Note on a family of surfaces with $p_g=q=2$ and $K^2=7$

We study a family of surfaces of general type with $p_g=q=2$ and $K^2=7$, originally constructed by C. Rito. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus $\mathcal{M}$ in the moduli space of the surfaces of general type. In particular we prove that $\mathcal{M}$ is an open subset, and it has three connected components, two dimensional, irreducible and generically smooth.


Introduction
In the last two decades, several authors worked intensively on the classification of irregular algebraic surfaces (i.e., surfaces S with q(S) > 0) and produced a considerable amount of results, see for example the survey papers [BaCaPi06,MP12,Pen13] for a detailed bibliography on the subject.
In particular, irregular surfaces of general type with χ(O S ) = 1, that is, p g (S) = q(S) ≥ 1 were investigated. By, nowadays classical, Debarre inequality [Deb81, Théorème 6.1] we have p g ≤ 4. Surfaces with p g = q = 4 and p g = q = 3 are completely classified, see [Bea82,CaCiML98,HP02,Pir02]. On the other hand, for the the case p g = q = 2, which presents a very rich and subtle geometry, we have so far only a partial understanding of the situation; we refer the reader to [Cat00,Cat11,Cat15,Pen09,PePol13a,PePol13b,PePol14,PiPol17,PRR20,Zuc03] for an account on this topic and recent results.
As the title suggest, in this paper we consider a family of minimal surfaces of general type with p g = q = 2 and K 2 = 7. The existence of these surfaces was originally established by Rito in [Rit18]; the present work provides an alternative construction of them, that allows us to describe their Albanese map and their moduli space.
Our results can be summarized as follows.
the degree of the Albanese map a topological invariant (see Proposition 5.1), these families provide a substantially new piece in the fine classification of minimal surfaces of general type with p g = q = 2 in the spirit of [Cat84,Cat89,Cat90].
The paper is organized as follows.
In Section 2 we explain our construction in details, pointing out the similarities and the differences with [Rit18], and computing the invariants of the resulting surfaces (Proposition 2.2). We study their Albanese map, giving a precise description of its image, isogenous to a product of two curves of genus 1, and of its branch curve.
In Section 3 we use our description to study the modular image of Rito's family, showing that it has three connected components, all irreducible of dimension 2.
The last two sections contain results of deformation theory headed to compute h 1 (S, T S ) = 2 (Proposition 5.7) from which it follows that each component is open and generically smooth in the moduli space.
Section 4 is devoted to a general result, Theorem 4.2, about the deformations of the blow up in a point, that was crucial for the proof and that we find of independent interest. The situation is the following: consider a point p in a smooth surface B, a curve D in B smooth at p and a vector v ∈ T p B. A standard exact sequence associates to v an infinitesimal deformation B of the blow-up of B in p. Then Theorem 4.2 says that B contains an infinitesimal deformation of the strict transform of D if and only if the class of v in the normal vector space T p B/T p D extends to a global section of the normal bundle of D in B.
Finally Section 5 is devoted to the study of the first-order deformations of the surfaces in M. To show h 1 (S, T S ) = 2, we show in fact that the map H 1 (S, T S ) → H 1 (A, T A ) is injective, and its image is given by the infinitesimal deformations of A that are still isogenous to a product.
Notation and conventions. We work over the field C of complex numbers. By surface we mean a projective, non-singular surface S, and for such a surface K S denotes the canonical class, p g (S) = h 0 (S, K S ) is the geometric genus, q(S) = h 1 (S, K S ) is the irregularity and χ(O S ) = 1 − q(S) + p g (S) is the Euler-Poincaré characteristic.

The construction
In this section we give an alternative, but equivalent, construction to the surface S of general type with p g = q = 2 and K 2 = 7 constructed by Rito in [Rit18].
Fix a square root c of ab and consider the following points on the curves we have just defined  Finally, let = (x 0 ) be the line through p 1 and p 2 and t = (2x 0 − x 1 − x 2 ) be the tangent line to C 1 through p 3 , see Figure 1 to have a visual representation of the situation.
Up to now, we followed Rito in [Rit18], changing the notation only for the curve t (R in Rito's notation). Now, we proceed a bit differently. Let us apply the following birational transformations of P 2 : 1. We blow up the point p 0 and we get σ 0 : Bl p 0 (P 2 ) −→ P 2 with exceptional divisor E 0 (see Figure 1 again).
