Bielliptic curves of genus three and the Torelli problem for certain elliptic surfaces

0.35 MB·

by  Atsushi Ikeda

Bielliptic curves of genus three and the Torelli problem for
certain elliptic surfaces
7
1
Atsushi Ikeda
0
2

n
a
J
Abstract

4
2 We study the Hodge structure of elliptic surfaces which are canonically defined from bielliptic
curvesofgenusthree. Weprovethattheperiodmapforthesecondcohomologyhasonedimensional

] fibers, and the period map for the total cohomology is of degree twelve, and moreover, by adding
G
theinformation of theHodge structureof thecanonical divisor, we provea generic Torelli theorem
A for these elliptic surfaces. Finally, we give explicit examples of the pair of non-isomorphic elliptic
surfaceswhichhavethesameHodgestructureonthemselvesandthesameHodgestructureontheir
.
h canonical divisors.
t
a
m
1 Introduction
[

1 Letf :Y →B beanellipticsurfacewithasection. TheTorelliproblemasksiftheisomorphismclassof
v the surface Y is determined by the isomorphism class of the Hodge structure Hi(Y,Z). In [8], Chakiris
4
provedthat generalsimply connectedelliptic surfacewith the geometricgenus p (Y)≥2 is determined
8 g
by the Hodge structure on H2(Y,Z), and it is called generic Torelli theorem. But we can not find this
9
6 kind of results in the case when Y has positive irregularityq(Y)≥1. In this paper, we consider certain
0 elliptic surfaces with p (Y)=q(Y)=1, which are canonically defined from bielliptic curves of genus 3.
g
. Let C be a nonsingular projective curve of genus 3. Then the symmetric square C(2) has the
1
0 involution κ : C(2) → C(2) which is defined as the extension of the birational involution given by
7 q+q +κ(q+q ) ∈ |Ω1| for q+q ∈ C(2) with h0(C,O (q+q )) < 2. In this paper, we consider the
′ ′ C ′ C ′
1 case when C has a bielliptic involution σ :C →C, whose quotient E =C/σ is a nonsingularprojective
:
v curve of genus 1. Then the involution σ(2) :C(2) →C(2) commutes with the involution κ, and we have
i several quotient surfaces of C(2);
X
r C(2)
a
ւ ↓ ց
A =C(2)/(σ(2)◦κ) X =C(2)/κ Y =C(2)/σ(2)
′ ′ ′
ց ↓ ւ ց
Z =C(2)/hσ(2),κi E(2).
′
The quotient X =C(2)/κ is a projective surface of general type, and it has 28 ordinary double points,
′
whichcomefromthe28bitangentlinesofthenon-hyperellipticcase,orthe28pairofdistinctWeierstrass
pointsofthehyperellipticcase. ThequotientY =C(2)/σ(2) isaprojectivesurfaceofKodairadimension
′
1,andithas6ordinarydoublepoints,whichcomefromthe6pairofdistinctfixedpointsoftheinvolution
σ. We denote by Y =Y(C/E) the minimal resolutionofsingularitiesofY , whichis the mainobjectof
′
this paper. The quotient A =C(2)/(σ(2)◦κ) is a nonsingular projective surface of Kodaira dimension
′
2010Mathematics Subject Classification. 14C34, 14J27,14H40.
1
0. We denote by A the minimal model of A. Then A is isomorphic to the dual abelian surface of the
′
Prym variety Prym(C/E) of the branched double covering C → E, and the Kummer surface of A is
isomorphic to the nonsingular minimal model of the quotient Z = C(2)/hσ(2),κi. Since the abelian
′
surface A and its Kummer surface are investigated by Barth in [2], we can apply their results to the
study of the surface Y.
The surface Y has the canonical elliptic fibration f : Y → B by Y → E(2) → Pic(2)(E)=B,
′
and the Hodge structure H1(Y,Z) ≃ H1(B,Z) recovers only the information of the base curve B.
We have to consider the Hodge structure on H2(Y,Z) for the Torelli theorem. Since the involution
κ acts trivially on the space H0(C(2),Ω2 ) of holomorphic 2-forms on C(2), we have the coincidence
C(2)
H0(C(2),Ω2 )σ(2) =H0(C(2),Ω2 )σ(2) κoftheinvariantsubspacesfortheinvolutionsσ(2)andσ(2)◦κ.
C(2) C(2) ◦
ItimpliesthattheHodgestructuresonH2(Y,Z)andonH2(A,Z)areessentiallyequivalent. Infact,we
have the isomorphism H2(Y,Q)≃H2(A,Q)⊕Q(−1) 8 of rational Hodge structures. We remark that
⊕
the integral cohomology H2(A,Z) is not a direct summand of H2(Y,Z), hence we need more technical
arguments to compare the integral Hodge structures. The detail is given in Theorem 4.1.
