Enumerating Exceptional Collections of Line Bundles

on Some Surfaces of General Type

Stephen Coughlan

Received: February 2, 2015 Revised: October 7, 2015

Communicated by Thomas Peternell

Abstract. We use constructions of surfaces as abelian covers to write down exceptional collections of line bundles of maximal length for every surface X in certain families of surfaces of general type with p_{g} = 0 and K_{X}^{2} = 3, 4, 5, 6, 8. We also compute the algebra of derived endomorphisms for an appropriately chosen exceptional collection, and the Hochschild cohomology of the corresponding quasiphantom category. As a consequence, we see that the subcategory generated by the exceptional collection does not vary in the family of surfaces. Finally, we describe the semigroup of effective divisors on each surface, answering a question of Alexeev.

2010 Mathematics Subject Classification: 14F05 (14J29) Keywords and Phrases: Derived category; Kulikov surface; Burniat surface; Beauville surface; Semiorthogonal decomposition; Exceptional sequence; Hochschild homology

Contents

1 Introduction  1256
2 Preliminaries  1259
3 Exceptional collections of line bundles on surfaces  1266
4 Heights of exceptional collections  1276
5 Secondary Burniat surfaces and effective divisors  1282
A Appendix: Acyclic bundles on the Kulikov surface  1287

B Appendix: Nodal Secondary Burniat surface with K^{2} = 4 1288

1 Introduction

Exceptional collections of maximal length on surfaces of general type with p_{g} = 0 have been constructed for Godeaux surfaces [13, 15], primary Burniat surfaces [2], and Beauville surfaces [24, 39]. Recently, progress has also been made for some fake projective planes [25, 23]. In this article, we present a method which can be applied uniformly to produce exceptional collections of line bundles on several surfaces with p_{g} = 0, including Burniat surfaces with K^{2} = 6 (cf. [2]), 5, 4, 3, Kulikov surfaces with K^{2} = 6 and some Beauville surfaces with K^{2} = 8 [24, 39]. In fact we do more: we enumerate all exceptional collections of line bundles corresponding to any choice of numerical exceptional collection. We can use this enumeration process to find those exceptional collections that are particularly well-suited to studying the surface itself, and possibly its moduli space. Both [2] and [24] hinted that it should be possible to produce exceptional collections of line bundles on a wide range of surfaces of general type with p_{g} = 0. This inspired us to build the approaches of [2, 24] into the larger framework of abelian covers (see especially Section 2), an important part of which is a new formula for the pushforward of certain line bundles on any abelian cover, generalising formulas of Pardini [43]. We believe that this work is a step in the right direction, even though there remain many families of surfaces which require further study (see Section 3.1 for more details). Let X be a surface of general type with p_{g} = 0, and let Y be a del Pezzo surface with K_{Y}^{2} = K_{X}^{2}. The lattices Pic X/Tors X and Pic Y are both isomorphic to ℤ^{1,N}, where N = 9−K_{X}^{2}, and moreover, the cohomology groups H^{2}(X, ℤ) and H^{2}(Y, ℤ) are completely algebraic. By exploiting this relationship between X and Y, we can study exceptional collections of line bundles on X. Indeed, exceptional collections on del Pezzo surfaces are well understood after [42], [33], and we sometimes refer to X as a fake del Pezzo surface, to emphasise this analogy. Suppose now that X is a fake del Pezzo surface that is constructed as a branched Galois abelian cover φ: X → Y, where Y is a (weak) del Pezzo surface with K_{Y}^{2} = K_{X}^{2}. Many fake del Pezzo surfaces can be constructed in this way [10], but we require certain additional assumptions on the branch locus and Galois group (see Section 3.1). These assumptions ensure that there is an appropriate choice of lattice isometry Pic Y → Pic X/Tors X. This isometry is combined with our pushforward formula to calculate the coherent cohomology of any line bundle on X.

Theorem 1.1 (Theorem 2.1) Let X be a fake del Pezzo surface satisfying our assumptions, and let L be any line bundle on X. We have an explicit formula for the line bundles M_{χ} appearing in the pushforward φ_{∗}L = ⨁_{χ∈G^{∗}} M_{χ}, where G is the Galois group of the cover φ: X → Y.

Working modulo torsion, we can use the above lattice isometry to lift any exceptional collection of line bundles on Y to a numerical exceptional collection on X. We then incorporate Theorem 1.1 into a systematic computer search, to find those combinations of torsion twists which correspond to an exceptional collection on X. The search for exceptional collections on fake del Pezzo surfaces, leads naturally to the following question, which was asked by Alexeev [1]:

Can we characterise effective divisors on X in terms of those on Y?

For example, in [1], Alexeev gives an explicit description of the semigroup of effective divisors on the Burniat surface with K^{2} = 6, and proposes similar descriptions for the other Burniat surfaces. We use our pushforward formula to prove these characterisations for the Burniat surfaces and other fake del Pezzo surfaces, cf. Theorems 3.2, 5.1.

Theorem 1.2 Let X be a fake del Pezzo surface satisfying our assumptions. Then the semigroup of effective divisors on X is generated by the reduced pullback of irreducible components of the branch divisor, together with pullbacks of certain (−1)-curves on Y.

Let 𝔼 be an exceptional collection on X, and suppose H_{1}(X, ℤ) is nontrivial. Then 𝔼 can not be full, for K-theoretic reasons (see Section 4). Hence we have a semiorthogonal decomposition of the bounded derived category of coherent sheaves on X:

D^{b}(X) = ⟨𝔼, A⟩.

If 𝔼 is of maximal length, then A is called a quasiphantom category; that is, K_{0}(A) is torsion and the Hochschild homology HH_{∗}(A) is trivial. Even when H_{1}(X, ℤ) vanishes, an exceptional collection of maximal length need not be full (see [15]), and in this case A is called a phantom category, because K_{0}(A) is trivial. On the other hand, the Hochschild cohomology does detect the quasiphantom category A; in fact, HH^{∗}(A) measures the formal deformations of A. We calculate HH^{∗}(A) by considering the A_{∞}-algebra of endomorphisms of 𝔼, together with the spectral sequence developed in [36]. Indeed, one of the advantages of our systematic search, is that we can find exceptional collections for which the higher multiplications in the A_{∞}-algebra of 𝔼 are as simple as possible. Theorem 1.3 below serves as a prototype statement of our results for a good exceptional collection on a fake del Pezzo surface. More precise statements can be found for the Kulikov surface in Section 4.7.

Theorem 1.3 Let X → T be a family of fake del Pezzo surfaces satisfying our assumptions. Then for any t in T, there is an exceptional collection 𝔼 of line bundles on X = X_{t} which has maximal length 12 − K_{X}^{2}. Moreover, the subcategory of D^{b}(X) generated by 𝔼 does not vary with t, and the Hochschild cohomology of X agrees with that of the quasi-phantom category A in degrees less than or equal to two.

The significance of Theorem 1.3 is amplified by the reconstruction theorem of [17]: if X and X′ are smooth, ±K_{X} is ample, and D^{b}(X) and D^{b}(X′) are equivalent bounded derived categories, then X ≅ X′. In conjunction with Theorem 1.3, we see that if K_{X} is ample, then X can be reconstructed from the quasi-phantom category A. The gluing between A and 𝔼 does not vary with X, because the statement about Hochschild cohomology implies that the formal deformation spaces of X are isomorphic to the formal deformation spaces of A. Currently, it is not clear whether there is any practical way to extract information about X from A, although some interesting ideas are discussed in [2]. It would be interesting to know whether this “rigidity” of 𝔼 is a general phenomenon, or just a coincidence for good choices of exceptional collection. In Section 2 we review abelian covers, and prove our result on pushforwards of line bundles, which is valid for any abelian cover, and is used throughout. In Section 3.1, we explain our assumptions on the fake del Pezzo surface X and its Galois covering structure φ: X → Y, and describe our approach to enumerating exceptional collections on the surface of general type. Section 3.2 is an extended treatment of the Kulikov surface with K^{2} = 6, which is an example of a fake del Pezzo surface. We give a cursory review of dg-categories and A_{∞}-algebras in Section 4, as background to our discussion of quasi-phantom categories and the theory of heights from [36]. We then show how to compute the A_{∞}-algebra and height of an exceptional collection on the Kulikov surface. In Section 5 we prove Theorem 1.2 for the secondary nodal Burniat surface with K^{2} = 4. Appendix A lists certain data relevant to the Kulikov surface example of Section 3.2, and Appendix B applies similarly to the secondary nodal Burniat surface of Section 5. With appropriate amendments, Theorems 1.2 and 1.3 hold for the Burniat surfaces with K^{2} = 6, 5, 4, 3 and some Beauville surfaces with K^{2} = 8. The arguments used are similar to those appearing in Sections 3.2 and 5.1, and we refer to [20] for details. We have exceptional collections of maximal length on the tertiary Burniat surface with K^{2} = 3. In this case it is necessary to use the Weyl group action on the Picard group to find exceptional collections. We can show that the A_{∞}-category is formal, but we do not yet know how to compute the Hochschild cohomology of the quasiphantom category. In order to use results on deformations of each fake del Pezzo surface, we work over ℂ.

Remark 1.1 The calculation of φ_{∗}L according to Theorem 1.1 is elementary but repetitive; we include a few sample calculations to illustrate how to do it by hand, but when the torsion group becomes large, it is more practical to use computer algebra. Our enumerations of exceptional collections are obtained by simple exhaustive computer searches. We use Magma [12], and the annotated scripts are available from [20].

Acknowledgements I would like to thank Valery Alexeev, Ingrid Bauer, Gavin Brown, Fabrizio Catanese, Paul Hacking, Al Kasprzyk, Anna Kazanova, Alexander Kuznetsov, Miles Reid and Jenia Tevelev for helpful conversations or comments about this work. I thank the DFG for support during part of this work through grant Hu 337-6/2.

2 Preliminaries

We collect together the relevant material on abelian covers. See especially [43],

[7] or [34] for details. Unless stated otherwise, X and Y are normal projective varieties, with Y nonsingular. Let G be a finite abelian group acting faithfully on X with quotient φ: X → Y. Write Δ = ∑ Δ_{i} for the branch locus of φ, where each Δ_{i} is a reduced, irreducible effective divisor on Y. The cover φ is determined by the group homomorphism

Φ: π_{1}(Y − Δ) → H_{1}(Y − Δ, ℤ) → G,

which assigns an element of G to the class of a loop around each irreducible component Δ_{i} of Δ. If Φ is surjective, then X is irreducible. The factorisation through H_{1}(Y − Δ, ℤ) arises because G is assumed to be abelian, so we only need to consider the map Φ: H_{1}(Y − Δ, ℤ) → G. For brevity, we refer to the loop around Δ_{i} by the same symbol, Δ_{i}. Let Ỹ be the blow up of Y at a point P where several branch components Δ_{i_{1}}, …, Δ_{i_{k}} intersect. Then there is an induced cover of Ỹ, and the image of the exceptional curve E under Φ is given by

Φ(E) = ∑_{j=1}^{k} Φ(Δ_{i_{j}}).  (1)

Fix an irreducible reduced component Γ of Δ and denote Φ(Γ) by γ. Then the inertia group of Γ is the cyclic group H ⊂ G generated by γ. Choosing the generator of H^{∗} = Hom(H, ℂ^{∗}) to be the dual character γ^{∗}, we may identify H^{∗} with ℤ/n, where n is the order of γ. Composing the restriction map res: G^{∗} → H^{∗} with this identification gives

G^{∗} → ℤ/n, χ ↦ k,

where χ|_{H} = (γ^{∗})^{k} for some 0 ≤ k ≤ n − 1. On the other hand, given χ in G^{∗} of order d, the evaluation map χ: G → ℤ/d satisfies

χ(γ) = \frac{d}{n}χ|_{H}(γ) = \frac{dk}{n}

as a residue class in ℤ/d (or as an integer between 0 and d − 1). The pushforward of φ_{∗}O_{X} breaks into a direct sum of eigensheaves

φ_{∗}O_{X} = ⨁_{χ∈G^{∗}} L_{χ}^{−1}.  (2)

Moreover, the L_{χ} are line bundles on Y and by Pardini [43], their associated (integral) divisors L_{χ} are given by the formula

dL_{χ} = ∑_{i} χ ∘ Φ(Δ_{i})Δ_{i}.  (3)

The line bundles L_{χ} play a pivotal role in the sequel, and we refer to them as the character sheaves of the cover φ: X → Y.

