AN EXAMPLE OF A SURFACE OF GENERAL TYPE WITH p_{g} = q = 2 AND K_{X}^{2} = 5

JUNGKAI A. CHEN AND CHRISTOPHER D. HACON

Abstract. We give an example of a minimal complex surface of general type with p_{g} = q = 2 and K_{X}^{2} = 5.

1. Introduction

Recently, there has been considerable interest in understanding the geometry of irregular complex projective surfaces with χ(X, ω_{X}) = 1 + p_{g}(X) − q(X) = 1, and in particular of surfaces with p_{g} = q = 2. Let X be a smooth minimal complex surface of general type. If χ(X, ω_{X}) = 1, then one has the bound 1 ≤ K_{X}^{2} ≤ 9. If, in addition the surface is irregular, i.e. q(X) = h^{0}(X, Ω_{X}^{1}) > 0, then one also has K_{X}^{2} ≥ 2p_{g}(X) and so p_{g}(X) ≤ 4. In [Be], it is shown that the case p_{g} = q = 4 corresponds to the product of two curves of genus 2. In [HP] and [Pi], surfaces with p_{g} = q = 3 are completely classified. When K_{X}^{2} = 2p_{g}(X) = 6 they are symmetric products of curves of genus 3 and when K_{X}^{2} = 8 they admit an irrational pencil. The case p_{g} = q = 2 seems considerably more delicate. At any rate Catanese suggests that, analogously to the p_{g} = q = 3 case, a surface of general type with p_{g} = q = 2 and with no fibration over an elliptic curve, is a degree 2 covering of a principally polarized abelian surface (A, Θ) branched along a divisor in the linear series |2Θ| (cf. [Zu]).

In [Zu], Zucconi has classified surfaces of general type with p_{g} = q = 2 which admit an irrational pencil. In [Ma], Manetti shows that a minimal surface of general type with K_{X} ample and K_{X}^{2} = 4, is a degree 2 covering of a principally polarized abelian surface (A, Θ) branched along a divisor D ∈ |2Θ|. Ciliberto and Mendes Lopes [CM], conjecture that this should be the case for any minimal surface of general type with p_{g} = q = 2 and K_{X}^{2} = 4.

The purpose of this note is to give a counter-example to Catanese’s conjecture above. The example we construct is birational to a triple cover of an abelian surface. Its canonical divisor K_{X} is ample, p_{g} = q = 2 and K_{X}^{2} = 5. The construction is motivated in §3, where we obtain restrictions on the structure of the sheaf alb_{X,∗}(ω_{X}). Acknowledgments We are indebted to Rita Pardini for useful conversations on this subject. The first author was supported by NSC 39151F, National Science Council, Taiwan. This work was completed during a visit of the first author to the University of California, Riverside. The first author would like to thank the University of California for its hospitality.

2000 MSC. 14C20, 14K12

2. Construction and Verification

We will need some results from the theory of Mukai transforms. Let Â be the dual abelian variety of A and P be the normalized Poincaré line bundle on A × Â. Following [Muk], define the functor Ŝ of O_{A}- modules into the category of OÂ-modules by

Ŝ(M) = π_{Â,∗}(P⊗π_{A}^{∗}M).

The derived functor RŜ of Ŝ then induces an equivalence of categories between the two derived categories D(A) and D(Â). More precisely, by [Muk]: There are isomorphisms of functors:

RS ∘ RŜ ≅ (−1_{A})^{∗}[−g]

and

RŜ ∘ RS ≅ (−1Â)^{∗}[−g],

where [−g] denotes ”shift the complex g places to the right”. The Weak Index Theorem (W.I.T.) holds for a coherent sheaf F on A if there exists an integer i(F) such that for all j ≠ i(F), one has R^{j}Ŝ(F) = 0. The coherent sheaf R^{i(F)}Ŝ(F) is denoted simply by F̂.

