WITH pg = q = 2 AND KX2 = 5
Jungkai A. Chen and Christopher D. Hacon
We give an example of a minimal complex surface of general type with pg = q = 2 and KX2 = 5.
1. Introduction
Recently, there has been considerable interest in understanding the geometry of irregular complex projective surfaces with χ(X, ωX) = 1+pg(X)−q(X) = 1, and in particular of surfaces with pg = q = 2. Let X be a smooth minimal
complex surface of general type. If χ(X, ωX) = 1, then one has the bound 1 ≤ KX2 ≤ 9. If, in addition the surface is irregular, i.e. q(X) = h0(X, Ω1X) > 0, then one also has KX2 ≥ 2pg(X) and so pg(X) ≤ 4. In [Be], it is shown
that the case pg= q = 4 corresponds to the product of two curves of genus 2.
In [HP] and [Pi], surfaces with pg = q = 3 are completely classified. When
K2
X = 2pg(X) = 6 they are symmetric products of curves of genus 3 and
when K2
X = 8 they admit an irrational pencil. The case pg = q = 2 seems
considerably more delicate. At any rate Catanese suggests that, analogously to the pg = q = 3 case, a surface of general type with pg = 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 pg = q = 2
which admit an irrational pencil. In [Ma], Manetti shows that a minimal surface of general type with KX ample and KX2 = 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 pg = q = 2 and
K2 X = 4.
The purpose of this note is to give a counter-example to Catanese’s con-jecture above. The example we construct is birational to a triple cover of an abelian surface. Its canonical divisor KX is ample, pg = q = 2 and K2
X = 5.
The construction is motivated in §3, where we obtain restrictions on the structure of the sheaf albX,∗(ωX).
Acknowledgments We are indebted to Rita Pardini for useful conversa-tions on this subject. This work was completed during a visit of the first
author to the University of California, Riverside and University of Utah. The first author would like to thank the Universities for their hospitality.
2. Construction and Verification
We will need some results from the theory of Mukai transforms. Let ˆA be the dual abelian variety of A and P be the normalized Poincar´e line bundle on A × ˆA. Following [Muk], define the functor ˆS of OA-modules into the
category of OAˆ-modules by
ˆ
S(M ) = πA,∗ˆ (P⊗π∗AM ).
The derived functor R ˆS of ˆS then induces an equivalence of categories be-tween the two derived categories D(A) and D( ˆA). More precisely, by [Muk]: There are isomorphisms of functors:
RS ◦ R ˆS ∼= (−1A)∗[−g]
and
R ˆS ◦ RS ∼= (−1Aˆ)∗[−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 6= i(F), one has RjS(F) = 0. The coherentˆ sheaf Ri(F)S(F) is denoted simply by ˆˆ F.
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 = dM∨ := R2S(Mˆ ∨)
be the Fourier-Mukai transform of M∨. The vector bundle F has rank 2.
Let E = F∨. One can check that,
dim Hom(S3E, Λ2E) = h0( ˆA, (S3E)∨⊗ ∧2E) = 2.
According to Miranda’s triple covering construction [Mi], there is a 2-dimensional family of triple coverings ˆf : ˆX → ˆA with ˆf∗OXˆ = OAˆ⊕ E.
The idea is as follows: in order to construct a triple covering ˆf : ˆX −→ ˆA over ˆA 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 : ˆX −→ ˆA. In Claim 2, we then verify that for a general such covering, the singularities of X are
rational. It follows that the singularities of ˆX are also rational. Finally, one can compute the invariants of ˆX via the invariants of X.
Let φM∨ : A −→ ˆA be the isogeny defined by M∨. We have the following
commutative diagram: X = A ×AˆXˆ −−−−→ ˆφ X f y fˆ y A φM ∨ −−−−→ ˆA
Where φ : X → ˆX is a 4 : 1 etale covering and f : X → A is a triple covering determined by a section of
φ∗M∨Hom(S3E, ∧2E) ⊂ Hom(S3φ∗M∨E, ∧2φ∗M∨E).
By [Muk], φ∗
M∨E ∼= M∨⊕ M∨. Thus
Hom(S3φ∗M∨E, ∧2φ∗M∨E) ∼= H0(A, M )⊕4.