Considering the pencil of lines through p 0 on Bl p 0 (P 2 ) we have a rational pencil of curves with self-intersection 0, which include the strict transforms of the four lines r i , i = 1, . . . , 4. We notice that on Bl p 0 (P 2 ) we can lift the natural involution on P 2 j : (x 0 : which has as fixed divisor E 0 + σ * 0 ( ).
2. The quotient by this involution Bl p 0 (P 2 )/j is the Segre-Hirzebruch surface F 2 .
The images of the four lines r i are fibres of the fibration on F 2 . Moreover, the only negative section of this fibration coincide with the image of E 0 .
3. We blow up on F 2 the images of the points p 1 and p 2 , introducing two exceptional divisors E 1 and E 2 .
We recall that the images of the lines r 1 and r 2 and of the conics C 1 and C 2 pass all through these points. Performing this operation the images of r 1 and r 2 became −1-curves (see Figure 2). 4. We contract the images of the curves r 1 and r 2 . The resulting surface is exactly P 1 × P 1 .
Summarizing we have obtained a rational map of degree 2 σ : P 2 P 1 × P 1 .
We denote with the same letters the strict transform on P 1 × P 1 of all the curves considered on P 2 , since no confusion arises (see Figure 3). The bidouble cover of P 1 × P 1 with ramification divisors is obviously the product T 1 × T 2 of two double covers φ j : T j → P 1 branched at 4 points, two curves of genus 1 (see Figure 3). The fibre product of the bidouble cover T 1 × T 2 → P 1 × P 1 with σ 0 gives the bidouble cover of P 2 studied in [Rit18][Section 3, Step 1], where it is shown that it is birational to an abelian surface that we denote by A (it was V in [Rit18]). We can summarize this construction with the following diagram.  We see (compare [Rit18][Section 3, Step 2]) that the strict transform of the curve C 1 is tangent to the curve t on A at a point p. This point is a tacnode (singularity of type (2, 2)) for the strict transform of the curve t. So the divisor t + C 1 is reduced and has a singularity of type (3, 3).
Remark 2.1. We see that we recover the construction due to Rito [Rit18] of an abelian surface with a (1, 2)-polarization, Please notice that in [Rit18] the abelian surface A was labelled by V and the curves C 1 and t byĈ 1 andR.
In [Rit18] it is shown that the divisor t + C 1 is even, i.e. there is a divisor L such that t + C 1 ≡ 2L, and that (t + C 1 ) 2 = 16.
So L is a polarization of type (1, 2). This is exactly the situation described by the first author and F. Polizzi in [PePol13a, Remark 2.2]. There the authors suggest how to construct a surface with p g = q = 2 and K 2 S = 7 as a generically finite double cover of A branched along a divisor as t + C 1 . We follow the suggestion slavishly and we summarize the situation with the following special case of [Rit18, Proposition 1] Proposition 2.2. Let A be an Abelian surface. Assume that A contains a reduced curve t + C 1 and a divisor L such that t + C 1 ≡ 2L, (t + C 1 ) 2 = 16 and t + C 1 contains a (3, 3)-point and no other singularity. Let S be the smooth minimal model of the double cover of A with branch locus t + C 1 . Then p g (S) = q(S) = 2 and K 2 S = 7. Let us now construct S step by step starting from A.
1. First, we resolve the singularity in p. To do that, we need to blow up A twice, first in p and then in a point infinitely close to p. Let us denote these two blow ups by On B , let us denote by F the exceptional divisor relative to σ 4 , by E the strict transform of the exceptional divisor E relative to σ 3 , by C 1 the strict transform of C 1 and, finally, by R the strict transform of t (see Figure 4). In addition, one gathers the following information: E ∼ = P 1 and (E ) 2 = −2, F ∼ = P 1 and F 2 = −1, g(C 1 ) = 1 and C 2 1 = −2.
We can summarize the construction of S with the following diagram.
We note that since α is the Albanese morphism of S, we obtained in particular that the Albanese variety of these surfaces is isogenous to a product of elliptic curves: Proposition 2.3. The Albanese variety A of the surface S is isogenous to a product via an isogeny ι : A → T 1 × T 2 of degree 2.
3 Rito's family has three components of moduli dimension 2 The surfaces S are constructed by a configuration of plane curves determined by two parameters (as noticed already in [Rit18, Section 3, Step 4]), that we denoted by a, b, and a choice of a linear system |L| such that |2L| contains the divisor |C 1 + t|. So there are 2 4 possible choice for L, since we can always add to L a 2−torsion line bundle. In this section we prove that the family has three connected components, all irreducible of moduli dimension 2.