The Torelli problem for the Prym map asks if the isomorphism class of the double covering C →E
is determined by the isomorphism class of the polarized abelian variety Prym(C/E). In the case for
bielliptic curves, it is studied in [14]. In our case, the Torelli theorem does not hold, because the
dimension of the moduli space of bielliptic curve of genus 3 is grater than the dimension of the moduli
spaceofpolarizedabeliansurfaces. Hence wehaveanontrivialdeformation{C →E } ofthebielliptic
t t t
curve C →E with the same Prym variety. By the above observation in Theorem 4.1, we have a family
{Y(C /E )} of surfaces with the constant Hodge structure H2(Y(C /E ),Z)≃H2(Y(C/E),Z). Since
t t t t t
the bielliptic curve C → E is recovered from the elliptic surface Y(C /E ) by Proposition 2.8, the
t t t t
family {Y(C /E )} is a nontrivialdeformation of the surface Y, and it forms a fiber of the period map
t t t
for the second cohomology.
Let F be the fiber of the elliptic fibration f : Y → B, and let D be the fiber of the composition
ξ ξ
C(2) → Y → B at ξ ∈ B. Then D → Y is a bielliptic curve of genus 3 for general ξ ∈ B. By
′ ξ ξ
Pantazis’ bigonal construction [2], [17], we can show that the Prym variety Prym(D /F ) is the dual
ξ ξ
abelian surface of Prym(C/E) (Corollary 3.7). Since the Hodge structure on the second cohomology
of the abelian surface is isomorphic to the Hodge structure of its dual abelian surface by [20], it gives
another family {Y(D /F )} of elliptic surfaces which satisfies H2(Y(D /F ),Z)≃H2(Y(C/E),Z). If
ξ ξ ξ ξ ξ
Prym(C/E)isnotaself-dualpolarizedabeliansurface,thisfamilydoesnotcontaintheoriginalsurface
Y =Y(C/E), hence the fiber of the period map for H2(Y,Z) has 2 connected components. Let M be
the set of isomorphism classes of bielliptic curves of genus 3, which distinguishes isomorphism classes
of corresponding elliptic surfaces, and let H be the set of isomorphism classes of Hodge structures with
additionalstructures(bilinearformsandsomeeffective classes). ThenwehavetheperiodmapM→H
bytheHodgestructureofY orthecanonicaldivisorK . Weremarkthattheeffectivecanonicaldivisor
Y
K of Y is uniquely determined because p (Y)=1.
Y g
Theorem 1.1. (1) The period map for the Hodge structure H2(Y,Z) has 1 dimensional fibers, and
the general fiber consists of 2 connected components (Theorem 5.5).
(2) The period map for the mixed Hodge structure 2 Hi(Y,Z) has finite fibers, and the general
i=1
fiber consists of 12 points (Theorem 5.6).
L
(3) TheperiodmapforthemixedHodgestructure 2 Hi(Y,Z) ⊕H1(K ,Z)isgenericallyinjective
i=1 Y
(Theorem 5.7).
(cid:0)L (cid:1)
We remark that the elliptic surface Y has a section and the non-constant j-function (Proposi-
tion 2.11). Our Theorem 1.1 (1) implies that the differential of the period map for H2(Y,Z) is not
injective, and it contradicts to the infinitesimal Torelli theorem by [18]. It seems that there is a gap in
2
the proofofMainTheorem(A) in[18]. Therefore,inProposition5.1,we givea directproofforthe fail-
ureofthe infinitesimalTorellitheoremwithout usingTheorem1.1(1). Aclassofsurfaceswhoseperiod
map has a positive dimensional fiber is known in [6], [7], [12], [21], [22]. Theorem 1.1 (3) is motivated
by the result on the mixed Torelli theorem for these surfaces in [19]. The global Torelli theorem for
the period map in Theorem 1.1 (3) is not true. We can find the pair of non-isomorphic elliptic surfaces
which have the same Hodge structure, and we give several types of examples which are defined over Q
in Example 5.11, Example 5.12 and Example 5.13.
This paper proceed as follows. In Section 2, we show that the pluri-canonical morphism of Y gives
theellipticfibrationY →B,andwedescribeitssingularfibers. Byusingtheinformationofthesingular
fibers and the canonical fiber, we give the way to reconstruct the original bielliptic curve C →E from
the elliptic surface Y. We also compute the j-function of the elliptic fibration. In Section 3, we review
some results in [2] on the Prym varieties for the bielliptic curve C → E of genus 3, and we show
that the Prym variety of the bielliptic curve D → F is the dual abelian surface of the Prym variety
ξ ξ
Prym(C/E). WecomputetheexplicitequationofthecanonicalmodelofthecurveD ,whichisusedin
ξ
the proofofTheorem1.1(3), andis usefultofindexamplesforthe failure ofthe globalTorellitheorem.
In Section 4, we compute the lattice structure on H2(Y,Z) using a basis of H1(C,Z). We explain the
waytoconstructtheHodgestructureH2(A,Z)fromtheHodgestructureH2(Y,Z),andalsoexplainthe
wayto constructH2(Y,Z) fromH2(A,Z). InSection 5, firstwe show thatthe differentialofthe period
map for H2(Y,Z) is not injective, but it is not used in the proof of the Main theorem (Theorem 1.1).
We prove the Main theorem by using results in Section 3 and Section 4, and we give examples for the
failure of the global Torelli theorem.