Consider now (A, M), a simple polarized abelian surface of type (1, 2). Assume that M is symmetric, i.e. (−1)^{∗}M ≅ M. The linear series |M| has 4 isolated base points {o, p, q, r}. We may assume that o is the identity of the abelian surface and p, q, r are 2-torsion with r = p + q (see, e.g. [Ba]). Each divisor D ∈ |M| is either a nonsingular curve of genus 3 or a singular curve with a simple node distinct from the base points. M^{∨} satisfies the W.I.T. of index 2. Let

F = M̂^{∨} := R^{2}Ŝ(M^{∨})

be the Fourier-Mukai transform of M^{∨}. The vector bundle F has rank 2. Let E = F^{∨}. One can check that,

dim Hom(S^{3}E, Λ^{2}E) = h^{0}(Â, (S^{3}E)^{∨}⊗ ∧^{2} E) = 2.

According to Miranda’s triple covering construction [Mi], there is a 2dimensional family of triple coverings f̂: X̂ → Â with f̂_{∗}OX̂ = OÂ ⊕E.

p_{g} = q = 2... 3

The idea is as follows: in order to construct a triple covering f̂ : X̂ ⟶ Â over Â with Tschirnhausen module E (cf. [Mi]), we first construct a triple covering f : X ⟶ A with Tschirnhausen module ϕ_{M^{∨}}^{∗}E. In Claim 1, we identify those that descend to a triple covering f : X̂ ⟶ Â. In Claim 2, we then verify that for a general such covering, the singularities of X are rational. It follows that the singularities of X̂ are also rational. Finally, one can compute the invariants of X̂ via the invariants of X.

Let ϕ_{M^{∨}} : A ⟶ Â be the isogeny defined by M^{∨}. We have the following commutative diagram:

X = A ×Â X̂ ⟶^{ϕ} X̂
f↓  f̂↓
A  ⟶^{ϕ_{M^{∨}}} Â

Where ϕ : X → X̂ is a 4 : 1 etale covering and f : X → A is a triple covering determined by a section of

ϕ_{M^{∨}}^{∗}Hom(S^{3}E, ∧^{2}E) ⊂ Hom(S^{3}ϕ_{M^{∨}}^{∗}E, ∧^{2}ϕ_{M^{∨}}^{∗}E).

By [Muk], ϕ_{M^{∨}}^{∗}E ≅ M^{∨} ⊕ M^{∨}. Thus

Hom(S^{3}ϕ_{M^{∨}}^{∗}E, ∧^{2}ϕ_{M^{∨}}^{∗}E) ≅ H^{0}(A, M)^{⊕4}.

In order to determine the corresponding 2-dimensional subspace, we consider the Heisenberg group action on H^{0}(A, M). The Heisenberg group can be identified as

G(δ) := {(α, t, l)|α ∈ k^{∗}, t ∈ ℤ_{2}, l ∈ ℤ̂_{2}}

with group law (α, t, l)(α, t′, l′) = (αα′l′(t), t + t′, l + l′). Moreover, H^{0}(A, M) corresponds to Hom(ℤ_{2}, k). The action of G(δ) on Hom(ℤ_{2}, k) is given as (α, t, l)f(x) = αl(x)f(t + x). Let X, Y be the sections in H^{0}(A, M) corresponding to the characteristic functions of 0, 1 in Hom(ℤ_{2}, k) respectively.

Claim 1: The 2-dimensional subspace is determined as

ϕ_{M^{∨}}^{∗}Hom(S^{3}E, ∧^{2}E) ≅ {(sX, tY, −tX, −sY)|s, t ∈ k} ⊂ H^{0}(A, M)^{⊕4}.

Grant the claim for the time being. Following Miranda (cf. [Mi]) we can then construct a triple covering f : X → A by using the data a = sX, b = tY, c = −tX, d = −sY. Over an affine open subset U of A, the triple covering can be described in U × 𝔸^{2} as by the 2 × 2 minors of

(\begin{matrix}Z + a & W − 2d & c \\ b & Z − 2a & W + d \end{matrix})

where Z, W are coordinates for 𝔸^{2}.

Following [Mi] §4, we have A = s^{2}X^{2} + stY^{2}, B = (t^{2} − s^{2})XY and C = s^{2}Y^{2} + stX^{2}. The branch locus is defined by D = B^{2} − 4AC ∈ H^{0}(M)^{⊗4} and one can see that it corresponds to a divisor D_{1} + D_{2} + D_{3} + D_{4} with D_{i} ∈ |M|. For general choice of s, t, the D_{i} are all distinct and nonsingular. It is easy to check (cf. [Mi] §5) that for general choices of s, t, the only possible singularities of X lie over the 4 base points {p, q, r, o}. We remark that f is totally ramified only over these 4 base points.

