Burniat Surfaces III: Deformations of

Automorphisms and Extended Burniat Surfaces

I. Bauer and F. Catanese

Received: September 26, 2012 Revised: April 22, 2013

Communicated by Thomas Peternell

Abstract. We continue our investigation of the connected components of the moduli space of surfaces of general type containing the Burniat surfaces, correcting a mistake in part II. We define the family of extended Burniat surfaces with K_{S}^{2} = 4, resp. 3, and prove that they are a deformation of the family of nodal Burniat surfaces with K_{S}^{2} = 4, resp. 3. We show that the extended Burniat surfaces together with the nodal Burniat surfaces with K_{S}^{2} = 4 form a connected component of the moduli space. We prove that the extended Burniat surfaces together with the nodal Burniat surfaces with K_{S}^{2} = 3 form an irreducible open set in the moduli space. Finally we point out an interesting pathology of the moduli space of surfaces of general type given together with a group of automorphisms G. In fact, we show that for the minimal model S of a nodal Burniat surface (G = (ℤ/2ℤ)^{2}) we have Def(S, G) ≠ Def(S), whereas for the canonical model X it holds Def(X, G) = Def(X). All deformations of S have a G-action, but there are different deformation types for the pairs (S, G) of the minimal models S together with the G-action, while the pairs (X, G) have a unique deformation type.

2010 Mathematics Subject Classification: 14J29, 14J25, 14J10, 14D22, 14H30, 32G05

Dedicated to David Mumford.

Introduction

In the present article we continue our investigation, begun in [BC09b] and [BC10], of the connected components of the moduli space (of minimal surfaces S of general type) which contain the Burniat surfaces. We also correct an error in [BC10]. Recall that Burniat surfaces have K_{S}^{2} = 6 − m, m = 0, …, 4, and the case m = 2 bifurcates in the subcases: the one of non-nodal Burniat surfaces, and the one of nodal Burniat surfaces. For m = 3 Burniat surfaces are three-nodal (this means that their canonical model has three nodes). The main goals that we achieve in this paper are the following:

(1) We define the family of extended Burniat surfaces for K_{S}^{2} = 3, resp. 4, and prove that they are a deformation of the family of nodal Burniat surfaces with K_{S}^{2} = 3, resp. 4.

(2) We show that the extended Burniat surfaces with K_{S}^{2} = 4, together with the nodal Burniat surfaces with K_{S}^{2} = 4, form a set N EB_{4} which is a connected component of the moduli space: thereby we correct Theorem 1.1 of [BC10] and simultaneously we answer a question posed on page 562 of [BC10]. ^{1}

(3) We show that the extended Burniat surfaces with K_{S}^{2} = 3, together with the nodal Burniat surfaces with K_{S}^{2} = 3 form an irreducible open set N EB_{3} of the moduli space, whose closure \overline{N EB_{3}} consists of bidouble covers of normal cubic surfaces in ℙ^{3} and is shown in Section 7 to be strictly larger than N EB_{3}.

(4) We point out a truly interesting pathology of the moduli space of varieties with a group G of automorphisms, which is the reason of our mistake mentioned above (Murphy’s law applies then, but in a different way than foreseen).

For nodal Burniat surfaces S, we have a group G ≅ (ℤ/2ℤ)^{2} of automorphisms, which is also the group of automorphisms of the canonical model X. But whereas Def(X) = Def(X, G), i.e., all deformations of X carry along a deformation of the G-action, Def(S) ≠ Def(S, G): thus even if all deformations of S have a G-action, the local moduli space Def(S, G) for the pairs yields a proper subvariety in the smooth germ Def(S).

We refer to [BC09b] and [BC10] for more details concerning investigation of the connected components of the moduli space containing the Burniat surfaces with K_{S}^{2} = 6, 5, 4, 2, which is fully achieved thanks to the results of the present article. What remains to be done in order to finish the investigation of Burniat surfaces is to decide, in the case K_{S}^{2} = 3 of tertiary Burniat surfaces, whether the irreducible component mentioned above is also a connected component, describing in detail all the surfaces which are in the closure and their local deformations.

1Namely, the integer m ≥ 2 in Theorem 1.1 is indeed = +∞, and the local moduli space of nodal Burniat surfaces is smooth.

The description of the closure of the irreducible component of the moduli space given by the extended nodal Burniat surfaces with K_{S}^{2} = 3 has been carried out by Y. Chen in his Ph. D. thesis (cf. [Ch12]). In [BC10] we proved that 3 of the 4 irreducible families of Burniat surfaces with K_{S}^{2} ≥ 4, i.e., of primary and secondary Burniat surfaces, are a connected component of the moduli space of surfaces of general type. In this paper we consider only nodal Burniat surfaces with K_{S}^{2} = 4, 3, showing that a general deformation of a nodal Burniat surface with K_{S}^{2} = 4, resp. with K_{S}^{2} = 3, is an extended Burniat surface, still a bidouble cover (through the bicanonical map) of a normal Del Pezzo surface of degree 4 with one ordinary double point, resp. of a cubic surface with three nodes. The main results of the present paper are the following:

Theorem 0.1. 1) The subset N EB_{4} of the moduli space of canonical surfaces of general type 𝔐_{1,4}^{can} given by the union of the open set corresponding to extended Burniat surfaces with K_{S}^{2} = 4 with the irreducible closed set parametrizing nodal Burniat surfaces with K_{S}^{2} = 4 is a three dimensional irreducible connected component, normal and unirational. Moreover the base of the Kuranishi family of deformations of any such a minimal model S is smooth.

2) The subset N EB_{3} of the moduli space of canonical surfaces of general type 𝔐_{1,3}^{can} corresponding to extended and nodal Burniat surfaces with K_{S}^{2} = 3 is an irreducible open set, normal, unirational of dimension 4. Moreover the base of the Kuranishi family of deformations of any such a minimal model S is smooth.

A very surprising and new phenomenon occurs for nodal surfaces, confirming Vakil’s ‘Murphy’s law’ philosophy ([Va06]). To explain what happens for the moduli spaces of extended and nodal Burniat surfaces, let us recall again an old result due to Burns and Wahl (cf. [BW74]). Let S be a minimal surface of general type and let X be its canonical model. Denote by Def(S), resp. Def(X), the base of the Kuranishi family of S, resp. of X. Their result explains the relation between Def(S) and Def(X).

Theorem (Burns-Wahl). Assume that K_{S} is not ample and let p : S → X be the canonical morphism. Denote by L_{X} the space of local deformations of the singularities of X and by L_{S} the space of deformations of a neighbourhood of the exceptional curves of p. Then Def(S) is realized as the fibre product associated to the Cartesian diagram

\xymatrix{Def(S) \ar[d] \ar[r] & L_{S} ≅ ℂ^{ν}, \ar[d]^{λ} \\ Def(X) \ar[r] & L_{X} ≅ ℂ^{ν},}

where ν is the number of rational (−2)-curves in S, and λ is a Galois covering with Galois group W := ⊕_{i=1}^{r}W_{i}, the direct sum of the Weyl groups W_{i} of the singular points of X.

An immediate consequence is the following

Corollary (Burns-Wahl).

1) ψ : Def(S) → Def(X) is a finite morphism, in particular, ψ is surjective.

2) If Def(X) → L_{X} is not surjective (i.e., the singularities of X cannot be smoothened independently by deformations of X), then Def(S) is singular.

Assume now that we have 1 ≠ G ≤ Aut(S) = Aut(X). Then we can consider the space Def(S, G) of local deformations of S together with the G-action (by [Cat88] this is the space of G-invariant local deformations of S), and similarly consider the space Def(X, G) of local deformations of X and its G-action; we have then a natural map Def(S, G) → Def(X, G). We indeed show here that, unlike the case for the corresponding morphism of local deformation spaces of the surfaces, this map needs not to be surjective; and, as far as we know, the following result gives the first global example of such a phenomenon.

Theorem 0.2. The deformations of nodal Burniat surfaces with K_{S}^{2} = 4, 3 to extended Burniat surfaces with K_{S}^{2} = 4, 3 yield examples where Def(S, (ℤ/2ℤ)^{2}) → Def(X, (ℤ/2ℤ)^{2}) is not surjective. Moreover, Def(S, (ℤ/2ℤ)^{2}) ⊊ Def(S), whereas for the canonical model we have: Def(X, (ℤ/2ℤ)^{2}) = Def(X). The moduli space of pairs, of an extended (or nodal) Burniat surface with K_{S}^{2} = 4, 3 and a (ℤ/2ℤ)^{2}-action, is disconnected; but its image in the moduli space is a connected open set.

The above phenomenon can already be seen locally around a node, as it will be explained in Section 2. Our results show that the local pathology does indeed globalize.

Our paper is organized as follows: in Section 1 we give the definition of extended Burniat surfaces and describe the respective branch loci of the bidouble covers yielding nodal Burniat surfaces, respectively extended Burniat surfaces. In the second chapter we analyse bidouble covers of a nodal surface singularity, explaining the phenomenon of Theorem 0.2 locally. In the third section we show that nodal Burniat surfaces with K_{S}^{2} = 4, 3 deform to extended Burniat surfaces with K_{S}^{2} = 4, 3. Section 4 is instead devoted to the calculation of H^{1}(S, Θ_{S}) for nodal and extended Burniat surfaces, and its eigenspaces for the G = (ℤ/2ℤ)^{2} action. In the course of doing this we need to amend a small mistake in [BC10] Lemma 2.10; this is done in an appendix, where we actually generalize this lemma substantially in order to make it appropriate for our present purposes and also applicable in other situations.

In the end we succeed to prove that the subset N EB_{4} of the moduli space of canonical surfaces of general type 𝔐_{1,4}^{can} corresponding to nodal and extended Burniat surfaces with K_{S}^{2} = 4 is an irreducible open set, normal, unirational of dimension 3 (similarly we show that the subset N EB_{3} of the moduli space of canonical surfaces of general type 𝔐_{1,3}^{can} corresponding to nodal and extended Burniat surfaces with K_{S}^{2} = 3 is an irreducible open set, normal, unirational of dimension 4). Section 5 is dedicated to the study of one-parameter limits of extended Burniat surfaces with K_{S}^{2} = 4, showing that the subset of the moduli space of canonical surfaces of general type 𝔐_{1,4}^{can} corresponding to nodal and extended Burniat surfaces with K_{S}^{2} = 4 is closed. In Section 6 we give examples of other surfaces which lie in the closure of the family of extended Burniat surfaces with K_{S}^{2} = 3. In another appendix we give an alternative proof of three of the four assertions of Proposition 4.1, by other methods which could be of independent interest.

1. Definition of extended and nodal Burniat surfaces

Burniat surfaces are minimal surfaces of general type with K^{2} = 6, 5, 4, 3, 2 and p_{g} = 0, which were constructed in [Bu66] as minimal resolutions of singular bidouble covers (that is, Galois covers with group (ℤ/2ℤ)^{2}) of the projective plane branched on 9 lines. We refer the reader to [BC10] for their construction, and we shall adhere to the notation introduced there.

Let P_{1}, P_{2}, P_{3} ∈ ℙ^{2} be three non-collinear points (which we assume to be the points (1 : 0 : 0), (0 : 1 : 0) and (0 : 0 : 1)), and let P_{4}, …, P_{3+m}, m = 2, 3, be further (distinct) points not lying on the sides of the triangle with vertices P_{1}, P_{2}, P_{3}. We make the further assumptions:

• for m = 2, the points P_{1}, P_{4}, P_{5} are collinear, while,

• for m = 3, we assume moreover that also P_{2}, P_{4}, P_{6} and P_{3}, P_{5}, P_{6} are collinear (in particular, no four points are collinear); we may also use the notation P′_{1} := P_{6}, P′_{2} := P_{5}, P′_{3} := P_{4}, so that P_{i}, P′_{i+1}, P′_{i+2} are collinear, where we use the convention i ∈ {1, 2, 3} mod 3.

Let’s denote by Ỹ := ℙ̂^{2}(P_{1}, P_{2}, …, P_{3+m}) the weak Del Pezzo surface of degree 6 − m, obtained blowing up ℙ^{2} in the points P_{1}, P_{2}, …, P_{3+m}. Saying that Ỹ is a weak Del Pezzo surface means that the anticanonical divisor −KỸ is nef and big; in our case it is not ample, because of the existence of (-2)-curves, i.e. curves N_{i} ≅ ℙ^{1}, with N_{i} · KỸ = 0: for m = 3 N_{i} is the strict transform of the line passing through P_{i}, P′_{i+1}, P′_{i+2}. Contracting the (-2)-curves N_{i} we obtain a normal singular Del Pezzo surface Y′ with −K_{Y′} very ample. We denote by L the divisor on Ỹ which is the total transform of a general line in ℙ^{2}, by E_{i} the exceptional curve lying over P_{i}, by E′_{i} the exceptional curve lying over P′_{i} (hence E′_{i} = E_{7−i}) and by D_{i,1} the strict transform of the line y_{i−1} = 0, side of the triangle joining the points P_{i}, P_{[i+1]}; that is, the unique effective divisor in |L − E_{i} − E_{[i+1]}|, where [i + 1] represents the residue class of i + 1 mod 3, an element of {1, 2, 3}. For m = 2 we have only one (-2)-curve N_{1}, such that {N_{1}} = |L−E_{1}−E_{4}−E_{5}|, while for m = 3 we also have the curves N_{2}, N_{3} such that {N_{2}} = |L − E_{2} − E_{4} − E_{6}|, {N_{3}} = |L − E_{3} − E_{5} − E_{6}|. Therefore the anticanonical image of Ỹ is a normal surface Y′ ⊂ ℙ^{6−m} of degree 6 − m, whose singularities are one node ν_{1} (an A_{1} singularity) in the case m = 2, and three nodes ν_{1}, ν_{2}, ν_{3} in the case m = 3 (the (-2)-curve N_{i} is the total transform of the point ν_{i}). In order to improve readability we separate the definitions for m = 2 and m = 3.

1.1. Nodal and extended Burniat surfaces with K_{S}^{2} = 4.

Definition 1.1. 1) Define the Burniat divisors for m = 2 as follows:

D_{1} ∈ |L − E_{1}| + |L − E_{1} − E_{2}| + |L − E_{1} − E_{4} − E_{5}| + E_{3},

i.e., D_{1} = D_{1,1}+N_{1}+C_{1}+E_{3}, where C_{1} ∈ |L−E_{1}| is assumed to be irreducible, whereas D_{2}, D_{3} are divisors such that

{D_{2}} = |L − E_{2} − E_{3}| + |L − E_{2} − E_{4}| + |L − E_{2} − E_{5}| + E_{1},

{D_{3}} = |L − E_{3} − E_{1}| + |L − E_{3} − E_{4}| + |L − E_{3} − E_{5}| + E_{2}.

P_{1}  P_{4} P_{3} P_{5}  P_{2}

2) The extended Burniat divisors for m = 2 are given as follows:

Δ_{1} ∈ |L − E_{1}| + |L − E_{1} − E_{2}| + E_{3},

i.e., Δ_{1} = D_{1,1} + C_{1} + E_{3}, where C_{1} ∈ |L − E_{1}| is assumed to be irreducible, and

Δ_{2} ∈ |L − E_{2} − E_{4}| + |L − E_{2} − E_{5}| + |2L − E_{2} − E_{3} − E_{4} − E_{5}|,

where we assume the divisor Γ_{2} ∈ |2L − E_{2} − E_{3} − E_{4} − E_{5}| to be irreducible; and Δ_{3} is the divisor such that

{Δ_{3}} = (|L−E_{3}−E_{4}|+|L−E_{3}−E_{5}|)+(|L−E_{3}−E_{1}|+E_{2})+|L−E_{1}−E_{4}−E_{5}|

=: Δ_{3,1} + Δ_{3,2} + N_{1}.

P_{1}  P_{3} P_{4}  P_{2}P_{5}

3) Define L_{i} := \frac{1}{2}(D_{[i−1]} + D_{[i+1]}) and observe that it is an integral divisor; define also Λ_{1} := L_{1} + N_{1} and Λ_{j} := L_{j} for j = 2, 3.

Remark 1.2. 1) Observe that (D_{1} + D_{2} + D_{3}) ∈ | − 3KỸ | is a reduced normal crossing divisor.

2) Similarly, (Δ_{1} + Δ_{2} + Δ_{3}) ∈ | − 3KỸ + N_{1}| is a reduced normal crossing divisor.