2.1 Line bundles on X

We develop tools for calculating with torsion line bundles on X. Let π′: A′ → X be the maximal abelian cover of X; that is, the étale cover of X associated to the subgroup π_{1}(X)^{ab} = H_{1}(X, ℤ) of π_{1}(X). Now let ψ′ be the composite map φ ∘ π′: A′ → Y. It is not always true that ψ′ is Galois and ramified over the same branch divisor Δ as φ: X → Y (see for example [45], [9]). So choose a maximal subgroup T of the torsion subgroup Tors X in Pic X whose associated cover ψ: A → Y is Galois and ramified over Δ. We have the following commutative diagram

\xymatrix{ & A \ar[dl]_{π} \ar[dr]^{ψ} \\ X \ar[rr]^{φ} &  & Y}

Let the Galois group of ψ be G̃. Then the original group G is the quotient G̃/T, so we get short exact sequences

0 → T → G̃ → G → 0  (4)

and

0 ← T^{∗} ← G̃^{∗} ← G^{∗} ← 0  (5)

where G^{∗} = Hom(G, ℂ^{∗}), etc. In fact, for each surface that we consider, these exact sequences are split, so that

G̃ = G ⊕ T, G̃^{∗} = G^{∗} ⊕ T^{∗}.  (6)

Let \overline{Γ} be a reduced irreducible component of the branch locus Δ of an abelian cover φ: X → Y and suppose the inertia group of \overline{Γ} is cyclic of order n. Then

Definition 2.1 (cf. [2]) The reduced pullback Γ of \overline{Γ} is the (integral) divisor Γ = \frac{1}{n}φ^{∗}(\overline{Γ}) on X.

Remark 2.1 The reduced pullback extends to arbitrary linear combinations ∑_{i} k_{i}Δ_{i} in the obvious way. We use a bar to denote divisors on Y and remove the bar when taking the reduced pullback. In other situations, it is convenient to use D_{i} to denote the reduced pullback of a branch divisor Δ_{i}.

The remainder of this section is dedicated to calculating the pushforward φ_{∗}(L⊗τ), where L = O_{X}(∑_{i} k_{i}D_{i}) is the line bundle associated to the reduced pullback of ∑_{i} k_{i}Δ_{i}, and τ is any torsion line bundle contained in T ⊂ Tors X. We do this by exploiting the association of the free part L with φ: X → Y, and the torsion part τ with π: A → X. The formulae that we obtain are a natural extension of results in [43]. It may be helpful to skip ahead to Examples 2.2.1 and 2.4.1 before reading this section in detail.

2.2 Free case

Until further notice, we write \overline{Γ} ⊂ Y for an irreducible component of the branch divisor Δ of φ: X → Y. By Pardini [43], the inertia group H ⊂ G of \overline{Γ} is cyclic, and H is generated by Φ(\overline{Γ}) of order n. Let Γ ⊂ X be the reduced pullback of \overline{Γ}, so that nΓ = φ^{∗}(\overline{Γ}). We start with cyclic covers.

Lemma 2.1 Let α: X → Y be a cyclic cover with group H ≅ ℤ/n, and suppose that \overline{Γ} is an irreducible reduced component of the branch divisor. Let Γ be the reduced pullback of \overline{Γ}, and suppose 0 ≤ k ≤ n − 1. Then

α_{∗}O_{X}(kΓ) = ⨁_{i∈H^{∗}−S} M_{i}^{−1} ⊕ ⨁_{i∈S} M_{i}^{−1}(\overline{Γ}),

where M_{i} is the character sheaf associated to α with character i ∈ H^{∗}, and

S = {n − k, …, n − 1} ⊂ H^{∗} ≅ ℤ/n.

Remark 2.2 If k is a multiple of n, say k = pn, the projection formula gives

α_{∗}O_{X}(kΓ) = α_{∗}(α^{∗}O_{Y} (p\overline{Γ})) = α_{∗}O_{X} ⊗ O_{Y} (p\overline{Γ}) = ⨁_{i∈H^{∗}} M_{i}^{−1}(p\overline{Γ}).

Thus the lemma extends to any integer multiple of Γ.

Proof After removing a finite number of points from \overline{Γ}, we may choose a neighbourhood U of \overline{Γ} such that U does not intersect any other irreducible components of Δ. Then since X and Y are normal we may calculate α_{∗}O_{X}(kΓ) locally on α^{−1}(U) and U. In what follows, we do not distinguish U (respectively α^{−1}(U)) from Y (resp. X). Let g = Φ(\overline{Γ}) so that H = ⟨g⟩ ≅ ℤ/n, and identify H^{∗} with ℤ/n via g^{∗} = 1. Locally, write α: α^{−1}(U) → U as z^{n} = b where b = 0 defines \overline{Γ} in U. Then

α_{∗}O_{X} = ⨁_{i=0}^{n−1} O_{Y} z^{i} = ⨁_{i=0}^{n−1} O_{Y} (−\frac{i}{n} \overline{Γ}) = ⨁_{i=0}^{n−1} M_{i}^{−1},

where the last equality is given by (3). Thus α_{∗}O_{X} is generated by 1, z, …, z^{n−1} as an O_{Y} -module, and the O_{Y} -algebra structure on α_{∗}O_{X} is induced by the equation z^{n} = b.

The calculation for O_{X}(kΓ) is similar,

α_{∗}O_{X}(kΓ) = α_{∗}O_{X} \frac{1}{z^{k}} = ⨁_{i=−k}^{n−k−1} O_{Y} z^{i} = ⨁_{i=0}^{n−k−1} O_{Y} z^{i} ⊕ ⨁_{i=−k}^{−1} O_{Y} \frac{z^{n+i}}{b}

where we use z^{n} = b to remove negative powers of z. Thus

α_{∗}O_{X}(kΓ) = ⨁_{i=0}^{n−k−1} O_{Y} (−\frac{i}{n} \overline{Γ}) ⊕ ⨁_{i=n−k}^{n−1} O_{Y} (−\frac{i}{n} \overline{Γ})(\overline{Γ})

= ⨁_{i∈H^{∗}−S} M_{i}^{−1} ⊕ ⨁_{i∈S} M_{i}^{−1}(\overline{Γ}),

where S = {n − k, …, n − 1}. □

The lemma can be extended to any abelian group using arguments inspired by Pardini [43] Sections 2 and 4.

Proposition 2.1 Let φ: X → Y be an abelian cover with group G, and let k = np + \overline{k}, where 0 ≤ \overline{k} ≤ n − 1. Then

φ_{∗}O_{X}(kΓ) = ⨁_{χ∈G^{∗}−S_{k\overline{Γ}}} L_{χ}^{−1}(p\overline{Γ}) ⊕ ⨁_{χ∈S_{k\overline{Γ}}} L_{χ}^{−1}((p + 1)\overline{Γ}),

where

S_{k\overline{Γ}} = \{χ ∈ G^{∗} : n − \overline{k} ≤ χ|_{H} ≤ n − 1\}.

Proof By the projection formula, we only need to consider the case k = \overline{k} (cf. Remark 2.2). As in the proof of Lemma 2.1, after removing a finite number of points, we may take a neighbourhood U of \overline{Γ} which does not intersect any other components of Δ. We work on U and its preimages φ^{−1}(U), β^{−1}(U). Factor φ: X → Y as

X ⟶^{α} Z ⟶^{β} Y,

where α is a cyclic cover ramified over Γ with group H = ⟨g⟩ ≅ ℤ/n, and β is unramified by our assumptions. As in Lemma 2.1 we denote the character sheaves of α by M_{i}, and those of the composite map φ = β ∘ α by L_{χ}. Now

β_{∗}M_{i} = ⨁_{χ∈[i]} L_{χ}  (7)

where the notation [i] means the preimage of i in H^{∗} under the restriction map res: G^{∗} → H^{∗}. That is,

[i] = {χ ∈ G^{∗} : χ|_{H} = i},

where d is the order of χ. Since β is not ramified we combine Lemma 2.1 and (7) to get

φ_{∗}O_{X}(kΓ) = ⨁_{χ∈G^{∗}−S_{k\overline{Γ}}} L_{χ}^{−1} ⊕ ⨁_{χ∈S_{k\overline{Γ}}} L_{χ}^{−1}(\overline{Γ})

where

S_{k\overline{Γ}} = \{χ ∈ G^{∗} : n − k ≤ χ|_{H} ≤ n − 1\}

is the preimage of S = {n − k, …, n − 1} ⊂ H^{∗} under res: G^{∗} → H^{∗}. □

2.2.1 Example (Campedelli surface)

Let φ: X → ℙ^{2} be a G = (ℤ/2)^{3}-cover branched over seven lines in general position. We label the lines Δ_{1}, …, Δ_{7}, and define Φ to induce a set-theoretic bijection between {Δ_{i}} and (ℤ/2)^{3} − {0}. We make the definition of Φ more precise later (see Example 2.4.1). It is well known ([34, §4]) that X is a surface of general type with p_{g} = 0, K^{2} = 2 and π_{1} = (ℤ/2)^{3}. Choose generators g_{1}, g_{2}, g_{3} for (ℤ/2)^{3} so that Φ(Δ_{1}) = g_{1}. There are eight character sheaves for the cover, which we calculate using formula (3),

L_{(0,0,0)} = O_{ℙ^{2}}, L_{χ} = O_{ℙ^{2}}(2) for χ ≠ (0, 0, 0).

Write D_{1} for the reduced pullback of Δ_{1}, so that φ^{∗}(Δ_{1}) = 2D_{1}. Then

S_{Δ_{1}} = {χ : χ|_{⟨g_{1}⟩} = 1} = {(1, 0, 0), (1, 1, 0), (1, 0, 1), (1, 1, 1)},

so that by Proposition 2.1, we have

φ_{∗}O_{X}(D_{1}) = O_{ℙ^{2}} ⊕ 4O_{ℙ^{2}}(−1) ⊕ 3O_{ℙ^{2}}(−2).

2.3 Torsion case

In this section we use the maximal abelian cover A to calculate the pushforward of a torsion line bundle on X. To simplify notation, we assume that the composite cover A → X → Y is Galois with group G̃, so that T = Tors X.

Proposition 2.2 Let τ be a torsion line bundle on X. Then

φ_{∗}O_{X}(−τ) = ⨁_{χ∈G^{∗}} L_{χ+τ}^{−1} .

where addition χ + τ takes place in G̃^{∗} = G^{∗} ⊕ T^{∗}.

Remark 2.3 Note that L_{χ+τ} is a character sheaf for the G̃-cover φ: A → Y, and the proposition allows us to interpret L_{χ+τ} as a character sheaf for the G-cover φ: X → Y. Unfortunately, there is still some ambiguity, because we do not determine which character in G^{∗} is associated to each L_{χ+τ} under the splitting of exact sequence (5). On the other hand, the special case τ = 0 gives

φ_{∗}O_{X} = ⨁_{χ∈G^{∗}} L_{χ}^{−1}.

Proof The structure sheaf O_{A} decomposes into a direct sum of the torsion line bundles when pushed forward to X

π_{∗}O_{A} = ⨁_{τ∈Tors X} O_{X}(−τ).

Thus O_{X}(τ) is the character sheaf with character τ under the identification T^{∗} ≅ Tors X. The composite φ_{∗}π_{∗}O_{A} breaks into character sheaves according to (2), and the image of O_{X}(−τ) is the direct sum of those character sheaves with character contained in the coset G^{∗} + τ of τ in G̃^{∗} under (6). □

2.4 General case

Now we combine Propositions 2.1 and 2.2 to give our formula for pushforward of line bundles O_{X}(∑_{i} D_{i}) ⊗ τ. The formula looks complicated, but most of the difficulty is in the notation.

