A rigid, not infinitesimally rigid surface with K ample

We produce an example of a rigid, but not infinitesimally rigid smooth compact complex surface with ample canonical bundle using results about arrangements of lines inspired by work of Hirzebruch, Kapovich & Millson, Manetti and Vakil.


Introduction
Let X be a compact complex space. A (small) deformation of X is a holomorphic proper flat map π : X → T where T is a (germ of a) complex space with a marked point o ∈ T such that π −1 (o) X . Recall that π is called complete if every other small deformation π : X → T , o ∈ T , (π ) −1 (o ) X , can be obtained from π via lifting along a map ϕ : T → T , ϕ(o ) = o, and that π is called semiuniversal if it is complete and the differential dϕ o is uniquely determined for the lifting map. As is well known, a semiuniversal π : X → T always exists and is unique up to isomorphism as was proven by Kuranishi [12] for X a complex manifold and Grauert [9] in the general case. The base germ (T , o) is called the Kuranishi space of X and sometimes denoted by Def(X ).
It was a long-standing problem of Morrow and Kodaira [14, p. 45] if there are compact complex manifolds X with Def(X ) a non-reduced point. Such X are called rigid, but not infinitesimally rigid. The same question was independently recently posed in [1, Question 1.5.B)]; it was suggested there to look for a rigid minimal surface of general type with canonical system not ample. The first examples of such X were given in [4]; these X are minimal surfaces of general type that arise as resolutions of certain nodal product-quotient We feel honoured to dedicate this article to our teacher, colleague and friend Fabrizio Catanese on the occasion of his 70th birthday.
surfaces. The nodal product-quotient surfaces are actually infinitesimally rigid, meaning that their Kuranishi space is a reduced point, but the Kuranishi spaces of the desingularisations acquire a non-reduced structure due to the presence of the (−2)-curves and their deformations.
It is therefore natural to ask if there are also minimal surfaces of general type with K ample such that the Kuranishi space is a non-reduced point. In this article we answer this in the affirmative. More precisely we prove the following Theorem 1.1 There exists a compact complex surfaceS ♥ with ample canonical bundle whose Kuranishi space is isomorphic to Spec C[x]/(x 2 ).
The surfaceS ♥ has invariants  [4] had h 1 (S ♥ , TS ♥ ) = 6 and in fact every example constructed with the same technique would have had h 1 (S ♥ , TS ♥ ) even by [19,Proposition 2.8], [3,Proposition 5.6]. We also note that the topological index τ S ♥ = K 2 S ♥ − 8χ(S) is positive, whereas the topogical index of all the examples in [4], as well as the topological index of any surface constructed with the same technique (see [18, page158]) is negative.
We use a totally different method as the basis of our construction: we construct a line arrangement L ♥ in P 2 whose associated incidence scheme is isomorphic to Spec C[x]/(x 2 ). We then associate a pair (S, B) to this line arrangement, where S is the blowup of P 2 in the points belonging to at least three lines of the arrangement, and B is a simple normal crossing divisor on S consisting of the strict transforms of the lines in L ♥ and the exceptional divisors. We prove that the deformations of the pair (S, B) are the same as the ones of L ♥ given by the incidence scheme. We then construct an abelian cover π :S ♥ → S branched in B with group G = (Z/7) 4 by a method due to Pardini [17]. Finally, we slightly refine methods introduced by Fantechi-Pardini [8] and Manetti [16] to prove that the Kuranishi space ofS ♥ is the same as the one of (S, B). Moreover, we show that the canonical bundle ofS ♥ is ample and compute its invariants.
It is not difficult to show that every rigid product-quotient surface is regular (q = 0), so this happens for all examples in [4]. Our example is also regular, and our proof shows that this follows from a condition we imposed on the construction to ensure that the Kuranishi space ofS ♥ is the same as the one of (S, B), and precisely condition a) in Theorem 3.19. However, irregular rigid surfaces that are abelian cover as above do exist, as shown in [2,10], so this method could produce irregular rigid not infinitesimally rigid surfaces, provided that one suitably replaces that condition. We refer to [7] for a rather complete treatement of known results about the rigidity of compact complex manifolds, including a chapter devoted to the abelian covers of the plane branched on a line configuration.
The roadmap of the paper is as follows. In Sect. 2 we recall the general theory of abelian covers and their deformations from Pardini [17], Fantechi-Pardini [8], Manetti [16]. For our purposes we need a slightly improved version of [16,Corollary 3.23], which we prove in Corollary 2.13 by the same methods.
In Sect. 3 we consider specifically abelian covers constructed from line arrangements and their deformation theory. In this situation we translate the conditions of Corollary 2.13 into a computationally accessible form given in Theorem 3.19. This is rather involved and one of the main technical ingredients of the proof of the main result.
In Sect. 4 we construct the line arrangement L ♥ and show that the associated incidence scheme is a non-reduced point isomorphic to Spec C[x]/(x 2 ). In Sect. 5 we constructS ♥ and show that its Kuranishi space is again isomorphic to Spec C[x]/(x 2 ). In Sect. 6 we prove thatS ♥ has ample canonical divisor and compute its invariants in Theorem 6.5.
Finally, Appendix A contains the data needed to constructS ♥ explicitly. The overall approach is inspired by [20]. However, Vakil's paper does not imply our result since he works with so-called singularity types, which are equivalence classes of pointed schemes for the equivalence relation generated by elementary equivalences (X , p) ∼ (Y , q) if there is a smooth morphism (X , p) → (Y , q); thus, loosely speaking, he only works up to addition of smooth parameters, and then proves that every singularity type of finte type over Z can be found on some moduli space of surfaces of general type with K ample.
Moreover, Kapovich-Millson [11] prove a version of Mnëv's universality theorem used by Vakil, where it is not necessary to add smooth parameters. However, the notion of incidence scheme used in [11] (the space of finite based realisations of an abstract arrangement) is not suited for our geometric construction since in [11] neither points nor lines need to be distinct in the realisations they consider.
The main goal of this section is Corollary 2.13, a variant of a criterion of Manetti giving, for a complex manifold X with an action of a finite abelian group G, necessary conditions for the small deformations of X to correspond exactly to the deformations of the pair (Y , D) where Y = X /G is the quotient manifold and D is the branch divisor suitably "decorated".