3) On the normal Del Pezzo surface Y′

• D_{1} yields a conic plus two lines, and Δ_{1} does the same (indeed D_{1} = Δ_{1} + N_{1} and N_{1} is the ‘nodal’ exceptional curve)

• D_{2} yields four lines, Δ_{2} yields a conic plus two lines (indeed Δ_{2} ≡ D_{2} + N_{1})

• D_{3} yields four lines, the same does Δ_{3} (indeed Δ_{3} = D_{3} + N_{1}). In particular, if the conic corresponding to Δ_{2} specializes to contain the line corresponding to E_{1}, we obtain then D_{2} subtracting the divisor N_{1} ≡ L − E_{1} − E_{4} − E_{5}. Finally, the four lines of Δ_{3} divide into two groups, i.e., we can write Δ_{3} = Δ_{3,1} + Δ_{3,2} + N_{1} so that, setting Γ_{1} := C_{1} and writing Δ_{i} = Γ_{i} + Δ′_{i}, for i = 1, 2, then

(∗) : Δ′_{i} + Δ_{3,i} ≡ −KỸ

(∗∗) : Γ_{1} + Γ_{2} ≡ −KỸ .

4) The divisors D_{i} enjoy the property (cf. [BC10]) that there are divisor classes L_{i} such that D_{[i−1]} + D_{[i+1]} ≡ 2L_{i}. Hence, in particular, Δ_{[i−1]} +Δ_{[i+1]} ≡ 2Λ_{i}, where, Λ_{1} := L_{1} +N_{1} and Λ_{j} := L_{j} for j = 2, 3.

5) Assume now that the conic corresponding to Γ_{2} becomes reducible: if the conic passes through P_{1}, then necessarily Γ_{2} splits as N_{1} + E_{1} + |L − E_{2} − E_{3}|, hence the conic is the union of two lines. If the conic is the union of two lines in another fashion, then necessarily either |L − E_{2} − E_{5}| or |L − E_{2} − E_{4}| is a component of Γ_{2}, hence Δ_{2} is not reduced.

We can now consider (cf. [Cat84b], [Cat99]) the associated bidouble covers S → Ỹ with branching divisors the Burniat divisors, respectively the extended Burniat divisors.

Definition 1.3. A secondary nodal Burniat surface is a bidouble cover S → Ỹ with branch divisors three Burniat divisors for m = 2. S is then a minimal surface of general type with p_{g}(S) = q(S) = 0, K_{S}^{2} = 6 − m = 4 (cf. [BC10]). If we let the three branch divisors be extended Burniat divisors, then we obtain a non minimal surface S̃ whose minimal model S is called a secondary extended Burniat surface.

1.2. Nodal and extended Burniat surfaces with K_{S}^{2} = 3.

Definition 1.4. 1) The Burniat divisors for m = 3 are defined to be the divisors D_{1}, D_{2}, D_{3} such that

D_{i} := D_{i,1} + N_{i} + E_{[i−1]} + |L − E_{i} − E′_{i}|.

More in detail,

{D_{1}} = |L − E_{1} − E_{2}| + |L − E_{1} − E_{4} − E_{5}| + |L − E_{1} − E_{6}| + E_{3},

{D_{2}} = |L − E_{2} − E_{3}| + |L − E_{2} − E_{4} − E_{6}| + |L − E_{2} − E_{5}| + E_{1},

{D_{3}} = |L − E_{3} − E_{1}| + |L − E_{3} − E_{5} − E_{6}| + |L − E_{3} − E_{4}| + E_{2}.

P_{3} P^{5} P_{4}  P_{2} P_{1}  P_{6}

2) The strictly extended Burniat divisors for m = 3 are defined as follows:

Δ_{i} := |L − E_{i} − E′_{i}| + N_{[i+1]} + Γ_{i},

where we assume Γ_{i} ∈ |2L − E_{i} − E_{[i+1]} − E′_{i} − E′_{[i+1]}| to be the strict transform of an irreducible conic. More in detail,

Δ_{1} ∈ |L − E_{1} − E_{6}| + |2L − E_{1} − E_{2} − E_{5} − E_{6}| + |L − E_{2} − E_{4} − E_{6}|,

Δ_{2} ∈ |L − E_{2} − E_{5}| + |2L − E_{2} − E_{3} − E_{4} − E_{5}| + |L − E_{3} − E_{5} − E_{6}|,

Δ_{3} ∈ |L − E_{3} − E_{4}| + |2L − E_{1} − E_{3} − E_{4} − E_{6}| + |L − E_{1} − E_{4} − E_{5}|.

P_{3}  P_{5} P_{4}  P_{2} P_{1} P_{6}

3) Define L_{i} := \frac{1}{2}(D_{[i−1]} + D_{[i+1]}) and observe that it is an integral divisor; define also Λ_{i} := L_{i} + N_{i}.

Remark 1.5. 1) Observe that (D_{1} + D_{2} + D_{3}) ∈ | − 3KỸ | is a reduced normal crossing divisor.

2) Similarly, (Δ_{1} + Δ_{2} + Δ_{3}) ∈ | − 3KỸ + ∑ N_{i}| is a reduced normal crossing divisor.

3) On the normal Del Pezzo surface Y′, for m = 3, Δ_{j} yields a conic and one line, D_{j} yields three lines (indeed Δ_{j} ≡ D_{j} − N_{j} + N_{j−1} + N_{j+1}). In particular, if the conic corresponding to Δ_{j} specializes to contain the line corresponding to E_{[j−1]} (here as before [j −1] ∈ ℤ/3ℤ), we obtain D_{2} subtracting the divisor N_{j−1} + N_{j+1} and adding the divisor N_{j}.

4) The divisors D_{i} enjoy the property (cf. [BC10]) that there are divisor classes L_{i} such that D_{[i−1]} + D_{[i+1]} ≡ 2L_{i}. Hence, in particular, Δ_{[i−1]} + Δ_{[i+1]} ≡ 2Λ_{i}, recalling that, for m = 3, Λ_{i} := L_{i} + N_{i}.

5) Assume that one or more of these conics become reducible. E.g., assume that the conic corresponding to Γ_{2} becomes reducible, and observe that this will be the case if the conic passes through P_{1} or P_{6}. We disregard this degeneration if the corresponding divisor Δ_{2} will be non reduced. The only possibility left over is that Γ_{2} splits as before, N_{1} + E_{1} + |L − E_{2} − E_{3}|. This degeneration will be considered admissible. This motivates the following definition:

Definition 1.6. Assume m = 3 and that one or two of these conics Γ_{j} become reducible in the admissible way Γ_{j} = N_{j−1} + E_{j−1} + |L − E_{j} − E_{j+1}| (here, as usual, j ∈ ℤ/3ℤ). In this case we define the (not strictly) extended Burniat divisors by subtracting to Γ_{j} the nodal divisor N_{j−1} it contains. Moreover, we replace

• Δ_{j+1} by (Δ_{j+1} − N_{j−1}), and

• Δ_{j−1} by (Δ_{j−1} + N_{j−1}).

For the convenience of the reader we have drawn the non strictly extended Burniat divisors in the case, where only Γ_{2} degenerates.

P_{3}  P_{5} P_{4}  P_{2} P_{1} P_{6}

Remark 1.7. If all three conics Γ_{j} become reducible in the admissible way and we define in the same way as in the previous definition the three divisors by subtracting to Γ_{j} the nodal divisor N_{j−1} it contains, by subtracting again the nodal divisor N_{j−1} from Δ_{j+1} and adding it to Δ_{j−1}, we get the Burniat divisors from Definition 1.4.

We can now consider (cf. [Cat84b], [Cat99]) the associated bidouble covers S → Ỹ with branching divisors the Burniat divisors, respectively the extended Burniat divisors.

Definition 1.8. A tertiary (three-)nodal Burniat surface S is a bidouble cover S → Ỹ with branch divisors three Burniat divisors for m = 3. S is then a minimal surface of general type with p_{g}(S) = q(S) = 0, K_{S}^{2} = 6 − m = 3 (cf. [BC10]). If we let the three branch divisors be extended Burniat divisors (i.e., either strictly extended or non strictly extended!), then we obtain a non minimal surface S̃ whose minimal model S is called a tertiary extended Burniat surface.

Remark 1.9. 1) In the nodal Burniat case the surface S does not have an ample canonical divisor K_{S}, due to the existence of (-2)-curves, which are exactly the inverse images of the (-2)-curves N_{i} ⊂ Ỹ. For this reason we call the above Burniat surfaces of nodal type. We denote their canonical model by X, and observe that X is a finite bidouble cover of the normal Del Pezzo surface Y′. If m = 2, then X has precisely one node (an A_{1}-singularity, corresponding to the contraction of the (-2)-curve) as singularity. While, for m = 3, the canonical model X has exactly three nodes as singularities.

2) In the extended Burniat case S̃ is not minimal. In the strictly extended Burniat case the inverse image of each N_{i} splits as the union of two disjoint (- 1)-curves. S has ample canonical divisor (hence S = X) exactly in the strictly extended case.

3) In all cases, the morphism X → Y′ is exactly the bicanonical map of X (see [BC10]).

4) Nodal Burniat surfaces are parametrized by a family with smooth base of dimension 2 for m = 2, of dimension 1 for m = 3. Strictly extended Burniat surfaces are parametrized by a family with smooth base of dimension 3 for m = 2, of dimension 4 for m = 3.

The key feature is that, both for nodal Burniat surfaces, and for extended Burniat surfaces, the canonical model X is a finite bidouble cover of a singular Del Pezzo surface Y′, which has one node in the case m = 2, and three nodes for m = 3 (in this case Y′ is a cubic surface in ℙ^{3}). In this case the direct image p_{∗}(O_{X}) splits as a direct sum of four reflexive character sheaves of generic rank 1. In the next section we shall describe how the covering behaves in the neighbourhood of a node in the two respective cases, and how these local coverings deform to each other (the Burniat case deforms to the extended Burniat case).

2. Local calculations around the nodes

In this section we consider finite bidouble covers of a node which are of Du Val type, i.e., yielding singularities which are at worst RDP ’s (rational double points). We obtain a classification which is a subset of the one made in [Cat87], classifying quotients of RDP’s by actions of ℤ/2ℤ or of G = (ℤ/2ℤ)^{2}.

We only need to look at Table 2, page 90, and Table 3, page 93, ibidem, to see which quotients of a rational double point by an involution, or by a pair of commuting involutions, yield an A_{1} singularity, i.e., a node. There are six cases for such coverings of Du Val type of a node Y, which in local holomorphic coordinates is given by

xy − z^{2} = 0.

In order to be more informative in our description, we denote by Ỹ the resolution of Y, which is the total space of a line bundle on N ≅ ℙ^{1} of degree −2 (hence N^{2} = −2). Denoting the bidouble cover of Y by X, we shall obtain, through the normalization of the fibre product, a finite bidouble cover of Ỹ, for which we shall give the three corresponding branch divisors. In the case where X is not irreducible, we shall describe a connected component X′ of X.

(1) X′ = Y (the covering is étale).

(2) X′ = ℂ^{2}, X has two components and the covering morphism is given

by

(u, v) ↦ (x = u^{2}, y = v^{2}, z = uv).

The branch divisor on Ỹ is just the (-2)-curve N.

(3) X′ = {w^{4} = xy}, X has two components and the covering morphism

is given by

(x, y, w) ↦ (x, y, z = w^{2}).

The branch divisor on Ỹ consists of the (-2)-curve N plus two fibres; the double cover of Ỹ has two nodes and resolving them we get the minimal resolution of the A_{3} singularity X′.

(4) X = {w^{2} = uv} and the covering morphism is given by

(u, v) ↦ (x = u^{2}, y = v^{2}, z = w^{2}).

The three intermediate ℤ/2ℤ covers are the two double covers (2), (3) described above, plus the intermediate cover (here a := uw, b := vw)

\{(x, y, z, a, b)| Rank (\begin{matrix}x, a, z, b \\ a, z, b, y \end{matrix}) = 1\},

which is the cone over a rational normal quartic (set x = t_{0}^{4}, a = t_{0}^{3}t_{1}, z = t_{0}^{2}t_{1}^{2}, z = t_{0}t_{1}^{3}, z = t_{1}^{4}).

The branch divisors on Ỹ are two: the (-2)-curve N and the divisor D formed by two fibres. The three intermediate double covers depend on the choice of the branch locus: N, respectively N + D, respectively D.

(5) X′ = {z^{2} = (w^{2}+y^{k+1})·y}, X has two components having a singularity of type D_{k+3}, and the covering morphism is given by

(y, z, w) ↦ (x = w^{2} + y^{k+1}, y, z).

The branch divisor on Ỹ is the total transform of the divisor C := {x = y^{k+1}, z^{2} = y^{k+2}} which is irreducible with a cusp for k odd, else it is reducible with a \frac{k}{2} -tacnode for k even.

In particular, N is part of the branch locus.

(6) X = {w^{2} = (u − v^{k+1})(u + v^{k+1})} = {w^{2} = u^{2} − v^{2k+2}} and the covering morphism is given by

(u, v, w) ↦ (x = u^{2}, y = v^{2}, z = uv).

X is a singularity of type A_{2k+1} and, in order to treat a new case, we make the assumption k ≥ 1.

The three intermediate ℤ/2ℤ covers are the smooth double cover (2), the double cover (5) {w^{2} = x − y^{k+1}}, and a third singularity which we omit to describe.

The branch divisors on Ỹ are two: the (-2)-curve N and the the total transform of the divisor C above.

The three intermediate covers depend on the choice of the branch locus: N, or N + C′, or C′, where C′ is the strict transform of C.

Letting p : X → Y be the finite bidouble cover, the direct image sheaf p_{∗}O_{X} splits as

O_{Y} ⊕ ( ⨁_{i=1,2,3} L_{i}),

where in the first case the reflexive sheaves L_{i} are locally free. To describe the other cases we use the reflexive sheaf F generated by u, v as O_{Y} -module, with relations

yu − zv = 0, zu − xv = 0.

We get (2) X′ = ℂ^{2}, (u, v) ↦ (x = u^{2}, y = v^{2}, z = uv),

p_{∗}O_{X} = (O_{Y} ⊕ F)^{⊕2}

(3) X′ = {w^{4} = xy}, (x, y, w) ↦ (x, y, z = w^{2})

p_{∗}O_{X} = (O_{Y} ⊕ O_{Y})^{⊕2}

(4) X = {w^{2} = uv} , (u, v) ↦ (x = u^{2}, y = v^{2}, z = w^{2})

p_{∗}O_{X} = (O_{Y} ⊕ F)^{⊕2},

with generators 1, {u, v}, w, {a = wu, b = vw}.

(5) X′ = {w^{2} = x − y^{k+1}}, (y, z, w) ↦ (x = w^{2} + y^{k+1}, y, z),

p_{∗}O_{X} = (O_{Y} ⊕ O_{Y})^{⊕2}.

(6) X = {w^{2} = u^{2} − v^{2k+2}}, (u, v, w) ↦ (x = u^{2}, y = v^{2}, z = uv)

p_{∗}O_{X} = (O_{Y} ⊕ F)^{⊕2}.

Remark 2.1. Cases 1, 3 and 5 are the case where we have a flat bidouble cover, i.e., p_{∗}O_{X} is locally free. In cases 2, 4 and 6 we have non-flat bidouble covers, but with the same character sheaves. We shall soon show how case 4 deforms to case 2.

Proposition 2.2. In case 2) X = Spec((O_{Y} ⊕ F) ⊕ (O_{Y} ⊕ F)), where the two addenda are orthogonal, and the algebra structure is determined by the nondegenerate pairing F × F → O_{Y}. In case 4) X = Spec((O_{Y} ⊕ F) ⊕ w(O_{Y} ⊕ F)), and the algebra structure is determined by the nondegenerate pairing F × F → O_{Y}, together with the assignment w^{2} = z.

We omit the simple proof. Case 4) deforms now to case 2) by changing the assignment w^{2} = z to w^{2} = z+t, t ≠ 0, so that w becomes then a local unit at the origin. We can relate the resulting picture with the local semiuniversal deformation of a node.

Proposition 2.3. Let t ∈ ℂ, and consider the action of G := (ℤ/2ℤ)^{2} on ℂ^{3} generated by σ_{1}(u, v, w) = (u, v, −w), σ_{2}(u, v, w) = (−u, −v, w). Then the hypersurfaces X_{t} = {(u, v, w)|w^{2} = uv + t} are G-invariant, and the quotient X_{t}/G is the hypersurface

Y_{t} = Y_{0} = {(x, y, z)|z^{2} = xy},

which has a nodal singularity at the point x = y = z = 0. X_{t} → Y_{t} is a bidouble covering of type 2 for t ≠ 0, and of type 4 for t = 0. We get in this way a flat family of (non flat) bidouble covers.