2 Construction and singular fibers of elliptic surfaces
2.1 Construction
Let E be a nonsingular projective curve of genus 1, and let π : C → E be a double covering branched
at distinct 4 points p ,...,p ∈E. Then C is a nonsingular projective curve of genus 3. In this paper,
1 4
the covering π : C → E is called bielliptic curve of genus 3. We denote by σ : C → C the covering
involution of π, and denote by σ(2) : C(2) →C(2) the induced involution on its symmetric square C(2).
Then the induced morphism π(2) : C(2) →E(2) on the symmetric squares factors through the quotient
C(2)/σ(2), and the morphism φ : C(2)/σ(2) → E(2) is a finite double covering of E(2) branched along
4 Γ , where Γ = {p +p ∈ E(2) | p∈E}. Let ν : Y → C(2)/σ(2) be the minimal resolution, and
i=1 i i i
let ψ : E(2) → Pic(2)(E) be the P1-bundle defined by p+p 7→ [O (p+p)]. Then the composition
S ′ E ′
f = ψ ◦φ◦ν : Y → Pic(2)(E) gives a fibration of curves of genus 1. We remark that the proper
transform of φ 1(Γ ) in Y gives a section of the fibration. Let η ∈Pic(2)(E) be the isomorphism class
− i
of the invertible sheaf with η 2 = [O (p +···+p )] which is determined by π η = [Ω1]. First, we
⊗ E 1 4 ∗ C
prove the following proposition.
Proposition 2.1. Y is a minimal surface with the numerical invariants p (Y) = 1, q(Y) = 1 and
g
K2 =0, and the effective canonical divisor K of Y is the fiber f 1(η) at η ∈Pic(2)(E).
Y Y −
For p∈E, we set Γ ={p+p ∈E(2) |p ∈E}, and we denote by Λ =ψ 1([O (2p)]) the fiber of
p ′ ′ p − E
ψ at [O (2p)]∈Pic(2)(E). We remark that Γ is a section of the P1-bundle ψ :E(2) →Pic(2)(E).
E p
Lemma 2.2. The canonical sheaf Ω2 is isomorphic to O (Λ −2Γ ) for any p∈E.
E(2) E(2) p p
Proof. We denote by ǫ : E ×E → E(2) the natural covering, denote by pr : E ×E → E the i-th
i
projection, and denote by ∆ the diagonal divisor on E×E. Then we have an isomorphism
E
OE E(ǫ−1(Λp)+∆E)|π−1(p′) ≃OE E(2pr−11(p)+2pr−21(p))|π−1(p′)
× i × i
3
for any p ∈E and i=1,2. By the seesaw theorem [15], we have
′
ǫ∗OE(2)(Λp)⊗OE E(∆E)≃OE E(ǫ−1(Λp)+∆E)≃OE E(2pr−11(p)+2pr−21(p))≃ǫ∗OE(2)(2Γp).
× × ×
Hence we have
ǫ O (Λ −2Γ )≃O (−∆ )≃Ω2 (−∆ )≃ǫ Ω2 .
∗ E(2) p p E E E E E E ∗ E(2)
× ×
By the injectivity of ǫ :Pic(E(2))→Pic(E×E), we have O (Λ −2Γ )≃Ω2 .
∗ E(2) p p E(2)
Let G ∈ Pic(E(2)) be the isomorphism class of the invertible sheaf with G 2 = [O ( 4 Γ )]
⊗ E(2) i=1 i
which determines the finite double covering φ:C(2)/σ(2) →E(2).
P
Lemma 2.3. ǫ G =pr η⊗pr η ∈Pic(E×E).
∗ ∗1 ∗2
Proof. The restriction of the covering φ:C(2)/σ(2) →E(2) to the divisor E ≃Γ ֒→E(2) is isomorphic
p
to the original covering π :C →E for p∈E\{p ,...,p }, hence we have
1 4
(ǫ∗G)|πi−1(p) =G|Γp =η =(pr∗1η⊗pr∗2η)|πi−1(p)
for p∈E\{p ,...,p } and i=1,2. By the seesaw theorem [15], we have ǫ G =pr η⊗pr η.
1 4 ∗ ∗1 ∗2
Proof of Proposition 2.1. Since the singularities of the branch divisor 4 Γ of the double covering
i=1 i
φ : C(2)/σ(2) → E(2) are at most nodes, the minimal resolution ν : Y → C(2)/σ(2) is the canonical
S
resolution in the sense of [10, Lemma 5]. Let p ∈ E be a point with η = [O (2p)]. By Lemma 2.3, we
E
have G =[O (2Γ )], and by Lemma 2.2, we have
E(2) p
Ω2 ⊗G ≃Ω2 ⊗O (2Γ )≃O (Λ ).
E(2) E(2) E(2) p E(2) p
By [10, Lemma 6], we can compute the numerical invariants
1
χ(Y,O )= (2Γ . Λ )+2χ(E(2),O )=1, K2 =2(Λ . Λ )=0.
Y 2 p p E(2) Y p p
Since
Ω2 ≃(φ◦ν) (Ω2 ⊗G)≃(φ◦ν) (O (Λ ))≃O (f 1(η)),
Y ∗ E(2) ∗ E(2) p Y −
we have
q(Y)=1+p (Y)−χ(Y,O )=p (Y)=h0(Y,O (f 1(η)))=h0(Pic(2)(E),O (η))=1,
g Y g Y − Pic(2)(E)
and the fiber f 1(η) is the canonical divisor of Y.