Remark 3.2. We label the the ramification points of φ 1 : T 1 → P 1 as a 1 , a 2 , a 3 , a 4 using Figure  3 as follows.
φ 1 (a 1 ) be the projection of the line labeled E 1 φ 1 (a 2 ) be the projection of the line labeled E 2 φ 1 (a 3 ) be the projection of the line labeled r 3 φ 1 (a 4 ) be the projection of the line labeled r 4 Similarly, we label the the ramification points of φ 2 : T 2 → P 1 as b 1 , b 2 , b 3 , b 4 so that φ 2 (b 1 ) be the projection of the line labeled C 1 φ 2 (b 2 ) be the projection of the line labeled C 2 φ 2 (b 3 ) be the projection of the line labeled l φ 2 (b 4 ) be the projection of the line labeled E 0 Both fibrations have been considered in [Rit18, Section 3, Step 3]. The fibration f 1 is the pull back of the pencil of the lines through the point p 0 and a j corresponds to the line r j . So, the branching points of φ 1 correspond to the lines r 1 , r 2 , r 3 and r 4 , that, in the natural coordinates, give the 4 points (1 : 0), (0 : 1), (1 : 1) and (a : b), with cross-ratio a b . The fibration f 2 is given by the pencil of conics tangent to the lines r i in the points p i , i = 1, 2: b 1 corresponding to the conic C 1 , b 2 corresponding to C 2 , b 3 corresponding to 2l, b 4 corresponding to r 1 + r 2 . Writing this pencil as x 2 0 , x 1 x 2 we get a parametrization of P 1 such that the branching points of φ 2 have coordinates (1 : 1), (1 : ab), (1 : 0) and (0 : 1), with cross-ratio ab.
We deduce the following Proof. The base of the family of the surfaces S has a finite proper map on an open subset of C 2 given by the parameters (a, b). So, if C is any irreducible component of it, dim C = 2. The relative Albanese morphism maps C to the moduli space of the Abelian surfaces with a polarization of type (1, 2). By Proposition 2.3 the image of C is contained in the 2−dimensional subvariety I of those isogenous to a product of curves. Since these curves are double covers of P 1 branched at 4 points with cross-ratio respectively a b and ab the general pair of curves of genus 1 appears in the image of C: the map C → I is generically finite and therefore dominant.
Since isomorphic manifolds have isomorphic Albanese varieties, C → I factors through the moduli space of the surfaces of general type, and then the moduli dimension of C is 2.
We can now determine the isogeny. Recall that anétale double cover of a variety is determined up to isomorphism by a 2−torsion line bundle on it, the antiinvariant part of the direct image of the structure sheaf of the cover. Moreover for {i, j, h, k} = {1, 2, 3, 4},

Proof. By Remark 3.2 we can write every line bundle of torsion 2 on
We compute separately each factor by restricting to a fibre of type Λ 1 resp. Λ 2 . In fact, restricting the isogeny to a fibre of type Λ 1 (respectively Λ 2 ) we obtain anétale double cover of T 2 (respectively T 1 ) given by the restriction of the above bundle We did the computation by using the fibres over a 3 and b 1 . First consider the curver 3 : It is invariant by the (Z/2Z) 2 action on A given by the bidouble cover π, and in factr 3 lies in the locus of the fixed points of one of the three involutions. Thus π induces a nontrivial involution on it, whose quotient is the double coverr 3 → r 3 branched on p 3 + p 5 + p 8 + p 10 (see Figure 1). The involution j acts on r 3 permuting those points as p 3 ↔ p 5 , p 8 ↔ p 10 lifting to an involution onr 3 without fixed points. Taking the quotient we get a commutative diagram Let us call q j the ramification point in A of π |r 3 mapping to p j . Then ι( The analogous computation for the elliptic curve C 1 = f −1 2 (b 1 ) leads to consider the 4 points on the corresponding conic cut by the lines r 3 and r 4 , permuted by j as p 3 ↔ p 5 , p 7 ↔ p 9 . A fully analogous computation leads to Recalling that the kernel of ι * : Pic(T 1 × T 2 ) → PicA is a subgroup of order 2 generated by the antiinvariant part of ι * O A , we deduce that can be written equivalently as Proposition 3.5.
It follows that we have the following description of the 16 possible linear systems L.
Proposition 3.6. |L| varies among the linear systems |f * 1ā ⊗ f * 2b | whereā andb solve one of the following Notice that each of the two systems of equations has 16 distinct solutions (ā,b), divided in pairs by the equivalence relation (1); so it gives 8 distinct linear systems. We get then 16 different possible choices of |L| as expected.