Proof. The invariants for the action of G on ℂ^{3} × ℂ are:

x := u^{2}, y := v^{2}, z := uv, s := w^{2}, t.

Hence the family 𝔛 of the hypersurfaces X_{t} is the inverse image of the family of hypersurfaces s = z + t on the product

Y′ × ℂ^{2} = {x, y, z, s, t)|xy = z^{2}}.

Hence the quotient of X_{t} is isomorphic to Y′. The rest was already explained before. □

Remark 2.4. i) The simplest way to view X_{t} is to see ℂ^{2} as a double cover of Y′ branched only at the origin, and then X_{t} as a family of double covers of ℂ^{2} branched on the curve uv + t = 0, which acquires a double point for t = 0. ii) The involution σ_{3}(u, v, w) = (−u, −v, −w) has only the origin as fixed point, which lies on X_{0}. Whereas σ_{3} acts freely on X_{t}, for t ≠ 0. F ix(σ_{1}) = {w = 0}, and {w = 0} ∩ X_{t} = {uv + t = w = 0}. Finally, F ix(σ_{2}) = {u = v = 0}, and {u = v = 0} ∩ X_{t} = {u = v = 0, w^{2} = t}, which consists of two points for t ≠ 0, one for t = 0. The corresponding branch loci are the origin, for t = 0, the divisor z = 0, and the point x = y = z − t = 0. iii) If we pull back the bidouble cover X_{t} to Ỹ, and we normalize it, we can see that

• D_{3} is, for t = 0, the nodal curve N, and is the empty divisor for t ≠ 0;

• D_{1} is, for t ≠ 0, the inverse image of the curve z+t = 0; while, for t = 0, it is only its strict transform, i.e. the divisor D considered previously, made up of two fibres;

• D_{2} is an empty divisor for t = 0, and the nodal curve N for t ≠ 0.

Remark 2.5. Part iii) of the previous remark shows that, as t → 0, one subtracts the nodal divisor N to D_{2}, and adds it to D_{3}; while for D_{1}, it specializes to D + N, and then we subtract N. This is precisely the algorithm which applies when passing from extended Burniat to Burniat divisors.

The really interesting part of the story comes now: the family X_{t} admits a simultaneous resolution only after that we perform a base change t = τ^{2} and the equation of X_{t} becomes

X_{τ} = {w^{2} − τ^{2} = uv}.

Definition 2.6. Let 𝔛 → T′ be the family where

𝔛 = {(u, v, w, τ)|w^{2} − τ^{2} = uv}

and T′ is the affine line with coordinate τ. Define S ⊂ 𝔛 × ℙ^{1} to be one of the small resolutions of 𝔛, and S′ to be the other one, namely:

S : \{(u, v, w, τ)(ξ) ∈ 𝔛 × ℙ^{1}| \frac{w − τ}{u} = \frac{v}{w + τ} = ξ\}

S′ : \{(u, v, w, τ)(η) ∈ 𝔛 × ℙ^{1}| \frac{w + τ}{u} = \frac{v}{w − τ} = η\}.

Let G be the group G ≅ (ℤ/2ℤ)^{2} acting on 𝔛 trivially on the variable τ, and else as in Proposition 2.3. Let further σ_{4} act by σ_{4}(u, v, w, τ) = (u, v, w, −τ), let G′ ≅ (ℤ/2ℤ)^{3} be the group generated by G and σ_{4}, and let H ≅ (ℤ/2ℤ)^{2} be the subgroup {Id, σ_{2}, σ_{1}σ_{4}, σ_{3}σ_{4}}.

The following is a rephrasing and a generalization of a discovery of Atiyah in our context: we omit the simple proof. For more details and a discussion of how these examples fit into the general theory of moduli spaces, see the ‘working guide’ written by the second author in [Cat11].

Lemma 2.7. The biregular action of G′ on 𝔛 lifts only to a birational action on S, respectively S′. The subgroup H acts on S, respectively S′, as a group of biregular automorphisms. The elements of G′ \ H = {σ_{1}, σ_{3}, σ_{4}, σ_{2}σ_{4}} yield isomorphisms between S and S′. The group G acts on the punctured family S \ S_{0}, in particular it acts on each fibre S_{τ}.

Since σ_{4} acts trivially on S_{0}, the group G′ acts on S_{0} through its direct summand G. The biregular actions of G on S \ S_{0} and on S_{0} do not patch together to a biregular action on S, in particular σ_{1} and σ_{3} yield birational maps which are not biregular: they are called Atiyah flops (cf. [At58]).

3. Nodal Burniat surfaces deform to extended Burniat surfaces

In this section we will show that

• the canonical models X of nodal Burniat surfaces with K_{X}^{2} = 4, together with the extended Burniat surfaces with K_{X}^{2} = 4 are parametrized by a family with smooth connected base of dimension 3, which maps to the moduli space via a finite morphism;

• the canonical models X of nodal Burniat surfaces with K_{X}^{2} = 3, together with the extended Burniat surfaces with K_{X}^{2} = 3 are parametrized by a family with smooth connected base of respective dimension 4, which maps to the moduli space via a finite morphism.

We shall treat first the easier case

3.1. (Extended) Burniat surfaces with K^{2} = 4.

Proposition 3.1. There exists a family, with connected smooth 3-dimensional base

B ⊂ {(P_{5}, C_{1}, Γ_{2})|C_{1} ∈ |L − E_{1}|, Γ_{2} ∈ |2L − E_{2} − E_{3} − E_{4} − E_{5}|}

parametrizing a flat family of canonical models, including exactly all the nodal Burniat surfaces and the extended Burniat surfaces with K_{X}^{2} = 4. The family maps to the moduli space via a quasi-finite morphism. Here, P_{1}, P_{2}, P_{3}, P_{4} are the standard projective basis in ℙ^{2}, the point P_{5} belongs to the line \overline{P_{1}P_{4}} and, in the blow up of the plane in the given five points P_{j}, j = 1, …, 5, C_{1}, Γ_{2} are as in Definition 1.1 (C_{1} is irreducible and either Γ_{2} is irreducible, or splits as N_{1} + E_{1} + |L − E_{2} − E_{3}|).

Proof. Recall that in this case D_{1} + D_{3} = Δ_{1} + Δ_{3}, and that N_{1} is a connected component of the above divisor D_{1} + D_{3} = Δ_{1} + Δ_{3}. We can therefore construct a family of double covers

W̃_{b} → Ỹ

such that the inverse image of N_{1} is a (-1)-curve. Blowing down this (-1)- curve we get a family of finite double covers W′_{b} → Y′, which are nodal and equisingular. Consider the pull back of the divisors Δ_{2} in the case where Γ_{2} is irreducible, and of the divisors D_{2} in the case where Γ_{2} is reducible. Since Δ_{2} ≡ D_{2}+N_{1}, and the divisor N_{1} becomes trivial on W′_{b}, since it contracts to a smooth point, it follows that all these divisors are linearly equivalent, and we have a family of divisors on the family W′_{b} We consider then the family of double covers X_{b} → W′_{b} branched on these divisors, and on the nodes of W′_{b}. Finally, assume that two surfaces S_{1}, S_{2} in the above family are isomorphic, equivalently that their canonical models X_{1}, X_{2} are isomorphic. Then this isomorphism would yield an isomorphism of the bicanonical morphisms of each X_{j}, hence we would have isomorphisms of the image normal Del Pezzo surfaces Y′_{1}, Y′_{2}, sending the respective triple of branch curves for X_{1} to the ones for X_{2}. Y′_{j} has exactly one node and contains exactly 12 lines (cf. Proposition 3.7 of [BC10]. Y′_{j} is obtained blowing down the (−2) curve N_{1} and conversely, blowing up the singular point of Y′_{j} and five disjoint (−1)-curves we obtain an isomorphism with the plane. Since these (−1)-curves correspond exactly to the lines in Y′_{j}, we have only a finite number of such birational contractions to the plane, and each of them determines a triple of branch curves, and five points in the plane. Therefore the number of surfaces S_{2} in the family which are isomorphic to S_{1} is finite.

□

3.2. (Extended) Burniat surfaces with K^{2} = 3. We argue similarly, but it may be useful to make right away a simple geometrical observation. Let P_{1}, P_{2}, P_{3}, P_{4} be the standard projective basis in ℙ^{2}, and consider a line L′ with P_{3} ∈ L′, different from te coordinate lines: then the line configuration of a ternary Burniat surface is completely determined by the line L′, since then

P_{5} := L′ ∩ \overline{P_{1}P_{4}}, P_{6} := L′ ∩ \overline{P_{2}P_{4}}.

Proposition 3.2. There exists a family, with connected smooth 4-dimensional base

T ⊂ {(L′, Γ_{1}, Γ_{2}, Γ_{3})}

where L′ is as above and Γ_{1}, Γ_{2}, Γ_{3} are as in Definitions 1.4 and 1.6, parametrizing a flat family of canonical models, including exactly all the nodal Burniat surfaces and the extended Burniat surfaces with K_{X}^{2} = 3. The family maps to the moduli space via a quasi-finite morphism.

Proof. Given a triple (Γ_{1}, Γ_{2}, Γ_{3}), according to the reducibility of each Γ_{i}, there corresponds either a Burniat divisor, or an extended Burniat divisor. We take the corresponding bidouble cover of Ỹ, hence we construct four families of smooth surfaces, which are not necessarily minimal. We take now the corresponding canonical models, which are finite bidouble covers of the normal Del Pezzo surface Y′. Observe that, given p′ : S̃ → Ỹ, and π : Ỹ → Y′,

X = Spec(π_{∗}(p′)_{∗}OS̃) = Spec(O_{Y′} ⨁(⊕_{i=1}^{3}F_{i})).

Now the reflexive sheaves F_{i} correspond to Weil divisors on Y′, and they are independent of t ∈ T by virtue of 4) of remark 1.5.

The multiplication maps correspond to a family of Weil divisors on Y′: whence we get a flat family on Y′ \ Sing(Y′). Locally around the nodes the structure of the deformation is as described in the previous section, therefore the family is flat everywhere. The assertion that for a given surface S_{1} in the family parametrized by T the number of surfaces S_{2} in the family which isomorphic to S_{1} is finite follows as in the case K^{2} = 4, using that the three nodal cubic surface Y′ contains only a finite number of lines.

□

Observe that proof given for the case K^{2} = 3 works also in the case K^{2} = 4.

4. Local deformations of the extended Burniat surfaces

The aim of this section is to calculate the dimension of H^{1}(S, Θ_{S}) for • nodal Burniat surfaces with K_{S}^{2} = 4,

• extended Burniat surfaces with K_{S}^{2} = 4,

• nodal Burniat surfaces with K_{S}^{2} = 3,

• extended Burniat surfaces with K_{S}^{2} = 3. The main results of this paragraph are the following (note that we use the notation introduced in Section 1:

Proposition 4.1. 1) Assume that S is a nodal Burniat surface with K_{S}^{2} = 4 (m = 2). Then the dimension of the vector space

H^{0}(Ω1 ; Ỹ (log(D_{i}))(KỸ + L_{i})) = H^{0}(Ω1 ; Ỹ (log(D_{i}))(E_{i} − E_{i+2}))

is 1 for i = 3, else it is 0.

2) Consider instead extended Burniat divisors for m = 2, and the corresponding vector spaces

H^{0}(Ω1 ; Ỹ (log(Δ_{i}))(KỸ + Λ_{i})).

Then their dimensions are the same as in the Burniat case, namely, 1 for i = 3, else 0.

3) Assume that S is a Burniat surface with K_{S}^{2} = 3 (m=3). Then each vector space

H^{0}(Ω1 ; Ỹ (log(D_{i}))(KỸ + L_{i})) = H^{0}(Ω1 ; Ỹ (log(D_{i}))(E_{i} − E_{i+2}))

is equal to 0.

4) In the case of (strictly or not strictly) extended Burniat divisors for m = 3 we have ∀i:

H^{0}(Ω1 ; Ỹ (log(Δ_{i}))(KỸ + Λ_{i})) = 0.

The proof of Proposition 4.1 is to be found in the second appendix. Using Proposition 4.1 we can explicitly determine the several G-character spaces of H^{i}(S, Θ_{S}) and their dimensions (here G = (ℤ/2ℤ)^{2} = {1, g_{1}, g_{2}, g_{3}}) . In the following, given a G-space V, we denote by V^{i}, for i ∈ 1, 2, 3, the eigenspace corresponding to the character whose kernel consists of {1, g_{i}}.

Proposition 4.2. 1) Let S be the minimal model of a Burniat surface. Then the dimensions of the eigenspaces of the cohomology groups of the tangent sheaf Θ_{S} (for the natural (ℤ/2ℤ)^{2}-action) are as follows.

(1) \underline{K^{2} = 4} of nodal type:

h^{1}(S, Θ_{S})^{inv} = 2, h^{2}(S, Θ_{S})^{inv} = 0,
h^{1}(S, Θ_{S})^{3} = 1 = h^{2}(S, Θ_{S})^{3},
h^{j}(S, Θ_{S})^{i} = 0, for i ∈ {1, 2}, ∀j;

(2) \underline{K^{2} = 3}:

h^{1}(S, Θ_{S})^{inv} = 1, h^{2}(S, Θ_{S})^{inv} = 0,
h^{1}(S, Θ_{S})^{i} = 1, h^{2}(S, Θ_{S})^{i} = 0, for i ∈ {1, 2, 3}.

2) Let S be a minimal model of an extended Burniat surface with K_{S}^{2} = 4. Then the dimensions of the eigenspaces of the cohomology groups of the tangent sheaf Θ_{S} (for the natural (ℤ/2ℤ)^{2}-action) are as follows.

• h^{1}(S, Θ_{S})^{inv} = 3, h^{2}(S, Θ_{S})^{inv} = 0,

• h^{1}(S, Θ_{S})^{i} = 0 = h^{2}(S, Θ_{S})^{i}, for i ∈ {1, 2},

• h^{1}(S, Θ_{S})^{3} = 1 = h^{2}(S, Θ_{S})^{3}.

3) Let S be the minimal model of an extended Burniat surface with K_{S}^{2} = 3. Then the dimensions of the eigenspaces of the cohomology groups of the tangent sheaf Θ_{S} (for the natural (ℤ/2ℤ)^{2}-action) are as follows:

(1) strictly extended case:

h^{1}(S, Θ_{S})^{inv} = 4, h^{2}(S, Θ_{S})^{inv} = 0, h^{j}(S, Θ_{S})^{i} = 0, for i ∈ {1, 2, 3};

(2) case where exactly one conic, w.l.o.g. the conic Γ_{1},because of symmetry,

degenerates to two lines: h^{1}(S, Θ_{S})^{inv} = 3, h^{2}(S, Θ_{S})^{inv} = 0, h^{1}(S, Θ_{S})^{i} = 0 = h^{2}(S, Θ_{S})^{i}, for i ∈ {1, 3}, h^{1}(S, Θ_{S})^{2} = 1, h^{2}(S, Θ_{S})^{2} = 0;

(3) case where exactly two conics, w.l.o.g. Γ_{1}, Γ_{2}, degenerate to two lines

each:

h^{1}(S, Θ_{S})^{inv} = 2, h^{2}(S, Θ_{S})^{inv} = 0,
h^{1}(S, Θ_{S})^{1} = 0 = h^{2}(S, Θ_{S})^{1},
h^{1}(S, Θ_{S})^{i} = 1, h^{2}(S, Θ_{S})^{i} = 0, for i ∈ {1, 3}.

For the reader’s convenience the above dimensions can also be found in Table 1. We begin with an easy but useful observation

Lemma 4.3. Assume that N is a connected component of a smooth divisor D ⊂ Y, where Y is a smooth projective surface. Moreover, let M be a divisor on Y. Then

H^{0}(Ω_{Y}^{1} (log(D − N))(N + M)) = H^{0}(Ω_{Y}^{1} (log(D))(M))

provided (K_{Y} + 2N + M) · N < 0.

Proof. The cokernel of Ω_{Y}^{1} (log(D))(M) → Ω_{Y}^{1} (log(D − N))(N + M) is supported on N and equal to Ω_{N}^{1}(N + M) = O_{N}(K_{Y} + 2N + M).

□

The lemma will be applied several times in the case where N ≅ ℙ^{1} and N^{2} < 0. Another useful lemma which will be crucial in some calculation is the following

Lemma 4.4. Assume that we have three linearly independent linear forms on ℙ^{2}, l_{1} := x_{1}, l_{2} := x_{2}, l_{3} := x_{3}. Then

(1) H^{0}(Ω_{ℙ^{2}}^{1}(2)) has as basis the three 1-forms, for j < i,

η_{ji} := x_{j}dx_{i} − x_{i}dx_{j} = −η_{ij}.