In order to determine the corresponding 2-dimensional subspace, we con-sider the Heisenberg group action on H0(A, M ). The Heisenberg group can
be identified as
G(δ) := {(α, t, l)|α ∈ k∗, t ∈ Z2, l ∈ ˆZ2}
with group law (α, t, l)(α, t0, l0) = (αα0l0(t), t+t0, l+l0). Moreover, H0(A, M ) corresponds to Hom(Z2, k). The action of G(δ) on Hom(Z2, k) is given as
(α, t, l)f (x) = αl(x)f (t + x). Let X, Y be the sections in H0(A, M )
corre-sponding to the characteristic functions of 0, 1 in Hom(Z2, k) respectively.
Claim 1: The 2-dimensional subspace is determined as
φ∗M∨Hom(S3E, ∧2E) ∼= {(sX, tY, −tX, −sY )|s, t ∈ k} ⊂ H0(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 × A2 as by the 2 × 2 minors of
µ
Z + a W − 2d c b Z − 2a W + d
¶
where Z, W are coordinates for A2.
Following [Mi] §4, we have A = s2X2 + stY2, B = (t2 − s2)XY and
C = s2Y2+stX2. The branch locus is defined by D = B2−4AC ∈ H0(M )⊗4
and one can see that it corresponds to a divisor D1+ D2+ D3+ D4 with
Di∈ |M |. For general choice of s, t, the Di 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.
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 ˆX. A resolution ˆ
µ : ˆX0 → ˆX can be obtained by blowing up along the singularity. The corresponding resolution µ : X0 → X is the blow up of X along the 4
points lying over {p, q, r, o}. Let {Ei}i=1,..,4 be the exceptional divisors and {Ri}i=1,...,4 be the proper transform of the Di, then
KX0 = X i=1,...,4 Ri+ X i=1,...,4 Ei.
Note that Ri· Rj = 0, Ri· Ej = 1 for all i, j and Ei2 = −3, Ei· Ej = 0 for
i 6= j. Thus we have K2
X0 = 20.
pg(X0) = h0(X0, ωX0) = h2(X0, OX0) = h2(X, OX) =
h2(A, OA) + 2h2(A, M∨) = 5. Similarly, q(X0) = 2 and χ(X0, ω
X0) = 4. One can also check that KX0 is
ample.
Since X0 → ˆX0 is an etale cover of degree 4, one has χ( ˆX0, ω ˆ
X0) = 1,
(KXˆ0)2= 5 and KXˆ0 is ample. ˆX has only rational singularity. It is easy to
see that q( ˆX0) = 2 and hence p
g( ˆX0) = 2. So ˆX0 is a surface of general type
with pg= q = 2 and K2= 5.
Proof. (Claim 1) We follow [Mum]. Let H(M∨) be the kernel of φM∨ :
A −→ ˆA, i.e. the set of points x ∈ A such that T∗
xM∨ ∼= 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 −→ Tx∗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 identifica-tion, the representation of G(M ) on H0(A, M ) corresponds to the unique representation of G(δ) on V = V (δ) := Hom(Z2, k), which is defined by
((α, t, l)f )(x) = α · l(x) · f (t + x). With respect to the ordered basis {χ0, χ1} (characteristic functions of 0, 1 respectively), this representation is induced by (1, 1, 1) → µ 0 1 −1 0 ¶ , (1, 1, 0) → µ 0 1 1 0 ¶ , (1, 0, 1) → µ 1 0 0 −1 ¶ . The corresponding G(δ) representation on S3V∨⊗Λ2V ⊗V can easily be
com-puted. By [Muk] Proposition 3.11, we have
One sees that
H0(A, S3φ∗M∨E∨⊗Λ2φ∗M∨E) ∼= S3H0(A, M )∨⊗Λ2H0(A, M )⊗H0(A, M ).
This vector space is in turn isomorphic to L4i=1H0(A, M ). We can now compute the corresponding G(M ) representation in terms of the above G(δ) representation.