Proof. If (ā,b) solves the system 2), then 2 (2). So all these linear systems are possible choices. Since they are 16, they are all possible choices.
We observe that a 3 ∈ T 1 and b 1 ∈ T 2 are the images of the essential singularity of the branching curve of the Albanese map of S.
Inspecting the linear equivalences in Proposition 3.6 we observe that in all cases O T 2 (b − b 1 ) is a 4− torsion line bundle. On the contrary O T 1 (ā − a 3 ) is a torsion line bundle whose torsion order may change: it is 4 in case 2) whereas in case 1) there are two possibilities: it may be 2 or 1. Recalling that we have two pairs (ā,b) for each choice of |L| (and then for each S) we deduce the following natural decomposition of M.
Definition 3.7. We say that a surface S ∈ M is of type j if the minimal (among the two possible choices ofā) torsion order of O T 1 (ā − a 3 ) is j.
By Proposition 3.6 the values that j assumes are 1, 2, 4. Setting M j for the subset of M of the surfaces of type j we observe that each M j is open and therefore we have decomposed as union of disjoint not empty open subsets. Now we prove that each M j is irreducible.
Definition 3.8. We denote (as usual) by M 1,3 the moduli space of the curves of genus 1 with three ordered marked points. We are not assuming the points to be distinct.
We denote an element of M 1,3 as (C, x, y, z) where C is a curve of genus 1 and x, y, z ∈ C. We denote by N j , j ∈ {1, 2, 4} the subvariety of M 1,3 × M 1,3 of the form Note the correspondence among conditions 1, 2, 3 and almost all solutions of the equations Proposition 3.6. The only solutions that do not have a counterpart here are those withā = a 4 . This is the reason for the map in the next statement to have a different degree for j = 1.
Proposition 3.9. For each j = 1, 2, 4 there is a proper finite surjective morphism Proof. We construct the map m j .
For every (T 1 , a 3 , a 4 ,ā) , Now we construct a bidouble cover π : A P 2 as in Rito's construction. We consider each T j with the group structure such that a 3 and b 1 are the respective neutral elements. This fixes an action of the Klein group K ∼ = (Z/2Z) 2 as group automorphisms of Then we choose a point p ∈ ι −1 (a 3 , b 1 ) and consider A with the group structure such that p is the neutral element, so that ι is a group homomorphism. Considering the analogous action of the Klein group on A we get a commutative diagram The bidouble cover T 1 × T 2 → P 1 × P 1 is ramified at the union of 8 elliptic curves, mapping to the four 2−torsion points on each factor, including a 3 , a 4 on T 1 and b 1 , b 2 on T 2 . We label the remaining points on T 1 as a 1 , a 2 and the remaining points on T 2 as b 3 , b 4 . We note that the 2-torsion bundle O T 1 (a 4 − a 3 ) O T 2 (b 2 − b 1 ), when restricted to them, is not trivial. This implies that their preimage on A, ramification locus of the 8 : 1 morphism A → P 1 × P 1 is again union of 8 elliptic curves naturally labeled as a 1 , . . . , a 4 , b 1 , . . . , b 4 .
A direct computation shows that the Klein group of A acts on each of them, action that is faithful exactly on the curves labeled a 1 , a 2 , b 3 , b 4 . So the ramification locus of the double cover D → P 1 × P 1 is the image of them, union of 4 rational curves, two on each ruling. Therefore D is a Del Pezzo surface of degree 4 with 4 nodes. Solving the 4 nodes we obtain a weak Del Pezzo surface with a configuration of 8 rational curves whose incidence graph is an octagon with alternating self intersections −1 and −2: the strict transforms of the ramification lines have self intersection −1 whereas the exceptional curves have self intersection −2.
Now we consider, among the −1 curves in the octagon, the one 'labeled' b 3 : contract first the other three −1 curves and then the two exceptional curves now of self intersection −1: the resulting surface is P 2 and the remaining three sides of the octagon map to three lines, let's call them l (the one coming from the −1-curve "'b 3 "), r 1 and r 2 . The preimages of the lines of P 1 × P 1 labeled a 3 , a 4 , b 1 , b 2 are respectively two lines r 3 and r 4 and two conics C 1 , C 2 forming the configuration of curves in Figure 1.
Notice that the two points in ι −1 (a 3 , b 1 ) map bijectively to the intersection points of r 3 and C 1 . We draw the tangent t to C 1 in the image of p. Pulling-back t and adding the elliptic curve dominating C 1 we obtain a divisor in A as in Proposition 2.2. We have recovered Rito's construction.