Let x ∈ X be a point lying over one of {p, q, r, o}.

Claim 2: For general s, t, the singularity of X at x, is locally isomorphic to a cone over a twisted cubic.

Therefore, X has only rational singularities and so does X̂. A resolution μ̂ : X̂′ → X̂ can be obtained by blowing up along the singularity. The corresponding resolution μ : X′ → X is the blow up of X along the 4 points lying over {p, q, r, o}. Let {E_{i}}_{i=1,..,4} be the exceptional divisors and {R_{i}}_{i=1,...,4} be the proper transform of the D_{i}, then

K_{X′} = ∑_{i=1,...,4} R_{i} + ∑_{i=1,...,4} E_{i}.

Note that R_{i} · R_{j} = 0, R_{i} · E_{j} = 1 for all i, j and E_{i}^{2} = −3, E_{i} · E_{j} = 0 for i ≠ j. Thus we have K_{X′}^{2} = 20.

p_{g}(X′) = h^{0}(X′, ω_{X′}) = h^{2}(X′, O_{X′}) = h^{2}(X, O_{X}) =
h^{2}(A, O_{A}) + 2h^{2}(A, M^{∨}) = 5.

Similarly, q(X′) = 2 and χ(X′, ω_{X′}) = 4. One can also check that K_{X′} is ample.

Since X′ → X̂′ is an etale cover of degree 4, one has χ(X̂′, ωX̂′) = 1, (KX̂′)^{2} = 5 and KX̂′ is ample. X̂ has only rational singularity. It is easy to see that q(X̂′) = 2 and hence p_{g}(X̂′) = 2. So X̂′ is a surface of general type with p_{g} = q = 2 and K^{2} = 5.

Proof. (Claim 1) We follow [Mum]. Let H(M^{∨}) be the kernel of ϕ_{M^{∨}} : A ⟶ Â, i.e. the set of points x ∈ A such that T_{x}^{∗}M^{∨} ≅ M^{∨}. Then H(M) = H(M^{∨}). Let G(M) be the set of pairs (x, φ) such that x ∈ H(M) and φ is an isomorphism φ : M ⟶ T_{x}^{∗}M. Then G(M) is a group sitting in the following exact sequence:

0 ⟶ k^{∗} ⟶ G(M) ⟶ H(M) ⟶ 0.

There is an isomorphism of groups G(M) ≅ G(δ). Under this identification, the representation of G(M) on H^{0}(A, M) corresponds to the unique representation of G(δ) on V = V (δ) := Hom(ℤ_{2}, k), which is defined by ((α, t, l)f)(x) = α · l(x) · f(t + x). With respect to the ordered basis {χ^{0}, χ^{1}} (characteristic functions of 0, 1 respectively), this representation is induced by

(1, 1, 1) → (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) , (1, 1, 0) → (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , (1, 0, 1) → (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) .

The corresponding G(δ) representation on S^{3}V^{∨}⊗Λ^{2}V ⊗V can easily be computed. By [Muk] Proposition 3.11, we have

ϕ_{M^{∨}}^{∗}F ≅ H^{2}(A, M)⊗M ≅ (H^{0}(A, M)⊗M^{∨})^{∨} ≅ ϕ_{M^{∨}}^{∗}E^{∨}.

One sees that

H^{0}(A, S^{3}ϕ_{M^{∨}}^{∗}E^{∨}⊗Λ^{2}ϕ_{M^{∨}}^{∗}E) ≅ S^{3}H^{0}(A, M)^{∨}⊗Λ^{2}H^{0}(A, M)⊗H^{0}(A, M).

This vector space is in turn isomorphic to ⨁_{i=1}^{4} H^{0}(A, M). We can now compute the corresponding G(M) representation in terms of the above G(δ) representation.