Definition 2.2 Let n_{i} be the order of Ψ(Δ_{i}) in G̃, and write k_{i} = n_{i}p_{i} + \overline{k}_{i}, where 0 ≤ \overline{k}_{i} ≤ n_{i} − 1. Then given any subset I ⊂ \{1, …, m\}, we define

S_{I}[τ] = ⋂_{i∈I} S_{k_{i}Δ_{i}}[τ] ∩ ⋂_{j∈I^{c}} S_{k_{j}Δ_{j}}[τ]^{c},

where

S_{k\overline{Γ}}[τ] = \{χ ∈ G^{∗} : n − \overline{k} ≤ \frac{n}{d}(χ + τ)(Ψ(\overline{Γ})) ≤ n − 1\}

for any reduced irreducible component \overline{Γ} of the branch locus Δ. Note that for fixed τ in T^{∗}, the collection of all S_{I}[τ] partitions G^{∗}.

Theorem 2.1 Let D = ∑_{i=1}^{m} k_{i}D_{i} be the reduced pullback of the linear combination of branch divisors ∑_{i=1}^{m} k_{i}Δ_{i} on Y. Then

φ_{∗}O_{X}(D − τ) = ⨁_{I} ⨁_{χ∈S_{I}[τ]} L_{χ+τ}^{−1} (Δ_{I}),

where I is any subset of {1, …, m} and Δ_{I} = ∑_{i∈I} Δ_{i}.

Remark 2.4 For simplicity, we have assumed that k_{i} = \overline{k}_{i} for all i in the statement and proof of the theorem. When this is not the case, by the projection formula (cf. Remark 2.2) we twist by O_{Y} (∑_{i=1}^{m} p_{i}Δ_{i}).

Proof Fix i and let D_{i} be the reduced pullback of an irreducible component Δ_{i} of the branch divisor. Choose a neighbourhood of Δ_{i} which does not intersect any other Δ_{j}. This may also require us to remove a finite number of points from D_{i}. We work locally in this neighbourhood and its preimages under φ, π. Now by the projection formula,

π_{∗}π^{∗}O_{X}(k_{i}D_{i}) = π_{∗}O_{A} ⊗ O_{X}(k_{i}D_{i}), and thus

ψ_{∗}π^{∗}O_{X}(k_{i}D_{i}) = ⨁_{τ∈T} φ_{∗}O_{X}(k_{i}D_{i} − τ).

Then we combine Propositions 2.1 and 2.2 to obtain

φ_{∗}O_{X}(k_{i}D_{i} − τ) = ⨁_{χ∈G^{∗}−S_{k_{i}Δ_{i}} [τ]} L_{χ+τ}^{−1} ⊕ ⨁_{χ∈S_{k_{i}Δ_{i}} [τ]} L_{χ+τ}^{−1} (Δ_{i}),

where the indexing is explained in Definition 2.2. To extend to the global setting and linear combinations ∑ k_{i}D_{i}, we just need to keep track of which components of Δ should appear as a twist of each L_{χ+τ}^{−1} in the direct sum. This book-keeping is precisely the purpose of Definition 2.2. □

Using the formula

K_{X} = φ^{∗}(K_{Y} + ∑_{i} \frac{n_{i}−1}{n_{i}}Δ_{i})  (8)

and the Theorem, we give an alternative proof of the decomposition of φ_{∗}O_{X}(K_{X}).

Corollary 2.1 [43, Proposition 4.1] We have

φ_{∗}O_{X}(K_{X}) = ⨁_{χ∈G^{∗}} L_{χ^{−1}}(K_{Y}).

Proof Let D_{i} be the reduced pullback of Δ_{i}. Then by (8) and the projection formula, we have

φ_{∗}(O_{X}(K_{X})) = φ_{∗}(φ^{∗}O_{Y} (K_{Y}) ⊗ O_{X}( ∑_{i}(n_{i} − 1)D_{i}))

= O_{Y} (K_{Y}) ⊗ φ_{∗}O_{X}( ∑_{i}(n_{i} − 1)D_{i}).

Now by definition,

S_{(n_{i}−1)Δ_{i}} = \{χ ∈ G^{∗} : 1 ≤ \frac{n_{i}}{d} χ(Φ(Δ_{i})) ≤ n_{i} − 1\} = \{χ ∈ G^{∗} : χ(Φ(Δ_{i})) ≠ 0\}.

Thus in the decomposition of φ_{∗}O_{X}( ∑_{i}(n_{i} − 1)D_{i}) given by Theorem 2.1, the summand L_{χ}^{−1} is twisted by ∑_{j∈J} Δ_{j}, where J is the set of indices j with χ(Φ(Δ_{j})) ≠ 0. Then by (3),

L_{χ}^{−1}( ∑_{i∈J} Δ_{i}) = ∑_{i}(1 − \frac{1}{d})χ(Φ(Δ_{i}))Δ_{i} = L_{χ^{−1}},

where the last equality is because χ^{−1}(g) = −χ(g) = d − χ(g) for any g in G. Thus we obtain

φ_{∗}(O_{X}( ∑_{i}(n_{i} − 1)D_{i})) = ⨁_{χ∈G^{∗}} L_{χ^{−1}},

and the Corollary follows. □

2.4.1 Example 2.2.1 continued

We resume our discussion of the Campedelli surface. The fundamental group of X is (ℤ/2)^{3}, and so the maximal abelian cover π: A → X is a (ℤ/2)^{6}-cover ψ: A → ℙ^{2} branched over Δ. Choose generators g_{1}, …, g_{6} of (ℤ/2)^{6}. As promised in Example 2.2.1, we now fix Φ and Ψ:

Δ_{i}  Δ_{1} Δ_{2} Δ_{3} Δ_{4}  Δ_{5}  Δ_{6}  Δ_{7}
Φ(Δ_{i})  g_{1} g_{2} g_{3} g_{1} + g_{2} g_{1} + g_{3} g_{2} + g_{3} g_{1} + g_{2} + g_{3}
Ψ(Δ_{i}) − Φ(Δ_{i}) 0 0 0  g_{4}  g_{5}  g_{6}  g_{4} + g_{5} + g_{6}

For clarity, the table displays the difference between Ψ(Δ_{i}) and Φ(Δ_{i}). In order that A be the maximal abelian cover, Ψ is defined so that each Ψ(Δ_{i}) generates a distinct summand of (ℤ/2)^{6}, excepting Ψ(Δ_{7}), which is chosen so that ∑_{i} Ψ(Δ_{i}) = 0. This last equality is induced by the relation ∑_{i} Δ_{i} = 0 in H_{1}(ℙ^{2} − Δ, ℤ). The torsion group Tors X is generated by g_{4}^{∗}, g_{5}^{∗}, g_{6}^{∗}. As an illustration of Theorem 2.1, we calculate φ_{∗}O_{X}(D_{1})⊗τ, where τ is the torsion line bundle on X associated to g_{4}^{∗}. Suppose φ_{∗}O_{X}(D_{1}) ⊗ τ = ⨁_{χ∈G^{∗}} M_{χ}, where M_{χ} are the line bundles to be calculated. In the table below, we collect the data relevant to Theorem 2.1.

χ  L_{χ+τ}^{−1} (χ + τ) ∘ Ψ(D_{1}) Twist by Δ_{1}? M_{χ}
(0, 0, 0) O_{ℙ^{2}}(−1)  0  No  O_{ℙ^{2}}(−1)
(1, 0, 0) O_{ℙ^{2}}(−1)  1  Yes  O_{ℙ^{2}}
(0, 1, 0) O_{ℙ^{2}}(−1)  0  No  O_{ℙ^{2}}(−1)
(0, 0, 1) O_{ℙ^{2}}(−2)  0  No  O_{ℙ^{2}}(−2)
(1, 1, 0) O_{ℙ^{2}}(−3)  1  Yes  O_{ℙ^{2}}(−2)
(1, 0, 1) O_{ℙ^{2}}(−2)  1  Yes  O_{ℙ^{2}}(−1)
(0, 1, 1) O_{ℙ^{2}}(−2)  0  No  O_{ℙ^{2}}(−2)
(1, 1, 1) O_{ℙ^{2}}(−2)  1  Yes  O_{ℙ^{2}}(−1)

Summing the last column of the table, we get

φ_{∗}O_{X}(D_{1}) ⊗ τ = O_{ℙ^{2}} ⊕ 4O_{ℙ^{2}}(−1) ⊕ 3O_{ℙ^{2}}(−2).

In particular, we see that the linear system on X associated to the line bundle O_{X}(D_{1}) ⊗ τ contains a single effective divisor.

3 Exceptional collections of line bundles on surfaces

3.1 Overview and definitions

We outline our method for producing exceptional collections, starting with some definitions and fundamental observations. A good reference for semiorthogonal decompositions is [37], and Proposition 3.1 is proved in [26].

Definition 3.1 An object E in D^{b}(X) is called exceptional if

Ext^{k}(E, E) = \{ \begin{matrix}ℂ & if k = 0, \\ 0 & otherwise. \end{matrix}

An exceptional collection 𝔼 ⊂ D^{b}(X) is a sequence of exceptional objects 𝔼 = (E_{0}, …, E_{n}) such that if 0 ≤ i < j ≤ n then

Ext^{k}(E_{j}, E_{i}) = 0 for all k.

Remark 3.1 Some authors prefer the term exceptional sequence rather than exceptional collection.

It follows from Definition 3.1 that a line bundle on a surface is exceptional if and only if p_{g} = q = 0. Moreover, if 𝔼 is an exceptional collection of line bundles, and L is any line bundle, then 𝔼 ⊗ L = (E_{0} ⊗ L, …, E_{n} ⊗ L) is again an exceptional collection, so we always normalise 𝔼 so that E_{0} = O_{X}. Let E = ⟨𝔼⟩ denote the smallest full triangulated subcategory of D^{b}(X) containing all objects in 𝔼. Then E is an admissible subcategory of D^{b}(X), and so we have a semiorthogonal decomposition

D^{b}(X) = ⟨E, A⟩,

where A is the left orthogonal to E. That is, A consists of all objects F in D^{b}(X) such that Ext^{k}(F, E) = 0 for all k and for all E in E. We say that the exceptional collection 𝔼 is full if D^{b}(X) = E. The K-theory is additive across semiorthogonal decompositions:

Proposition 3.1 If D^{b}(X) = ⟨A, B⟩ is a semiorthogonal decomposition, then

K_{0}(X) = K_{0}(A) ⊕ K_{0}(B).

Moreover, if 𝔼 is an exceptional collection of length n, then K_{0}(E) = ℤ^{n}. Thus if K_{0}(X) is not free, then X can never have a full exceptional collection. The maximal length of an exceptional collection on X is less than or equal to the rank of K(X).

3.1.1 Exceptional collections on del Pezzo surfaces

Let Y be the blow up of ℙ^{2} in n points, and write H for the pullback of the hyperplane section, \overline{E}_{i} for the ith exceptional curve. Then by work of Kuleshov and Orlov [42], [33] there is an exceptional collection of sheaves on Y

O_{\overline{E}_{1}}(−1), …, O_{\overline{E}_{n}}(−1), O_{Y} , O_{Y} (H), O_{Y} (2H).

Note that the blown up points do not need to be in general position, and can even be infinitely near. We prefer an exceptional collection of line bundles on Y, so we mutate past O_{Y} to get

O_{Y} , O_{Y} (\overline{E}_{1}), …, O_{Y} (\overline{E}_{n}), O_{Y} (H), O_{Y} (2H).  (9)

In fact, we only use the numerical properties of a given exceptional collection of line bundles on Y. Choose a basis e_{0}, …, e_{n} for the lattice Pic Y ≅ ℤ^{1,n} with intersection form diag(1, −1^{n}). Then we write equation (9) numerically as

0, e_{1}, …, e_{n}, e_{0}, 2e_{0}.

3.1.2 From del Pezzo to general type

Let X be a surface of general type with p_{g} = 0 which admits an abelian cover φ: X → Y of a del Pezzo surface Y with K_{Y}^{2} = K_{X}^{2}. In addition, we suppose that the maximal abelian cover A → X → Y is also Galois. Otherwise choose a maximal subgroup T ⊂ Tors X for which the associated cover is Galois, and replace A, as in Section 2. The branch divisor is Δ = ∑_{i} Δ_{i} and we assume that Δ is sufficiently reducible so that

(A1) Pic Y is generated by integral linear combinations of Δ_{i}.

Now the Picard lattices of X and Y are isomorphic. Thus if G is not too complicated, e.g. of the form ℤ/p × ℤ/q, we might hope to have:

(A2) The reduced pullbacks D_{i} of Δ_{i} (see Definition 2.1) generate Pic X/Tors X.