Definition 2.1
Let Y be a variety. An abelian cover of Y with group G is a finite map π : X → Y together with a faithful action of G on X such that π exhibits Y as the quotient of X via G.
For every finite abelian group set G * = Hom(G, C * ) for its dual group. In the sequel we will always make the standing assumption that X is normal and Y is smooth.
Then π is flat and the action of G induces a splitting π Let R, D denote the ramification locus and the branch locus of π respectively. R consists of the points of X that have nontrivial stabilizer. [17]  Since the group G is commutative, if E is a component of D, then all the components of π −1 (E) have the same inertia group H and character ψ ∈ H * . This splits the branch locus as a sum of (reduced effective but still possibly reducible) divisors D = H ,ψ D H ,ψ .
Following [8,Section 2] we set, for all m ∈ N, ζ m := e 2πi m and we observe that there is a bijection among G and the set of the pairs (H , ψ) where H is a cyclic subgroup of G and ψ is a generator of H * , the bijection being given by (H , ψ) → g where g ∈ H is the generator of H such that ψ(g) = ζ # H . This allows to set D g := D H ,ψ and write We note D 0 = 0. We find this notation more convenient than the one with the D H ,ψ so will formulate all results in this notation.
Following again [8, Section 2], for χ ∈ G * , g ∈ G, let a g χ be the unique integer 0 ≤ a g The following is Pardini's structure theorem for abelian covers.

Theorem 2.4 Let G be an abelian group.
Let Y be a smooth variety, X a normal one, and let π : X → Y be an abelian cover with group G. We have associated to π two functions where Div + (Y ) is the subset of the group Div(Y ) formed by the effective divisors. Setting D(g) = D g and L (χ) = L χ we have D 0 = L 1 = 0 and D = D g reduced. Then the following set of linear equivalences is satisfied: Conversely, for every smooth Y , to any two functions D : The smoothness of X is easy to translate in this setting.