Let p_{i}, i = 1, ..., 4 denote the projection onto the i-th factor. With respect to the ordered basis

{e_{1}, e_{2}, ..., e_{8}} = {p_{1}^{∗}X, p_{1}^{∗}Y, ..., p_{4}^{∗}X, p_{4}^{∗}Y } =

{X̂^{3}⊗X ∧ Y ⊗X, X̂^{3}⊗X ∧ Y ⊗Y, X̂^{2}Ŷ ⊗X ∧ Y ⊗X, X̂^{2}Ŷ ⊗X ∧ Y ⊗Y,

X̂Ŷ^{2}⊗X ∧ Y ⊗X, X̂Ŷ^{2}⊗X ∧ Y ⊗Y, Ŷ^{3}⊗X ∧ Y ⊗X, Ŷ^{3}⊗X ∧ Y ⊗Y }, we have that (1, 1, 1) → R ∈ M_{8}(k) with R_{i,j} = 0 if i + j ≠ 8 and R_{i,8−i} = {−1, 1, 1, −1, −1, 1, 1, −1}, respectively (1, 1, 0) → M ∈ M_{8}(k) with M_{i,j} = 0 if i + j ≠ 8 and M_{i,8−i} = {1, 1, 1, 1, 1, 1, 1, 1}. In particular R^{2} = M^{2} = 1 and RM = MR. There is an induced representation of H(M^{∨}) ≅ ℤ_{2} × ℤ_{2}. It is easy to see that the ℤ_{2} × ℤ_{2} invariant elements are just the subspace

{s(e_{1} − e_{8}) + t(e_{4} − e_{5})|s, t ∈ k} = {(sX, tY, −tX, −sY)|s, t ∈ k}.

These invariant elements correspond to the subspace ϕ_{M^{∨}}^{∗}Hom(S^{3}E, Λ^{2}E). □

Proof. (Claim 2) On a neighborhood of one of the base loci o, p, q, r we may assume that X, Y (or any two distinct sections of H^{0}(A, M)) are local coordinates. By [Ha] pg. 14 exercise 1.25, the above mentioned 2 × 2 minors define a twisted cubic if and only if for all [u : v] ∈ ℙ^{1}, the linear forms

u(Z + sX) − vtY, u(W + 2sY) − v(Z − 2sX), −utX − v(W − sY)

are linearly independent. In other words, if and only if the matrix

(\begin{matrix}us & −vt & u & 0 \\ 2vs & 2us & −v & u \\ −ut & vs & 0 & −v \end{matrix})

has a nonzero 3 × 3 minor for every value of u, v By inspection one sees that this is the case for general s, t (more precisely for t ≠ 0 and t^{2} ≠ 9s^{2}). □

3. Computation of alb_{X,∗}(ω_{X})

In this section, using the techniques of [HP], we find restrictons on the structure of the coherent sheaf alb_{X,∗}(ω_{X}). It was this computation that suggested to us the possibility of constructing the example of §2. Proposition 3.1. ([CM] Proposition 2.3) Let X be a minimal surface of general type with p_{g} = q = 2. Then a := alb_{X} : X ⟶ Alb(X) =: A is not surjective if and only if B := a(X) is a curve of genus 2 and a : X ⟶ B has smooth connected fibers of genus 2 with constant modulus and K_{X}^{2} = 8.

We now therefore consider the situation where a : X ⟶ Alb(X) =: A is surjective. For any coherent sheaf F on X, define

V^{i}(X, F) := {P ∈ Pic^{0}(X)|h^{i}(X, F⊗P) ≠ 0}.

Since a is generically finite, R^{i}a_{∗}ω_{X} = 0 for all i > 0 and so V^{i}(X, ω_{X}) = V^{i}(A, a_{∗}ω_{X}) for all i. Lemma 3.2. Let X be a minimal surface with p_{g} = q = 2 and surjective Albanese map. If dim V^{1}(X, ω_{X}) ≥ 1, then there exists an elliptic pencil X ⟶ E with g(E) = 1.