(2) H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(1)) has as basis the six 1-forms

ω_{ij} := \frac{x_{j}dx_{i} − x_{i}dx_{j}}{x_{i}} .

(3) H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(2)) has as basis the three 1-forms η_{ji}, for j < i, plus the six 1-forms x_{j}ω_{ij} and the three 1-forms x_{1}ω_{23}, x_{2}ω_{31}, x_{3}ω_{12}.

Proof. 1) is well known and follows from the Euler sequence.

2) Take the chart x_{i} ≠ 0 ⇔ x_{i} = 1: then in this chart ω_{ij} := −dx_{j} is a regular 1-form. In the chart x_{j} = 1 we have ω_{ij} := \frac{dx_{i}}{x_{i}} , while in the chart x_{h} = 1 we have ω_{ij} := x_{j} \frac{dx_{i}}{x_{i}} − dx_{j}. Hence ω_{ij} has logarithmic poles on x_{i} = 0, and the coefficient of the logarithmic term vanishes for x_{i} = x_{j} = 0, and is equal to 1 in x_{i} = x_{h} = 0. The above observation shows the linear independence of the above 6 forms. Moreover, ω_{ij} is an eigenvector with character λ for the ℂ^{∗}-action x_{i} ↦ λx_{i}, hence ω_{ij} ∈ H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(1)). It suffices to show that this space has vector dimension equal to 6. This follows however from the exact sequence

0 → Ω_{ℙ^{2}}^{1}(1) → Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(1) → ⊕_{i=1}^{3}O_{l_{i}}(1) → 0

and the vanishing of H^{j}(Ω_{ℙ^{2}}^{1}(1)) for j = 0, 1.

3) Observe that ω_{ij} = \frac{1}{x_{i}} η_{ji}, so that x_{i}ω_{ij} = η_{ji} = −η_{ij} = x_{j}ω_{ji}. Moreover, if h ≠ i, j, x_{h}ω_{ij} − x_{j}ω_{ih} = η_{jh}, so that the products x_{r}ω_{ij} generate a subspace of dimension at most 12. By the exact sequence

0 → Ω_{ℙ^{2}}^{1}(2) → Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(2) → ⊕_{i=1}^{3}O_{l_{i}}(2) → 0

and since H^{1}(Ω_{ℙ^{2}}^{1}(2)) = 0, h^{0}(O_{l_{i}}(2)) = 3 we infer that the dimension is indeed 12. Since H^{0}(O_{l_{i}}(2)) is generated by H^{0}(O_{ℙ^{2}}(1)) ⊗_{ℂ} H^{0}(O_{l_{i}}(1)) we conclude that the twelve 1-forms are a basis. □

Lemma 4.5. Assume that we have two linearly independent linear forms on ℙ^{2}, l_{1} := x_{1}, l_{2} := x_{2}.

1110 I. Bauer and F. Catanese

(1) H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2})(1)) has as basis the 4 forms

ω_{ij} := \frac{x_{j}dx_{i} − x_{i}dx_{j}}{x_{i}} , 1 ≤ i, j ≤ 3, i ≠ 3.

(2) H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2})(2)) has as basis the 3 forms η_{ji}, for j < i, plus

the 6 forms x_{2}ω_{12}, x_{1}ω_{21}, x_{3}ω_{13}, x_{3}ω_{23}, x_{2}ω_{12}, x_{1}ω_{23}.

Proof. Follows from Lemma 4.4 observing that H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2})(i)) is a subspace of H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(i)). The above two sets of vectors are linearly independent and the dimensions are 4, resp. 9. □ Corollary 4.6. 1) Let ω ∈ H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2})(1)). Then there are complex numbers a_{ij} such that:

ω = a_{12}ω_{12} + a_{21}ω_{21} + a_{13}ω_{13} + a_{23}ω_{23} = \frac{dx_{1}}{x_{1}} (a_{12}x_{2} − a_{21}x_{1} + a_{13}x_{3})+

+ \frac{dx_{2}}{x_{2}} (−a_{12}x_{2} + a_{21}x_{1} + a_{23}x_{3}) + dx_{3}(−a_{13} − a_{23}).

2) Let ω ∈ H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2})(2)): then we can write

ω = a_{12}η_{12} + a_{13}η_{13} + a_{23}η_{23} + a_{212}x_{2}ω_{12} + a_{121}x_{1}ω_{21} + a_{313}x_{3}ω_{13}+ + a_{323}x_{3}ω_{23} + a_{213}x_{2}ω_{13} + a_{123}x_{1}ω_{23} =

= \frac{dx_{1}}{x_{1}} (−a_{12}x_{1}x_{2} − a_{13}x_{3}x_{1} + a_{212}x_{2}^{2} − a_{121}x_{1}^{2} + a_{313}x_{3}^{2} + a_{213}x_{2}x_{3})+

+ \frac{dx_{2}}{x_{2}} (a_{12}x_{1}x_{2} − a_{23}x_{3}x_{2} − a_{212}x_{2}^{2} + a_{121}x_{1}^{2} + a_{323}x_{3}^{2} + a_{123}x_{1}x_{3})+

+ dx_{3}(a_{13}x_{1} + a_{23}x_{2} − a_{313}x_{3} − a_{323}x_{3} − a_{213}x_{2} − a_{123}x_{1}).

3) Any ω ∈ H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(1)) can be written as:

ω = a_{12}ω_{12} + a_{13}ω_{13} + a_{23}ω_{23} + a_{21}ω_{21} + a_{31}ω_{31} + a_{32}ω_{32} =

= \frac{dx_{1}}{x_{1}} (a_{12}x_{2} − a_{21}x_{1} + a_{13}x_{3} − a_{31}x_{1})+

+ \frac{dx_{2}}{x_{2}} (−a_{12}x_{2} + a_{21}x_{1} + a_{23}x_{3} − a_{32}x_{2})+

+ \frac{dx_{3}}{x_{3}} (−a_{13}x_{3} + a_{31}x_{1} + a_{32}x_{2} − a_{23}x_{3}).

4) Any ω ∈ H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(2)) can be written as:

ω = a_{12}η_{12} + a_{13}η_{13} + a_{23}η_{23} + a_{212}x_{2}ω_{12} + a_{313}x_{3}ω_{13} + a_{323}x_{3}ω_{23}+ + a_{121}x_{1}ω_{21} + a_{131}x_{1}ω_{31} + a_{232}x_{2}ω_{32} + a_{123}x_{1}ω_{23} + a_{231}x_{2}ω_{31} + a_{312}x_{3}ω_{12} =

Documenta Mathematica 18 (2013) 1089–1136

= \frac{dx_{1}}{x_{1}} (−a_{12}x_{1}x_{2} − a_{13}x_{3}x_{1} + a_{212}x_{2}^{2} + a_{313}x_{3}^{2}−

− a_{121}x_{1}^{2} − a_{131}x_{1}^{2} − a_{231}x_{2}x_{1} + a_{312}x_{3}x_{2})+

+ \frac{dx_{2}}{x_{2}} (a_{12}x_{1}x_{2} − a_{23}x_{3}x_{2} − a_{212}x_{2}^{2} + a_{121}x_{1}^{2} + a_{323}x_{3}^{2}+

+ a_{123}x_{1}x_{3} − a_{232}x_{2}^{2} − a_{312}x_{3}x_{2})+

+ \frac{dx_{3}}{x_{3}} (a_{13}x_{1}x_{3} + a_{23}x_{2}x_{3} − a_{313}x_{3}^{2} − a_{323}x_{3}^{2} + a_{131}x_{1}^{2}+

+ a_{232}x_{2}^{2} − a_{123}x_{1}x_{3} + a_{231}x_{1}x_{2}).

Proof. This is an easy verification. □

Now we have prepared everything for the proof of Proposition 4.1. Since the proof is long and technical we prefer to put it in a separate section (cf. Appendix 8).

Proof of Proposition 4.2. For the invariant part, the calculation goes exactly as the proof of Lemma 2.9. of [BC10], using that h^{i}(ΘS̃)^{inv} = h^{i}(Θ_{S})^{inv}. For the other character spaces, we use the same argument as in Lemma 2.12. in [BC10] to calculate χ(Ω1 ; Ỹ (log D_{i})(KỸ +L_{i})) (resp. χ(Ω1 ; Ỹ (log Δ_{i})(KỸ +Λ_{i})) for extended Burniat surfaces). We first observe that

χ(Ω1 ; Ỹ (log D_{i})(KỸ + L_{i})) = χ(Ω1 ; Ỹ (KỸ + L_{i})) + χ(O_{D_{i}}(KỸ + L_{i})),

(and analogously for χ(Ω1 ; Ỹ (log Δ_{i})(KỸ + Λ_{i})) for extended Burniat surfaces). Moreover, note that with the same calculation as in Lemma 2.12. of [BC10], we see that χ(Ω1 ; Ỹ (KỸ + L_{i})) = K2 ; Ỹ − 12. Each D_{i} (resp. Δ_{i}) consists of k_{i} irreducible connected components, each of them being a smooth rational curve. Write D_{i} = D_{i,1} + … + D_{i,k_{i}} as disjoint union of smooth rational curves and let n_{j} := D_{i,j} · (KỸ + L_{i}). Then

χ(O_{D_{i}}(log D_{i})(KỸ + L_{i})) = ∑_{j=1}^{k_{i}} max(0, n_{j} + 1).

Therefore

χ(Ω1 ; Ỹ (log D_{i})(KỸ + L_{i})) = K2 ; Ỹ − 12 + ∑_{j=1}^{k_{i}} max(0, n_{j} + 1).

We summarize the calculations in the following table (note that we write χ_{i} for χ(Ω1 ; Ỹ (log D_{i})(KỸ + L_{i}))). The values for h^{2}(ΘS̃)^{i} have been calculated in prop. 4.1. The notation: extended case (2), resp. (3), refers to Proposition 4.2. Moreover, we use Lemma 9.22 of [Cat88] to compare h^{1}(ΘS̃) and h^{1}(Θ_{S}): it asserts that for a single blow up of a point P

π_{∗}ΘS̃ = 𝔪_{P} Θ_{S}, R^{1}π_{∗}ΘS̃ = 0.

Table 1.

K_{S}^{2}  i  (n_{1}, …, n_{k_{i}}) χ_{i} h^{2}(ΘS̃)^{i} h^{1}(ΘS̃)^{i} h^{2}(Θ_{S})^{i} h^{1}(Θ_{S})^{i}
0  −2  0  2  0  2
4  1  (1, 1, 1, 1)  0  0  0  0  0
n.B.  2  (1, 1, 1, 1)  0  0  0  0  0
3  (1, 1, 1, 1)  0  1  1  1  1
0  −3  0  3  0  3
4  1  (1, 1, 1)  −2  0  2  0  0
ext.  2  (1, 1, 1)  −2  0  2  0  0
3  (1, 1, 1, 0, 1) 1  1  0  1  0
0  −1  0  1  0  1
3  1  (1, 1, 1, 1) −1  0  1  0  1
B.  2  (1, 1, 1, 1) −1  0  1  0  1
3  (1, 1, 1, 1) −1  0  1  0  1
0  −4  0  4  0  4
3  1, 2, 3  (1, 1, 0)  −4  0  4  0  0

str.ext.

0  −3  0  3  0  3
3  1  (1, 1, 0, 1) −2  0  2  0  0
ext. (2)  2  (1, 1)  −5  0  5  0  1
3  (1, 1, 0, 1) −2  0  2  0  0
0  −2  0  2  0  2
3  1  (1, 1, 1, 1, 0) 0  0  0  0  0
ext. (3). 2  (1, 1, 1)  −3  0  3  0  1
3  (1, 1, 1)  −3  0  3  0  1
□

From the above calculations and from Propositions 3.1, 3.2 follow all the statements of our first main theorem, with the exception of the statement that N EB_{4} is a connected component. It follows that N EB_{4} is open, while the statement that N EB_{4} is closed will be shown in the forthcoming section.

Theorem 0.1 1) The subset N EB_{4} of the moduli space of canonical surfaces of general type 𝔐_{1,4}^{can} given by the union of the open set corresponding to extended Burniat surfaces with K_{S}^{2} = 4 with the irreducible closed set parametrizing nodal Burniat surfaces with K_{S}^{2} = 4 is an irreducible connected component, normal, unirational of dimension 3. Moreover the base of the Kuranishi family of deformations of such a minimal model S is smooth.

2) The subset N EB_{3} of the moduli space of canonical surfaces of general type 𝔐_{1,3}^{can} corresponding to extended and nodal Burniat surfaces with K_{S}^{2} = 3 is an irreducible open set, normal, unirational of dimension 4. Moreover the base of the Kuranishi family of S is smooth.

Proof. We show here the smoothness of the base 𝔅 of the Kuranishi family. This follows, in the case K^{2} = 4, from the fact that we have shown (Proposition 3.1) the existence of a family with smooth three dimensional base B mapping to the moduli space, hence also to the base 𝔅 of the Kuranishi family, in a quasi-finite way. Since the tangent dimension of 𝔅 is also equal to three, it follows that the dimension of 𝔅 is exactly three, and that 𝔅 is smooth. The argument in the case K^{2} = 3 is identical, using Proposition 3.2.

□

We are also almost done with the proof of our second main theorem

Theorem 0.2 The deformations of nodal Burniat surfaces with K_{S}^{2} = 4, 3 to extended Burniat surfaces with K_{S}^{2} = 4, 3 yield examples where Def(S, (ℤ/2ℤ)^{2}) → Def(X, (ℤ/2ℤ)^{2}) is not surjective. Moreover, Def(S, (ℤ/2ℤ)^{2}) ⊊ Def(S), whereas for the canonical model we have: Def(X, (ℤ/2ℤ)^{2}) = Def(X). The moduli space of pairs, of an extended (or nodal) Burniat surface with K_{S}^{2} = 4, 3 and a (ℤ/2ℤ)^{2}-action, is disconnected; but its image in the moduli space is a connected open set.

Proof. By Propositions 3.1 and 3.2 we have two families with smooth connected rational base of dimension 3, resp. 4, parametrizing all the canonical models X of the surfaces in N EB_{4}, resp. N EB_{3}. In the previous Theorem 0.1 we showed that the base of the Kuranishi family of S is smooth, hence base change of these families yield the Kuranishi family of S. The above families of canonical models X yield the Kuranishi family of X, e.g., by the Theorem of Burns and Wahl. Propositions 3.1 and 3.2, exhibiting all the canonical models as bidouble covers of normal Del Pezzo surfaces, immediately show that Def(X, (ℤ/2ℤ)^{2}) = Def(X). Let now S be a nodal Burniat surface. Since, by (7.1), page 23, of [Cat88] Def(S, (ℤ/2ℤ)^{2}) ⊊ Def(S) is the intersection with H^{1}(Θ_{S})^{0}, which is the smooth subvariety corresponding to the nodal Burniat surfaces, we obtain that Def(S, (ℤ/2ℤ)^{2}) ⊊ Def(S). On the other hand, for instance in the case K_{S}^{2} = 4, we explicitly see that N EB_{4} is the union of two families of bidouble covers, the family of nodal Burniat surfaces, respectively the family of extended Burniat surfaces: hence the moduli space of pairs (S, (ℤ/2ℤ)^{2}) has exactly two connected components.

□

5. One parameter limits of extended Burniat surfaces with K_{S}^{2} = 4

In this section we shall prove the following:

Theorem 5.1. The family of extended Burniat surfaces with K_{S}^{2} = 4 yields, together with the family of nodal Burniat surfaces with K_{S}^{2} = 4, a closed subset N EB_{4} of the moduli space.

This will be accomplished through the study of limits of one parameter families of such extended Burniat surfaces: we shall indeed show that only nodal Burniat surfaces (or extended Burniat surfaces) occur. Let Y′ be a normal ℚ- Gorenstein surface and denote the dualizing sheaf of Y′ by ω_{Y′}. Then there is a minimal natural number m such that ω_{Y′}^{⊗m} is an invertible sheaf and it makes sense to define ω_{Y′} to be ample, respectively anti-ample; Y′ is Gorenstein iff m = 1. We recall the following result which was shown in [BC10].