Proposition 2.5 In the situation of Theorem 2.4 X is smooth if and only if
(a) the divisor D = D g is a smooth normal crossing divisor; (b) if D g 1 ∩ D g 2 ∩ · · · ∩ D g r = ∅ then the map g 1 ⊕ g 2 ⊕ · · · ⊕ g r → G is injective.
Proof [17,Prop. 3.1] Now we want to study infinitesimal deformations of π : X → Y obtained by "moving" the branch divisors, following [16]. For the rest of this Section, we assume for simplicity from now on X and Y are smooth and π is totally ramified in the following sense: Definition 2.6 A cover is said to be totally ramified if the inertia groups of all the components of R generate G. We introduce the following notation.
Then G acts on π * T X , where T X denotes the sheaves of the holomorphic vector fields on X , splitting it as follows.
It follows among other things where as usual we denote by T 1 the tangent space and by T 2 the obstruction space arising from the cotangent complex. Please note that G acts on T i Def X and we can write the corresponding eigenspaces as cohomology groups on Y by Lemma 2. Comparing it with Lemma 2.8 and 3 we see that T i De f (Y ,D ) is isomorphic to the invariant part of T i Def X . Indeed this corresponds to the natural relation among the functors De f (Y ,D ) and De f X associating to every infinitesimal deformation of (Y , D) the infinitesimal deformation of X obtained by "moving" Y and the branching divisors in it. This has been formalized as follows Then there is a natural transformation of functors η : De f (Y ,D ) → De f X acting on the T j s by mapping isomorphically T i De f (Y ,D ) in the invariant part of T i De f X .
More precisely, M. Manetti states the result assuming G of the form (Z/2Z) r . However his proof works for every abelian group.
It follows the following Proposition 2.12 Let π : X → Y be a totally ramified abelian cover, with X and Y smooth and H 0 (X , Proof This is essentially explained in [16, page 58], we sketch the argument here for the convenience of the reader. The natural maps T i De f (Y ,D ) → T i De f X are injective by Lemma 2.11 and the assumed cohomological vanishing ensures by Lemma 2.8 the surjectivity on T 1 s. By the standard smoothness criterion η is smooth.
We deduce the following useful criterion, a slight modification of [16,Corollary 3.23].

Corollary 2.13
Let π : X → Y be a totally ramified abelian cover, with X and Y smooth. Assume induced by the natural maps among the tangent bundle of Y and the normal bundles of the curves D g , is injective.
Proof We have assumed the vanishing of H 0 (Y , T Y ) (condition a) for χ = 1); the vanishing of H 0 (X , T X ) follows by Lemma 2.9.
We only need then to prove the cohomological vanishing assumed in the statement of Proposition 2.12. This follows by considering the cohomology exact sequence associated to the short exact sequence of O Y -modules The main difference with [16,Corollary 3.23] is that our condition c) is weaker than Manetti's condition is easier to check but it fails in the examples in the next sections.
A second difference is that Manetti's criterion is stated for G of the form (Z/2Z) r : [20] had already noticed that his results could be easily extended to any abelian group. Finally, condition (b) is replaced in Manetti's criterion by some conditions on Y that are used in his proof to show exactly the vanishing (b). Since in our examples the divisors D g are smooth curves with all components rational, it is easier to check directly (b).

Properties of abelian covers from line arrangements
Now we specialize the results of the previous Section to the case of interest in the sequel.
From now on we are going to consider only elementary p−groups: G := (Z/ p) r . For this class of abelian groups we find convenient to switch to the additive notation. To mark this difference with the previous section, we are going to denote the dual of G by G ∨ .

Definition 3.1 Denote by
the natural pairing between characters and group elements. We write for the natural lift of the preceding pairing to Z taking values in the set {0, 1, . . . , p − 1}.
The abelian covers will be constructed from certain line arrangements in P 2 .
Our aim is to construct a smooth abelian G-cover π :S → S with branch locus B. We now introduce a compact way to produce building data in our special case. be a map (the letter λ is chosen to suggest "label"). We say that λ satisfies the divisibility condition if for every χ ∈ G ∨ D∈D χ, λ(D) D is divisible by p in Pic(S). In this case we define for every χ ∈ G ∨ the line bundle: on S. This uniquely determines L χ since Pic(S) has no torsion. Proof Fix χ ∈ G ∨ and let λ be a map satisfying the divisibility condition.
Recall that Pic(S) is a free Z−module of rank m + 1, generated by the classes of the exceptional divisors E ν and by σ * H . To determine the divisibility of an element of Pic(S) we calculate its coefficients with respect to this basis and we impose their divisibility.
Since σ * H occurs in the class ofL i with coefficient 1 and in the class of E ν with coefficient