Then Proposition 3.5 applies and we have 16 double covers S → A branched on this divisor, determined by the 32 solutions of the equations in Proposition 3.6. Since the pair (ā,b) is a solution by assumption, we can define as image of our element in N j the surface S of the corresponding double cover. Then S belongs to M j by construction.
It is important to recall here that we have done an arbitrary choice in this construction, when we chose p ∈ ι −1 (a 3 , b 1 ). We notice now that the isomorphism class of the surface S does not depend on this choice, since the two corresponding double covers of A are conjugated by the involution of A given by the isogeny. So we have a well defined morphism N j → M j .
Finally since there are two pairs of possible (ā,b) for each |L| the maps m j are proper of degree 2 for j ≥ 2. For j = 1 the degree is 1 because we are not considering the solutions with a = a 4 . The surjectivity is obvious. Now, we shall deal with the problem of irreducibility of N j for j = 1, 2 and 4, to do that we need to introduce some notation.
Let us recall some well known fact about modular curves, see e.g. [DS05, Section 1.5]. The principal congruence subgroup of level N is The most important congruence subgroups are The modular curve Y(Γ) for Γ is defined as and the special cases of modular curves for Γ 1 (N ) denoted by Y 1 [N ] = H/Γ 1 (N ).
Theorem 3.10. Points of Y 1 [N ] correspond to pairs (E, P ), where E is an elliptic curve and P ∈ E is a point of exact order N . Two such pairs (E, P ) and (E 0 , P 0 ) are identified when there is an isomorphism of E onto E 0 taking P to P 0 .
We are interested in the case when N = 4 and in the special modular curve Y 1 [4] which parametrizes elliptic curves with 4-torsion points. Now, let Y 1 [2, 4] the space parametrizing an elliptic curves with a 2-line bundle point Q and a 4-torsion line bundle T such that T 2 = Q, than we have the following proposition. induced over a character χ is as follows: Therefore we have

And by equations (3) one can compute the induced action of M over a character ψ.
Thus, to prove the first part of the proposition it is sufficient to check that the monodromy action of G is transitive on the set This is a straightforward computation which can be carried out as the one in the proof of [PePol13a, Proposition A1] and it is left to the reader.
Therefore we can consider the set of triples The group G acts on the set of triple (z, χ, ψ), with the natural action of the modular group on H and by the induced monodromy action on the second two ones. The corresponding quotient Y 1 [2, 4] is a quasi-projective variety. Moreover π : Y 1 [2, 4] −→ H/G given by the forgetful map, is anétale covers on a smooth Zariski open set Y 0 1 ⊂ H/G; then it is generically smooth. Finally, by construction Y 1 [2, 4] is a normal varieties, because it only has quotient singularities. Then, since it is connected, it must be also irreducible.
Finally, let Y 1 [2, 2] the space parametrizing an elliptic curves with a 2-line bundle Q and a second 2-torsion line bundle T such that T = Q, than we have the following proposition.
Proof. The proof is analogous to the one for Proposition 3.11. One has to be careful, again, in checking that the monodromy action of G is transitive on the set But, again, this follows from the actions (4), from which one sees right away that the image of T is always different form the image of Q.
Proof. First of all, we mean by an elliptic curve marked with a point that we have fixed a group structure on the curve of genus 1 for which that point is the neutral element. We always choose for T 1 the point a 3 and for T 2 the point b 1 as neutral elements.
After this global consideration, we prove the claim case by case as j varies.
Case j=4: By Definition 3.8 the variety N 4 depends on the following data: • one elliptic curve T 1 marked with a point a 3 and a 4−torsion line bundle T 1 = O T 1 (ā − a 3 ) which is not 2−torsion -its square determines the last point on T 1 (a 4 ); • one elliptic curve T 2 marked with a point b 1 , a 4−torsion line bundle In other words there is a dominant morphism We observe that Y 1 [4] is a generically smooth quasi-projective variety, connected, and irreducible of dimension 1, [DS05, Chapter 2]. By Proposition 3.11 Y 1 [2, 4] is irreducible and generically smooth of dimension 1. This concludes the proof since dim N 4 = 2 by Proposition 3.3 and Proposition 3.9.