−
Let D be a nonsingular rational curve on Y. Since Pic(2)(E) is not rational, D is contained in a
fiber of f. Then we have (K . D)=0 and (D2)=−2, hence Y is a minimal surface.
Y
Corollary 2.4. There is an isomorphism of Hodge structures H1(Y,Z)≃H1(E,Z).
Proof. By Proposition2.1, the cokernel of the pull-back f :H1(E,Z)≃H1(Pic(2)(E),Z)→H1(Y,Z)
∗
by f :Y →Pic(2)(E)≃E is finite. Since f is connected, it is an isomorphism.
In the following, we denote by B = Pic(2)(E) the base space of the elliptic surface f : Y →
B =Pic(2)(E), and we call the point η ∈B the canonical point of the covering π :C →E.
Corollary 2.5. The pluri-canonical morphism Φ : Y → Pm 1 factors through the elliptic
surface f :Y →B. |OY(mKY)| −
Proof. By Proposition 2.1, we have (Ω2) m = f O (mη) and h0(Y,(Ω2) m)= h0(B,O (mη)) = m.
Y ⊗ ∗ B Y ⊗ B
Hence the composition
Y −f→B Φ|O−B→(mη)| Pm 1
−
is defined by the pluri-canonical morphism Φ .
|OY(mKY)|
4
2.2 Singular fibers
We explain about fibers of f : Y → B. If ξ ∈ B = Pic(2)(E) is not the class [O (p + p )] for
E i j
1 ≤ i < j ≤ 4, then the fiber f 1(ξ) is the double covering of ψ 1(ξ) ≃ P1 branched at 4-points
− −
ψ 1(ξ)∩ 4 Γ , hence f 1(ξ) is a nonsingular curve of genus 1. The fiber of f at [O (p +p )] ∈
− i=1 i − E 1 2
B = Pic(2)(E) is not irreducible, and it is of type I if [O (p +p )] 6= [O (p +p )], or of type I
S 1 2 E 1 2 E 3 4 1 4
if [O (p +p )]= [O (p +p )], by the notation of Kodaira [11]. Let Σ ⊂ B be the set of the critical
E 1 2 E 3 4
points of f : Y → B. Then we have 3 ≤ ♯Σ ≤ 6. The following Lemma is used for recovering the
bielliptic curve C →E from the elliptic surface Y in Proposition 2.8.
Lemma 2.6. The image of Σ by the morphism Φ :B →P1 is a set of distinct 3 points in P1.
|OB(2η)|
Proof. It follows from
[O (p +p )]+[O (p +p )]∼[O (p +p )]+[O (p +p )]∼[O (p +p )]+[O (p +p )]∼2η,
E 1 2 E 3 4 E 1 3 E 2 4 E 1 4 E 2 3
where ∼ denotes the linear equivalence on the curve B =Pic(2)(E).
Remark 2.7. Let ∆ ⊂C(2) be the curve defined by
h
∆ ={q+q ∈C(2) |h0(C,O (q+q ))=2},
h ′ C ′
which is empty if C is a non-hyperelliptic curve. Since C is a nonsingular projective curve of genus 3,
the symmetric square C(2) has a birational involution
κ:C(2)\∆ −→C(2)\∆ ; q+q 7−→κ(q+q ),
h h ′ ′
where κ(q +q ) ∈ C(2) is defined as the unique member of the linear system |Ω1(−q −q )|. If C is
′ C ′
hyperelliptic, then it extends to the regular involutionκ:C(2) →C(2), because κ(q+q )=h(q)+h(q )
′ ′
for q+q ∈ C(2)\∆ , where h : C → C denotes the hyperelliptic involution. Then the involution κ
′ h
commutes with the involution on the base B defined by the double covering Φ :B →P1, hence
we have a fibration Y/κ→P1 of curves of genus 1 in the commutative diagram|OB(2η)|
Y −→ Y/κ
↓ ↓
B −→ P1.
Φ|OB(2η)|
Let Y =Y(C/E) be the surface constructed from a bielliptic curve π :C →E.
Proposition2.8. The isomorphism class of thebielliptic curveπ :C →E is recovered from thesurface
Y.
Proof. We set E = Φ (Y) ⊂ P2. By Corollary 2.5, E is isomorphic to E, and by Proposi-
′ |OY(3KY)| ′
tion2.1,theimageoftheeffectivecanonicaldivisorK ofY byΦ isapointt∈E . LetΣ ⊂E
be the set of critical points of the fibration Φ Y :Y →E|.OBY(y3KCYo)r|ollary 2.5 and L′emma 2′.6, we′
writeasΣ ={r ,...,r ,s ,...,s }withthec|OonYd(3itKioYn)|Φ ′(r )=Φ (s )∈P1 fori=1,2,3.
We remar′k that1it is p3oss1ible tha3t r = s , if it has a s|iOngEu′(l2atr)|fibier of |tOyEp′e(2t)I| .iWe define the point
i i 1 4
r ∈E by O (s +s +s +r )≃O (4t). Then r ,...,r ∈E are distinct 4 points, and we have
4 E′ 1 2 3 4 E′ 1 4 ′
OE′(r1+···+r4)≃OE′(t+r4)⊗2,
because O (r +s )≃O (2t) for i=1,2,3. Proposition 2.8 is proved by the following Lemma.