Since this holds for all
As above we obtain Hence, all values of λ are determined by the g i , and conversely prescribing values λ(L i ) = g i for i ∈ {1, . . . , n − 1} arbitrarily, we obtain an λ satisfying the divisibility condition using the preceding assignments.
In particular, the coefficient of each E ν in every L χ is nonpositive.
In particular The formula of the lemma follows. Similarly, the coefficient of The second formula of the lemma follows.

Definition 3.10
In the situation of Definition 3.6 let be a map (note that we now exclude 0 ∈ G as a permissible value for λ). We say, that λ satisfies the injectivity condition if the values of λ define distinct projective points in P r −1 (F p ). Also we say that λ satisfies the spanning condition if the image of λ spans G.
If a λ (with values in G\{0}) satisfies the injectivity and divisibility conditions, one obtains building data for an abelian G-cover in the sense of Sect. 2 as follows: we get maps by putting D(g) = D g = D if λ(D) = g and D g = 0 otherwise. Moreover, L (χ) = L χ is then as in Definition 3.7.

Theorem 3.11 Let
be a map satisfying the divisibility, injectivity and spanning conditions. Then there exists a finite flat totally ramified Galois cover π :S λ → S with group G, branch locus D∈D D, and with the covering surfaceS λ smooth, with the property that From now on we assume for the rest of the section that we are given a cover π :S λ → S constructed by the method in Theorem 3.11, and to simplify notation we writeS =S λ . Both the incidence scheme and the Kuranishi space above are universal objects representing two deformation functors. The above reasoning shows that these two functors are isomorphic, hence the representing germs of analytic spaces are isomorphic, together with their potentially nonreduced structure.
Our ultimate goal now in the rest of the Section is to produce a computationally checkable criterion that implies the conditions of Corollary 2.13. For this, we first need to compute some cohomology groups.
where the equality is in the derived category D b (P 2 ), I p ν is the ideal sheaf of the point p ν , and the intersection is taken inside O P 2 .
Proof Since all fibres of σ have dimension ≤ 1, R k σ * L = 0 for k ≥ 2 anyway. It suffices to check the remaining assertions for the case where there is just one exceptional divisor E mapping to one point p ∈ P 2 by working stalk-wise and gluing.
We have the exact sequence The fact that σ * O(−h E) = I h p for h ≥ 0 is a local calculation. Let us now prove We prove this by induction on h, the case h = 0 being clear. The exact sequence The map α is surjective by a dimension count. Hence Definition 3.14 In the situation of Theorem 3.11, write

Corollary 3.15
For all i ≥ 0 we have Proof By Lemma 3.9 each h χ ν ≥ −1 since E ν occurs with nonpositive coefficient in L χ and coefficient 1 in K S . The result then follows from Lemma 3.13 and the derived projection formula for L locally free and F any coherent sheaf.
The Castelnuovo-Mumford regularity of F , denoted by reg(F ), is the smallest integer r such that F is r -regular.