In very good cases, reduced pullback actually induces an isometry of lattices

(A3) f: Pic Y → Pic X/Tors X, such that f(K_{Y}) = −K_{X} modulo Tors X.

We say that a surface satisfies assumption (A) if (A1), (A2) and (A3) hold. These conditions are quite strong, and are not strictly necessary for our methods. For example, we could replace (A3) with an isometry of lattices from the abstract lattice ℤ^{1,n} to Pic X/Tors X.

Definition 3.2 A sequence 𝔼 = (E_{0}, …, E_{n}) of line bundles on X is called numerically exceptional if χ(E_{j}, E_{i}) = 0 whenever 0 ≤ i < j ≤ n.

Assume X satisfies (A), and let (Λ_{i}) = (Λ_{0}, …, Λ_{n}) be an exceptional collection on Y. Now define (L_{i}) = (L_{0}, …, L_{n}) by L_{i} = f(Λ_{i})^{−1}. A calculation with the Riemann–Roch formula shows that (L_{i}) is a numerically exceptional collection on X. This is explained in [2]. Given a numerically exceptional collection (L_{i}) of line bundles on X, the remaining obstacle is to determine whether (L_{i}) is genuinely exceptional rather than just numerically so. Indeed, most numerically exceptional collections on X are not exceptional. The standard trick (see [13]) is to choose torsion line bundles τ_{i} in such a way that the twisted sequence (L_{i} ⊗ τ_{i}) is an exceptional collection. We examine these choices of τ_{i} more carefully in what follows.

3.1.3 Acyclic line bundles

We discuss acyclic line bundles following [24].

Definition 3.3 Let L be a line bundle on X. If H^{i}(X, L) = 0 for all i, then we call L an acyclic line bundle. We define the acyclic set associated to L to be

A(L) = {τ ∈ Tors X : L ⊗ τ is acyclic} .

We call L numerically acyclic if χ(X, L) = 0. Clearly, an acyclic line bundle must be numerically acyclic.

Remark 3.2 In the notation of [24], τ = −χ.

Lemma 3.1 ([24], Lemma 3.4) A numerically exceptional collection L_{0} = O_{X}, L_{1} ⊗ τ_{1}, …, L_{n} ⊗ τ_{n} on X is exceptional if and only if

−τ_{i} ∈ A(L_{i}^{−1}) for all i, and
τ_{i} − τ_{j} ∈ A(L_{j}^{−1} ⊗ L_{i}) for all j > i. (10)

Thus to enumerate all exceptional collections on X of a particular numerical type, it suffices to calculate the relevant acyclic sets, and systematically test the above conditions (10) on all possible combinations of τ_{i}.

3.1.4 Calculating cohomology of line bundles

Given a torsion twist L ⊗ τ, Theorem 2.1 gives a decomposition

φ_{∗}(L ⊗ τ) = ⨁_{χ∈G^{∗}} M_{χ},

for some line bundles M_{χ} on Y, which may be computed explicitly. Since φ is finite, we have

h^{p}(L ⊗ τ) = ∑_{χ∈G^{∗}} h^{p}(M_{χ})

for all p. Thus L ⊗ τ is acyclic if and only if each summand M_{χ} is acyclic on Y. Now if χ(Y, M_{χ}) = 0 and h^{0}(M_{χ}) = h^{2}(M_{χ}) = 0, we see that h^{1}(M_{χ}) = 0. Thus by Serre duality and the Riemann–Roch theorem, we are reduced to calculating Euler characteristics and determining effectivity for (lots of) divisor classes on the del Pezzo surface Y.

3.1.5 Coordinates on Pic X/Tors X

Under assumption (A), we make the following definition.

Definition 3.4 Choose a basis B_{1}, …, B_{n} for Pic X/Tors X consisting of linear combinations of reduced pullbacks. Then any line bundle L on X may be written uniquely as

L = O_{X}(d_{1}, …, d_{n}) ⊗ τ

so that L = O_{X}( ∑_{i=1}^{n} d_{i}B_{i}) ⊗ τ. We call d (respectively τ) the multidegree (resp. torsion twist) of L with respect to the chosen basis.

The torsion twist associated to any line bundle on X may be calculated using Theorem 2.1 and the following immediate lemma. See Lemma 3.4 for an example.

Lemma 3.2 If τ is a torsion line bundle, then h^{0}(τ) ≠ 0 implies τ = 0.

Remark 3.3 Definition 3.4 fixes a basis for Pic Y = ℤ^{1,9−K^{2}} via the isometry with Pic X/Tors X. This basis corresponds to a geometric marking on the del Pezzo surface Y, and the multidegree d of L is just the image of L in Pic Y under the isometry. In fixing our basis, we break some of the symmetry of the coordinates. This is necessary in order to use the computer to search for exceptional collections. We can recover the symmetry later using the Weyl group action (see Section 3.1.7).

3.1.6 Determining effectivity of divisor classes

For each fake del Pezzo surface, we have the following theorem.

Theorem 3.1 Suppose X is a fake del Pezzo surface satisfying assumption (A) and with T = Tors X. Let 𝔈 denote the semigroup generated by the reduced pullbacks D_{i} of the irreducible branch components Δ_{i}, and pullbacks of the other (−1)- and (−2)-curves on Y. Then 𝔈 is the semigroup of all effective divisors on X.

We prove this theorem for the secondary nodal Burniat surface with K^{2} = 4 in Section 5 (cf. [1] for the Burniat surface with K^{2} = 6). The other fake del Pezzo surfaces work in the same way, see [20]. Moreover, 𝔈 is graded by multidegree, and we define a homomorphism

t: 𝔈 → Tors X

sending D_{i} to its torsion twist under Definition 3.4. The image under t of the graded summand 𝔈_{d} of multidegree d is the set of torsion twists τ for which O_{X}(∑ d_{i}B_{i}) ⊗ τ is effective.

3.1.7 Group actions on the set of exceptional collections

We consider a dihedral group action and the Weyl group action on the set of exceptional collections on X. Mutations are not considered systematically in this article, since a mutation of a line bundle need not be a line bundle.

Let 𝔼 = (E_{1}, …, E_{n}) be an exceptional collection of line bundles on X. If we normalise the first line bundle of any exceptional collection to be O_{X}, then there is an obvious dihedral group action on the set of exceptional collections of length n on X, generated by 𝔼 ↦ (E_{2}, …, E_{n}, E_{1}(−K_{X})) and 𝔼 ↦ 𝔼^{−1} = (E_{n}^{−1}, …, E_{1}^{−1}). The Weyl group of Pic Y is generated by reflections in (−2)-classes. That is, suppose α is a class in Pic Y with K_{Y} · α = 0 and α^{2} = −2. Then

r_{α}: L ↦ L + (L · α)α

is a reflection on Pic Y which fixes K_{Y}. Any reflection sends an exceptional collection on Y to another exceptional collection. Thus the Weyl group action on numerical exceptional collections on Y induces an action on numerical exceptional collections on X under assumption (A). This action accounts for the choices made in giving Y a geometric marking (see Definition 3.4).

3.2 The Kulikov surface with K^{2} = 6

For details on the Kulikov surface (first described in [34]), its torsion group and moduli space, see [19]. The Kulikov surface X is a (ℤ/3)^{2}-cover of the del Pezzo surface Y of degree 6. Figure 1 shows the associated cover of ℙ^{2} branched over six lines in special position. The configuration has just one free parameter, and in fact, the Kulikov surfaces form a 1-dimensional, irreducible, connected component of the moduli space of surfaces of general type with p_{g} = 0 and K^{2} = 6.

[figure: Δ_{1}; Δ_{2}; Δ_{3}; Δ_{4}; Δ_{5}; Δ_{6}; P_{1}; P_{2}; P_{3}]

Figure 1: The Kulikov configuration

To obtain a nonsingular cover, we blow up the plane at three points P_{1}, P_{2}, P_{3}, giving a (ℤ/3)^{2}-cover of a del Pezzo surface of degree 6. The exceptional curves are denoted \overline{E}_{i}. By results of [19], the torsion group Tors X is isomorphic to (ℤ/3)^{3}, so the maximal abelian cover ψ: A → Y has group G̃ ≅ (ℤ/3)^{5}. Let g_{i} generate G̃, and write g_{i}^{∗} for the dual generators of G̃^{∗}. As explained in Section 2, the covers are determined by Φ: H_{1}(ℙ^{2}−Δ, ℤ) → G and Ψ: H_{1}(ℙ^{2}−Δ, ℤ) → G̃, which are defined in the table below.

D  Δ_{1} Δ_{2}  Δ_{3}  Δ_{4}  Δ_{5}  Δ_{6}
Φ(D)  g_{1} g_{1}  g_{1}  g_{2} g_{1} + g_{2} 2g_{1} + g_{2}
Ψ(D) − Φ(D) 0 g_{3} 2g_{3} + g_{4} 2g_{4}  g_{5}  2g_{5}

The images of the exceptional curves \overline{E}_{i} under Φ and Ψ are computed using formula (1):

Φ(\overline{E}_{1}) = 2g_{1} + g_{2}, Φ(\overline{E}_{2}) = g_{2}, Φ(\overline{E}_{3}) = g_{1} + g_{2}, etc.

Lemma 3.3 The Kulikov surface satisfies assumptions (A1) and (A2). That is, the free part of Pic X is generated by the reduced pullbacks of Δ_{1} + \overline{E}_{2} + \overline{E}_{3}, \overline{E}_{1}, \overline{E}_{2}, \overline{E}_{3}, and the intersection pairing diag(1, −1, −1, −1) is inherited from Y.

Proof Define e_{0} = D_{1} +E_{2} +E_{3}, e_{1} = E_{1}, e_{2} = E_{2}, e_{3} = E_{3} in Pic X. These are integral divisors, since they are reduced pullbacks, and the intersection pairing is diag(1, −1, −1, −1), which is unimodular. For example, by definition of reduced pullback, 3e_{0} = φ^{∗}(Δ_{1} + \overline{E}_{2} + \overline{E}_{3}), and so

(3e_{0})^{2} = φ^{∗}(Δ_{1} + \overline{E}_{2} + \overline{E}_{3})^{2} = 9 · 1,

or e_{0}^{2} = 1. Hence we have an isomorphism of lattices. □

Using the basis chosen in this lemma, we compute the coordinates (Definition 3.4) of the reduced pullback D_{i} of each irreducible branch component Δ_{i}.

Lemma 3.4 We have

O_{X}(D_{1}) = O_{X}(1, 0, −1, −1),  O_{X}(D_{4}) = O_{X}(1, −1, 0, 0)[2, 1, 2],
O_{X}(D_{2}) = O_{X}(1, −1, 0, −1)[1, 0, 2], O_{X}(D_{5}) = O_{X}(1, 0, −1, 0)[2, 1, 0],
O_{X}(D_{3}) = O_{X}(1, −1, −1, 0)[2, 0, 2], O_{X}(D_{6}) = O_{X}(1, 0, 0, −1)[2, 1, 1],

where [a, b, c] in (ℤ/3)^{3} denotes a torsion line bundle on X.

Proof We prove that O_{X}(D_{2}) = O_{X}(1, −1, 0, −1)[1, 0, 2]. The other cases are similar. It is clear that Δ_{2} ∼ Δ_{1} − \overline{E}_{1} + \overline{E}_{2} on Y, so the multidegree is correct. It remains to check the torsion twist, by showing that F = O_{X}(D_{2} − D_{1} + E_{1} − E_{2} − τ) has a global section when τ = [1, 0, 2]. Then by Lemma 3.2, we have the desired equality. The pushforward φ_{∗}F splits into a direct sum of line bundles ⨁ M_{χ}, one for each character χ = (a, b) in G^{∗}. The following table collects the data required to calculate each M_{χ} via Theorem 2.1. The second column is calculated using equation (3), and the next four columns evaluate χ + τ on each Ψ(Γ), where Γ is any one of Δ_{1}, Δ_{2}, \overline{E}_{1} and \overline{E}_{2}. The final column is explained below.