Let pi, i = 1, ..., 4 denote the projection onto the i-th factor. With respect
to the ordered basis
{e1, e2, ..., e8} = {p∗1X, p∗1Y, ..., p∗4X, p∗4Y } =
{ ˆX3⊗X ∧ Y ⊗X, ˆX3⊗X ∧ Y ⊗Y, ˆX2Y ⊗X ∧ Y ⊗X, ˆˆ X2Y ⊗X ∧ Y ⊗Y,ˆ ˆ
X ˆY2⊗X ∧ Y ⊗X, ˆX ˆY2⊗X ∧ Y ⊗Y, ˆY3⊗X ∧ Y ⊗X, ˆY3⊗X ∧ Y ⊗Y }, we have that (1, 1, 1) → R ∈ M8(k) with Ri,j = 0 if i + j 6= 8 and Ri,8−i =
{−1, 1, 1, −1, −1, 1, 1, −1}, respectively (1, 1, 0) → M ∈ M8(k) with Mi,j = 0 if i + j 6= 8 and Mi,8−i = {1, 1, 1, 1, 1, 1, 1, 1}. In particular R2 = M2 = 1
and RM = M R. There is an induced representation of H(M∨) ∼= Z2× Z2.
It is easy to see that the Z2× Z2 invariant elements are just the subspace
{s(e1− e8) + t(e4− e5)|s, t ∈ k} = {(sX, tY, −tX, −sY )|s, t ∈ k}. These invariant elements correspond to the subspace φ∗M∨Hom(S3E, Λ2E).
¤ 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 H0(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] ∈ P1, 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
2vsus −vt2us −vu u0
−ut vs 0 −v
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 6= 0 and t2 6= 9s2). ¤
3. Computation of albX,∗(ωX)
In this section, using the techniques of [HP], we find restrictons on the structure of the coherent sheaf albX,∗(ωX). It was this computation that
Proposition 3.1. ([CM] Proposition 2.3) Let X be a minimal surface of general type with pg = q = 2. Then a := albX : 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 K2
X = 8.
We now therefore consider the situation where a : X −→ Alb(X) =: A is surjective. For any coherent sheaf F on X, define
Vi(X, F) := {P ∈ Pic0(X)|hi(X, F⊗P ) 6= 0}. Since a is generically finite, Ria
∗ωX = 0 for all i > 0 and so Vi(X, ωX) =
Vi(A, a∗ωX) for all i.
Lemma 3.2. Let X be a minimal surface with pg = q = 2 and surjective
Albanese map. If dim V1(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 V1(X, ωX) <
2 and if T is a component of V1(X, ω
X) of dimension 1, then T is a
trans-late of an elliptic curve T0 ⊂ Pic0(X). The pencil X −→ E is induced
by a : X −→ Alb(X) composed with the dual map of abelian varieties
Alb(X) −→ E := T0∨. ¤
An immediate consequence is the following:
Corollary 3.3. Let X be a minimal surface of general type with pg = q = 2 without irrational pencils. Then a : X −→ A is surjective with V1(A, a
∗ωX)
supported on finitely many points.
A vector bundle U on an abelian variety A is unipotent if it has a filtration 0 = U0 ⊂ U1 ⊂ ... ⊂ Un−1⊂ Un= U
such that Ui/Ui−1∼= OA. A vector bundle is homogeneous if and only if it
is isomorphic to ⊕n
i=1(Pi⊗Ui) with Pi ∈ Pic0(A) and Ui 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 RiSRjS(F) =ˆ 0 for (i, j) ∈ {(1, 2), (2, 2), (0, 0), (1, 0)} and there is an injection (resp. sur-jection) d : R0SR1SF −→ Rˆ 2SR0SF (resp. dˆ 0: R0SR2SF −→ Rˆ 2SR1SF).ˆ In particular R0SF (resp. Rˆ 2SF) satisfies the W.I.T. of index 2 (resp. 0).ˆ Proof. As mentioned above, by [Muk], there is an isomorphism of functors
RS ◦ R ˆS ∼= (−1A)∗[−2].
In particular there is a spectral sequence E2p,q= RpSRqSF with Eˆ ∞p,q= 0 if
p + q 6= 2. The only possibly non-vanishing differentials d2 are
One sees that E2p,q= E∞p,q= 0 for (p, q) ∈ {(1, 2), (2, 2), (0, 0), (1, 0)}.
More-over, ker(d) = E30,1 = E∞0,1 = 0. So d is an injection. Similarly d0 is a
surjection. ¤
Theorem 3.5. Let X be a minimal surface of general type with pg = q = 2
without any irrational pencil. Then there exist homogeneous vector bundles H, and a negative definite line bundle L on ˆA = Pic0(A) (i.e. L∨ is ample)
such that a∗ωX fits into the following exact sequences
0 −→ OA−→ a∗ωX −→ F −→ 0, 0 −→ H −→ ˆL −→ (−1A)∗F −→ 0.