Case j=2: By Definition 3.8 the variety N 2 depends on the following data: • one elliptic curve T 1 marked with a point a 3 and two 2−torsion line bundles • one elliptic curve T 2 marked with a point b 1 , a 4−torsion line bundle In other words there is a dominant morphism By Proposition 3.12, we have that Y 1 [2, 2] is irreducible and generically smooth of dimension 1. This concludes the proof since dim N 2 = 2 by Proposition 3.3 and Proposition 3.9.
Case j=1: Finally, we have by Definition 3.8 that the variety N 1 depends on the following data: • one elliptic curve T 1 marked with a point a 3 and one 2−torsion line bundles In other words there is a dominant morphism We observe that Y 1 [2] is a generically smooth quasi-projective variety, connected, and irreducible of dimension 1, [DS05, Chapter 2]. Ad we conclude as the previous cases. Proof. By Proposition 3.9 we have that m j : N j → M j is a proper finte surjective morphism for each j = 1, 2 and 4. Moreover, by Proposition 3.14 we have that each N j is irreducible of dimension 2 for each j = 1, 2 and 4.

Some remarks on the deformations of a blown up surface
In this section we shall present some classicall results on deformation of a pairs. The main result is Theorem 4.2, possibly known to the experts, although we could not find it in the literature. This section will be employed systematically in the Moduli Space Section 5 and Theorem 4.2 mainly for the Remark 5.6.
Let us first recall some basic definition. Let B an algebraic nonsingular variety over an algebraically closed field k. The first order deformation of B is a commutative diagram where π is a flat morphism, Spec(k[ ]) = Spec(k[t]/t 2 ) and such that the induced morphism is an isomorphism. There is a natural notion of isomorphism between first order deformations, see [Ser06, Section 1.2]. The set of first order deformations, up to isomorphisms, is usually denoted by T 1 (B) and it has a natural structure of complex vector space (see [Sch68]). If B has a semiuniversal deformationB → Def (B) then every first order deformation is induced by a unique map Spec(k[ ]) → Def (B) and then there exists an isomorphisms of vector spaces for the last isomorphism see e.g. [Ser06, Proposition 1.2.9]. Now, we look at deformations of subvarieties in a given variety. Given a closed embedding D ⊂ B, the first order deformation of D in B is a cartesian diagram where π is flat and it is induced by the projection from B × Spec(k[ ]). Again we can give a cohomological interpretation to these deformations, indeed there is a natural identification between the first order deformations of D in   Let D 1 , . . . , D k be divisors in a smooth manifold X and x 1 , . . . , x k equations for them. Define Ω 1 S (log D 1 , . . . , log D k ) to be the subsheaf (as O X -module) of Ω 1 X (D 1 + . . . + D k ) generated by Ω 1 X and by dxj x j for j = 1, . . . k.
The next situation we want to look at is the case of deformation of a pair (B, D) where j : D → B is a closed embedding. The deformation theory of morphisms is more subtle if we want to allow both the domain and the target to deform nontrivially. A first order deformation of the pair (D, B) is a commutative diagram where π D and π B come from first deformations of D and B respectively and J is a closed embedding. There is a natural notion of isomorphism between first order deformations of pairs see e.g. [Ser06,Section 3.4]. And, we denote by Def j the set if isomorphism classes of first order deformations of the pair (B, D), which are locally trivial. Also in this case we have a cohomological interpretation, by [Ser06,Proposition 3.4.17], Def j has a formal semiuniversal deformation and its tangent space is isomorphic to H 1 (T B (− log D )), where T B (− log D ) is the sheaf of germs of tangent vectors to B which are tangent to D .
Finally, let us consider the following situation. Let B be a compact complex smooth surface, p ∈ B and σ : B → B the blow up of B in p with exceptional divisor E. Let D be an effective divisor on B which has multiplicity c in p. Moreover, let us denote by D = σ * (D) − cE the strict transform of D in B and assume that D is a smooth normal crossing divisor. We want to describe the relations between the deformations of the pair (B , D ) with those of D in B.
We know that the first order deformations of the pair (B , D ) are parameterized by the vector space H 1 (T B (− log D )). The natural map corresponds to the forgetful map, which forget the deformation of D . By [Har10, Exercise 10.5] we have an exact sequence where T p B ∼ = C 2 is the tangent space of B in p seen as skyscreaper sheaf concentrated in p. Then we consider the long exact sequence in cohomology and in particular the connecting homomorphism ψ : The next result give us a better understanding of the intersection between the images of the maps ϑ and ψ in H 1 (T B ).
Theorem 4.2. Keeping the same notation as before, assume that D is smooth at p, so c = 1, and choose an element v ∈ T p B. Then ψ(v) is contained in Im(ϑ) if and only if the class of v in the normal vector space T p B/T p D extends to a global section of the normal bundle v D ∈ H 0 (D, N D|B ).