Proof. By the generic vanishing theorems of Green and Lazarsfeld, dim V^{1}(X, ω_{X}) < 2 and if T is a component of V^{1}(X, ω_{X}) of dimension 1, then T is a translate of an elliptic curve T_{0} ⊂ Pic^{0}(X). The pencil X ⟶ E is induced by a : X ⟶ Alb(X) composed with the dual map of abelian varieties Alb(X) ⟶ E := T_{0}^{∨}. □

An immediate consequence is the following: Corollary 3.3. Let X be a minimal surface of general type with p_{g} = q = 2 without irrational pencils. Then a : X ⟶ A is surjective with V^{1}(A, a_{∗}ω_{X}) supported on finitely many points.

A vector bundle U on an abelian variety A is unipotent if it has a filtration

0 = U_{0} ⊂ U_{1} ⊂ ... ⊂ U_{n−1} ⊂ U_{n} = U

such that U_{i}/U_{i−1} ≅ O_{A}. A vector bundle is homogeneous if and only if it is isomorphic to ⊕_{i=1}^{n}(P_{i}⊗U_{i}) with P_{i} ∈ Pic^{0}(A) and U_{i} unipotent vector bundles. By [Muk], there is an one-to-one corresponce between sheaves supported on finitely many points and homogeneous vector bundles via the Fourier-Mukai transform. Lemma 3.4. Let F be a coherent sheaf on an abelian surface, then R^{i}SR^{j}Ŝ(F) = 0 for (i, j) ∈ {(1, 2), (2, 2), (0, 0), (1, 0)} and there is an injection (resp. surjection) d : R^{0}SR^{1}ŜF ⟶ R^{2}SR^{0}ŜF (resp. d′ : R^{0}SR^{2}ŜF ⟶ R^{2}SR^{1}ŜF). In particular R^{0}ŜF (resp. R^{2}ŜF) satisfies the W.I.T. of index 2 (resp. 0).

Proof. As mentioned above, by [Muk], there is an isomorphism of functors

RS ∘ RŜ ≅ (−1_{A})^{∗}[−2].

In particular there is a spectral sequence E_{2}^{p,q} = R^{p}SR^{q}ŜF with E_{∞}^{p,q} = 0 if p + q ≠ 2. The only possibly non-vanishing differentials d_{2} are

d : R^{0}SR^{1}ŜF ⟶ R^{2}SR^{0}ŜF and d′ : R^{0}SR^{2}ŜF ⟶ R^{2}SR^{1}ŜF.

One sees that E_{2}^{p,q} = E_{∞}^{p,q} = 0 for (p, q) ∈ {(1, 2), (2, 2), (0, 0), (1, 0)}. Moreover, ker(d) = E_{3}^{0,1} = E_{∞}^{0,1} = 0. So d is an injection. Similarly d′ is a surjection. □

Theorem 3.5. Let X be a minimal surface of general type with p_{g} = q = 2 without any irrational pencil. Then there exist homogeneous vector bundles H, and a negative definite line bundle L on Â = Pic^{0}(A) (i.e. L^{∨} is ample) such that a_{∗}ω_{X} fits into the following exact sequences

0 ⟶ O_{A} ⟶ a_{∗}ω_{X} ⟶ F ⟶ 0,
0 ⟶ H ⟶ L̂ ⟶ (−1_{A})^{∗}F ⟶ 0.

Proof. Notice that ω_{A} = O_{A}. By assumption X has no irrational pencils, therefore a : X ⟶ A is surjective and dim V^{1}(X, ω_{X}) = 0, hence V^{1}(X, ω_{X}) = {O_{X}, P_{1}, ..., P_{n}}. Let F be the coherent sheaf defined by the short exact sequence

0 ⟶ O_{A} ⟶ a_{∗}ω_{X} ⟶ F ⟶ 0.

Since R^{i}a_{∗}ω_{X} = 0 for i > 0, one sees that for i ≥ 0,

H^{i}(A, a_{∗}ω_{X}) ≅ H^{i}(X, ω_{X}) ≅ H^{i}(A, ω_{A})

and therefore h^{1}(F) = h^{2}(F) = 0. Moreover, for all O_{X} ≠ P ∈ Pic^{0}(A), one has h^{i}(A, F⊗P) = h^{i}(X, ω_{X}⊗P) for all i. In particular V^{2}(A, F) = ∅ and V^{1}(A, F) = {P_{1}, ..., P_{n}}. We have that R^{2}ŜF = 0 and R^{1}ŜF = ⊕B_{i} where the sheaves B_{i} are supported at the points P_{i} (and are Artinian O_{Â,P_{i}}-modules cf. [Muk] Example 2.9). In particular, R^{1}ŜF satisfies the W.I.T. of index 0. Consider now the spectral sequence of the proof of Lemma 3.4. The only non-zero E_{2} terms are E_{2}^{0,1} and E_{2}^{2,0}. Therefore, one has the following exact sequence

0 ⟶ R^{0}SR^{1}ŜF ⟶ R^{2}SR^{0}ŜF ⟶ (−1_{A})^{∗}F ⟶ 0.

