The Prym map of degree-7 cyclic coverings

0.27 MB·English

by  Herbert Lange

THE PRYM MAP OF DEGREE-7 CYCLIC COVERINGS
HERBERTLANGEAND ANGELA ORTEGA
Abstract. We study the Prym map for degree-7 ´etale cyclic coverings over a curve of genus
2. We extend this map to a proper map on a partial compactification of the moduli space and
5
provethat the Prym map is generically finiteonto its image of degree 10.
1
0
2

n
a 1. Introduction
J
9 Consider an ´etale finite covering f : Y X of degree p of a smooth complex projective curve
→
2 X of genus g 2. Let Nm : JY JX denote the norm map of the corresponding Jacobians.
f
≥ →
One can associate to the covering f its Prym variety
]
G
P(f):= (KerNm )0,
f
A
. theconnected componentcontaining 0ofthekernelofthenormmap,whichisanabelianvariety
h
of dimension
t
a
dimP(f)= g(Y) g(X) = (p 1)(g 1).
m
− − −
[ The variety P(f) carries a natural polarization namely, the restriction of the principal polariza-

tion Θ of JY to P(f). Let D denote the type of this polarization. If moreover f : Y X is
1 Y →
v a cyclic covering of degree p, then the group action induces an action on the Prym variety. Let
1 denote the moduli space of abelian varieties of dimension (p 1)(g 1) with a polarization
D
1 B − −
of type D and an automorphism of order p compatible with the polarization. If denotes
5 Rg,p
7 the moduli space of ´etale cyclic coverings of degree p of curves of genus g, we get a map
0
Pr :
. g,p g,p D
1 R → B
0 associating to every covering in its Prym variety, called the Prym map.
g,p
5 R
Particularly interesting are the cases where dim = dim . For instance, for p = 2 this
1 Rg,p BD
: occurs only if g = 6. In this case the Prym map Pr6,2 : 6 5 is generically finite of degree
v R → A
27 (see [6]) and the fibers carry the structure of the 27 lines on a smooth cubic surface. For
i
X (g,p) = (4,3), it is also known that Pr is generically finite of degree 16 onto its 9-dimensional
4,3
r image (see [7]) .
a BD
In this paper we investigate the case (g,p) = (2,7), where dim = dim . The main
g,p D
R B
result of the paper is the following theorem. Let G be the cyclic group of order 7.
Theorem 1.1. For any ´etale G-cover f : C C of a curve C of genus 2, the Prym variety
→
Pr(f) is an abelian variety of dimension 6 with a polarization of type D = (1,1,1,1,1,7) and a
e
G-action. The Prym map
Pr :
2,7 2,7 D
R → B
is generically finite of degree 10.
1991 Mathematics Subject Classification. 14H40, 14H30.
Key words and phrases. Prym variety,Prym map.
The second author was supported byDeutscheForschungsgemeinschaft, SFB647.
1
2 HERBERTLANGEANDANGELAORTEGA
The paper is organized as follows. First we compute in Section 2 the dimension of the moduli
space when (g,p) = (2,7). We also show that the case (g,p) = (2,6), mentioned in [7] as
D
B
one where the dimension of equals the dimension of the image of the Prym map, does not
2,7
R
have this property. In Sections 3-5, we extend the Prym map to a partial compactification of
admissible coverings such that Pr : is a proper map. We prove the generic
2,7 2,7 2,7 D
R R → B
finiteness of the Prym map in Section 6 by specializing to a curve in the boundary. In order to
e e
compute the degree of the Prym map we describe in Section 7 a complete fiber over a special
abelian sixfold with polarization type (1,1,1,1,1,7), and in Section 8 we give a basis for the
Prym differentials for the different types of admissible coverings appearing in the special fiber.
Finally, in Section 9 we determine the degree of the Prym map by computing the local degrees
along the special fiber.
We would like to thank E. Esteves for his useful suggestions for the proof of Theorem 6.2.
The second author is thankful to G. Farkas for stimulating discussions.
2. Dimension of the moduli space
D
B
As in the introduction, let denote the moduli space of non-trivial cyclic ´etale coverings
2,7
R
f : C C of degree 7 of curves of genus 2. The Hurwitz formula gives g(C) = 8. Hence
→
the Prym variety P = P(f) is of dimension 6 and the canonical polarization of the Jacobian
e e
JC induces a polarization of type (1,1,1,1,1,7) on P. Let σ denote an automorphism of JC
generating the group of automorphisms of C/C. It induces an automorphism of P, also of
e e
order 7, which is compatible with the polarization. The Prym map Pr : is the
