. To prove the theorem, it suffices to show that (3.1) is a semiuniversal Q-Gorenstein deformation of X.
Since all the possible singularities on X are of type (U 1a ), (U 1b ), (U 2a ) or (U 2b ), for any P ∈ X and Q ∈ π −1 (P ), the germ Q ∈ Z is the canonical covering of P ∈ X. So (3.1) is in fact a Q-Gorenstein deformation of X.
By the infinitesimal lifting property of a smooth variety ([Har77, Ex. II.8.6]), if we can show that the natural map dλ : (T 1 Z ) G → T 1 QG,X is an isomorphism, then f G : f −1 ((Def Z ) G )/G → (Def Z ) G is an unobstructed semiuniversal Q-Gorenstein deformation of X. Consider the following commutative diagram QG,X ) → H 2 (X, T X ) in which the rows are exact. We will prove that α, β, γ are isomorphisms, so that dλ : (T 1 Z ) G → T 1 QG,X is also an isomorphism by the Five Lemma. Let F = T Z or T 1 Z in Lemma 3.1, we get the following equations H 1 (Z, T Z ) G = H 1 (X, π G * T Z ), H 0 (Z, T 1 Z ) G = H 0 (X, π G * T 1 Z ), H 2 (Z, T Z ) G = H 2 (X, π G * T Z ). By Lemma 3.2, we have π G * T Z = T X , π G * T 1 Z = T 1 QG,X . Therefore H 1 (Z, T Z ) G = H 1 (X, T X ), H 0 (Z, T 1 Z ) G = H 0 (X, T 1 QG,X ), H 2 (Z, T Z ) G = H 2 (X, T X ), and α, β, γ are isomorphisms.
Remark 3.4. It remains to address the case where X has singularities of type (U 2c ) or (M ). The canonical coverings of these two types of singularities are not local complete intersections, which results in a more difficult Q-Gorenstein deformation theory. In contrast to the infinitesimal consideration, there might be some hope for good properties of a one-parameter family of such singularities. corresponds to a surface S ′ isogenous to a product with representation (C ′ × D ′ )/G. In our case, since a triangle curve is rigid, we can assume (C ′ , G) is also a triangle curve. In fact, (C ′ , G) = (C, G) or (C, G). For the same reason, (C ′ , G) remains the same in the process of degeneration. If [X] is in M top S , then X = (C ′ × D)/G, where D is a stable curve. By Prop. 2.9 and Cor. 2.11, the possible singularities of X are of type (U 1a ) or (U 1b ). Therefore Def QG X is defined and is smooth by Theorem 3.3. This implies that M st a,b is irreducible at [X] and the assertion of the corollary follows.