Theorem 3.17 If a coherent sheaf is r -regular then F (r ) is generated by its global sections and the natural map
Proof This is [15, Prop. on page 99, parts a) and a')].
We now define, for a given χ = 0 two maps α χ and β χ that will later be used in the proof of Theorem 3.19.
We first consider the following diagram for each D ∈ D that is not an exceptional divisor: The upper and middle rows are tensored with appropriate line bundles, and the morphism between those rows is induced by the ideal sheaf sequence The left and middle columns are exact because the result from tensoring this ideal sheaf sequence by vector bundles. Tensoring the middle row with O D gives the third row; note that exactness on the left follows from the vanishing of T or 1 (O E ν , O D ) because D and E ν intersect transversely. The snake lemma shows that the kernel and cokernel of ϕ D are the same as the kernel and cokernel of ψ D , hence completes the diagram. We get the commutative diagram where β D is induced by η D , and γ D is induced by ζ D . The map μ D is induced by the normal bundle sequence, and α D is defined by the diagram. Then we define the maps α χ , β χ as in the following diagram: Theorem 3. 19 Keeping the hypotheses of Theorem 3.11, assume the following: (iii) For all χ ∈ G ∨ , with χ = 0, the natural map induced by the natural maps T S ⊗ L −1 χ → N D|S ⊗ L −1 χ given by restricting germs of tangent vector fields to the normal bundle of D, is injective.
We now proceed to deduce (i), (ii), (iii) from the assumptions of the Theorem.
Step 3: Proving the injectivity of the map in (iii). This is slightly more involved and we divide the proof into three sub-steps.
Step 3. 1 We first show δ χ : is an isomorphism. By the sequence (5), it is sufficient to show h i (σ * T P 2 ⊗ L −1 χ ) = 0. for i = 0, 1. We have already proved the vanishing of h 0 (σ * T P 2 ⊗ L −1 χ ) in Step 1 using (7). The same argument shows h 1 (L −1 χ (σ * H )) = 0 and therefore we only need to prove that the map is injective. By Serre duality and Corollary 3.15, this is equivalent to the surjectivity of the map guaranteed by assumption (a) and Theorem 3.17.
Hence to prove that the map in (iii) is injective it suffices to prove that both α χ and β χ are injective.
Step 3. 2 We show that α χ is injective: It suffices to show that for a given D ∈ A (χ), the map Then we consider the diagram 0 0 The middle row is obtained by tensoring (5)  Tensoring the bottom row by L −1 χ , we find that ν H 0 (O E ν ∩D ) embeds into the space The latter vanishing follows since D ∈ A (χ).
Step 3. 3 We show that is injective. Indeed, we show that for a given, but arbitrary ν is injective.
Hence the left hand side in (8) will only be nonzero if either E ν occurs with coefficient 0 in L χ , in which case the left hand side is a two-dimensional vector space; or if E ν occurs with coefficient −1, in which case the dimension of that vector space is 1. By assumption (c), we have at least two D's, D ∈ A (χ), intersecting E ν in the first case, and at least one such D in the second case. Hence the map in (8) is injective in both cases.

An example of a rigid, but not infinitesimally rigid incidence scheme of points and lines in P 2
In this Section we show how to construct an arrangement of lines L having q 1 = (1 : 0 : 0), q 2 = (0 : 1 : 0), q 3 = (0 : 0 : 1), q 4 = (1 : 1 : 1) as singular points, and such that the associated incidence scheme I(L) in the sense of Definition 3.5 is a double point. We will first consider certain generalised incidence schemes that we call triangle schemes. Their construction is very simple, with fixed data given by three points and three lines, and variable data also given by three points and three lines. We will show that under certain conditions the triangle scheme is a double point and we will choose a suitable triangle scheme T ♥ with this property.
Then we will produce from T ♥ a line arrangement L ♥ with special properties, listed in Remark 4.7. Using these properties we prove that its associated incidence scheme I(L ♥ ) is isomorphic to T ♥ , and therefore a double point as well.
At the end of the section we will discuss these special properties, and how difficult is to find a triangle scheme such that the induced line arrangement has them.
In fact, this line arrangement does not seem to be at all unique: we believe that via a similar construction method, many other such arrangements can be found; but we confine ourselves to giving one particular example for the sake of definiteness and because already that needs quite a bit of work to construct. Definition 4.1 A triangle scheme T(P, Q, R) ⊂ (P 2 ) 3 × ((P 2 ) * ) 3 is a generalised incidence scheme defined as follows.
We take as associated fixed data the lines L X , L Y , L Z through the coordinate points and three points P, Q, R ∈ P 2 (Fig. 1). Then, denoting the variable points by (X , Y , Z ) ∈ (P 2 ) 3 and the variable lines by (L P , L Q , L R ) ∈ ((P 2 ) * ) 3