In particular v is tangent to D if and only if v D vanishes in p.
Proof. We start constructing a family of first order deformations of B . Let U be an affine chart of B centered in p with local coordinates x, y such that D = {x = 0}. We consider a section s a,b of the trivial family B × Spec( a a obtained by mapping (x, y, ) to (a , b , ), so that the image is locally the complete intersection Blowing up this section we obtain the following families over Spec(C[ ]) where Φ a,b is a first-order deformation of B . The Kodaira-Spencer correspondence associates to Φ a,b a class in κ (Φ a,b ) ∈ H 1 (B , T B ), its Kodaira-Spencer class. This can be explicitly computed: following e.g. the proof of [Ser06, Proposition 1.2.9] we find The blown up chart U a,b is the subscheme of where (X, Y ) are homogeneous coordinates on the factor P 1 . It is the union of two affine charts, given respectively by imposing X = 0 and Y = 0. Let us work locally and restrict to the affine chart of U a,b given by Y = 0, and let us introduce the new coordinate z = X Y . Then, we can eliminate x by Since the class of v = a ∂ ∂x + b ∂ ∂y in T p B/T p D equals the class of a ∂ ∂x , then ψ (v) is in the image of θ if and only if there is some δ ∈ H 0 (B, N D|B ) whose value at p is the class of v.
The situation is even simpler if D is a rigid divisor. Let v ∈ T p B such that ψ(v) ∈ Im(ϑ). Then v is tangent to D.
Keeping the same notation as above, the application we have in mind for the next can be summarized in (5) Proof. We have that ϑ factors through the analogous map for D j , j = 1, 2: Hence, the image of ϑ is contained in the image of both H 1 (T B (− log(D j ))) for j = 1, 2. Than we apply the Corollary 4.3 and we obtain a vector v which is tangent to both D 1 and D 2 . Finally, observe that if a vector is tangent to two transversal curves must vanish. is injective.
Proof. The proof follows directly from the Proposition 4.4 and the following diagram with exact row and column.
We conclude the section with the following general result.
Lemma 4.6. Let A be an abelian surface isogenous to a product of elliptic curves T 1 × T 2 . Let H ⊂ H 1 (T A ) be the linear subspace corresponding to the projective deformations of A and let H j ⊂ H 1 (T A ) be the linear subspaces corresponding to the deformations preserving the fibration A −→ T j for j = 1, 2. Then H, H 1 and H 2 are three different hyperplanes such that the intersection of any two of them is contained in the third.
Proof. The isogeny maps H 1 (T A ) isomorphically to H 1 (T T 1 ×T 2 ) by a map preserving H, H 1 and H 2 . Therefore we may assume without loss of generality A = T 1 × T 2 . For a product of curves the period matrix assumes the form It is well known that one can identify the deformation space H 1 (T A ) of a polarized abelian surface A = V /Λ with the space of the square matrices τ (see [HKW93, Chapter 1]). For τ = a b c d we obtain the deformation given by The Riemann-Conditions for an abelian surface with a principal polarization yields the existence of an integral basis {λ i } i for Λ and a complex basis {e i } for V such that the period matrix can be normalized so that the matrix τ is symmetric with positive imaginary part (see [GH94] p.306), so The subspaces H j are respectively b = 0 c = 0 and this concludes the lemma.

The moduli space
The following result can be found in [Cat11, Section 5].
Proposition 5.1. Let S be a minimal surface of general type with q(S) ≥ 2 and Albanese map α : S → A, and assume that α(S) is a surface. Then this is a topological property. If in addition q(S) = 2, then the degree of α is a topological invariant.
Proof. By [Cat91] the Albanese map α induces a homomorphism of cohomology algebras and H * (Alb(S), Z) is isomorphic to the full exterior algebra * In particular, if q = 2 the degree of the Albanese map equals the index of the image of 4 H 1 (S, Z) inside H 4 (S, Z) and it is therefore a topological invariant.
Consider a surface S in M. By Proposition 5.1 it follows that one may study the deformations of S by relating them to those of the flat double cover β : S → B . By [Ser06, p. 162] we have an exact sequence where N β is a coherent sheaf supported on the ramification divisorR +Ĉ 1 +Ê called the normal sheaf of β.