Proposition 5.2. Let T be a smooth affine curve, t_{0} ∈ T, and let f: X → T be a flat family of canonical surfaces. Suppose that X_{t} is the canonical model of a Burniat surface with 4 ≤ K_{X_{t}}^{2} for t ≠ t_{0} ∈ T. Then there is a biregular action of G := (ℤ/2ℤ)^{2} on X yielding a one parameter family of finite (ℤ/2ℤ)^{2}-covers

\xymatrix{X \ar[dr]_{f} \ar[rr] &  & Y \ar[dl] \\  & T & ,}

(i.e., X_{t} → Y_{t} is a finite (ℤ/2ℤ)^{2}-cover), such that Y_{t} is a Gorenstein Del Pezzo surface for each t ∈ T.

Observe that the above result remains true if we replace “Burniat surface” by “extended Burniat surface”. This implies immediately the following:

Corollary 5.3. Consider a one parameter family of bidouble covers X → Y as in prop. 5.2. Then the branch locus of X_{t_{0}} → Y_{t_{0}} is the limit of the branch locus of X_{t} → Y_{t}, and it is reduced.

Note that the limit of a line on the del Pezzo surfaces Y_{t} is a line on the del Pezzo surface Y_{t_{0}}, and, as a consequence of the above assertion, two lines in the branch locus in Y_{t} cannot tend to the same line in Y_{t_{0}}.

Remark 5.4. Let X be the canonical model of an extended or nodal Burniat surface with K_{X}^{2} = 4. Recall that X is smooth for a general member of the family of extended Burniat surfaces, whereas X has one ordinary node if X is the canonical model of a nodal Burniat surface with K^{2} = 4. In the extended case the branch locus consists of the union of 3 hyperplane sections, containing 8 lines, 2 conics and the node. In the nodal Burniat case one of the conics degenerates to two lines, hence the branch locus consists instead of 10 lines and one conic.

The first step towards proving Theorem 5.1 is the following:

Proposition 5.5. Consider a one parameter family of bidouble covers X → Y as in prop. 5.2 except that X_{t} is an extended Burniat surface with K_{X_{t}}^{2} = 4 for t ≠ 0. Then Y_{0} is a normal Del Pezzo surface with exactly one node as singularity.

Lemma 5.6. A normal singular Del Pezzo surface with K_{Y_{0}}^{2} = 4 containing at least 8 lines has as singularities either

(1) one node, and then it contains 12 lines, or

(2) two nodes, and then it contains 9 lines, or

(3) an A_{2} singularity, and then it contains 8 lines, 4 of which pass through the singular point.

Proof. The assertion is a generalization of Proposition 3.6 of [BC10], page 581, see especially the proof in the appendix ibidem, pages 585-587. We blow up r = 5 points in the plane. By [BC10], p.586, we have that the loss of number of lines when one has a chain of k infinitely near points is bounded by

(k − 1)(r − (k − 1)) + \frac{1}{2}(k + 1)(k − 2) = (k − 1)(5 − (k − 1)) + \frac{1}{2}(k + 1)(k − 2).

Since a smooth Del Pezzo surface of degree four has 16 lines, k ≥ 4 implies that the number of lines is less than or equal to 16 − 11 = 5. If there is a chain with k = 3, the same estimate gives a loss of 8, and we cannot then have other (-2)-curves, else the number would be strictly smaller than 16 − 8 = 8. In this case we get an A_{2} singularity and 8 lines. In fact, in the chosen plane model we have 5 points lying on an irreducible conic C, of which P_{2} infinitely near to P_{1}, and P_{3} infinitely near to P_{2}. The lines are given by

E_{3}, E_{4}, E_{5}, |L − E_{1} − E_{4}|, |L − E_{1} − E_{5}|, |L − E_{1} − E_{2}|, |L − E_{4} − E_{5}|, C′,

where C′ is the strict transform of C. In this case the 4 lines passing through the singular point are

E_{3}, |L − E_{1} − E_{4}|, |L − E_{1} − E_{5}|, |L − E_{1} − E_{2}|.

In the case where there is no chain of three infinitely near points by a standard Cremona transformation as in [BC10], ibidem, we may reduce to the case where there are no infinitely near points and then we have that the weak Del Pezzo surface is Ŷ_{0} := ℙ̂^{2}(P_{1}, …, P_{5}), where P_{1}, P_{2}, P_{3} and P_{1}, P_{4}, P_{5} are collinear. Then Ŷ_{0} contains nine lines. In fact, the set of lines of Ŷ_{0} is:

L := {E_{1}, …, E_{5}, L − E_{2} − E_{4}, L − E_{2} − E_{5}, L − E_{3} − E_{4}, L − E_{3} − E_{5}}.

□

Proof of prop. 5.5. Since the branch locus of X_{t} → Y_{t} contains eight lines for t ≠ 0, also the branch locus of X_{0} → Y_{0} contains eight lines. We want to show that cases (2) and (3) of the previous lemma cannot occur.

We start by eliminating case (3). Here, the A_{2} singularity must be a limit of the node of Y_{t}, hence the bidouble cover is branched at the singular point. The bidouble cover is a RDP, hence, looking at table 2, page 90 of [Cat87], and table 3, page 93 ibidem, we see that the branch locus is analytically isomorphic to

• an ordinary cusp {y = 0 = z^{2} + x^{3} = 0} for E_{6} = {z^{2} + x^{3} + t^{4} = 0} → A_{2} = {z^{2} + x^{3} + y^{2} = 0},

• two lines {x = 0 = z^{2} + y^{2} = 0} for A_{5} = {z^{2} + w^{6} + y^{2} = 0} → A_{2} = {z^{2} + x^{3} + y^{2} = 0},

• two lines {x = 0 = z^{2} + y^{2} = 0} for the composition of A_{2} → A_{5} (ramified only at the singular point) with the previous A_{5} = {z^{2} + w^{6} + y^{2} = 0} → A_{2} = {z^{2} + x^{3} + y^{2} = 0}.

We observe however that by our previous arguments the branch locus contains the 8 lines, 4 of which pass through the A_{2} singularity, contradicting the above local description of the branch locus. Assume now by contradiction that we have case (2), i.e., Y_{0} has two nodes. Then

Claim 5.7. E_{1} is not a component of the total branch locus Δ of X̂_{0} → Ŷ_{0} , i.e.,

E_{2}, …, E_{5}, L − E_{2} − E_{4}, L − E_{2} − E_{5}, L − E_{3} − E_{4}, L − E_{3} − E_{5}

are exactly the 8 lines contained in Δ.

Proof of the claim. Assume that E_{1} is contained in the total branch locus Δ of the bidouble cover X̂_{0} → Ŷ_{0}. Then Δ contains three lines intersecting one of the two (−2) curves. But a bidouble cover of a node branched in at least three lines does not give a rational double point, as shown by the classification recalled in Section 2. A contradiction. □

Since for each node ν_{1}, ν_{2} there are two lines in the total branch divisor passing through ν_{i}, it follows by the classification given in Section 2, that N_{1}, N_{2} ≤ Δ and that (Δ − N_{i})N_{i} = 2. Denote by π: Ŷ_{0} → Y′ be the desingularization map. Then π_{∗}(Δ) ≡ −3K_{Y′}, whence

Δ ≡ −3KŶ_{0} + n_{1}N_{1} + n_{2}N_{2}.

Then 2 = (Δ − N_{i})N_{i} = (n_{i} − 1)N_{i}^{2} = 2(1 − n_{i}) ⇔ n_{i} = 0. We conclude that

Δ ≡ −3KŶ_{0}.

Observe that

−3KŶ_{0} − ∑_{l∈L\{E_{1}}} l − N_{1} − N_{2} ≡ 3L − E_{1} − E_{2} − … − E_{5}.

Since no other component of Δ can intersect the (−2)-curves, we see immediately that the remaining two components of Δ are:

L − E_{1}, 2L − E_{2} − E_{3} − E_{4} − E_{5}.

We write now

Δ_{1} = λ_{1}L − E_{1} − a_{2}E_{2} − a_{3}E_{3} − a_{4}E_{4} − a_{5}E_{5},
Δ_{2} = λ_{2}L − E_{1} − b_{2}E_{2} − b_{3}E_{3} − b_{4}E_{4} − b_{5}E_{5},
Δ_{3} = λ_{3}L − E_{1} − c_{2}E_{2} − c_{3}E_{3} − c_{4}E_{4} − c_{5}E_{5}.

Here we have used that, since E_{1} is not a component of Δ and since Δ_{i} + Δ_{j} has to be divisible by two, the only possibility is

E_{1} · (Δ_{1}, Δ_{2}, Δ_{3}) = (1, 1, 1).

Note that, since λ_{1} + λ_{2} + λ_{3} = 9 (and again since Δ_{i} + Δ_{j} is divisible by two) we have:

(λ_{1}, λ_{2}, λ_{3}) ∈ {(3, 3, 3), (1, 3, 5), (1, 1, 7)}.

Moreover, since the branch divisor is reduced, for each i it happens that, among the three numbers a_{i}, b_{i}, c_{i}, there cannot be two which are negative, and if one such a number is negative, then it is = −1; hence the only possibilities are: {a_{i}, b_{i}, c_{i}} = {1, 1, 1} or {−1, 1, 3}, for i ∈ {2, …, 5}.

\underline{(λ_{1}, λ_{2}, λ_{3}) = (3, 3, 3) :} then we get for the character sheaves:

L_{1} = O(3L − E_{1} − \frac{b_{2} + c_{2}}{2} E_{2} − \frac{b_{3} + c_{3}}{2} E_{3} − \frac{b_{4} + c_{4}}{2} E_{4} − \frac{b_{5} + c_{5}}{2} E_{5}),

L_{2} = O(3L − E_{1} − \frac{a_{2} + c_{2}}{2} E_{2} − \frac{a_{3} + c_{3}}{2} E_{3} − \frac{a_{4} + c_{4}}{2} E_{4} − \frac{a_{5} + c_{5}}{2} E_{5}),

L_{3} = O(3L − E_{1} − \frac{a_{2} + b_{2}}{2} E_{2} − \frac{a_{3} + b_{3}}{2} E_{3} − \frac{a_{4} + b_{4}}{2} E_{4} − \frac{a_{5} + b_{5}}{2} E_{5}).

Note that (a_{i}, b_{i}, c_{i}) = (1, 1, 1) for all i ∈ {2, …, 5} implies that p_{g}(X_{0}) ≠ 0, whence w.l.o.g.

(a_{2}, b_{2}, c_{2}) = (−1, 1, 3).

Then E_{2} ≤ Δ_{1}, and by the local calculations in Section 2 this implies that also E_{3} ≤ Δ_{1} (since the two lines of the branch locus intersecting a (−2)-curve belong to the same Δ_{i}). Therefore

(a_{3}, b_{3}, c_{3}) ∈ {(−1, ∗, ∗), (1, 1, 1)}.

Again using p_{g}(X_{0}) = 0, we conclude (looking at L_{3}) that (up to exchanging P_{4} with P_{5})

(a_{4}, b_{4}, c_{4}) ∈ {(3, 1, −1), (1, 3, −1)},

and again this implies that

(a_{5}, b_{5}, c_{5}) ∈ {(∗, ∗, −1), (1, 1, 1)}.

But in all of these cases we have

\frac{a_{i} + c_{i}}{2} ∈ \{0, 1\} ∀ ∈ \{2, …, 5\},

contradicting p_{g} = 0.

\underline{(λ_{1}, λ_{2}, λ_{3}) = (1, 3, 5) :} here we have

L_{1} = O(4L − E_{1} − \frac{b_{2} + c_{2}}{2} E_{2} − \frac{b_{3} + c_{3}}{2} E_{3} − \frac{b_{4} + c_{4}}{2} E_{4} − \frac{b_{5} + c_{5}}{2} E_{5}),

L_{2} = O(3L − E_{1} − \frac{a_{2} + c_{2}}{2} E_{2} − \frac{a_{3} + c_{3}}{2} E_{3} − \frac{a_{4} + c_{4}}{2} E_{4} − \frac{a_{5} + c_{5}}{2} E_{5}),

L_{3} = O(2L − E_{1} − \frac{a_{2} + b_{2}}{2} E_{2} − \frac{a_{3} + b_{3}}{2} E_{3} − \frac{a_{4} + b_{4}}{2} E_{4} − \frac{a_{5} + b_{5}}{2} E_{5}).

Again, p_{g} = 0 implies that there is an i ∈ \{2, …, 5\} such that \frac{a_{i}+c_{i}}{2} = 2. W.l.o.g. we can assume that \frac{a_{2}+c_{2}}{2} = 2. Therefore

(a_{2}, b_{2}, c_{2}) ∈ {(3, −1, 1), (1, −1, 3)},

whence

(a_{3}, b_{3}, c_{3}) ∈ {(3, −1, 1), (1, −1, 3), (1, 1, 1)}.

Then \frac{b_{2}+c_{2}}{2} , \frac{b_{3}+c_{3}}{2} ≤ 1 and \frac{b_{4}+c_{4}}{2} , \frac{b_{5}+c_{5}}{2} ≤ 2, which implies that O(L − E_{2} − E_{4}) ⊂ O(KŶ_{0}) ⊗ L_{1}, contradicting p_{g}(X_{0}) = 0.

\underline{(λ_{1}, λ_{2}, λ_{3}) = (1, 1, 7) :} this case can be excluded since

4 = Δ_{3} · (−KŶ_{0}) = 3λ_{3} − 1 − ∑_{i=2}^{5} c_{i} ⇒ 12 ≥ ∑_{i=2}^{5} c_{i} = 16,

a contradiction. This proves the proposition. □

Consider a one parameter family of bidouble covers X → Y as in prop. 5.5. Then Y′ := Y_{0} is a normal Del Pezzo surface with exactly one node. Let Ỹ be the blow up of Y′ in the node and denote the exceptional (−2)-curve of Ỹ over the node by A. The following result concludes the proof of Theorem 5.1.

Proposition 5.8. For the limit of a one parameter family of extended Burniat surfaces with K_{S}^{2} = 4 we have:

(1) if A does not intersect Δ − A, then X_{0} is an extended Burniat surface with K_{S}^{2} = 4 ;

(2) if A intersects Δ − A, then X_{0} is a nodal Burniat surface with K_{S}^{2} = 4 .

Proof. We can assume that Ỹ = ℙ̂^{2}(P_{1}, …, P_{5}), and w.l.o.g. P_{1}, P_{4}, P_{5} collinear, i.e., A ≡ L − E_{1} − E_{4} − E_{5}. Recall that we have shown that in both cases A is contained in the branch locus, hence the two alternatives are that A is a connected component of the branch locus, or not.

1) In the first case, argueing as in Proposition 5.5, we get that the total branch locus is Δ ≡ −3KỸ + A.

It is easy to see that Ỹ contains exactly 8 lines l_{1}, …, l_{8} which do not intersect A. Then these 8 lines have to be contained in Δ. Then Δ − A − ∑_{i=1}^{8} l_{i} ≡ 3L − ∑ E_{i}, which has to split into two Del Pezzo conics, which then have to be L − E_{1} and 2L − ∑_{i=2}^{5} E_{i}. Hence we get an extended Burniat surface.

2) Here L − E_{1} − E_{4} − E_{5} ≡ A ≤ Δ ≡ −KỸ. Observe that Ỹ contains exactly 4 lines intersecting A: E_{1}, E_{4}, E_{5}, L−E_{2} −E_{3}. By our local calculations in Section 2 two of these four lines are components of the total branch divisor and the two other not. W.l.o.g. we can assume E_{1}, L−E_{2}−E_{3} ≤ Δ. Since E_{4} and E_{5} are not contained in the branch divisor, we see (writing Δ_{i} as in the proof of Proposition 5.5) that(a_{4}, b_{4}, c_{4}) = (a_{5}, b_{5}, c_{5}) = (1, 1, 1). Now it is straightforward that (λ_{1}, λ_{2}, λ_{3}) = (3, 3, 3) (use the same argument as in the proof of prop. 5.5 to exclude the cases (1, 3, 5) and (1, 1, 7)). Since p_{g} = 0, we have (up to a permutation of {1,2,3})

\frac{b_{1} + c_{1}}{2} = \frac{a_{2} + c_{2}}{2} = \frac{a_{3} + b_{3}}{2} = 2.

W.l.o.g. we can assume (a_{1}, b_{1}, c_{1}) = (−1, 1, 3); then E_{1}, L − E_{2} − E_{3} ≤ Δ_{1}. Therefore

(a_{2}, b_{2}, c_{2}) ∈ {(3, −1, 1), (1, −1, 3)}

and

(a_{3}, b_{3}, c_{3}) ∈ {(3, 1, −1), (1, 3, −1)}.