(χ + τ) ∘ Ψ(Γ)
χ  L_{χ+τ}^{−1}  Δ_{1} Δ_{2} \overline{E}_{1} \overline{E}_{2}  M_{χ}
(0, 0) O_{Y} (−2, 1, 1, 0) 0 1 0 1 O_{Y} (−3, 1, 2, 1)
(1, 0) O_{Y} (−1, 0, 0, 1) 1 2 2 1  O_{Y}
(0, 1) O_{Y} (−2, 1, 0, 1) 0 1 1 2 O_{Y} (−3, 1, 1, 2)
(2, 0) O_{Y} (−2, 0, 1, 1) 2 0 1 1 O_{Y} (−2, 0, 1, 1)
(1, 1) O_{Y} (−2, 1, 0, 1) 1 2 0 2 O_{Y} (−1, 0, 0, 0)
(0, 2) O_{Y} (−2, 1, 1, 0) 0 1 2 0 O_{Y} (−3, 2, 1, 1)
(2, 1) O_{Y} (−2, 0, 1, 0) 2 0 2 2 O_{Y} (−2, 1, 1, 0)
(1, 2) O_{Y} (−3, 1, 1, 1) 1 2 1 0 O_{Y} (−2, 0, 0, 0)
(2, 2) O_{Y} (−2, 1, 1, 1) 2 0 0 0 O_{Y} (−2, 1, 0, 1)

Now by the projection formula (cf. Remark 2.2),

φ_{∗}F = φ_{∗}O_{X}(2D_{1} + D_{2} + E_{1} + 2E_{2} − τ) ⊗ O_{Y} (−Δ_{1} − \overline{E}_{2}).

So according to Theorem 2.1 and the remark following it, each M_{χ} is a twist of L_{χ+τ}^{−1} (−Δ_{1} − \overline{E}_{2}) by a certain combination of Δ_{1}, Δ_{2}, \overline{E}_{1} and \overline{E}_{2}. By Definition 2.2, the rules governing the twists are:

twist by Δ_{1} ⟺ (χ + τ) ∘ Ψ(Δ_{1}) = 1 or 2
twist by Δ_{2} ⟺ (χ + τ) ∘ Ψ(Δ_{2}) = 2
twist by \overline{E}_{1} ⟺ (χ + τ) ∘ Ψ(\overline{E}_{1}) = 2
twist by \overline{E}_{2} ⟺ (χ + τ) ∘ Ψ(\overline{E}_{2}) = 1 or 2.

Thus φ_{∗}F is given by the direct sum of the line bundles M_{χ} listed in the final column. Note that M_{(1,0)} = O_{Y}, so h^{0}(φ_{∗}F) = 1. Hence D_{2} − D_{1} + E_{1} − E_{2} − τ ∼ 0. □

Corollary 3.1 By formula (8), we have

O_{X}(K_{X}) = O_{X}(3, −1, −1, −1)[0, 0, 2].

Thus the Kulikov surface satisfies (A3).

Proof The multidegree is clear by (8), but the torsion twist requires some care. Since K_{X} is the pullback of an integral divisor on Y, it should be torsionneutral with respect to our coordinate system on Pic X. Thus by Lemma 3.4, we see that the required twist is [0, 0, 2]. □

Theorem 3.2 The semigroup 𝔈 of effective divisors on the Kulikov surface is generated by the nine reduced pullbacks of components of the branch divisor D_{1}, …, D_{6}, E_{1}, E_{2}, E_{3}. □

This Theorem is proved using an easier variant of the proof of Theorem 5.1. The situation here is easier, because all of the (−1)-curves on Y are branch divisors, and there are no (−2)-curves. Thus we have a homomorphism of semigroups t: 𝔈 → Tors X, which sends an effective divisor to its associated torsion twist (see Lemma 3.3), under the choice of basis (from Lemma 3.4).

3.2.1 Acyclic line bundles on the Kulikov surface

Let us start with the following numerical exceptional collection on Y:

Λ: 0, e_{0} − e_{1}, e_{0} − e_{2}, e_{0} − e_{3}, 2e_{0} − ∑_{i=1}^{3}e_{i}, e_{0}.

Given assumptions (A), we see that Λ corresponds to the following numerically exceptional sequence of line bundles on X:

\begin{matrix}L_{0} = O_{X}, L_{1} = O_{X}(−1, 1, 0, 0), L_{2} = O_{X}(−1, 0, 1, 0), \\ L_{3} = & O_{X}(−1, 0, 0, 1), L_{4} = O_{X}(−2, 1, 1, 1), L_{5} = O_{X}(−1, 0, 0, 0). \end{matrix}  (11)

We find all collections of torsion twists L_{i} ⊗τ_{i} which are exceptional collections on X. The first step is to find the acyclic sets associated to the various L_{j}^{−1}⊗L_{i}.

Proposition 3.2 The acyclic sets A(L_{j}^{−1} ⊗ L_{i}) for j > i ≥ 0 are listed in Appendix A.

First Proof By Theorem 3.2, it is an easy exercise to check each entry in the table. As an illustration, we calculate A(L_{1}^{−1}). The effective divisors on X of multidegree (1, −1, 0, 0) are D_{2} + E_{3}, D_{3} + E_{2}, D_{4}. Thus applying the homomorphism t to each of these effective divisors, we see that [1, 0, 2], [2, 0, 2], [2, 1, 2] do not appear in A(L_{1}^{−1}). Next we consider degree two cohomology via Serre duality. The effective divisors of multidegree (2, 0, −1, −1) are

2D_{1} + E_{2} + E_{3}, D_{1} + D_{2} + E_{1} + E_{3}, D_{1} + D_{3} + E_{1} + E_{2},
D_{2} + D_{3} + 2E_{1}, D_{1} + D_{4} + E_{1}, D_{1} + D_{5} + E_{2}, D_{1} + D_{6} + E_{3},
D_{2} + D_{5} + E_{1}, D_{3} + D_{6} + E_{1}, D_{5} + D_{6}.

Again, applying t we find that [0, 0, 2], [2, 0, 0], [1, 0, 0], [0, 0, 1], [1, 2, 0], [1, 2, 2], [1, 2, 1], [0, 2, 0], [2, 2, 2], [2, 1, 1] can not appear in A(L_{1}^{−1}). The acyclic set is made up of those elements of Tors X which do not appear in either of the two lists above. □

Second Proof As a sanity check, an alternative proof is to use Theorem 2.1 repeatedly, to calculate the cohomology of all possible torsion twists of L_{1}. □

Both methods are implemented in our computer script [20].

3.2.2 Exceptional collections on the Kulikov surface

We now find all exceptional collections on X which are numerically of the form (11). Lemma 3.1 reduces us to a simple search, which can be done systematically [20].

Theorem 3.3 The surface X has nine exceptional collections L_{0} = O_{X}, L_{1} ⊗ τ_{1}, …, L_{5} ⊗τ_{5} which are numerically of the form (11). They are given in Table 1 below. Each row lists the required torsion twists τ_{i} for i = 1, …, 5 as elements of (ℤ/3)^{3}.

τ_{1}  τ_{2}  τ_{3}  τ_{4}  τ_{5}
1 [0, 0, 0] [0, 2, 2] [2, 2, 1] [2, 2, 1] [0, 0, 1]
2 [2, 2, 0] [2, 1, 2] [0, 0, 1] [1, 1, 1] [2, 2, 1]
3 [2, 2, 1] [2, 1, 2] [0, 0, 1] [1, 1, 1] [2, 0, 2]
4 [2, 2, 0] [2, 0, 1] [0, 2, 0] [2, 2, 1] [2, 1, 2]
5 [1, 1, 0] [1, 0, 2] [2, 2, 0] [1, 1, 1] [2, 2, 1]
6 [1, 1, 0] [1, 0, 2] [0, 0, 1] [1, 1, 1] [2, 2, 1]
7 [1, 1, 0] [1, 0, 2] [2, 2, 1] [1, 1, 1] [0, 0, 1]
8 [2, 0, 2] [2, 2, 0] [0, 1, 2] [1, 1, 1] [2, 2, 1]
9 [2, 0, 2] [2, 2, 1] [0, 1, 2] [1, 1, 1] [1, 0, 2]

Table 1: Exceptional collections on the Kulikov surface

Remark 3.4 1. The precise number of exceptional collections is not important. Rather, the fact that we have definitively enumerated all exceptional collections of numerical type Λ, means that we can sift through the list to find one with the most desirable properties.

2. Let Λ′ be any translation of Λ under the Weyl group action of A_{1} × A_{2} on Pic Y. Then Λ′ is another numerical exceptional collection on X (see Section 3.1.7), so we may enumerate exceptional collections on X of numerical type Λ′. For the Kulikov surface, each element of the orbit corresponds to either 9, 14, 18 or 24 exceptional collections on X. Thus, the Weyl group action does not “lift” to X in a way which is compatible with the covering X → Y. On occasion, this incompatibility is used to our advantage (see [20]). We return to these exceptional collections in Section 4.

4 Heights of exceptional collections

Let X be a surface of general type with p_{g} = q = 0, Tors X ≠ 0 with an exceptional collection of line bundles 𝔼 = (E_{0}, …, E_{n−1}). Write E for the smallest full triangulated subcategory of D^{b}(X) containing 𝔼. In this section we calculate some invariants of 𝔼. The invariants we consider are essentially determined by the derived category, but we must enhance the derived category in order to make computations. For completeness, we discuss some background first.

4.1 Motivation from del Pezzo surfaces

Let Y be a del Pezzo surface and let 𝔼 be a strong exceptional collection of line bundles on Y. Recall that 𝔼 is strong if Ext^{k}(E_{i}, E_{j}) = 0 for all i, j and for all k > 0. We define the partial tilting bundle of 𝔼 to be T = ⨁_{i} E_{i}. Then the derived endomorphism ring Ext^{∗}(T, T) = ⨁_{i,j} Hom(E_{i}, E_{j}) is an associative algebra, and we have an equivalence of categories E ≅ D^{b}(mod- Ext^{∗}(T, T)) (see [16]). From now on, we assume that 𝔼 is an exceptional collection on a fake del Pezzo surface X, so that we do not have the luxury of choosing a strong exceptional collection. Instead, we recover E by studying the higher multiplications coming from the A_{∞}-algebra structure on Ext^{∗}(T, T).

4.2 Digression on dg-categories

We sketch the construction of a differential graded (or dg) enhancement D of D^{b}(X). Objects in D are the same as those in D^{b}(X), but morphisms Hom_{D}^{•}(F, G) form a chain complex, with differential d of degree +1. Composition of maps Hom_{D}^{•}(F, G) ⊗ Hom_{D}^{•}(G, H) → Hom_{D}^{•}(F, H) is a morphism of complexes (the Leibniz rule), and for any object F in D, we require d(id_{F}) = 0. For a precise definition of Hom_{D}^{•}(F, G), one could use the Čech complex, and we refer to [36] for details. The main point is that the cohomology of Hom_{D}^{•}(F, G) in degree k is Ext_{D^{b}(X)}^{k}(F, G), so in particular, we have H^{0}(Hom_{D}^{•}(F, G)) = Hom_{D^{b}(X)}(F, G).

4.3 Hochschild homology

We first compute some additive invariants, only making implicit use of the dgstructure. The Hochschild homology of X is given by the Hochschild–Kostant– Rosenberg isomorphism

HH_{k}(X) ≅ ⨁_{p} H^{p+k}(X, Ω_{X}^{p}),

so HH_{0}(X) = ℂ^{12−K^{2}} and HH_{k}(X) = 0 in all other degrees. Moreover, Hochschild homology is additive over semiorthogonal decompositions.

Theorem 4.1 [35] If D^{b}(X) = ⟨A, B⟩ is a semiorthogonal decomposition, then

HH_{k}(X) = HH_{k}(A) ⊕ HH_{k}(B).

Assuming the Bloch conjecture on algebraic zero-cycles, we have

K_{0}(X) = ℤ^{12−K^{2}} ⊕ Tors X,

and we note that K-theory is also additive over semiorthogonal decompositions (see Proposition 3.1). Now for an exceptional collection of length n, K_{0}(E) = ℤ^{n} and

HH_{k}(E) = \{ \begin{matrix}ℂ^{n} & if k = 0 \\ 0 & otherwise. \end{matrix}

Thus the maximal length of 𝔼 is at most 12 − K_{X}^{2}, and such an exceptional sequence of maximal length effects a semiorthogonal decomposition D^{b}(X) = ⟨E, A⟩ with nontrivial semiorthogonal complement A. We say that A is a quasiphantom category; by additivity, the Hochschild homology vanishes, but K_{0}(A) ⊇ Tors X ≠ 0, so A can not be trivial.