We notice that H 0 (OR(2R)) = H 0 (O R (R)) = H 0 (N R|B ). Recall that by adjunction the normal bundle of a curve in an abelian surface equals its canonical bundle, so N t|A = ω t . The map ν = (σ 4 • σ 3 )| R : R −→ t is the normalization of t. Let q 1 , q 2 ∈ R such that ν(q i ) = p with i = 1, 2 and recall that p is the tacnode of t. We have By construction R is a smooth irreducible curve of genus 3 with a (Z/2Z) 2 -action, by [BO20, Lemma 2.15] R is not hyperelliptic. Thus, R is a plane quartic curve invariant under the action (x 0 : x 1 : x 2 ) → (±x 0 , ±x 1 , ±x 2 ), the equation defining it is biquadratic, and the divisor q 1 + q 2 is invariant. This means that q 1 and q 2 have a stabilizer of order 2 and lie on a coordinate line x j .
Since the quartic equation defining R is biquadratic, this would imply that R is singular in q 1 and q 2 , but this is absurd.
Recall that S is a surfaces of general type, hence h 0 (T S ) = 0 and using the bit of information of the previous lemma, the sequence (6) induces the following long sequence in cohomology.
Proposition 5.3. Keeping the notation as above, then the sheaf β * T B satisfies Proof. Since β : S → B is a finite map, by using projection formula and the Leray spectral sequence we deduce Recall that p g (S ) = q(S ) = 2 and B is an abelian surface blown up twice, then we have By the same argument above we have We look first at σ 3 . There is a short exact sequence The analogous computation for σ 4 , for the exact sequence Therefore the claim follows.
Let us consider the exact sequence where the last sheaf is supported on E and F . We tensor (12) by L −1 B and we obtain the sequence 0 −→ T B ⊗ L −1 B → (L −1 B ) ⊕2 → N σ 4 •σ 3 ⊗ L −1 B → 0. Considering the induced long exact sequence in cohomology, by (9) and Lemma 5.4. It holds h 0 N σ 4 •σ 3 ⊗ L −1 sheaf σ * 4 (O E (−E)) is locally free and it is supported on F ∪ E . Its restriction to the irreducible components are We tensor the last horizontal sequence in (16) by L −1 B ∼ = O B (R + E + C 1 ) and we get The long exact sequence in cohomology yields By the intersection computation (R + E + C 1 )E = −2 and (R + E + C 1 )F = 4 the sheaf σ * 4 (O E (−E)) ⊗ L −1 B is a locally free sheaf on E ∪ F which has degree −1 on F and degree 2 on E . Hence its global sections vanish on F and, fixing an isomorphism E ∼ = P 1 , correspond to the sections of H 0 (O P 1 (2)) which vanish on the point E ∪ F . Thus Remark 5.5. Let q ∈ S be the point blown-up by S → S. The short exact sequence obtained pushing forward (6) produces a cohomology exact sequence Recall that if β : S −→ B is a finite two to one cover, then H 1 (S , T S ) = H 1 (B , β * T S ) splits as invariant and anti-invariant part. Since q is an isolated fixed point of the involution induced by the Albanese map, it acts as the multiplication by −1 on T q S and then the image of T q S is contained in H 1 (S , T S ) − . By is injective as well.
Hence we have a commutative diagram H 1 (T B (− log((R + E + C 1 ))) The left vertical map is an isomorphism by Remark 5.5. The composition of the top horizontal arrow and the right vertical arrow is the map in Remark 5.6, so injective, and therefore the lower horizontal map is injective.
Proof. The image of the map H 1 (T S ) → H 1 (T A ) is contained in the hyperplane H of Lemma 4.6, since the Albanese variety of every surface of general type is an abelian variety. We proved that H 1 (T B (− log((R+E +C 1 ))) ∼ = H 1 (T S ) and the induced map ϕ : H 1 (T B (− log(R+ E + C 1 ))) → H 1 (T A ) is injective. So it is enough to prove dim Im(ϕ) = 2 The function ϕ factorizes as in the following commutative diagram.
H 1 (T B (− log(R + E + C 1 )) where C 1 is the elliptic curve in Figure 4. We recall that A is isogenous to the product of two elliptic curves T 1 × T 2 and C 1 is a fibre of the induced elliptic fibration f 2 on T 2 . So the image of is contained in H 2 . Then Im(ϕ) ⊂ H ∩ H 2 has, by Lemma 4.6, dimension at most 2. On the other hand it is at least 2 by Proposition 3.3, and therefore it equals 2.
Proposition 5.8. The following holds: for all j ∈ {1, 2, 4} M j is a generically smooth irreducible component of the moduli space of the surfaces of general type of dimension 2.