But only (a_{2}, b_{2}, c_{2}) = (3, −1, 1) and (a_{3}, b_{3}, c_{3}) = (1, 3, −1) is possible (since a cubic cannot have two triple points, i.e., this would contradict the effectivity of Δ_{i} for some i). Therefore we get a nodal Burniat surface. □

6. Nodal and extended Burniat surfaces do not form a closed set for K_{S}^{2} = 3

We are going to exhibit surfaces which are in the closure of the family of nodal and extended Burniat surfaces, but for which the image of the bicanonical map is a normal cubic with other singularities than 3 nodes. In our first example we exhibit a 3 -dimensional family with a 4-nodal cubic as image. Consider a specialization of the 6 points P_{1}, …, P_{6} in ℙ^{2} so that P_{1}, P_{2}, P_{3} become collinear, and, more precisely, the point P_{2} moves in the line joining P_{4} and P_{6} till it reaches the line joining P_{1} and P_{3}. Then P_{1}, …, P_{6} are the vertices of a complete quadrilateral with sides N_{1}, N_{2}, N_{3}, N_{4}: here we identify N_{4} to the (-2) curve N_{4} ≡ L − E_{1} − E_{2} − E_{3} on the weak Del Pezzo Ỹ of degree 3 obtained blowing up the 6 points. Our notation for N_{1}, N_{2}, N_{3} remains the same, and Ỹ is the minimal resolution of the 4-nodal cubic surface Y′ := Σ.

We consider exactly the same divisors as the strictly extended Burniat divisors in 4) of Definition 1.4. We obtain a three dimensional family of bidouble covers X of Σ, with total branch locus consisting of 9 connected components, namely:

N_{1}, N_{2}, N_{3}; Γ_{1}, Γ_{2}, Γ_{3}; G_{1}, G_{2}, G_{3}.

G_{1}, G_{2}, G_{3} correspond to the three diagonals of the quadrilateral, and are the 3 lines of Σ not passing through the nodes, whereas Γ_{1}, Γ_{2}, Γ_{3} are conics as in Definition 1.4. The canonical models X have therefore 4 nodes lying over the node of Σ corresponding to N_{4}. We have therefore proven:

Proposition 6.1. The closure of the (4-dimensional) open set corresponding to nodal and extended Burniat surfaces with K_{X}^{2} = 3 contains a 3-dimensional family of canonical models which are bidouble covers of a 4-nodal cubic surface Σ. Each such surface X has 4 nodes, lying over one fixed node of Σ, and where the bicanonical map Φ_{2} : X → Σ is unramified.

In our second example we obtain a 3-dimensional family of bidouble covers of a cubic surface Y′ with a singularity of type D_{4}. We give this example using the different planar realization which was indeed the way we found our first description of the deformation of nodal Burniat surfaces with K_{S}^{2} = 3 to extended Burniat surfaces. To do this, we relabel the 6 points in the plane as follows:

P′_{3} := P_{4}, P′_{2} := P_{5}, P′_{1} := P_{6}.

We have therefore irreducible rational curves

D_{i,1} := L − E_{i} − E_{i+1}, D_{i,2} := N_{i} = L − E_{i} − E′_{i+1} − E′_{i+2},

D_{i,3} := G_{i} = L − E_{i} − E′_{i}

on the weak Del Pezzo Ŷ of degree 3. Blowing down the 3 (-1) curves D_{i,1} (i = 1, 2, 3) first, and then the strict transform of the 3 (-2) curves D_{i,2} ( i = 1, 2, 3) we obtain another copy of the projective plane where one has blown up three points Q_{i} (i = 1, 2, 3) and three points Q′_{i} (i = 1, 2, 3), where Q′_{i} is infinitely near to Q_{i}. We denote by slight abuse of notation by Q_{i} the full transform of the point Q_{i}, namely, the divisor D_{i,1} + D_{i−1,2}, and by Q′_{i} the full transform of the point Q′_{i},namely, the divisor D_{i,1}. The pull back of the system of lines in the new ℙ^{2} is, by the Hurwitz formula, the linear system

L := 4L − 2 ∑_{i} E_{i} − ∑_{i} E′_{i}.

And the curve D_{i,3} is linearly equivalent to

D_{i,3} ≡ L − 2D_{i+1,1} − D_{i,2} = L − Q_{i+1} − Q′_{i+1}.

Hence D_{i−1,2} = Q_{i} − Q′_{i}, and we can write the branch loci for the extended Burniat surfaces as:

Δ_{i} ∈ D_{i,3} + D_{i+1,2} + |D_{i,3} + D_{i+1,3}| = D_{i,3} + |Q_{i−1} − Q′_{i−1}| + |D_{i,3} + D_{i+1,3}| =

= |L − Q_{i+1} − Q′_{i+1}| + |Q_{i−1} − Q′_{i−1}| + |2L − Q_{i+1} − Q′_{i+1} − Q_{i−1} − Q′_{i−1}| =

= |L − Q_{i+1} − Q′_{i+1}| + N_{i−1} + |2L − Q_{i+1} − Q′_{i+1} − Q_{i−1} − Q′_{i−1}|.

Now, we simply let the three points Q_{1}, Q_{2}, Q_{3} become collinear, but we let the tangent directions Q′_{i} remain general. The blow up of the plane in the 6 points possesses now 4 (-2) curves, the three curves N_{1}, N_{2}, N_{3} and the strict transform N of the line through Q_{1}, Q_{2}, Q_{3}. Since N intersects each N_{i} and these are disjoint, the corresponding normal Del Pezzo surface Y′ has a singularity of type D_{4}. Letting the branch divisor be as before (namely, take pull backs of general conics in |2L − Q_{i+1} − Q′_{i+1} − Q_{i−1} − Q′_{i−1}|), we obtain

Proposition 6.2. The closure of the (4-dimensional) open set corresponding to nodal and extended Burniat surfaces with K_{X}^{2} = 3 contains a 3-dimensional family of canonical models which are bidouble covers of a normal cubic surface Y′ with a singularity of type D_{4}. The branch locus on Y′ has the singular point as an isolated point, and the local covering is determined by the epimorphism D_{4} → (ℤ/2ℤ)^{2} = (D_{4})^{ab} of the local fundamental group of the singularity to its abelianization.

Proof. The inverse image of the (-2) curves in the bidouble cover are: the inverse image N′ of N, which is a (-8) curve, and, for each N_{i}, there is a pair of (-1) curves meeting N′. After contracting the 6 (-1) curves we obtain a (-2) curves.

□

7. Appendix 1: a corrigendum to Burniat surfaces II

Parts 1), 2) and 3 ) of the following lemma were contained in Lemma 2.10 of [BC10], while 4) corrects a wrongly stated assertion of 2) of loc. cit. We also amend the proof for the correct assertions.

Lemma 7.1. Consider a finite set of distinct linear forms

l_{α} := y − c_{α}x, α ∈ A

vanishing at the origin in ℂ^{2}. Let p: Z → ℂ^{2} be the blow up of the origin, let D_{α} be the strict transform of the line L_{α} := {l_{α} = 0}, and let E be the exceptional divisor. Let Ω_{ℂ^{2}}^{1}((dlog l_{α})_{α∈A}) be the sheaf of rational 1-forms η generated by Ω_{ℂ^{2}}^{1} and by the differential forms d log l_{α} as an O_{ℂ^{2}}-module and define similarly Ω_{Z}^{1}((log D_{α})_{α∈A}). Then:

1) p_{∗}Ω_{Z}^{1}(log E)(−E) = Ω_{ℂ^{2}}^{1},

2) p_{∗}Ω_{Z}^{1}(log E, (log D_{α})_{α∈A}) = Ω_{ℂ^{2}}^{1}((dlog l_{α})_{α∈A}), 3) p_{∗}Ω_{Z}^{1}((log D_{α})_{α∈A}) = {η ∈ Ω_{ℂ^{2}}^{1}((dlog l_{α})_{α∈A})|η = ∑_{α} g_{α} dlog l_{α} + ω, ω ∈ Ω_{ℂ^{2}}^{1}, ∑_{α} g_{α}(0) = 0}.

4) p_{∗}Ω_{Z}^{1}((log D_{α})_{α∈A})(E) ⊃ Ω_{ℂ^{2}}^{1}((dlog l_{α})_{α∈A}) and

dim_{ℂ}[p_{∗}Ω_{Z}^{1}((log D_{α})_{α∈A})(E)/Ω_{ℂ^{2}}^{1}((dlog l_{α})_{α∈A})] = d − 2

is supported at the origin, where d := |A|. More precisely, we have an exact sequence

0 → Ω_{ℂ^{2}}^{1} → p_{∗}Ω_{Z}^{1}((log D_{α})_{α∈A})(E) → ⨁_{α=1}^{d} O_{D_{α}}(0) → ℂ_{0}^{2} → 0.

5) Assume w.l.o.g. c_{1} = 0 in the following formulae: then p_{∗}Ω_{Z}^{1}(log D_{1})(−E) ⊂ Ω_{ℂ^{2}}^{1}(dlog l_{1}) is the subsheaf of forms

\{ω = αdx + β \frac{dy}{y} |β(0) = 0, \frac{∂β}{∂y} (0) = 0, \frac{∂β}{∂x} (0) + α(0) = 0\}.

6) p_{∗}Ω_{Z}^{1}(−E) = 𝔪_{0}Ω_{ℂ^{2}}^{1}, where 𝔪_{0} is the maximal ideal of the origin 0 in the sheaf O_{ℂ^{2}}.

7) p_{∗}Ω_{Z}^{1}(log D_{1}, log D_{2})(−E) ⊂ Ω_{ℂ^{2}}^{1}(dlog l_{1}, dlog l_{2}) is the subsheaf of forms

\{ω = α\frac{dx}{x} + β \frac{dy}{y} |α(0) = 0, β(0) = 0, \frac{∂(α + β)}{∂x}(0) = 0, \frac{∂(α + β)}{∂y}(0) = 0\}.

Proof. We show 2), 3), 4), 5) and 7). Observe that

p_{∗}Ω_{Z}^{1}((log D_{α})_{α∈A})(mE)

consists of rational differential 1-forms ω which, when restricted to ℂ^{2} \ {0}, yield sections of Ω_{ℂ^{2}}^{1}((dlog l_{α})_{α∈A}). Therefore, in particular, ω ∏_{α∈A} l_{α} is a regular holomorphic 1-form on ℂ^{2}. Hence ω, modulo holomorphic 1-forms, can be written as

ω = \frac{f}{∏_{α∈A} l_{α}} dx + \frac{g}{∏_{α∈A} l_{α}} dy,

where f, g are Weierstrass pseudopolynomials of degree in y strictly less than d := card(A). Since dy = dl_{α} + c_{α}dx, the condition that ω restricted to ℂ^{2} \ {0} yields a section of Ω_{ℂ^{2}}^{1}((dlog l_{α})_{α∈A}) implies that l_{α}|(f + c_{α}g). Whence l_{α} divides fx + yg, and we conclude, since ∏_{α∈A} l_{α} is a pseudo polynomial of degree d, that

fx + yg = c(x) ∏_{α∈A} l_{α}.

This allows us to write, modulo holomorphic 1-forms,

ω = \frac{g(dy − \frac{y}{x}dx)}{∏_{α∈A} l_{α}} + \frac{c}{x}dx,

where now c ∈ ℂ. Let us pull back ω to Z, using local coordinates (x, t) such that y = xt, and where we make the assumption c_{α} ≠ 0, ∀α.

p^{∗}ω = \frac{x^{−d}g(x, xt)(xdt)}{∏_{α∈A}(t − c_{α})} + \frac{c}{x}dx.

The pull back form has logarithmic poles along E = {x = 0} iff g(x, y) has multiplicity at least d − 1 at the origin, and poles of order at most one along E iff g(x, y) has multiplicity at least d − 2 at the origin. Observe that the d polynomials P_{β} := ∏_{α∈A,α≠β} l_{α} are linearly independent and homogeneous of degree d − 1, hence they generate the vector space of homogeneous polynomials of degree d − 1, hence they generate the ideal of holomorphic functions vanishing at the origin of multiplicity at least d − 1. Hence g(x, y) has multiplicity at least d − 1 iff we can write

g = ∑_{α∈A} g_{α}P_{α}.

And since g is a pseudo polynomial of degree ≤ d−1, the g_{α}’s are just functions of x. In this case we can write

ω = \frac{c}{x}dx + ∑_{α∈A} \frac{g_{α}}{l_{α}} (dy − \frac{y}{x}dx) = \frac{1}{x} (cdx + ∑_{α∈A} \frac{g_{α}}{l_{α}} (xdy − ydx)) .

= \frac{1}{x} (cdx + ∑_{α∈A} \frac{g_{α}}{l_{α}} (xdl_{α} + xc_{α}dx − ydx)) = ∑_{α∈A} \frac{g_{α}}{l_{α}} dl_{α} + \frac{1}{x}dx (c −∑_{α∈A} g_{α}) .

The above form ω does not have poles on the line x = 0 if and only if c = (∑_{α∈A} g_{α}(0)). Observing that the strict transform of the line x = 0 is not among the divisors D_{α}, we establish claim (2), while (3) follows since c = 0 iff there are no poles along E. The very first assertion of (4) follows by (2), so let’s proceed to verify the other assertions. Assume now that p^{∗}ω has poles of order 1 along E; equivalently, assume that g(x, y) has multiplicity at least d − 2 at the origin. Since we already dealt with the case where this multiplicity is at least d − 1, we may assume that g(x, y) is homogeneous of degree d − 2, and that c = 0. Argueing as before, the space of homogeneous polynomials of degree d − 2 has as basis the d − 1 polynomials (β = 1, …, d − 1)

Q_{β} := ∏_{α∈A,α≠β,α≠d} l_{α}.

Whence g = ∑_{α∈A,α≠d} g_{α}Q_{α}, where g_{α} ∈ ℂ, and we may write:

ω = ∑_{α=1}^{d−1} \frac{g_{α}}{l_{α}l_{d}} (dy − \frac{y}{x}dx).

Since we want no poles on the line x = 0, we must have

∑_{α=1}^{d−1} \frac{g_{α}y}{y^{2}} = 0 ⇔ ∑_{α=1}^{d−1} g_{α} = 0.

Under this condition we may then write

ω = ∑_{α=1}^{d−1} \frac{g_{α}}{l_{α}l_{d}} (dl_{α}),

which has logarithmic poles along l_{α} = 0. Logarithmic poles along l_{d} = 0 follow by writing

ω = ∑_{α=1}^{d−1} \frac{g_{α}}{l_{α}l_{d}} (dl_{d}) + ∑_{α=1}^{d−1} \frac{g_{α}(c_{d} − c_{α})}{l_{α}l_{d}}dx,

and observing that ∑_{α=1}^{d−1} \frac{g_{α}(c_{d}−c_{α})}{y−c_{α}x} vanishes for l_{d} = 0 since on \{l_{d} = 0\} we have y = c_{d}x. Applying the residue sequence, we see that each such form ω has as residue on D_{α} a function with a single pole at most at the origin O, and with coefficient of \frac{1}{x} respectively equal to r_{d} := ∑_{α=1}^{d−1} \frac{g_{α}}{(c_{d}−c_{α})} in the case of D_{d}, and r_{α} := −\frac{g_{α}}{(c_{d}−c_{α})} in the case of D_{α}. In other words, the sum of the ‘double’ residues is 0, and the other condition ∑_{α=1}^{d−1} g_{α} = 0 can be also written down as ∑_{α=1}^{d} c_{α}r_{α} = 0. To show 5), observe that

p_{∗}Ω_{Z}^{1}(log D_{1})(−E) ⊂ p_{∗}Ω_{Z}^{1}(log D_{1}) ⊂ Ω_{ℂ^{2}}^{1}(dlog l_{1}).

Take coordinates x, y such that l_{1} = y, and write ω = αdx + β \frac{dy}{y}. We just pull back ω on the blow up Z in the chart where we have y = tx, and impose that it lies in the span of

x\frac{dt}{t}, xdx.

We have

ω = α(x, tx)dx + β(x, tx)(\frac{dt}{t} + \frac{dx}{x})

and we must clearly have β(0) = 0. Then β(x, tx)\frac{dt}{t} is a multiple of x\frac{dt}{t}, and it suffices to require that α(x, tx) + \frac{1}{x}β(x, tx) be divisible by x. Writing β(x, y) = β_{1}x + β_{2}y + …, our condition boils down to the divisibility by x of

α(0) + β_{1} + β_{2}t ⇔ β_{2} = 0, α(0) + β_{1} = 0.

Finally, let us show 7). Write

ω = α\frac{dx}{x} + β \frac{dy}{y}

and pull back to the blow up in the chart where y = tx: we get

(α + β)\frac{dx}{x} + β \frac{dt}{t},

which must be divisible by x, hence in particular β(0) = 0. Looking at the other chart we get symmetrically α(0) = 0. Now, α + β must vanish of order two, in order that its pull back be divisible by x^{2}.