First note that R^{1}ŜF is supported on finitely many points. It follows that R^{0}SR^{1}ŜF = RSR^{1}ŜF is a homogeneous vector bundle, call it H. It suffices to show that R^{0}ŜF is a negative line bundle.

Let U = Pic^{0}(A) − {O_{A}, P_{1}, ..., P_{n}}, then for all P ∈ U

h^{0}(A, F⊗P) = h^{0}(A, a_{∗}ω_{X}⊗P) = χ(X, ω_{X}) = 1.

Thus R^{0}ŜF|U is locally free of rank 1. Let L = (R^{0}ŜF)^{∨∨}. Then L is a reflexive sheaf of rank 1 on a non-singular surface and hence a line bundle. Since R^{0}ŜF = R^{0}Ŝa_{∗}ω_{X} is torsion free, we have an exact sequence of coherent sheaves on Â:

0 ⟶ R^{0}ŜF ⟶ L ⟶ Q ⟶ 0

where Q is supported at most on the points P_{i}.

We claim that Q = 0. Suppose on the contrary that Q ≠ 0. By Lemma 3.4, R^{i}SR^{0}ŜF = 0 for i = 0, 1, hence R^{0}L ≅ R^{0}Q. So for general P ∈ A = Pic^{0}(Â) one has h^{0}(L⊗P) = h^{0}(Q⊗P) ≠ 0 since Q ≠ 0 is supported on points. It follows that L is an ample line bundle and therefore satisfying I.T of index 0. In particular, R^{2}SL = 0. On the other hand, since Q is supported on points, we have that R^{1}SQ = 0. The exact sequence

R^{1}SQ → R^{2}SR^{0}ŜF → R^{2}SL,

yields R^{2}SR^{0}ŜF = 0. It follows that F = 0 since there is a surjection R^{2}SR^{0}ŜF → (−1)^{∗}F. One concludes that O_{A} = a_{∗}ω_{X}, and in particular X ⟶ A is birational, which is the required contradiction.

We may therefore assume that Q = 0 and hence L = R^{0}ŜF is a line bundle. By Lemma 3.4, L satifies W.I.T of index 2, hence it is a negative definite line bundle. □

Remark. It follows that if the degree of X ⟶ A is 2, then rk(F) = 1 The only possibility is that a_{∗}ω_{X} = O_{A}⊕O_{A}(−Θ) where Θ is a principal polarization. This is a 2 : 1 covering branched along a divisor D ∈ |2Θ|.

We have given an example with a_{∗}ω_{X} = O_{A} ⊕ L̂, where L^{∨} is an ample line bundle of type (1, 2). Unluckily we have not been able to rule out the cases in which H ≠ 0. For example is it possible to have a_{∗}ω_{X} = O_{A} ⊕ F with F as follows? Example. Let (A, L) be a general polarized abelian surface of type (1, 3) and x ∈ A a closed point. Then h^{i}(A, L⊗I_{x}⊗P) = 0 for all i > 0 and all P ∈ Pic^{0}(A). Let E = { \atop L⊗I_{x}} and F = E^{∨}. Then we have an exact sequence

0 ⟶ P_{x}^{∨} ⟶ L̂^{∨} ⟶ F ⟶ 0.

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Jungkai Alfred Chen Department of Mathematics National Chung Cheng University Ming Hsiung, Chia Yi, 621, Taiwan jkchen@math.ccu.edu.tw Christopher D. Hacon Department of Mathematics University of Utah 155 South 1400 East, JWB 233 Salt Lake City, Utah 84112-0090, USA hacon@math.ucr.edu