Remark 4.2
For general choices of P, Q, R, the triangle scheme T(P, Q, R) consists of two reduced points. This can be easily computed with Macaulay 2, as we did.
However, we would like to give here some geometrical interpretation of it. We note that Z is the image of Y in L Z by the projection of center R. Similarly X is the image of Z in L X by the projection of center Q, and Y is the image of X in L Y by the projection of center R. So Y is a fixed point for the projectivity of L Y obtained by composing these three projections, a projectivity that depends on the choice of the points P, Q and R.
For general choice of P, Q, R we obtain a general projectivity of L Y ∼ = P 1 , associated to a general invertible 2 × 2 matrix, with two distinct eigenvalues and then exactly two fixed points. In this case the projection (P 2 ) 3 × ((P 2 ) * ) 3 → P 2 corresponding to Y (forgetting X , Z , L P , L Q , L R ) embeds T(P, Q, R) in L Y , giving exactly the reduced scheme of the fixed points of this projectivity.
However for special choice of P, Q and R the matrix may have only one eigenvalue. For example, if we choose

Remark 4.3
One can also see by intersection arguments that T(P, Q, R) is of length 2 when zero-dimensional: let P 1 X , P 1 Y , P 1 Z be the lines on which the points X , Y , Z lie, and let P 1 P , P 1 Q , P 1 R be the pencils of lines through P, Q, R. Then the parameter space for (X , Y , Z , L P , L Q , L R ) is Each point of P 1 X determines a divisor in P, fibre of the point by the natural projection, whose class in Pic(P) do not depend on the choice of the point: we denote it by x. Similarly we obtain classes y, z, p, q, r by considering projections on the P 1 labeled by the corresponding uppercase letter.
The locus of 6-tuples (X , Y , Z , L P , L Q , L R ) with X ∈ L P is a divisor on P of class x + p, and similarly for the other incidences. Thus the triangle scheme has class

Remark 4.4 Consider the discriminant
, the Zariski closure of the set of triples P, Q, R such that T(P, Q, R) is a double point. One can compute that is a divisor of multidegree (2, 2, 2). If we write P = (P 0 : P 1 : The following is the triple we chose. Proof This is checked in [5]. This is the triangle scheme we are going to use. So, from now on, we will denote by P, Q, R ∈ P 2 the points whose coordinates are given in Proposition 4.5.
We consider the two lines through P defined by the polynomial the two lines through Q defined by the polynomial and the two lines through R defined by the polynomial These six lines form a line arrangement L having P, Q and R as double points. Consider the following iterative construction of point sets and line arrangements: (a) Let P 0 = {q 1 , q 2 , q 3 , q 4 } (b) Let L i be the set of lines that contain at least 2 points of P i−1 .
(c) Let P i be the set of points that lie on at least 2 lines of L i . L 1 contains 6 lines, L 3 contains 25 lines and P 3 contains 97 points. From this we construct our line arrangement, by adding L and the triangle arrangement above:

Definition 4.6 Set
is the line arrangement corresponding to the unique geometric point of T ♥ . Furthermore let P + be the set of intersection points of the line arrangement L + .

Remark 4.7
The explicit coordinates of all 34 lines in L ♥ are listed in Table 1 of Appendix A.
One can check that (a) the 6 lines forming L each pass through at least 2 points of P 3 .
(b) two of them intersects in P, other two of them intersects in in Q and the last two in R.
(c) L P , L Q and L R contain none of the points in P + except for P, Q and R respectively. see [5] for a Macaulay2 script doing this computation.
Our goal is to show that I(L ♥ ) ∼ = T ♥ , For this we need the following Lemma 4.8 Suppose that I is a generalised incidence scheme with associated fixed data q 1 , . . . , q m and M 1 , . . . , M n . Assume that there is a variable line L k such that the incidence q i ∈ L k , q j ∈ L k (9) for q i = q j is part of the defining set of equations. Then I is isomorphic to the generalised incidence scheme I with associated fixed data q 1 , . . . , q m and M 1 , . . . , M n , M n +1 where M n +1 is the unique line passing through the fixed points q i , q j , defined by the same set of equations as I, omitting the incidences in (9) and replacing L k by M n +1 whenever it occurs. Hence I is a closed subscheme of (P 2 ) m × ((P 2 ) * ) n−1 . We say that I becomes isomorphic to I by eliminating L k .
A similar result holds for the elimination of a variable point that lies on two fixed lines.
Since q i = q j these define a reduced point in (P 2 ) * whose coordinates are M n +1 . Hence projecting (P 2 ) m × ((P 2 ) * ) n onto (P 2 ) m × ((P 2 ) * ) n−1 by omitting the k-th copy of (P 2 ) * gives an isomorphism between I and I .