□

8. Appendix 2: Proof of Proposition 4.1

Proof of Proposition 4.1. We can prove 1) and 2) simultaneously for i = 1. Observe that D_{1} = Δ_{1} + N_{1}, that Λ_{1} = L_{1} + N_{1}, and apply Lemma 4.3 (observing that (KỸ + 2N_{1} + (E_{1} − E_{3}))N_{1} = −4 + 1 < 0) in order to conclude that

H^{0}(Ω1 ; Ỹ (log(Δ_{1}))(E_{1} − E_{3} + N_{1})) ≅ H^{0}(Ω1 ; Ỹ (log(D_{1}))(E_{1} − E_{3})).

Moreover we observe that, again by Lemma 4.3,

H^{0}(Ω1 ; Ỹ (log(D_{1}))(E_{1} − E_{3})) = H^{0}(Ω1 ; Ỹ (log(D_{1} − (L − E_{1})))(L − E_{3})).

Let f: Ỹ → ℙ^{2} be the blow down of E_{1}, …, E_{5}. Then f_{∗}(D_{1} − (L − E_{1})) splits as the sum of two lines l_{1}, l_{2} in ℙ^{2} intersecting in P_{1}. W.l.o.g. we can assume that P_{1} = (0 : 0 : 1), P_{2} = (0 : 1 : 0), P_{4} = (1 : 0 : 0) and P_{5} = (1 : 0 : λ), with λ ≠ 0. Applying Proposition 7.1 several times for each blow down we get that

H^{0}(Ω1 ; Ỹ (log(D_{1} −(L−E_{1})))(L−E_{3})) = H^{0}(f_{∗}Ω1 ; Ỹ (log(D_{1} −(L−E_{1})))(L−E_{3}))

is the subspace V_{1} of H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2})(1)) consisting of sections satisfying several linear conditions. We write these conditions using the basis provided by Lemma 4.5 and its corollary, in order to show that V_{1} = 0. By prop. 7.1, 3) we get for P_{1}:

a_{13} + a_{23} = 0;

for P_{2}, P_{4} and P_{5} the three equations

a_{12} = a_{21} = a_{21} + λa_{23} = 0.

This shows that V_{1} = 0. We continue with the proof of 1). For i = 2, again by Lemma 4.3 we have to calculate

V_{2} = H^{0}(Ω1 ; Ỹ (log(L − E_{2} − E_{5}), log(L − E_{2} − E_{4}))(L − E_{3})),

which after blowing down E_{1}, …, E_{5} corresponds to a subspace of H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2})(1)).

W.l.o.g. we can assume that P_{2} = (0 : 0 : 1), P_{5} = (0 : 1 : 0), P_{4} = (1 : 0 : 0) and P_{3} = (1 : 1 : 1). By prop. 7.1, 3), we get for P_{2},P_{4}, P_{5} the three linear equations:

a_{13} + a_{23} = 0, a_{21} = 0, a_{12} = 0.

We evaluate ω in P_{3}, and get (using the above equalities)

ω(P_{3}) = a_{13}dx_{1} + a_{23}dx_{2},

whence by Proposition 7.1, 6) a_{13} = a_{23} = 0 and therefore we have verified that V_{2} = 0. For i = 3, using Lemma 4.3, we have to calculate:

V_{3} := H^{0}(Ω1 ; Ỹ (log(L − E_{3} − E_{4}), log(L − E_{3} − E_{5}))(L − E_{1})),

which, after blowing down E_{1}, …, E_{5}, becomes a linear subspace of H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2})(1)). W.l.o.g. we can assume that P_{3} = (0 : 0 : 1), P_{4} = (0 : 1 : 0), P_{5} = (1 : 0 : 0), P_{1} = (1 : 1 : 0). By prop. 7.1, 3), we get for P_{3},P_{4}, P_{5} the three linear equations:

a_{13} + a_{23} = 0, a_{12} = 0, a_{21} = 0.

Setting the evaluation of ω in P_{1} equal to zero is easily seen to give no new conditions, hence V_{3} ≅ ℂ. Let’s proceed to prove 2) for i = 2, 3. For i = 2, 3, by 4) of remark 1.5,

H^{0}(Ω1 ; Ỹ (log Δ_{i})(KỸ + Λ_{i})) = H^{0}(Ω1 ; Ỹ (log Δ_{i})(E_{i} − E_{i+2})).

For i = 2, using again Lemma 4.3, observing that

(KỸ + 2Γ_{2} + (E_{2} − E_{1}))Γ_{2} < 0,

we see that we have to calculate

V_{2} := H^{0}(Ω1 ; Ỹ (log(L − E_{2} − E_{4}), log(L − E_{2} − E_{5}))(2L − E_{1} − E_{3} − E_{4} − E_{5})),

which, after blowing down E_{1}, …, E_{5}, becomes a linear subspace of H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2})(2)). W.l.o.g. we can assume that P_{2} = (0 : 0 : 1), P_{4} = (0 : 1 : 0), P_{5} = (1 : 0 : 0), P_{3} = (1 : 1 : 1), and then P_{1} = (1 : λ : 0), where λ ≠ 0, 1. Using cor. 4.6, we get by prop. 7.1, 3) for P_{2} the linear equation

a_{313} + a_{323} = 0.

By prop. 7.1, 5) the conditions for P_{4} are

a_{212} = 0, a_{12} = 0, a_{23} = 0;

whereas the conditions for P_{5} are

a_{121} = 0, a_{12} = 0, a_{13} = 0.

Imposing that ω vanishes in P_{3}, we get

ω(P_{3}) = dx_{1}(a_{313} + 2a_{213} + a_{123}) + dx_{2}(a_{323} + 2a_{123} + a_{213}) = 0.

The above conditions yield:

a_{123} = a_{313} = −a_{213} = −a_{323}.

Finally, imposing that ω vanishes in P_{1} we obtain:

ω(P_{1}) = −dx_{3}(λa_{213} + a_{123}) = 0,

whence (λ − 1)a_{213} = 0. Since λ ≠ 0, 1 this implies a_{213} = 0, and we have shown that V_{2} = 0. We are left with the case i = 3. Using repeatedly Lemma 4.3 and Proposition 7.1, we see that we have to calculate

V_{3} := H^{0}(Ω1 ; Ỹ (log Δ_{3})(E_{3} − E_{2})) =

= H^{0}(Ω1 ; Ỹ (log(L − E_{1} − E_{3}), log(L − E_{1} − E_{4} − E_{5}))(2L − E_{3} − E_{4} − E_{5})).

After blowing down E_{1}, …, E_{5}, we can assume w.l.o.g. that P_{1} = (0 : 0 : 1), P_{4} = (0 : 1 : 0), P_{3} = (1 : 0 : 0), P_{5} = (0 : 1 : 1), and V_{3} becomes a linear subspace of H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2})(2)). Using cor. 4.6, we get by prop. 7.1, 3) for P_{1} the linear equation

a_{313} + a_{323} = 0.

By prop. 7.1, 5) the conditions for P_{4} are

a_{212} = 0, a_{12} = 0, a_{23} = 0;

whereas the conditions for P_{3} are

a_{121} = 0, a_{12} = 0, a_{13} = 0.

For P_{5}, we get instead (again by prop. 7.1, 5)), the two linear equations (the third is trivial):

a_{213} + a_{313} = 0, 2a_{313} = 0.

This implies that a_{313} = a_{213} = a_{323} = 0, but a_{123} is arbitrary. This shows that V_{3} ≅ ℂ. Thus 2) is proven. To prove 3), by symmetry, we may assume without loss of generality that i = 1. We have to calculate V_{1} := H^{0}(Ω1 ; Ỹ (log(D_{1})(E_{1} − E_{3})), which by Lemma 4.3 is equal to

H^{0}(Ω1 ; Ỹ (log(L − E_{1} − E_{2}), log(L − E_{1} − E_{4} − E_{5}))(L − E_{6})),

which, after blowing down E_{1}, …, E_{5}, becomes a linear subspace of H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2})(1)). W.l.o.g. we can assume that P_{1} = (0 : 0 : 1), P_{5} = (0 : 1 : 0), P_{2} = (1 : 0 : 0), P_{4} = (0 : 1 : 1). Since P_{2}, P_{4}, P_{6} are collinear, P_{6} = (1 : μ : μ), where μ ≠ 0. By prop. 7.1, 3), we get for P_{1},P_{2}, P_{4} and P_{5} the linear equations:

a_{13} + a_{23} = 0, a_{21} = 0, a_{12} + a_{13} = 0, a_{12} = 0.

This already shows that V_{1} = 0. Thus 3) is proven.

Let us treat the subcase of 4) where we have strictly extended Burniat divisors: the situation is here symmetric in the indices i, hence it suffices to show the vanishing of

H^{0}(Ω1 ; Ỹ (log(Δ_{1}))(KỸ + Λ_{1})).

Recall that we have the decomposition in irreducible connected components Δ_{1} = G_{1} + Γ_{1} + N_{2}, where G_{1} is the del Pezzo line G_{1} ≡ L − E_{1} − E_{6}. By Lemma 4.3 we get:

H^{0}(Ω1 ; Ỹ (log(Δ_{1}))(KỸ + Λ_{1})) = H^{0}(Ω1 ; Ỹ (log(Δ_{1} + N_{1}))(E_{1} − E_{3})),

since (KỸ + 2N_{1} + (E_{1} − E_{3}))N_{1} < 0. Using again Lemma 4.3 we see that

H^{0}(Ω1 ; Ỹ (log(Δ_{1} + N_{1}))(E_{1} − E_{3})) =

= H^{0}(Ω1 ; Ỹ (log(Δ_{1} + N_{1} − Γ_{1}))((E_{1} − E_{3}) + Γ_{1})) =

= H^{0}(Ω1 ; Ỹ (log(G_{1} + N_{1} + N_{2}))(2L − E_{2} − E_{3} − E_{5} − E_{6})),

because (KỸ + 2Γ_{1} + (E_{1} − E_{3}))Γ_{1} < 0. Let f: Ỹ → ℙ^{2} be the blow down of E_{1}, …, E_{6}. Then f_{∗}(G_{1}+N_{1}+N_{2}) splits as the sum of three lines l_{1}, l_{2}, l_{3} in ℙ^{2} forming a triangle. W.l.o.g. we can assume that P_{6} = (1 : 0 : 0), P_{1} = (0 : 1 : 0), P_{4} = (0 : 0 : 1) and P_{3} = (1 : 1 : 1). Then P_{5} = (0 : 1 : 1), whereas P_{2} is collinear with P_{6}, P_{4}, whence P_{2} = (1 : 0 : λ), with λ ≠ 0, 1. Then

H^{0}(Ω1 ; Ỹ (log(G_{1} + N_{1} + N_{2}))(2L − E_{2} − E_{3} − E_{5} − E_{6})) =

H^{0}(f_{∗}Ω1 ; Ỹ (log(G_{1} + N_{1} + N_{2}))(2L − E_{2} − E_{3} − E_{5} − E_{6}))

is a subspace of

H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(2)),

where l_{i} = x_{i}, whence P_{1}, P_{4}, P_{5} ∈ {l_{1} = 0}, P_{6}, P_{4}, P_{2} ∈ {l_{2} = 0}, P_{1}, P_{6} ∈ {l_{3} = 0}, consisting of sections satisfying fourteen linear conditions described in Proposition 7.1. We explicitly write these conditions using Lemma 4.4 and Lemma 4.6 in order to show that this subspace must be trivial. Let ω ∈ H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(2)) and we write ω in the basis of Lemma 4.4:

ω = a_{12}η_{12} + a_{13}η_{13} + a_{23}η_{23} + a_{212}x_{2}ω_{12} + a_{313}x_{3}ω_{13} + a_{323}x_{3}ω_{23}+ + a_{121}x_{1}ω_{21} + a_{131}x_{1}ω_{31} + a_{232}x_{2}ω_{32} + a_{123}x_{1}ω_{23} + a_{231}x_{2}ω_{31} + a_{312}x_{3}ω_{12}.

Then by prop. 7.1, 3) the condition for P_{1} = (0 : 1 : 0) is

(1)  a_{212} + a_{232} = 0.

The same argument shows that the linear condition for P_{4} = (0 : 0 : 1) is

(2)  a_{313} + a_{323} = 0.

Next we work out the conditions for P_{5}, P_{2} using prop. 7.1, 5). For P_{5} := (0 : 1 : 1) we work in the chart x_{3} = 1 and write ω locally around (0, 1) as α(x_{1}, x_{2})dx_{2} + β(x_{1}, x_{2})\frac{dx_{1}}{x_{1}} . Then we get (using Lemma 4.6):

(3)  β(0, 1) = a_{212} + a_{313} + a_{312} = 0;

(4)  \frac{∂β}{∂x_{1}} (0, 1) = −a_{12} − a_{13} − a_{231} = 0;

(5)  \frac{∂β}{∂x_{2}} (0, 1) + α(0, 1) = −a_{23} + 2a_{212} + a_{323} = 0.

The same argument for P_{2} = (1 : 0 : λ) (working in the chart x_{1} = 1 and writing ω locally around (0, λ) as α(x_{2}, x_{3})dx_{3} +β(x_{2}, x_{3})\frac{dx_{2}}{x_{2}} ) gives the following three linear equations (λ ≠ 0, 1):

(6)  β(0, λ) = a_{323}λ^{2} + a_{121} + a_{123}λ = 0;
(7)  \frac{∂β}{∂x_{2}} (0, λ) = a_{12} − λa_{23} − λa_{312} = 0;

(8) \frac{∂β}{∂x_{3}} (0, λ) + α(0, λ) = a_{13} + \frac{1}{λ}a_{131} + λa_{323} − λa_{313} =

= a_{13} + \frac{1}{λ}a_{131} + 2λa_{323} = 0.

There are four linear conditions coming from P_{6} = (1 : 0 : 0), given in prop. 7.1, 7). We work in the chart x_{1} = 1 and write ω = α(x_{2}, x_{3})\frac{dx_{2}}{x_{2}} +β(x_{2}, x_{3})\frac{dx_{3}}{x_{3}} . Then we get:

(9)  α(0, 0) = a_{121} = 0;
(10)  β(0, 0) = a_{131} = 0;
(11)  \frac{∂(α + β)}{∂x_{2}}(0, 0) = a_{12} + a_{231} = 0;

(12)  \frac{∂(α + β)}{∂x_{3}}(0, 0) = a_{13} = 0.

From equation (11) we get: a_{12} = −a_{231}. Since a_{13} = a_{131} = 0, equation (8) implies a_{323} = 0, whence by (2) also a_{313} = 0. Moreover, by (6), we get a_{123} = 0. We write finally the conditions coming from P_{3} = (1 : 1 : 1) (using again that certain coefficients are zero). We evaluate ω in P_{3} and work in the affine chart x_{2} = 1 to obtain

(13) ω(P_{3}) = (−a_{12} + a_{212} − a_{231} + a_{312})dx_{1}+

(a_{23} + a_{232} + a_{231})dx_{3} = 0.

Since:

ω(P_{3}) = (a_{212} + a_{312})dx_{1} + ((a_{23} + a_{232} + a_{231})dx_{3}

we get the last two linear equations:

(14)  a_{231} + a_{23} + a_{232} = 0;

(15)  a_{212} + a_{312} = 0.

These immediately imply that

a_{312} = a_{232} = −a_{212}.

By (14) a_{23} = −a_{232} − a_{231}, and using (5), we see that a_{23} = 2a_{212}. Again by (14) we get then that a_{212} = −a_{231}. Therefore, we have:

a_{212} = −a_{231} = −a_{312} = −a_{232} = \frac{a_{23}}{2} .

By (7):

0 = a_{12} − λa_{23} − λa_{231} = a_{12} + λa_{231},

whence by (4) λ = 1, which gives a contradiction, or a_{12} = a_{231} = 0. Hence the claim for strictly extended Burniat surfaces with K_{S}^{2} = 3 is established. Next we come to the case of (non strictly) extended Burniat surfaces. Here we have to consider two cases:

a) only one of the three conics Γ_{i} degenerates to two lines;

b) exactly two of the three conics Γ_{i} degenerate to two lines.

a) W.l.o.g. and by remark 1.5, (5) we may assume that Γ_{1} splits as

Γ_{1} ≡ (L − E_{1} − E_{2}) + N_{3} + E_{3}.