4.4 Height

The Hochschild cohomology groups of X may be computed via the other Hochschild–Kostant–Rosenberg isomorphism (cf. [35]):

HH^{k}(X) = ⨁_{p+q=k} H^{q}(X, Λ^{p}T_{X}).

Thus for a surface of general type with p_{g} = 0, we have

HH^{0}(X) ≅ H^{0}(O_{X}) = ℂ, HH^{1}(X) = 0, HH^{2}(X) ≅ H^{1}(T_{X}),
HH^{3}(X) ≅ H^{2}(T_{X}), HH^{4}(X) ≅ H^{0}(2K_{X}) = ℂ^{1+K^{2}}.

Recall that the degree two (respectively three) Hochschild cohomology is the tangent space (resp. obstruction space) to the formal deformations of a category [32]. In principle, [36] gives an algorithm for computing HH^{∗}(A) using a spectral sequence and the notion of height of an exceptional collection. Moreover, by [36, Prop. 6.1], for an exceptional collection to be full, its height must vanish. Thus the height may be used to prove existence of phantom categories without reference to the K-theory. We outline the algorithm of [36] below. Given an exceptional collection 𝔼 on X, there is a long exact sequence (induced by a distinguished triangle)

… → NHH^{k}(𝔼, X) → HH^{k}(X) → HH^{k}(A) → NHH^{k+1}(𝔼, X) → …

where NHH(𝔼, X) is the normal Hochschild cohomology of the exceptional collection 𝔼. The normal Hochschild cohomology can be computed using a spectral sequence with first page

E_{−p,q}^{1} = ⨁_{\begin{matrix}0≤a_{0}<···<a_{p}≤n−1 \\ k_{0}+···+k_{p}=q \end{matrix}} Ext^{k_{0}}(E_{a_{0}}, E_{a_{1}}) ⊗ ⋯

⋯ ⊗ Ext^{k_{p−1}}(E_{a_{p−1}}, E_{a_{p}}) ⊗ Ext^{k_{p}}(E_{a_{p}}, S^{−1}(E_{a_{0}})).

The spectral sequence relies on the dg-structure on D; the initial differentials d′ and d′′ are induced by the differential on D and the composition map respectively, while the higher differentials are related to the A_{∞}-algebra structure on Ext-groups, (see Section 4.6). The existing examples of exceptional collections on surfaces of general type with p_{g} = 0 suggest that NHH^{k}(𝔼, X) vanishes for small k. Thus the height h(𝔼) of an exceptional collection 𝔼 = (E_{0}, …, E_{n−1}) is defined to be the smallest integer m for which NHH^{m}(𝔼, X) is nonzero. Alternatively, m is the largest integer such that the canonical restriction morphism HH^{k}(X) → HH^{k}(A) is an isomorphism for all k ≤ m − 2 and injective for k = m − 1.

4.5 Pseudoheight

The height may be rather difficult to compute in practice, requiring a careful analysis of the Ext-groups of 𝔼 and the maps in the spectral sequence. The pseudoheight is easier to compute and sometimes gives a good lower bound for the height.

Definition 4.1 The pseudoheight ph(𝔼) of an exceptional collection 𝔼 = (E_{0}, …, E_{n−1}) is

ph(𝔼) = min_{0≤a_{0}<···<a_{p}≤n−1} (e(E_{a_{0}}, E_{a_{1}}) + ⋯
+ e(E_{a_{p−1}}, E_{a_{p}}) + e(E_{a_{p}}, E_{a_{0}}(−K_{X})) − p + 2),

where e(F, F′) = min{i : Ext^{i}(F, F′) ≠ 0}.

The pseudoheight is just the total degree of the first nonzero term in the first page of the spectral sequence, where the shift by 2 takes care of the Serre functor. Consider the length 2n anticanonical extension of the sequence 𝔼 (see also Section 3.1.7):

E_{0}, …, E_{n−1}, E_{n} = E_{0}(−K_{X}), …, E_{2n−1} = E_{n−1}(−K_{X}).  (12)

If the E_{i} are line bundles, then we have a numerical lower bound for the pseudoheight.

Lemma 4.1 [36, Lem. 4.10, Lem. 5.1] If K_{X} is ample and E_{i} · K_{X} ≥ E_{j} · K_{X} for all i < j and for all E_{i}, E_{j} in the anticanonically extended sequence (12), then ph(𝔼) ≥ 3.

The numerical conditions required by the Lemma are not particularly stringent. For example, all the exceptional collections we have exhibited on the Kulikov surface in Section 3.2 have pseudoheight at least 3, even before we consider the Ext-groups more carefully.

Remark 4.1 If L is a line bundle, then dim Ext^{k}(L, L(−K_{X})) = h^{2−k}(2K_{X}) by Serre duality, which is the case p = 0 in Definition 4.1. Thus any exceptional collection of line bundles on a surface of general type with p_{g} = 0 has pseudoheight at most 4. Moreover, if ph(𝔼) = 4, then h(𝔼) = 4 by [36].

4.6 The A_{∞}-algebra of an exceptional collection

Let 𝔼 = (E_{0}, …, E_{n−1}) be an exceptional collection on X, and define T = ⊕_{i=0}^{n−1}E_{i}. Then B = Hom_{D}^{•}(T, T) is a differential graded algebra via the dgstructure on D (see Section 4.2). It can be difficult to compute the dg-algebra structure on B directly, so we pass to the A_{∞}-algebra H^{∗}B. We discuss A_{∞}-algebras, referring to [29] for details and further references. An A_{∞}-algebra is a graded vector space A = ⨁_{p∈ℤ} A^{p}, together with graded multiplication maps m_{n}: A^{⊗n} → A of degree 2 − n, for each n ≥ 1. These multiplication maps satisfy an infinite sequence of relations, starting with

m_{1}m_{1} = 0,
m_{1}m_{2} = m_{2}(m_{1} ⊗ id_{A} + id_{A} ⊗ m_{1}).

These first two relations ensure that m_{1} is a differential on A, satisfying the Leibniz rule with respect to m_{2}. The third relation is

m_{2}(id_{A} ⊗ m_{2} − m_{2} ⊗ id_{A}) =
m_{1}m_{3} + m_{3}(m_{1} ⊗ id_{A} ⊗ id_{A} + id_{A} ⊗ m_{1} ⊗ id_{A} + id_{A} ⊗ id_{A} ⊗ m_{1}),

which shows that m_{2} is not associative in general, but if m_{n} = 0 for all n ≥ 3, then A is an ordinary associative differential graded algebra. In fact, by the above discussion, we can view B as an A_{∞}-algebra, with m_{1} being the differential, m_{2} the multiplication, and m_{n} = 0 for n ≥ 3. By a theorem of Kadeishvili (cf. [29]), the homology H^{∗}B = H^{∗}(B, m_{1}) has a canonical A_{∞}-algebra structure, for which m_{1} = 0, m_{2} is induced by the multiplication on B, and H^{∗}B and B are quasi-isomorphic as A_{∞}-algebras. This canonical A_{∞}-structure is unique, and H^{∗}B is called a minimal model for B. We say that B is formal if it has a minimal model H^{∗}B for which m_{n} = 0 for all n ≥ 3, so that H^{∗}B is just an associative graded algebra. The A_{∞}-algebra of 𝔼 is

H^{∗}B = Ext^{∗}(T, T) = ⨁_{k} ⨁_{0≤i,j≤n−1} Ext^{k}(E_{i}, E_{j}),

and m_{2} coincides with the Yoneda product on Ext-groups. Clearly, if the exceptional collection 𝔼 consists of sheaves, then H^{∗}B has only three nontrivial graded summands, in degrees 0, 1 and 2. Since m_{n} has degree 2 − n, the summands of degree 0 and 1 are crucial in determining the A_{∞}-algebra structure.

4.6.1 Recovering E from H^{∗}B

According to [16], [30], the subcategory E of D generated by the exceptional collection 𝔼 is equivalent to the triangulated subcategory Perf(B) ⊂ D^{b}(mod-B) of perfect objects over the dg-algebra B. A perfect object is a differential graded B-module that is quasi-isomorphic to a bounded chain complex of projective and finitely generated modules. As mentioned above, it is preferable to consider the A_{∞}-algebra H^{∗}B instead, noting that E is in turn equivalent to the triangulated category of perfect A_{∞}-modules over H^{∗}B. If B is formal, the equivalence reduces to E ≅ D^{b}(mod-H^{∗}B), which should be compared with Section 4.1. We search for exceptional collections whose Hom- and Ext^{1}-groups are mostly zero. In good cases, this implies that B is formal, and H^{∗}B has no deformations. It then follows that E is rigid, i.e. constant in families.

4.7 Quasiphantoms on the Kulikov surface

We study some properties of the exceptional collections on the Kulikov surface from Section 3.2. For the purposes of the discussion, we fix the following exceptional collection

𝔼: O, L_{1}[2, 2, 0], L_{2}[2, 1, 2], L_{3}[0, 0, 1], L_{4}[1, 1, 1], L_{5}[2, 2, 1],

which can be found in the second row of Table 1 in Section 3.2. Using Theorem 2.1, we may compute the Ext-groups of the extended sequence (12). We present the results in Table 2 below. The ijth entry of the table is the following formal polynomial in q

∑_{k∈ℤ} dim Ext^{k}(E_{i}, E_{i+j})q^{k},

where 0 ≤ i, j ≤ 5, and the zigzag delineates those entries whose target E_{i+j} is in the anticanonically extended part of (12).

Lemma 4.2 The only nonzero Ext^{1}-groups are Ext^{1}(E_{1}, E_{4}) which is 2dimensional, and Ext^{1}(E_{1}, E_{5}) which is 1-dimensional. □

Remark 4.2 The lemma shows that 𝔼 does not have 3-block structure. A 3block structure means the exceptional collection can be split into three mutually orthogonal blocks (cf. [28]). In fact, every exceptional collection in Table 1, and every exceptional collection in the Weyl group orbit (cf. Section 3.1.7), has some non-zero Ext^{1}-groups. This is in contrast with the exceptional collections on the Burniat surface exhibited in [2], which are of the same numerical type, and have 3-block structure.

0 1 2  3  4  5
0 1 2q^{2} 2q^{2}  2q^{2}  3q^{2} 3q^{2}
1 1 0 0 2q + 3q^{2} q + 2q^{2} 4q^{2}
2 1 0 q^{2}  q^{2}  4q^{2} 6q^{2}
3 1 q^{2} q^{2}  4q^{2}  6q^{2} 6q^{2}
4 1 0 3q^{2}  5q^{2}  5q^{2} 5q^{2}
5 1 3q^{2} 5q^{2}  5q^{2}  5q^{2} 6q^{2}

Table 2: Ext-table of an exceptional collection on the Kulikov surface

Proposition 4.1 The A_{∞}-algebra of 𝔼 is formal, and the product m_{2} of any two elements with strictly positive degree is trivial.

Proof The A_{∞}-algebra H^{∗}B of 𝔼, is the direct sum of all Ext-groups appearing above the zigzag in the table. By [46, Lemma 2.1] or [38, Theorem 3.2.1.1], we may assume that m_{n}(…, id_{E_{i}}, …) = 0 for all E_{i} and all n > 2. We show that every product m_{3} must be zero for degree reasons. By Lemma 4.2, there are only two nonzero arrows in degree 1, and they can not be composed with one another, since they have the same source. Thus the product m_{3} of any 3 composable elements of H^{∗}B has degree at least deg m_{3} + 1 + 2 + 2 = 4, and is therefore identically zero, because the graded piece H^{4}B is trivial. The same argument applies for all products m_{n} with n ≥ 3. Thus H^{∗}B is a formal A_{∞}-algebra. In fact, we see from the table that any product m_{2} of two elements of nonzero degree also vanishes for degree reasons. □

Moreover, we calculate the Hochschild cohomology of A using heights.

Proposition 4.2 We have HH^{0}(A) = ℂ, HH^{1}(A) = 0, HH^{2}(A) = ℂ, and HH^{3}(A) contains a copy of ℂ^{3}.