Proposition 4.9 The incidence scheme I(L ♥ ) is a double point.
Proof We now show that I(L ♥ ) is isomorphic to T ♥ . First we recall that by Definition 3.5 the incidence scheme I(L ♥ ) is the generalized incidence scheme associated to, as fixed data, the set of points P 0 = {q 1 , q 2 , q 3 , q 4 } and, as variable data, the set of all the lines of the arrangement and the set of all its singular points. In particular the variable data contain the lines {L P , L Q , L R } and the points {X , Y , Z }.
First we observe that I(L ♥ ) is isomorphic to the generalised incidence scheme I obtained from I(L ♥ ) by adding additional variable points for each point of P + .
By Lemma 4.8 we can "eliminate" all lines in L 1 since each contains two points belonging to the fixed data. So we move the lines in L 1 from the variable data to the fixed data obtaining an isomorphic incidence scheme. Then, since the lines in L 1 are now fixed, we can apply Lemma 4.8 to eliminate the points of P 1 (including those of P 0 that were already in the fixed data). By the same argument we eliminate successively the points and lines L 2 , P 2 , L 3 , P 3 from I using Lemma 4.8.
Because of condition (a) in Remark 4.7. we can now eliminate the six lines in L , and then all remaining points of P + .
The variable data of the resulting generalized incidence scheme I ♥ is given exactly by the variable points {X , Y , Z } and the variable lines {L P , L Q , L R } (the fixed data being given by all the lines and points that we have eliminated). It also counts the incidences defining T ♥ among its relations. It remains to check that there are no further incidence relations in I ♥ . This follows from condition (c) in Remark 4.7.

Remark 4.10
Let's discuss the conditions in Remark 4.7, starting from conditions (a) and (b). The lines in L are chosen in such a way that the first two intersect in P, the second two intersect in Q and the last two intersect in R. Furthermore each of these lines contains two points of P 3 . Such a situation is remarkably easy to arrange due to the following heuristic involving heights of point and lines in P 2 . Here by the height of a rational point in P 2 we mean the minimum absolute value of an entry in a set of integer homogeneous coordinates for the given point that are chosen such that their greatest common divisor is 1. Similarly the height of a rational line in P 2 is the height of the point in (P 2 ) ∨ representing the line.
First observe that the points of P 3 all have height at most 5. Therefore a line L containing two of the points of P 3 has height at most 25 (the coefficients of the equation of L are determinants of 2 × 2-matrices in the coefficients of the points). Evaluating such a line in a point P of height h gives a number between −25h and 25h. One of these numbers is 0 so the probability of a point of height h lying on one of these lines is approximately 1/(50h). Now we have about 97 2 5000 such lines. So if the height of P is less than 100 we have a good chance of finding one (or sometimes 2) lines passing though P and 2 points of P 3 .
In other words, P 4 is so big that it already contains a considerable portion of the rational points of height at most 100 in P 2 .

Remark 4.11
The difficult condition for a random construction such as ours to satisfy is condition (c). Here Remark 4.10 works against us: Since the points of P 3 ⊂ P + have small height, the probability that one of them lies on L P , L Q or L R is quite high. This probability is reduced if we choose P, Q and R of large height. If the height is too large we do not get the advantages described in Remark 4.10 so we are forced to choose P, Q and R of somewhat intermediate height (around 20) as we have done above.

An example of a rigid, not infinitesimally rigid surface of general type
We have constructed a line arrangement L ♥ in the preceding Section whose associated incidence scheme I(L ♥ ) is a nonreduced point. In this Section we show how to associate building data for an abelian cover to the initial datum of this line arrangement so that we can carry out the program set out in Sect. 3.  Table 1 of Appendix A.