E′ i i E′
5
Lemma 2.9. Let π : C → E be the double covering branched at the 4 points r ,...,r which is
′ ′ ′ 1 4
determined by [O (t+r )] ∈ Pic(2)(E ). Then π : C → E is isomorphic to the original covering
E′ 4 ′ ′ ′ ′
π :C →E.
Proof. By Corollary2.5,there is anisomorphismϕ:E →Pic(2)(E) withϕ◦Φ =f. We have
to consider two cases for the proof, which depends on′the choice of r from {r|O,Ys(3}K.YF)|irst we assume
i i i
that ϕ(r )=[O (p +p )] for i=1,2,3. Then we have
i E i 4
ϕ(s )⊗ϕ(s )⊗ϕ(s )=[O (2(p +p +p ))]=η 4⊗[O (−2p )]=ϕ(t) 4⊗[O (−2p )],
1 2 3 E 1 2 3 ⊗ E 4 ⊗ E 4
hence ϕ(r ) = [O (2p )]. Let ϕ : E → Pic(2)(E) be the isomorphism defined by p 7→ [O (p+p )].
4 E 4 4 E 4
Then the isomorphism ϕ−41◦ϕ:E′ →E satisfies (ϕ−41◦ϕ)(ri)=pi for i=1,...,4, and
η =[OE(ϕ−41(η)+p4)]=[OE((ϕ−41◦ϕ)(t)+(ϕ−41◦ϕ)(r4))]=(ϕ−41◦ϕ) [OE′(t+r4)].
∗
Next we assume that ϕ(s )=[O (p +p )] for i=1,2,3. Then we have
i E i 4
ϕ(s1)⊗ϕ(s2)⊗ϕ(s3)=[OE(p1+p2+p3+3p4)]=ϕ(t)⊗4⊗[OE(−p1−p2−p3+p4)],
hence ϕ(r4) = [OE(p1 + p2 + p3 − p4)]. Let ϕ¯4 : E → Pic(2)(E) be the isomorphism defined by
p 7→ [OE(p1+p2+p3−p)]. Then the isomorphism ϕ−¯41 ◦ϕ : E′ → E satisfies (ϕ−¯41 ◦ϕ)(ri) = pi for
i=1,...,4, and
η =η⊗2⊗η∨ =η⊗2⊗[OE(−p1−p2−p3+ϕ−¯41(η))]=[OE(ϕ−¯41(η)+p4)]=(ϕ−¯41◦ϕ)∗[OE′(t+r4)],
where η denotes the class of the dual invertible sheaf of η.
∨
Proposition 2.10. The effective canonical divisor K = f 1(η) is singular if and only if C is a
Y −
hyperelliptic curve. In this case, the fiber K =f 1(η) is of type I .
Y − 1 4
Proof. Let q ∈ C be the ramification point of π : C → E with π(q ) = p . If C is a hyperelliptic
i i i
curve, then the hyperelliptic involution h commutes with the involution σ, hence h acts on the set
{q ,...,q } ⊂ C. By [2, Lemma (1.9)], we have h(q ) 6= q for 1 ≤ i ≤ 4. We may assume that
1 4 i i
q =h(q ) andq =h(q ). ThenwehaveO (q +q )≃O (q +q ),hence π η =[O (q +···+q )]=
2 1 4 3 C 1 2 C 3 4 ∗ C 1 4
[O (2q +2q )]=π [O (p +p )]. Sinceπ :Pic(E)→Pic(C)isinjective,wehaveη =[O (p +p )]=
C 1 2 ∗ E 1 2 ∗ E 1 2
[O (p +p )], and the fiber K =f 1(η) is singular of type I .
E 3 4 Y − 1 4
Conversely, if K = f 1(η) is singular, then η = [O (p + p )] for some 1 ≤ i < j ≤ 4. We
Y − E i j
may assume that η = [O (p + p )]. Since [O (2q +2q )] = π η = [O (q +··· +q )], we have
E 1 2 C 1 2 ∗ C 1 4
O (q +q )≃O (q +q ). Itmeans that the linearsystem|O (q +q )| givesthe morphismofdegree
C 1 2 C 3 4 C 1 2
2 to the projective line.
2.3 The j-function
Proposition 2.11. The j-function j :B →P1 for theelliptic surface f :Y →B is a covering of degree
12, and it is ramified at the canonical point η ∈B.
The statement that the j-function is ramified at η ∈ B follows from Remark 2.7. For a point
ξ ∈B\Σ,thefiberf 1(ξ)isthedoublecoveringofψ 1(ξ)≃P1 branchedat4pointsψ 1(ξ)∩ 4 Γ .
− − − i=1 i
Let i :ψ 1(ξ)→P1 =C∪{∞} be the isomorphism defined by
ξ −
S
i (ψ 1(ξ)∩Γ )=∞, i (ψ 1(ξ)∩Γ )=0, i (ψ 1(ξ)∩Γ )=1.