Then we get the extended Burniat divisors:

{D′_{1}} = |L − E_{1} − E_{6}| + |L − E_{1} − E_{2}| + E_{3} + N_{2},
D′_{2} ∈ |L − E_{2} − E_{5}| + |2L − E_{2} − E_{3} − E_{4} − E_{5}|,
D′_{3} ∈ |L − E_{3} − E_{4}| + |2L − E_{1} − E_{3} − E_{4} − E_{6}| + N_{1} + N_{3}.

We make the assumption, for each D′_{i}, i = 2, 3 that the strict transform of the conic Γ_{i} is irreducible. Then we have

KỸ + L′_{1} ≡ L − E_{3} − E_{4} − E_{5} ≡ KỸ + Λ_{1},
KỸ + L′_{2} ≡ L − E_{1} − E_{4} − E_{6} ≡ KỸ + Λ_{2},
KỸ + L′_{3} ≡ E_{3} − E_{2} ≡ KỸ + Λ_{3} − N_{3},

where the Λ_{i} are as for the strictly extended Burniat divisors. Observe that D′_{2} + N_{3} = Δ_{2}, whence

H^{0}(Ω1 ; Ỹ (log D′_{2})(KỸ + L′_{2})) ⊂ H^{0}(Ω1 ; Ỹ (log D′_{2})(KỸ + L′_{2} + N_{3})) =

H^{0}(Ω1 ; Ỹ (log(D′_{2} + N_{3}))(KỸ + L′_{2})) = H^{0}(Ω1 ; Ỹ (log Δ_{2})(KỸ + Λ_{2})) = 0, where the first equality holds by Lemma 4.3 and the last holds by our previous computations for strictly extended Burniat surfaces. Moreover, D′_{3} = Δ_{3} + N_{3}, L′_{3} = Λ_{3} − N_{3}, whence the vanishing of H^{0}(Ω1 ; Ỹ (log D′_{3})(KỸ + L′_{3})) follows again using Lemma 4.3 from the analogous vanishing for strictly extended Burniat surfaces. It remains to prove the following

Claim 8.1. H^{0}(Ω1 ; Ỹ (log D′_{1})(KỸ + L′_{1})) = 0

Proof of the claim. By Lemma 4.3 we see that

H^{0}(Ω1 ; Ỹ (log D′_{1})(KỸ + L′_{1})) = H^{0}(Ω1 ; Ỹ (log(D′_{1} − E_{3}))(L − E_{4} − E_{5})).

Let f: Ỹ → ℙ^{2} be the blow down of E_{1}, …, E_{6}. Then f_{∗}(D′_{1} − E_{3}) splits as the sum of three lines l_{1}, l_{2}, l_{3} in ℙ^{2} forming a triangle. W.l.o.g. we can assume that P_{1} = (1 : 0 : 0), P_{2} = (0 : 1 : 0), P_{6} = (0 : 0 : 1) and P_{5} = (1 : 1 : 1). Then P_{4} = (0 : 1 : 1). We conclude that H^{0}(Ω1 ; Ỹ (log(D′_{1} − E_{3}))(L − E_{4} − E_{5})) = H^{0}(f_{∗}Ω1 ; Ỹ (log(D′_{1} − E_{3}))(L − E_{4} − E_{5})) is the subspace of H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(1)) consisting of sections satisfying one linear condition for P_{1}, P_{2}, P_{6} each, two linear conditions for P_{5} and three linear conditions for P_{4}, described in Proposition 7.1. We write these conditions using Lemma 4.4 in order to show that this subspace must be trivial. By Lemma 4.4 we write ω ∈ H^{0}(Ω_{ℙ^{2}}^{1}(dlog l_{1}, dlog l_{2}, dlog l_{3})(1) as

ω = ∑_{i≠j} a_{ij}ω_{ij}.

Then the three equations for P_{1}, P_{2}, P_{6} (cf. prop. 7.1, 3)) are

a_{21} + a_{31} = 0, a_{12} + a_{32} = 0, a_{13} + a_{23} = 0.

By prop. 7.1, 5), we get for P_{4} the linear equations:

a_{12} + a_{13} = 0, −a_{21} − a_{31} = 0, a_{23} − a_{32} = 0.

The above conditions already imply:

a_{13} = a_{12} = a_{23} = a_{32} = 0, a_{21} = −a_{31}.

We impose the vanishing of ω in P_{5} = (1 : 1 : 1) working in the affine chart x_{3} = 1 and obtain

ω(1 : 1 : 1) = (−a_{21} − a_{31})dx_{1} + (a_{21})dx_{2} = 0,

whence a_{21} = a_{31} = 0. □

b) W.l.o.g. we can assume that each of the two conics Γ_{1} and Γ_{2} degenerate to two lines. Then we get the extended Burniat divisors:

{D′′_{1}} = |L − E_{1} − E_{6}| + |L − E_{1} − E_{2}| + E_{3} + N_{1} + N_{2},

{D′′_{2}} = |L − E_{2} − E_{5}| + |L − E_{2} − E_{3}| + E_{1},

D′′_{3} ∈ |L − E_{3} − E_{4}| + |2L − E_{1} − E_{3} − E_{4} − E_{6}| + N_{3}.

We make the assumption for D′′_{3} that the strict transform of the conic Γ_{3} passing through P_{1}, P_{3}, P_{4}, P_{6} is irreducible. Then we have

KỸ + L′′_{1} ≡ E_{1} − E_{3} = KỸ + L′_{1} − N_{1}, D′′_{1} = D′_{1} + N_{1};
KỸ + L′′_{2} ≡ L − E_{1} − E_{4} − E_{6} ≡ KỸ + L_{2} + N_{2}, D′′_{2} = D_{2} − N_{2};
KỸ + L′′_{3} ≡ E_{3} − E_{2} ≡ KỸ + Λ′_{3}, D′′_{3} = D′_{3} − N_{3}.

Therefore for i = 1, 2, 3 the vanishing of H^{0}(Ω1 ; Ỹ (log D′′_{i})(KỸ + L′′_{i})) can be reduced via Lemma 4.3 to the analogous vanishing for extended Burniat surfaces of case a) (i = 1, 3) and to the analogous vanishing for Burniat divisors (for i = 2), which was already proved in part 3).

□

9. Appendix 3: an alternative proof of statements 1), 2), 3) of Proposition 4.1

In this appendix we present other methods to calculate the space of sections of twisted logarithmic sheaves, in particular a fibration method. Assume that we have d smooth rational curves C_{α} ⊂ Y contained in a smooth algebraic surface Y, meeting with distinct tangents in a point O, a divisor B_{α} on C_{α} of degree 0, 1 or 2, and disjoint from O, and let Z be the blow up of Y in the point O. Denote by D_{α} the strict transform of C_{α}, and denote by Ω_{Y}^{1} ((log C_{α}(−B_{α}))_{α∈A}) the sheaf which is the inverse image, under the residue sequence, of ⊕_{α∈A}O_{C_{α}}(−B_{α}). Then by 4) of Proposition 7.1 we have an exact sequence

0 → Ω_{Y}^{1} ((log C_{α}(−B_{α}))_{α∈A}) →

→ p_{∗}Ω_{Z}^{1}((log D_{α}(−B_{α}))_{α∈A})(E) → ℂ_{O}^{d−2} → 0

which is exact on global sections if

h := dim_{ℂ}H^{1}(Ω_{Y}^{1} ((log C_{α}(−B_{α}))_{α∈A})) = 0.

Or, more generally, iff h = h′, where

h′ := dim_{ℂ}H^{1}(Ω_{Z}^{1}((log D_{α}(−B_{α}))_{α∈A})).

Consider the exact sequence

0 → Ω_{Y}^{1} → Ω_{Y}^{1} ((log C_{α}(−B_{α}))_{α∈A}) → ⨁_{α=1}^{d} O_{C_{α}}(−B_{α}) → 0,

and assume that H^{2}(Ω_{Y}^{1} ) = 0. Then h = a + b, where a is the number of α’s such that B_{α} has degree 2, while b is the difference of the dimensions between H^{1}(Ω_{Y}^{1} ) and the subspace generated by the Chern classes of the C_{α}’s such that B_{α} has degree 0. If we choose Y = ℙ^{2} then h = 0 as soon as no B_{α} has degree 2, and some B_{α} has degree 0.

Otherwise, one can calculate h′ in a similar way. We assume for simplicity that Y = ℙ^{2}. We have a similar exact sequence

0 → Ω_{Z}^{1}(E) → Ω_{Z}^{1}((log D_{α}(−B_{α}))_{α∈A})(E) → ⨁_{α=1}^{d} O_{C_{α}}(O − B_{α}) → 0,

and since H^{1}(O_{C_{α}}(O − B_{α})) = 0 by our assumption, we get that h′ is the dimension of the cokernel of

⨁_{α=1}^{d} H^{0}(O_{C_{α}}(O − B_{α})) → H^{1}(Ω_{Z}^{1}(E)).

To calculate the last space, observe that

Ω_{Z}^{1} ⊗ O_{E} = O_{E}(−2) ⊕ O_{E}(1)

whence h^{1}(Ω_{Z}^{1}(E)) = h^{1}(Ω_{Z}^{1}) + 1 = 3. These criteria can now be used in order to prove statements 1), 2), 3) of Proposition 4.1. We can prove 1) and 2) simultaneously for i = 1. Observe that D_{1} = Δ_{1} + N_{1}, that Λ_{1} = L_{1} + N_{1}, and apply Lemma 4.3 in order to conclude that

H^{0}(Ω1 ; Ỹ (log(Δ_{1}))(E_{1} − E_{3} + N_{1})) ≅ H^{0}(Ω1 ; Ỹ (log(D_{1}))(E_{1} − E_{3})).

By Lemma 7.1 we can blow down E_{3} and obtain H^{0}(Ω1 ; Ỹ′(log(D′_{1}))(E_{1})). In this case the respective degrees of the divisors B_{α} are 0, 1, 2 hence h = 1. We have to decide whether h′ is 0 or 1. We contract E_{3}, E_{4}, E_{5}, and we let Z be the blow up of the plane in P_{1}. We must calculate h^{0}(Ω_{Z}^{1}(log(C_{α}(−B_{α})))(E_{1})). Here the curves C_{α} are fibres of the ruling of Z, f: Z → ℙ^{1}. Using the exact sequence

(∗∗) 0 → f^{∗}Ω_{ℙ^{1}}^{1} → Ω_{Z}^{1} → ω_{Z|ℙ^{1}} = O_{Z}(−F − 2E_{1}) → 0

we obtain the analogous sequence

0 → f^{∗}O_{ℙ^{1}}(1)(E_{1}) → Ω_{Z}^{1}(log(C_{α})(E_{1}) → O_{Z}(−F − E_{1}) → 0

to infer that

H^{0}(f^{∗}O_{ℙ^{1}}(1)(E_{1})) = H^{0}(Ω_{Z}^{1}(log(C_{α}))(E_{1})).

We are imposing some vanishing on three points lying in two fibres, hence we get the sections of H^{0}(O_{Z}(F + E_{1})) = p^{∗}H^{0}(O_{ℙ^{2}}(1)) vanishing in the three points P_{4}, P_{5}, P_{2}, whence we conclude that this space has dimension = 0. This argument shows 1) also for i = 2, 3. For a nodal Burniat with m = 2 the space H^{0}(Ω1 ; Ỹ (log(D_{i}))(E_{i} − E_{i+2})), vanishes for i = 2, but it has dimension equal to 1 for i = 3, since then the three points P_{4}, P_{5}, P_{1} are collinear. Let’s proceed with 2). For i = 2, 3

H^{0}(Ω1 ; Ỹ (log Δ_{i})(KỸ + Λ_{i})) = H^{0}(Ω1 ; Ỹ (log Δ_{i})(E_{i} − E_{i+2})).

By applying again Lemma 7.1 for i = 3 we can blow down the curve E_{2} and the curve E_{3} and apply the residue sequence to the sheaf Ω1 ; Ỹ′(log Δ′_{3}). Since each component is smooth and rational, we find that H^{0}(Ω1 ; Ỹ′(log Δ′_{3})) = ker(ℂ^{4} → H^{1}(Ω1 ; Ỹ′)), while H^{1}(Ω1 ; Ỹ′(log Δ′_{3})) = Coker(ℂ^{4} → H^{1}(Ω1 ; Ỹ′)). The map is given by the Chern classes of L − E_{1}, L − E_{4}, L − E_{5}, L − E_{1} − E_{4} − E_{5}. These generate a rank 4 subspace of the 4-dimensional space H^{1}(Ω1 ; Ỹ′) (we are blowing up 3 points in the plane), whence h^{0}(Ω1 ; Ỹ′(log Δ′_{3})) =

0, h^{1}(Ω1 ; Ỹ′(log Δ′_{3})) = 0.

We conclude, by the exact cohomology sequence associated to

0 → Ω1 ; Ỹ′(log Δ′_{3}) → f_{∗}(Ω1 ; Ỹ (log Δ_{3})(KỸ + Λ_{3})) → ℂ_{P_{3}} → 0,

that h^{0}(Ω1 ; Ỹ (log Δ_{3})(KỸ + Λ_{3})) = 1 + h^{0}(Ω1 ; Ỹ′(log Δ′_{3})) = 1. For the case i = 2 recall that Δ_{2} ∈ |L − E_{2} − E_{4}| + |L − E_{2} − E_{5}| + |2L − E_{2} − E_{3} − E_{4} − E_{5}| consists of three smooth connected components. Blow down E_{1}, E_{3}, E_{4}, E_{5} and obtain the ruled surface Z equal to the blow up of the plane in P_{2}. Denote by f : Z → ℙ^{1} the standard fibration. The direct image Δ′_{2} := f_{∗}Δ_{2} decomposes as the union of two fibres F_{4} and F_{5} and a section C with C · E_{2} = 1. We have to calculate the space of global sections of

F := 𝔪_{P_{1}}Ω_{Z}^{1}(log F_{4}, log F_{5}, log C(−P_{3}))(E_{2})

satisfying two linear conditions imposed by the points P_{4}, P_{5}. Using the exact sequence (**) we get the exact sequence

0 → 𝔪_{P_{1}}O_{Z}(E_{2}) → F → 𝔐_{P_{3}}𝔪_{P_{1}}O_{Z}(−F − E_{2} + C) → 0.

Observe that O_{Z}(−F − E_{2} + C) has degree 0 on each fibre and degree 1 on E_{2}. If D ≡ −F − E_{2} + C ≡ L − E_{2} is effective, then D is a fibre. Since no fibre contains both P_{1}, P_{3}, we obtain

H^{0}(𝔪_{P_{3}}𝔪_{P_{1}}O_{Z}(−F − E_{2} + C)) = 0.

Since |E_{2}| consists of the curve E_{2}, which does not contain P_{1}, we conclude that H^{0}(M_{P_{1}}O_{Z}(E_{2})) = H^{0}(F) = 0. To prove 3), by symmetry, we may assume without loss of generality that i = 1. Blow down all the curves E_{j} excet E_{1}, so that , as usual, we have the blow up Z of the plane in a point (P_{1}) and the standard fibration f : Z → ℙ^{1}. By Lemma 7.1 and since E_{3} is a connected component of D_{1}, the direct image F of Ω1 ; Ỹ (log(D_{1})(E_{1} − E_{3}) is contained in Ω_{Z}^{1}(log(F_{2} + F_{6} + F_{4,5}))(E_{1}) where F_{j} denotes the unique fibre of f passing through the point P_{j}. More precisely, we have an exact sequence

0 → 𝔪_{P_{2}}𝔪_{P_{4}}𝔪_{P_{5}}𝔪_{P_{6}}O_{Z}(F + E_{1}) → F → O_{Z}(−F − E_{1}) → 0.

Clearly H^{0}(O_{Z}(−F − E_{1})) = 0 since F · (F + E_{1}) = 1. On the other hand H^{0}(O_{Z}(F + E_{1})) = H^{0}(O_{ℙ^{2}}(1)), hence the fact that the points P_{2}, P_{4}, P_{5}, P_{6} are not collinear implies the desired vanishing

H^{0}(𝔪_{P_{2}}𝔪_{P_{4}}𝔪_{P_{5}}𝔪_{P_{6}}O_{Z}(F + E_{1})) = 0.

Thus 3) is proven.

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I. Bauer Lehrstuhl Mathematik VIII Mathematisches Institut der Universität Bayreuth NW II, Universitätsstr. 30 95447 Bayreuth ingrid.bauer@uni-bayreuth.de F. Catanese Lehrstuhl Mathematik VIII Mathematisches Institut der Universität Bayreuth NW II, Universitätsstr. 30 95447 Bayreuth fabrizio.catanese@uni-bayreuth.de