Lemma 5.2 λ ♥ also satisfies the injectivity and the spanning condition.
Proof This can be checked easily with a computer algebra program, for example with the Macaulay 2 script available at [5]. Remark 5.3 λ ♥ was found by a random search. We chose the values λ(L i ), i = 1, . . . , 33 such that they represented distinct points in P 4 (F 7 ). We then computed λ(L 34 ) and λ(E i ), . . . λ(E 51 ) using the formulas in the proof of Lemma 3.8. Finally we checked if all 85 image points represented distinct points in P 4 (F 7 ). If not we started with new random values.
The chances of success of this scheme can be estimated by modelling the computed values as equally distributed random variables. Since This was one of our criteria for choosing G. For smaller groups the birthday problem takes too long to solve. For larger groups we have to check too many χ's in what follows below.

Theorem 5.5 The Kuranishi space ofS ♥ is a non-reduced point.
Proof By Propositions 4.9, 3.12 and Theorem 3.19, we only have to check that

Ampleness of the canonical class
Now we show that the canonical class KS ♥ is ample. We start by recalling the ramification formula for the behaviour of the canonical class under a covering.
Theorem 6.1 Let π :S → S be a smooth Galois cover with group G = (Z/ p) r obtained via building data as in Theorem 3.11. Set B = g∈G D g for the branch divisor, R = π −1 (B) for the ramification divisor (both effective and reduced). Then Moreover KS is numerically equivalent, as Q−divisor, to Proof As is well known (cf. e.g. [6, Anhang A.1, A.]), if π :S → S is a surjective holomomorphic mapping of smooth projective complex surfaces, then KS = π * K S +R whereR is the ramification or Jacobi-divisor of π (given by the zeroes of the Jacobian determinant of the mapping π in local coordinates on the base and covering). Now by construction, the cover is regularly ramified with constant local ramification order p: this means that if y ∈ B and x ∈ π −1 (y), then there are local coordinates (u, v) centred at y, and local coordinates Proposition 6.2 With the notation of Definition 3.6 let μ ν be the number of lines L i of the line arrangement L passing through p ν . Assume that there is a λ satisfying the hypotheses of Theorem 3.11. Assume furthermore that

Then the canonical bundle KS is ample.
Proof We use the fact that by Theorem 6.1 where B ⊂ S is the branch divisor given by Moreover, We need to show that := pK S + ( p − 1)B is ample (since π is finite). We use the Nakai-Moishezon criterion for this, meaning we will show 2 > 0 and ·C > 0 for every irreducible curveC on S. Now and hence Since each μ ν ≥ 3 Moreover, by assumption (b). If nowC is an irreducible curve on S that is not contracted by σ , let σ (C) =: C be its image in P 2 . We can writeC where s ν =C · E ν is the multiplicity with which C passes through p ν . Let d be the degree of C in P 2 . We compute that We need to show that this is bigger than zero, which will finish the proof. For this consider the n × m matrix (whose rows we imagine to be indexed by the lines, and columns indexed by the points we blow up) with entries a iν defined by Then we have (the first inequality because of Bezout, the latter equalities from the definition). Hence we can estimate In other words, Using formula (15), we can use this to estimate ·C: This is clearly strictly greater than zero if for all ν μ ν < (2 p − 1)n 3 p (assumption c)) and one of the s ν is nonzero. If all s ν are zero, ·C > 0 by formula (15) since in that case ·C = [( p − 1)n − 3 p]d > 0 by assumption d).

Theorem 6.3
The canonical class of the surfaceS ♥ is ample. In particular,S ♥ is of general type and coincides with its canonical model which is rigid, but not infinitesimally rigid.
Proof It suffices to check that the hypotheses of Proposition 6.2 are satisfied for L ♥ and p = 7. This is done in [5] using Macaulay2.
As a sanity check and to locate ourS ♥ in the geography of the surfaces of general type, we want to compute K 2 S ♥ and χ(S ♥ ).
We give a direct proof of the vanishing of q(S ♥ ). We have already checked that, for all χ = 0, reg(I χ ) < d χ , that implies by Corollary 3.15 and Serre duality that all h 1 (S, L −1 χ ) vanish. Since π is finite, q(S ♥ ) = h 1 (S, π * OS) = χ ∈G ∨ h 1 (S, L −1 χ ) = 0. Remark 6.6 Note that for a minimal surface X of general type over C, the dimension of the Kuranishi space of X is bounded below by 10χ(X ) − 2K 2 X , see [13]. Moreover, any such surface satisfies K 2 X ≤ 9χ(X ) by the Bogomolov-Miyaoka-Yau inequality.