ξ − 1 ξ − 2 ξ − 3
We defines the λ-function by
λ:B\Σ−→C; ξ 7−→i (ψ 1(ξ)∩Γ ).
ξ − 4
6
Then f 1(ξ) is isomorphic to the compactification of
−
{(x,y)∈A2 |y2 =x(x−1)(x−λ(ξ))}.
Lemma2.12. Theλ-functioncoincideswiththemorphism Φ :B →P1,bytakingthecoordinate
of P1 =C∪{∞} as |OB(2η)|
Φ ([O (p +p )])=∞, Φ ([O (p +p )])=0, Φ ([O (p +p )])=1.
|OB(2η)| E 1 4 |OB(2η)| E 2 4 |OB(2η)| E 3 4
Proof. We fix a point p ∈E with η =[O (2p )]. Let E\{p } be defined by the equationy2 =g(x) in
0 E 0 0
A2, where g(x) is a separated monic cubic polynomial. We assume that p ,...,p ∈E\{p }, because
1 4 0
the case p =p is easier. We denote the coordinate ofthe point p ∈E by (x,y)=(a ,b ). For a point
1 0 i i i
p=(a,b)∈E\{p }, we define a rational function by
0
y+b
f : E −→P1; (x,y)7−→ ∈P1 =C∪{∞},
p
x−a
which is the morphism given by the linear system |O (p+p )|. Hence we have
E 0
f (p )−f (p ) f (p )−f (p ) b4+b − b2+b b3+b − b1+b
λ(ξ)= p 4 p 2 · p 3 p 1 = a4−a a2−a a3−a a1−a
fp(p4)−fp(p1) fp(p3)−fp(p2) b4+b − b1+b b3+b − b2+b
a4−a a1−a a3−a a2−a
for general ξ =[O (p+p )]∈Pic(2)(E). Let p ∈E be the point with O (p +p +p )≃O (3p ).
E 0 ij E 1 j ij E 0
We assume that p ∈ E\{p }, because the case p = p is easier. We denote the coordinate of the
ij 0 12 0
point pij ∈E by (x,y)=(aij,bij), and we set r = ab44−−ba11 + ab33−−ba22. Then we have
(rr2==ab44(a−−4ab−11a+3)(aab233−−−aba212)==ab(44a−−4−ba22a2+)(aab333−−−aba111) == a(ba444−−−aba331)+(a3ab−22−−a2ba)11,,
a24−a14 a34−a14 a34−a24
where these equalities are proved from O (p +···+p ) ≃ O (4p ) by the elementary computation.
E 1 4 E 0
We set
b −b a b −a b b −b a b −a b
4 2 4 2 2 4 3 1 3 1 1 3
A(x)= x+ , B(x)= x+ ,
a −a a −a a −a a −a
4 2 4 2 3 1 3 1
b −b a b −a b b −b a b −a b
4 1 4 1 1 4 3 2 3 2 2 3
C(x)= x+ , D(x)= x+ .
a −a a −a a −a a −a
4 1 4 1 3 2 3 2
Then
(a −a )(a −a ) (A(a)+b)(B(a)+b) b2+A(a)B(a)+(A(a)+B(a))b
4 2 3 1
λ(ξ)= · =c· ,
(a −a )(a −a ) (C(a)+b)(D(a)+b) b2+C(a)D(a)+(C(a)+D(a))b
4 1 3 2
where we set c= ((aa44−−aa21))((aa33−−aa12)). Since (a24,b24)=(a13,−b13), we have
A(x)+B(x)=r(x−a )
24
and
g(x)=A(x)2+(x−a )(x−a )(x−a )=B(x)2+(x−a )(x−a )(x−a ),
2 4 24 1 3 24
hence
2b2+2A(a)B(a)=2g(a)+2A(a)B(a)
=A(a)2+(a−a )(a−a )(a−a )+B(a)2+(a−a )(a−a )(a−a )+2A(a)B(a)
2 4 24 1 3 24
=(A(a)+B(a))2+(a2−(a +a +a +a )a+a a +a a )(a−a )
1 2 3 4 1 3 2 4 24
=r2(a−a )2+(a2−(a +a +a +a )a+a a +a a )(a−a ).
24 1 2 3 4 1 3 2 4 24
7
Since (a ,b )=(a ,−b ), by the same way, we have
14 14 23 23
C(x)+D(x)=r(x−a )
14
and
2b2+2C(a)D(a)=r2(a−a )2+(a2−(a +a +a +a )a+a a +a a )(a−a ).
14 1 2 3 4 1 4 2 3 14
Hence we have
a−a r2(a−a )+(a2−(a +a +a +a )a+a a +a a )+2rb
24 24 1 2 3 4 1 3 2 4
λ(ξ)=c· · .
a−a r2(a−a )+(a2−(a +a +a +a )a+a a +a a )+2rb
14 14 1 2 3 4 1 4 2 3
Since c= aa3344−−aa1244 and
r2(a−a )+(a2−(a +a +a +a )a+a a +a a )=r2(a−a )+(a2−(a +a +a +a )a+a a +a a ),
24 1 2 3 4 1 3 2 4 14 1 2 3 4 1 4 2 3
we have
a −a a−a
34 14 24
λ(ξ)= · =Φ (ξ).
a34−a24 a−a14 |2η|
Proof of Proposition 2.11. By Lemma 2.12, the j-function j(ξ)=28(λλ((ξξ))22(−λλ(ξ(ξ))+11))23 is of degree 12, and
it is ramified at η ∈B. −
Corollary 2.13. If C is not a hyperelliptic, then the Kodaira-Spencer map T | → H1(K ,T ) at
E η Y KY
the canonical fiber K =f 1(η) is zero.