Proof The pseudoheight of 𝔼 may also be computed from the table, where now we also need the portion below the zigzag. The minimal contribution to the pseudoheight is achieved by incorporating one of the nonzero Ext^{1}-groups. For example,

e(E_{1}, E_{4}) + e(E_{4}, E_{1} ⊗ ω_{X}) − 1 + 2 = 1 + 2 − 1 + 2 = 4,

so ph(𝔼) = 4. In this case, by [36], the height and pseudoheight are equal. Hence HH^{k}(A) = HH^{k}(X) for k ≤ 2, and HH^{3}(A) ⊃ HH^{3}(X). By the Hochschild–Kostant–Rosenberg isomorphism, the dimensions of HH^{k}(X) follow from the infinitesimal deformation theory of the Kulikov surface, which was studied in [19]: H^{1}(T_{X}) = 1 and H^{2}(T_{X}) = 3. □

In summary, we have Theorem 4.2 Every Kulikov surface X has a semiorthogonal decomposition

D^{b}(X) = ⟨E, A⟩

where E is generated by the exceptional collection 𝔼, and E is rigid, i.e. E does not vary with X. The semiorthogonal complement A is a quasiphantom category whose formal deformation space is isomorphic to that of D^{b}(X), and therefore X may be reconstructed from A.

5 Secondary Burniat surfaces and effective divisors

Burniat surfaces were discovered in [18], and an alternate construction is given in [27]. There are several cases X_{k}, with K^{2} = k for 2 ≤ k ≤ 6. For details we refer to [44], [4]. Exceptional collections on primary Burniat surfaces X_{6} with K^{2} = 6 were first constructed and studied in [2], where two 3-block exceptional collections are exhibited. Burniat surfaces with K^{2} = 3, 4, 5, 6 can be constructed as abelian covers satisfying assumptions (A), and so we are able to enumerate exceptional collections on all these Burniat surfaces. We do not reproduce these computations here, but see [20]. Exceptional collections of line bundles of maximal length on the Burniat–Campedelli surface X_{2} with K^{2} = 2 remain elusive, because this surface does not satisfy assumption (A1). In computing exceptional collections on fake del Pezzo surfaces, it becomes clear that a characterisation of effective line bundles is very useful. In this section we prove the following theorem for the secondary nodal Burniat surface.

Theorem 5.1 Let X be a nodal secondary Burniat surface with K^{2} = 4. Then the semigroup of effective divisors on X is generated by the reduced pullbacks of irreducible components of the branch divisor, together with the pullbacks E_{4}, E_{5} of two (−1)-curves on Y.

With appropriate changes, the same proof works for the other surfaces satisfying assumptions (A). Indeed, Theorem 3.2 above for the Kulikov surface is an easier case of this result. The additional complexity here arises from two sources: some of the exceptional curves on Y are not branch divisors, and there is a (−2)-curve.

5.1 Burniat surfaces revisited

We first describe the nodal secondary Burniat line configuration. Take the three coordinate points P_{1}, P_{2}, P_{3} in ℙ^{2}, and label the edges \overline{A}_{0} = P_{1}P_{2}, \overline{B}_{0} = P_{2}P_{3}, \overline{C}_{0} = P_{3}P_{1}. Then let \overline{A}_{1}, \overline{A}_{2} (respectively \overline{B}_{i}, \overline{C}_{i}) be two lines passing through P_{1} (resp. P_{2}, P_{3}). We require that \overline{A}_{1}, \overline{B}_{1}, \overline{C}_{2} are concurrent in P_{4} (respectively \overline{A}_{1}, \overline{B}_{2}, \overline{C}_{1} in P_{5}). This gives nine lines in total, four passing through each of P_{1}, P_{2}, P_{3} and three passing through each of P_{4}, P_{5}. Moreover, \overline{A}_{1} passes through three triple points. Blow up the five points P_{i} to obtain a weak del Pezzo surface Y of degree 4. The strict transforms of these nine lines (for which we use the same labels) together with the three exceptional curves \overline{E}_{i} for i = 1, 2, 3, are called the nodal secondary Burniat configuration (see Figure 2).

[figure: ^{\overline{A}^{1}}; ^{\overline{A}^{2}}; ^{\overline{B}^{1}}; ^{\overline{B}^{2}}; _{\overline{C}_{2}}; _{\overline{C}_{1}}; P_{1}; P_{2}; P_{3}; P_{4}; P_{5}; ^{\overline{A}^{1}}; ^{\overline{A}^{2}}; ^{\overline{B}^{1}}; ^{\overline{B}^{2}}; _{\overline{C}_{2}}; _{\overline{C}_{1}}; P_{1}; P_{2}; P_{3}; P_{4}; P_{5}]

Figure 2: The secondary Burniat configurations with K^{2} = 4 (nodal configuration is on the right)

The nodal secondary Burniat surface X_{4} with K_{X}^{2} = 4 is a (ℤ/2)^{2}-cover of Y branched in the configuration of Figure 2, and X_{4} is a surface of general type with p_{g} = 0, K^{2} = 4 and Tors X = (ℤ/2)^{4}. The cover is not ramified in \overline{E}_{4}, \overline{E}_{5}. The weak del Pezzo surface Y has a (−2)-curve, \overline{A}_{1} and the canonical model of X_{4} is a (ℤ/2)^{2}-cover of a nodal quartic. Following the description of [5, 6], the nodal secondary Burniat surfaces form an irreducible closed family, inside a 3-dimensional irreducible connected component of the moduli space. This component is given by the union of the family of nodal secondary Burniat surfaces with the extended secondary Burniat surfaces, which form an open subset. We do not directly consider extended Burniat surfaces here. In Appendix B, we show that the secondary Burniat surface satisfies assumptions (A). More precisely, we exhibit an explicit basis e_{0}, …, e_{5} for Pic X/Tors X, in terms of reduced pullbacks of irreducible branch divisors. The appendix also lists coordinates for the reduced pullback of each irreducible component of the branch divisor, according to Definition 3.4. We define 𝔈 to be the semigroup generated by the reduced pullbacks A_{0}, …, C_{2}, E_{1}, E_{2}, E_{3} together with ordinary pullbacks E_{4}, E_{5}. There is a multigrading on 𝔈 by multidegree in Pic X/Tors X, and we write 𝔈(d) for graded piece of multidegree d. Using the coordinates from Appendix B, we define a homomorphism t: 𝔈 → Tors X, sending each generator of 𝔈 to its associated torsion twist. Remember that t(E_{4}) = t(E_{5}) = 0 because these are pulled back from Y.

5.2 Proof of Theorem 5.1.

The stategy of proof is similar to that of Alexeev, [1], but for completeness, we outline the whole proof. The main differences are the (−2)-curve on Y and the (−1)-curves which are not branch divisors. These introduce new complications which are not present in [1]. We are able to resolve these issues because we can use the pushforward formula and Appendix B to check effectivity in a systematic manner.

Suppose D is an effective divisor and C is an effective curve class on X for which D · C < 0. Then C^{2} < 0 and C is in the base locus of D, so we define D′ = D−aC where a is the smallest positive integer for which (D−aC)·C ≥ 0. In this way, we can reduce an effective divisor on X to one which has positive intersection with all curve classes in 𝔈. Such divisor classes form a rational polyhedral cone P in N^{1}(X, ℝ). To describe the generators of P in the most geometric way, we first construct certain divisor classes on X in terms of reduced pullbacks and the birational transformations the del Pezzo surface Y. Suppose we take a standard Cremona transformation of ℙ^{2} centred on any three non-collinear triple points P_{i}, P_{j} and P_{k}. The numerical class of the hyperplane section of the image ℙ^{2} is h_{ijk} = 2e_{0} − e_{i} − e_{j} − e_{k} for any {i, j, k} ≠ {1, 4, 5}, or h_{0} = e_{0}. There are also natural fibrations on Y which arise from the pencil of hyperplanes passing through a fixed P_{k} on some copy of ℙ^{2}. The numerical classes of these fibrations are denoted f_{i} = e_{0}−e_{i} or f_{ijkl} = 2e_{0}−e_{i}−e_{j}−e_{k}−e_{l} with {1, 4, 5} ⊄ {i, j, k, l}.

Lemma 5.1 The polyhedron P is generated by the ten hyperplane classes h_{0}, h_{ijk} and eight fibrations f_{i}, f_{ijkl} defined above, together with the four additional classes

g_{1} = 3e_{0} − e_{1} − 2e_{2} − e_{3} − e_{4} − e_{5}, g_{2} = 3e_{0} − e_{1} − e_{2} − 2e_{3} − e_{4} − e_{5},
g_{3} = 3e_{0} − e_{1} − 2e_{2} − e_{4} − e_{5},  g_{4} = 3e_{0} − e_{1} − 2e_{3} − e_{4} − e_{5}.

Proof Any generator D of 𝔈 determines a linear function ·D, which in turn defines a collection of hyperplanes supporting the polyhedron P. We use the computer [20] to calculate the integral generators of the cone. We examine the additional generators. The class g_{1} is the hyperplane section of the copy of ℙ^{2} obtained by contracting \overline{A}_{0}, \overline{A}_{1}, \overline{B}_{0}, \overline{B}_{1} and \overline{B}_{2} on Y, and g_{3} is the hyperplane section of the quadric cone given by contracting \overline{A}_{0}, \overline{A}_{1}, \overline{E}_{3}, \overline{B}_{1} and \overline{B}_{2}. There are similar descriptions of g_{2} and g_{4}. □

Lemma 5.2 Suppose D is an effective divisor on X with K_{X} · D ≤ 4. Then D is in 𝔈.

Proof We may assume that D is in P. This is a finite (and small) number of classes to check, and we do this directly using the computer implementation [20] of our pushforward formula Theorem 2.1. □

Proposition 5.1 Suppose D is an effective divisor on X with K_{X} ·D > 4 and χ(D) > 0. Then D is in 𝔈.

Proof Since K_{X} · D > 4 we have that (K_{X} − D) · K_{X} < 0 and so K_{X} − D can not be effective. By Serre duality, h^{2}(D) = h^{0}(K_{X} − D) = 0, hence D is effective.

Choose D̄ in Pic Y such that the numerical class of K_{Y} + D̄ in Pic Y is the same as that of D − K_{X} in Pic X/Tors X by assumption (A). Then

χ(K_{Y} + D̄) = 1 + \frac{1}{2}(K_{Y} + D̄)D̄ = 1 + \frac{1}{2}(D − K_{X})D = χ(D) > 0.

Moreover, h^{2}(K_{Y} + D̄) = h^{0}(−D̄) = 0 because −K_{Y} · −D̄ < 0, so by the same argument as above, we see that K_{Y} + D̄ is effective on Y. Now, any effective divisor on Y is a positive linear combination of branch divisors \overline{A}_{0}, …, \overline{C}_{2}, \overline{E}_{1}, \overline{E}_{2}, \overline{E}_{3} and exceptional curves \overline{E}_{4} and \overline{E}_{5}. So taking the reduced pullback, we get the following expression for the numerical class of D in Pic X/Tors X:

D = K_{X} + (combination of A_{0}, …, E_{3}) + \frac{1}{2}(combination of E_{4}, E_{5}).

The coefficient of \frac{1}{2} appears because \overline{E}_{4} and \overline{E}_{5} are not branch divisors. It remains to show that D + τ is in 𝔈 for any τ such that D + τ is effective. This is implied by the following lemma:

Lemma 5.3 (1) Let L be any of the following line bundles on X:

O_{X}(K_{X} + γ) ⊗ τ, O_{X}(K_{X} + \frac{1}{2}E_{4}) ⊗ τ, O_{X}(K_{X} + \frac{1}{2}E_{5}) ⊗ τ, or

O_{X}(K_{X} + \frac{1}{2}(E_{4} + E_{5})) ⊗ τ

where γ is any generator of 𝔈 and τ is any element of Tors X. Then L is effective and in 𝔈 unless L = O_{X}(K_{X} + A_{1}).

(2) The line bundles L = O_{X}(K_{X} + kA_{1}) are not effective for any k > 0.

Proof

(1) Suppose L = O_{X}(K_{X} + A_{0}) ⊗ τ, and take the graded piece of 𝔈 with multidegree d = (4, −2, −2, −1, −1, −1). We use the computer [20] to check that the image of 𝔈(d) under t is all of Tors X. This proves that L is effective and in 𝔈 for any τ. The same computation works for all multidegrees listed in the statement, except when L = O_{X}(K_{X} + A_{1}) ⊗ τ, for which we refer to the proof of part (2).