Proof. Notice that ωA = OA. By assumption X has no irrational
pen-cils, therefore a : X −→ A is surjective and dim V1(X, ωX) = 0, hence
V1(X, ω
X) = {OX, P1, ..., Pn}. Let F be the coherent sheaf defined by the
short exact sequence
0 −→ OA−→ a∗ωX −→ F −→ 0.
Since Ria∗ωX = 0 for i > 0, one sees that for i ≥ 0,
Hi(A, a∗ωX) ∼= Hi(X, ωX) ∼= Hi(A, ωA)
and therefore h1(F) = h2(F) = 0. Moreover, for all O
X 6= P ∈ Pic0(A),
one has hi(A, F⊗P ) = hi(X, ω
X⊗P ) for all i. In particular V2(A, F) = ∅
and V1(A, F) = {P1, ..., Pn}. We have that R2SF = 0 and Rˆ 1SF = ⊕Bˆ i
where the sheaves Biare supported at the points Pi (and are Artinian OA,Pˆ i
-modules cf. [Muk] Example 2.9). In particular, R1SF satisfies the W.I.T.ˆ of index 0. Consider now the spectral sequence of the proof of Lemma 3.4. The only non-zero E2 terms are E20,1 and E22,0. Therefore, one has the
following exact sequence
0 −→ R0SR1SF −→ Rˆ 2SR0SF −→ (−1ˆ A)∗F −→ 0.
First note that R1SF is supported on finitely many points. It followsˆ that R0SR1SF = RSRˆ 1SF is a homogeneous vector bundle, call it H. Itˆ suffices to show that R0SF is a negative line bundle.ˆ
Let U = Pic0(A) − {OA, P1, ..., Pn}, then for all P ∈ U
h0(A, F⊗P ) = h0(A, a∗ωX⊗P ) = χ(X, ωX) = 1.
Thus R0SF|U is locally free of rank 1. Let L = (Rˆ 0SF)ˆ ∨∨. Then L is a
reflexive sheaf of rank 1 on a non-singular surface and hence a line bun-dle. Since R0SF = Rˆ 0Saˆ ∗ωX is torsion free, we have an exact sequence of
coherent sheaves on ˆA:
0 −→ R0SF −→ L −→ Q −→ 0ˆ where Q is supported at most on the points Pi.
We claim that Q = 0. Suppose on the contrary that Q 6= 0. By Lemma 3.4, RiSR0SF = 0 for i = 0, 1, hence Rˆ 0L ∼= R0Q. So for general P ∈
A = Pic0( ˆA) one has h0(L⊗P ) = h0(Q⊗P ) 6= 0 since Q 6= 0 is supported
on points. It follows that L is an ample line bundle and therefore satisfying I.T of index 0. In particular, R2SL = 0. On the other hand, since Q is
supported on points, we have that R1SQ = 0. The exact sequence
R1SQ → R2SR0SF → Rˆ 2SL,
yields R2SR0SF = 0. It follows that F = 0 since there is a surjectionˆ R2SR0SF → (−1)ˆ ∗F. One concludes that O
A = a∗ωX, and in particular
X −→ A is birational, which is the required contradiction.
We may therefore assume that Q = 0 and hence L = R0SF is a lineˆ 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∗ωX = OA⊕ OA(−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∗ωX = OA⊕ ˆL, 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 6= 0. For example is it possible to have a∗ωX = OA⊕ 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 hi(A, L⊗I
x⊗P ) = 0 for all i > 0 and all
P ∈ Pic0(A). Let E = \L⊗Ix and F = E∨. Then we have an exact sequence
0 −→ Px∨ −→ ˆL∨ −→ F −→ 0. References
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Received Received date / Revised version date Jungkai Alfred Chen
Department of Mathematics National Taiwan University 1 Sec. 4, Roosevelt Rd. Taipei, 106, Taiwan
E-mail address: [email protected]
Christopher Derek Hacon Department of Mathematics University of Utah
155 South 1400 East, JWB 233
Salt Lake City, Utah 84112-0090, USA
E-mail address: [email protected]
The first author was partially supported by NSC 92-2115-M-002-029, Taiwan.
The first author is a member of Mathematics Division, National Center for Theoretical Sciences at Taipei.