Y −
Proof. By Proposition 2.10, the canonical divisor K is nonsingular, and by Proposition 2.11, the
Y
differential of the j-function is zero at η ∈B, hence the Kodaira-Spencer map is zero.
3 Prym varieties and duality
3.1 Prym varieties
In this section, we review some results on the Prym varieties [16] for the case when they are defined
frombielliptic curvesofgenus 3. In this case,the Prymvarietyis a(1,2)-polarizedabeliansurface,and
it is intensively studied by Barth in [2]. Particularly, we introduce the duality of bielliptic curves of
genus3,whichis givenin[17], andiscalledPantazis’bigonalconstruction. We willgiveanexplanation
for the duality through the elliptic surface Y.
Let π : C → E be a bielliptic curve of genus 3, and let σ : C → C be its bielliptic involution. The
Prym variety P =Prym(C/E) is defined as the image of the homomorphism
id σ∗
J(C) −−→ J(C),
where J(C) denotes the Jacobian variety of C. Since π : C → E has ramification points, the kernel of
the norm homomorphism
J(C)≃Pic(0)(C)−N→Pic(0)(E)≃J(E)
is connected, and it coincides with P. The Prym variety P is a (1,2)-polarized abelian surface by the
restriction Θ | of the theta divisor Θ on J(C). Let A be the abelian surface defined as the cokernel
C P C
of the homomorphism π : J(E) → J(C). Then A is isomorphic to the dual abelian variety Pic(0)(P)
∗
8
of P. The natural composition P ֒→ J(C) ։ A is an isogeny, which coincides with the polarization
isogeny
P −→A≃Pic(0)(P); x7−→[t O (Θ | )⊗O (−Θ | )],
∗x P C P P C P
wheret :P →P denotesthetranslationbyx∈P. Letι:C →Abe thecompositionC ֒→J(C)։A,
x
whereC ֒→J(C)istheAbel-JacobiembeddingdefinedbyfixingapointofC. By[2,Proposition(1.8)],
the morphism ι is a closed immersion, and it gives a (1,2)-polarization on A. By [2, Duality theorem
(1.12)], the class [ι(C)] ∈ H2(A,Z) corresponds to 2[Θ | ] ∈ H2(P,Z) by the pull-back H2(A,Z) ֒→
C P
H2(P,Z), which means that the polarizations [ι(C)] ∈ H2(A,Z) and [Θ | ] ∈ H2(P,Z) are dual to
C P
each other in the sense of [4, Section 14.4].
Lemma 3.1. The morphism
C(2) −→A; q+q′ 7−→ι(q)+ι(q′)
is a generically finite double covering, and it induces a birational morphism C(2)/(σ(2)◦κ)→A, where
κ denotes the involution given in Remark 2.7.
Proof. Let ∆ ⊂C(2) be the curve defined by
σ
∆ ={q+q ∈C(2) |q =σ(q)}.
σ ′ ′
Then ∆ and ∆ are contracted by the morphism C(2) → A. Since the involution σ(2) ◦κ does not
σ h
have isolated fixed points, the quotient C(2)/(σ(2)◦κ) is a nonsingular surface, and the image of ∆ in
σ
C(2)/(σ(2)◦κ) is a (−1)-curve. If ∆ 6=∅, then the image of ∆ in C(2)/(σ(2)◦κ) is also a (−1)-curve
h h
disjoint from the image of ∆ . Since
σ
OC(q+q′)⊗π∗η ≃OC(σ(2)◦κ(q+q′))⊗π∗OE(π(q)+π(q′))
forq+q ∈C(2),themorphismC(2) →AfactorsthroughthequotientC(2)/(σ(2)◦κ). Forq+q ,r+r ∈
′ ′ ′
C(2)\(∆ ∪∆ ), we assume that ι(q)+ι(q )=ι(r)+ι(r ) in A. Then
σ h ′ ′
OC(q+q′−r−r′)=π∗OE(π(x)−π(r))
forsomex∈C,hencewehaveO (q+q +σ(r))=O (x+σ(x)+r ). Ifh0(C,O (q+q +σ(r)))=1,then
C ′ C ′ C ′
q+q =r+r ,becauseq+q ∈/ ∆ . Ifh0(C,O (q+q +σ(r))) =2,thenh1(C,Ω1(−q−q −σ(r)))) =1,
′ ′ ′ σ C ′ C ′
hence there is a unique point y ∈C such that q+q +σ(r)+y ∈|Ω1|. Since x+σ(x)+r +y, σ(x)+
′ C ′
x+σ(r )+σ(y)∈|Ω1|, we have O (r +y)≃O (σ(r )+σ(y)). If y 6=σ(r ), then σ(x)=h(x), but it
′ C C ′ C ′ ′
is a contradictionto [2, Lemma (1.9)]. Hence we have y =σ(r ) and q+q +σ(r)+σ(r )∈|Ω1|. Since
′ ′ ′ C
q+q ∈/ ∆ , we have r+r = σ(2) ◦κ(q+q ). Hence C(2)/(σ(2)◦κ) → A should be the blow-down of
′ h ′ ′
the above (−1)-curves.