(2) When L = O_{X}(K_{X} +A_{1})⊗τ, the same computation as above shows that the image of 𝔈(4, −2, −1, −1, −2, −2) under t is Tors X − {[1, 0, 0, 0]}. Thus O_{X}(K_{X} + A_{1}) is not in 𝔈. Indeed, the pushforward is

φ_{∗}L = O_{Y} (e_{2} −e_{1})⊕O_{Y} (e_{3} −e_{2})⊕O_{Y} (−2e_{0} +e_{2} +e_{3})⊕O_{Y} (e_{0} −e_{3} −e_{4} −e_{5}),

which is not effective. Moreover, by the projection formula, we have

φ_{∗}L(2mA_{2}) = φ_{∗}L ⊗ O_{Y} (m\overline{A}_{2}) = φ_{∗}L ⊗ O_{Y} (m(e_{0} − e_{1} − e_{4} − e_{5})),

which is not effective for any m, and so O_{X}(K_{X} + kA_{2}) is not effective for any odd k = 1 + 2m. For even k, the proof is similar, starting from φ_{∗}O_{X}(K_{X}). □

Remark 5.1 Since A_{2} is a (−2)-curve, we have K_{X} · (K_{X} + kA_{1}) = K_{X}^{2} = 4 for all k. Thus we do not need part (2) of the above lemma, because K_{X} + kA_{1} does not satisfy the assumptions of Proposition 5.1.

Finally, we take care of the cases with χ(D) ≤ 0.

Lemma 5.4 Suppose D is an effective divisor on X with numerical class in P and χ(D) ≤ 0. Then D is in one of the following classes:

(1) h or 2h for any hyperplane generator h;

(2) g or 2g_{1} or 2g_{3}, where g refers to any of the additional generators described in Lemma 5.1;

(3) nf, nf + f′, nf + h, nf + g for any n ≥ 1 where f is a fibration and f′ is another fibration with intersection f · f′ = 1, h is a hyperplane generator with f · h = 1, g is an additional generator with f · g = 1.

Proof This is a systematic induction. We note that each generator γ of P has χ(γ) = 0. Moreover, if D = D_{1} +D_{2} then χ(D) = χ(D_{1})+χ(D_{2})+D_{1} ·D_{2} −1. So for example, starting from f_{1}, we choose another fibration generator f′. Either f_{1} · f′ = 0, in which case f_{1} = f′ and χ(2f_{1}) = −1, or f_{1} · f′ = 1, so that χ(f_{1} + f′) = 0. Now adding a further generator γ to f_{1} + f′ yields χ(f_{1} +f′ +γ) > 0 by simple consideration of the intersection numbers, unless γ is one of f_{1} or f′. We continue in this way, to produce the list of possibilities.□

Lemma 5.5 Suppose L is an effective line bundle with numerical class one of the exceptional cases from Lemma 5.4. Then L is in 𝔈.

Proof We give a proof for nf_{1}. The other possibilities listed in Lemma 5.4(3) work in the same way, and cases (1) and (2) can be checked by a direct computation [20]. As in the proof of Lemma 5.3, we split into even and odd cases and make use of the projection formula. Let L = O_{X}(2f_{1})⊗τ for some τ in the image of t(𝔈(2f_{1})), so that in particular, L is effective. Since C_{0} +E_{3} is a section of O_{X}(f_{1}), it follows that O_{X}(nf_{1})⊗τ is effective and in 𝔈 for any n ≥ 2. Now suppose τ is any torsion element in Tors X − t(𝔈(2f_{1})), so that L is not in 𝔈. For example, τ = [0, 0, 0, 1]. Then we compute

φ_{∗}L = O_{Y} (0, −1, 0, 0, 0, 1) ⊕ O_{Y} (−2, 1, 1, 1, 1, 1)

⊕ O_{Y} (−1, −1, 1, 1, 1, 0) ⊕ O_{Y} (−2, 0, 1, 1, 1, 1),

which is clearly not effective. Moreover, by the projection formula, we see that φ_{∗}L ⊗ O_{X}(2mf_{1}) = φ_{∗}L ⊗ O_{Y} (mf_{1}) is not effective for any m either, for degree reasons. This completes the proof for any even multiple of f_{1}. A similar computation proves the odd case, starting from 3f_{1}. □

A Appendix: Acyclic bundles on the Kulikov surface

For reference, here are the acyclic line bundles on the Kulikov surface used in Section 3.2.

L  A(L)
L_{1}^{−1}  [0, 0, 0], [0, 1, 0], [1, 1, 0], [2, 1, 0], [2, 2, 0], [1, 0, 1], [2, 0, 1], [0, 1, 1],

[1, 1, 1], [0, 2, 1], [2, 2, 1], [0, 1, 2], [1, 1, 2], [0, 2, 2]

L_{2}^{−1}  [0, 1, 0], [1, 1, 0], [2, 2, 0], [2, 0, 1], [0, 1, 1], [1, 1, 1], [2, 1, 1], [1, 2, 1],

[2, 2, 1], [0, 0, 2], [1, 0, 2], [0, 1, 2], [1, 1, 2], [1, 2, 2]

L_{3}^{−1}  [0, 1, 0], [1, 1, 0], [1, 0, 1], [0, 1, 1], [1, 1, 1], [0, 2, 1], [1, 2, 1], [0, 0, 2],

[2, 0, 2], [0, 1, 2], [1, 1, 2], [2, 1, 2], [0, 2, 2], [1, 2, 2]

L_{4}^{−1}  [0, 0, 0], [0, 1, 0], [2, 1, 0], [0, 2, 0], [2, 2, 0], [1, 0, 1], [2, 0, 1], [0, 1, 1],

[1, 1, 1], [2, 1, 1], [0, 2, 1], [2, 2, 1], [1, 1, 2], [0, 2, 2], [2, 2, 2]

L_{5}^{−1}  [0, 1, 0], [1, 1, 0], [2, 2, 0], [1, 0, 1], [2, 0, 1], [0, 1, 1], [1, 1, 1], [0, 2, 1],

[1, 2, 1], [2, 2, 1], [0, 0, 2], [0, 1, 2], [1, 1, 2], [0, 2, 2], [1, 2, 2]

L_{2}^{−1} ⊗ L_{1} [1, 0, 0], [2, 0, 0], [2, 1, 0], [0, 1, 1], [0, 1, 2], [2, 1, 2], [0, 2, 2]
L_{3}^{−1} ⊗ L_{1} [0, 0, 0], [1, 0, 0], [2, 0, 0], [1, 1, 0], [2, 1, 0], [2, 2, 0], [1, 1, 2], [2, 1, 2],

[2, 2, 2]

L_{4}^{−1} ⊗ L_{1} [0, 1, 0], [1, 1, 0], [0, 1, 1], [1, 1, 1], [1, 2, 1], [0, 0, 2], [1, 0, 2], [2, 0, 2],

[0, 1, 2], [1, 1, 2], [1, 2, 2]

L_{5}^{−1} ⊗ L_{1} [1, 0, 0], [2, 0, 0], [1, 1, 0], [2, 1, 0], [2, 2, 0], [0, 1, 1], [0, 0, 2], [0, 1, 2],

[1, 1, 2], [2, 1, 2], [0, 2, 2], [2, 2, 2]

L_{3}^{−1} ⊗ L_{2} [1, 0, 1], [1, 1, 1], [2, 1, 1], [2, 0, 2], [1, 1, 2], [2, 1, 2], [1, 2, 2]
L_{4}^{−1} ⊗ L_{2} [0, 0, 0], [0, 1, 0], [1, 1, 0], [1, 0, 1], [0, 1, 1], [1, 1, 1], [0, 2, 1], [2, 0, 2],

[0, 1, 2], [1, 1, 2], [0, 2, 2]

L_{5}^{−1} ⊗ L_{2} [0, 1, 0], [1, 0, 1], [0, 1, 1], [1, 1, 1], [2, 1, 1], [0, 2, 1], [0, 0, 2], [2, 0, 2],

[1, 1, 2], [2, 1, 2], [0, 2, 2], [1, 2, 2]

L_{4}^{−1} ⊗ L_{3} [0, 0, 0], [0, 1, 0], [1, 1, 0], [2, 2, 0], [2, 0, 1], [0, 1, 1], [1, 1, 1], [2, 2, 1],

[1, 0, 2], [0, 1, 2], [1, 1, 2]

L_{5}^{−1} ⊗ L_{3} [0, 1, 0], [1, 1, 0], [2, 1, 0], [2, 2, 0], [2, 0, 1], [1, 1, 1], [2, 1, 1], [1, 2, 1],

[0, 0, 2], [1, 0, 2], [0, 1, 2], [1, 2, 2]

L_{5}^{−1} ⊗ L_{4} [1, 0, 0], [2, 0, 0], [1, 1, 0], [2, 2, 0], [0, 0, 2], [0, 1, 2], [2, 1, 2], [2, 2, 2]

B Appendix: Nodal Secondary Burniat surface with K^{2} = 4

The maps Ψ_{4}, Ψ_{4}^{n}: H_{1}(Y − Δ, ℤ) → (ℤ/2)^{6} determining respectively the nonnodal and nodal Burniat surfaces, differ from one another slightly. We tabulate them below.

Γ  \overline{A}_{0} \overline{A}_{1} \overline{A}_{2} \overline{B}_{0} \overline{B}_{1} \overline{B}_{2}  \overline{C}_{0}  \overline{C}_{1}  \overline{C}_{2}
Ψ_{4}(Γ) − Φ(Γ) 0 g_{3} g_{4} 0 g_{5} g_{6}  0  g_{4} + g_{6} g_{3} + g_{5}
Ψ_{4}^{n}(Γ) − Φ(Γ) 0 g_{3} g_{4} 0 g_{5} g_{6} g_{3} + g_{4} g_{3} + g_{6} g_{3} + g_{5}

The restriction imposed by P_{5} is Ψ_{4}(\overline{A}_{2} + \overline{B}_{2} + \overline{C}_{1}) = 0 in the non-nodal case, and Ψ_{4}^{n}(\overline{A}_{1} + \overline{B}_{2} + \overline{C}_{1}) = 0 in the nodal case. Either way, g_{7} is eliminated, so the torsion group is (ℤ/2)^{4}, generated by g_{3}^{∗}, …, g_{6}^{∗}. We extend the basis chosen for the free part of Pic(X_{5}). The basis is the same for non-nodal and nodal surfaces

e_{0} = C_{0} + E_{1} + E_{3}, e_{1} = E_{1}, e_{2} = E_{2}, e_{3} = E_{3},
e_{4} = C_{0} − C_{2} + E_{1}, e_{5} = B_{0} − B_{2} + E_{3}.

Coordinates for non-nodal surface:

Multidegree  Torsion
O_{X}(A_{0}) 1 −1 −1 0 0 0 [1, 1, 0, 0]
O_{X}(A_{1}) 1 −1 0 0 −1 0 [1, 0, 0, 0]
O_{X}(A_{2}) 1 −1 0 0 0 −1 [0, 1, 1, 0]
O_{X}(B_{0}) 1 0 −1 −1 0 0 [0, 0, 1, 1]
O_{X}(B_{1}) 1 0 −1 0 −1 0 [0, 0, 1, 0]
O_{X}(B_{2}) 1 0 −1 0 0 −1 [0, 0, 1, 1]
O_{X}(C_{0}) 1 −1 0 −1 0 0  0
O_{X}(C_{1}) 1 0 0 −1 0 −1 [0, 0, 1, 0]
O_{X}(C_{2}) 1 0 0 −1 −1 0  0

Coordinates for nodal surface are the same (with same multidegrees) except for the following:

Multidegree  Torsion
O_{X}(A_{1}) 1 −1 0 0 −1 −1 [1, 0, 1, 0]
O_{X}(A_{2}) 1 −1 0 0 0 0 [0, 1, 0, 0]

In both cases, O_{X}(K_{X}) = O(3, −1, −1, −1, −1, −1)[0, 0, 1, 0].

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Former: Department of Mathematics and Statistics Lederle Graduate Research Tower University of Massachusetts Amherst, MA 01003-9305 coughlan@math.umass.edu Current: Institute of Algebraic Geometry, Leibniz Universität Hannover Welfengarten 1 30167 Hannover, Germany coughlan@math.uni-hannover.de 1292

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