Remark 3.2. The fixed locus of the involutionσ(2)◦κ on C(2) is givenas a component D of the fiber of
ψ◦π(2) :C(2) →Pic(2)(E) at the canonical point η ∈B =Pic(2)(E);
D (if C is non-hyperelliptic),
(ψ◦π(2)) 1(η)=
−
(D+∆h (if C is hyperelliptic).
We remark that D is the normalization of the “dual” C of C, which is defined later.
∨
Remark 3.3. When we take a ramification point q of π :C →E as the base point of the embedding
i
ι: C −→A; q 7−→O (q−q ) mod π Pic(0)(E),
C i ∗
9
the involution σ(2) : C(2)/(σ(2) ◦κ) → C(2)/(σ(2)◦κ) is compatible with the involution (−1) on A.
A
HencethequotientC(2)/hσ(2),κiisbirationaltotheKummersurfaceKm(A)ofA. Infact,thequotient
Y/κ has 12 ordinary double points, and Km(A) is the minimal resolution of Y/κ;
Y −→ C(2)/σ(2)
↓ ↓
Km(A) −→ Y/κ −→ C(2)/hσ(2),κi −→ A/(−1) .
A
By Remark 2.7, we have a fibration g : Km(A) → Y/κ → P1 = |O (2η)| of curves of genus 1. If
1 B
f : Y → B does not have a singular fiber of type I , then g : Km(A) → P1 has 6 singular fibers.
1 4 1
They contains 3 singular fibers of type I which come from the singular fibers of f : Y → B, and the
1 2
other 3 singular fibers are of type I which appear at ξ ∈B with O (2ξ)≃O (2η) and ξ 6=η. On the
0∗ B B
other hand, the linear pencil |O (ι(C))| defines a fibration
A
Φ:A∼ −→P1 =|OA(ι(C))|,
ofcurvesofgenus3bytheeliminationofthebasepointofthecompletelinearsystem|O (ι(C))|, where
A
A →A is the blow-up at 4 points ι(q ),...,ι(q ). If the fiber of C =Φ 1(t) is not singular, then C
∼ 1 4 t − t
is a bielliptic curve of genus 3, and the bielliptic involution σ :C →C is defined from the involution
t t t
(−1) by [2, Proposition (1.6)]. Hence the fibration Φ factors through the quotient A /(−1) ≃Y/κ,
A ∼ A
anditgivesanotherfibrationg :Km(A)→P1 =|O (ι(C))|ofcurvesofgenus1. LetE bethefiberof
2 A t
g att∈P1. ThenthePrymvarietyP(C /E )isisomorphictoP =P(C/E)by[2,Proposition(1.10)].
2 t t
Let π : C →E be a bielliptic curve of genus 3. We denote by W ⊂ Pic(2)(C) the image of natural
morphism C(2) → Pic(2)(C), and denote by P ⊂ Pic(2)(C) the fiber of the norm homomorphism
ξ
N : Pic(2)(C) → Pic(2)(E) at ξ ∈ Pic(2)(E). We set D = W ∩P . Since the divisors W and P on
ξ ξ ξ
Pic(2)(C) is stable under the involution
Pic(2)(C)−→Pic(2)(C); [L]7−→[σ L],
∗
it acts on D . If D is nonsingular, then the quotient morphism
ξ ξ
Dξ =W ∩Pξ ≃(ψ◦π(2))−1(ξ)−→(ψ◦φ)−1(ξ)
is a bielliptic curve of genus 3, where we recall the notation in Section 2;
C(2) −→ C(2)/σ(2)
ψ◦π(2) ց ւψ◦φ
Pic(2)(E).
We define the dual of C by C = D , and denote by π : C → E the quotient by the involution,
∨ η ∨ ∨ ∨
where η ∈ Pic(2)(E) is the canonical point of the covering π : C → E. If C is not hyperelliptic, then
by Proposition2.10, the dual π :C →E is a nonsingularbielliptic curve. If C is hyperelliptic, then
∨ ∨ ∨
E is an irreducible rational curve with one node, and C is an irreducible curve of geometric genus 2
∨ ∨
with one node, which is given by contracting the hyperelliptic locus ∆ form (ψ◦π(2)) 1(η)=D∪∆
h − h
in Remark 3.2. We remark that P is isomorphic to the Prym variety P =Prym(C/E)⊂J(C) by the
ξ
translation
Pic(2)(C)⊃Pξ −≃→P ⊂Pic(0)(C); [L]7−→[L⊗π∗OE(−p)],
where p ∈ E is a point with ξ = [O (2p)]. Let D ⊂ P be the image of D = W ∩P by the above
E p ξ ξ
translation, by abusing the notation. We fix a point p ∈ E with η = [O (2p )]. Then the divisor
0 E 0
C ≃ D ⊂ P defines the (1,2)-polarization on the Prym variety P. We recall that the embedding
∨ p0
10