2,7 2,7 D
e R → B
morphism defined by f P(f). Here is the moduli space of abelian varieties of dimension
D
7→ B
6 with a polarization of type (1,1,1,1,1,7) and an automorphism of order 7 compatible with
the polarization. The main result of this section is the following proposition.
Proposition 2.1.
dim = dim = 3.
D 2,7
B R
Proof. Clearly dim = dim = 3. So we have to show that also dim = 3. For this we
2,7 2 D
R M B
use Shimura’s theory of abelian varieties with endomorphism structure (see [12] or [5, Chapter
9]).
Let K = Q(ρ ) denote the cyclotomic field generated by a primitive 7-th root of unity ρ .
7 7
Clearly coincides with one of Shimura’s moduli spaces of polarized abelian varieties with
D
B
endomorphism structure in K. The field K is a totally complex quadratic extension of a totally
real number field of degree e =3. Denote
0
dimP
m := = 2.
e
0
Thepolarization of P dependson the lattice of P and a skew-hermitian matrix T M (Q(ρ )).
m 7
∈
For each of the e real embeddings of the totally real subfield of Q(ρ ) consider an extension
0 7
Q(ρ ) ֒ C and let (r ,s ) be the signature of T considered as a matrix in the extension C. The
7 ν ν
→
signature of T is defined to be the e -tuple ((r ,s )....,(r ,s )) with
0 1 1 e0 s0
r +s = m = 2.
ν ν
for all i. Then, according to [12, p. 162] or [5, p. 266, lines 6-8] we have
e0
(2.1) dim = r s 3
D ν ν
B ≤
Xν=1
with equality if and only if r = s = 1 for all ν.
ν ν
3
On the other hand, in Section 6 we will see that the map Pr is generically injective. This
2,7
implies that
dim dim = 3
D 2,7
B ≥ R
which completes the proof of the proposition. (cid:3)
Remark 2.2. According to [11] we know that P is isogenous to the product of a Jacobian of
dimension 3 with itself. Then End (P) is not a simple algebra. Hence, if one knew that Pr
Q 2,7
is dominant onto the component , then [5, Proposition 9.9.1] implies that r = s = 1 for
D ν ν
B
ν = 1,2,3 which also gives dim = 3.
D
B
Remark 2.3. In [7] it is claimed that also the Prym map Pr : satisfies dim =
2,6 2,6 D D
R → B B
dim = 3. However, we claim that the dimension of in this case cannot be 3.
2,6 D
R B
For the proof note that the cyclotomic field of the 6-th roots of unity Q(ρ ) is the imaginary
6
quadratic field Q(√ 3). So with the notation of the proof of Proposition 2.1 we have in this
−
case
dimP
e = 1 and m = =5
0
e
0
and we have that
dim = r s with r +s = 5.
D 1 1 1 1
B
So for (r ,s ) there are the following possibilities (up to exchanging r and s which does not
1 1 1 1
modify dim ):
D
B
(r ,s ) = (5,0),(4,1) or (3,2)
1 1
giving respectively
dim = 0,4 or 6
D
B
which in any case is different from 3.
3. The condition (*)
In this section we study the Prym map for coverings of degree 7 between stable curves. Let
G = Z/7Z be the the cyclic group of order 7 with generator σ and f : C C be a G-cover of
→
a connected stable curve C of arithmetic genus g. We fix in the sequel a primitive 7-th root of
e
the unity ρ. In this section we assume the following condition for the covering f.
The fixed points of σ are exactly the nodes of C
( ) and at each node one local parameter is multiplied by
∗  ρδ and the other by ρ−δ for some δ, 1 δ 3.e
≤ ≤
As in [4] we have f∗ωC ≃ ωCe which implies
p (C) = 7g 6.
a
−
Let N respectively N be the normalizationeof C, respectively C, and f : N N the induced
→
map. At each node s of C we make the usual identification
e e e e
e ∗/ ∗ C∗ Z Z.
Ks Os ≃ × ×
Then the action of σ on ∗/ ∗ is:
Ks Os
σ∗((z,m,n) )= (ρδ(m+n)z,m,n)
s
4 HERBERTLANGEANDANGELAORTEGA
for some δ, 1 δ 3. Here we label the branches at the node s such that a local parameter at
≤ ≤
the first branch (corresponding to m) is multiplied by ρδ with 1 δ 3. Then we have
≤ ≤
6
f ((z,m,n) ) = ( (σk)∗z,m,n) = (z7,m,n) .
∗ s f(s) f(s)
kY=0
We define the multidegree of a line bundle L on C by
degL = (d1e,...,dv)
where v is the number of components of C and d is the degree of L on the i-th component of
i
C.
e
Leemma 3.1. Let L PicC with NmL e. Then
∈ ≃ OC
e L M σ∗M−1
≃ ⊗
for some M PicC. Moreover, M can be chosen of multidegree (k,0,...,0) with 0 k 6.
∈ ≤ ≤
Proof. As in [9, Lemema 1] using Tsen’s theorem, there is a divisor D such that L e(D) and
≃ OC
f∗D = 0. Writing D = x∈Ceregx+ s∈Cesing(zs,ms,ns), we see that, at singular points s ∈ C,
D is a linear combinatioPn of divisorsPx σ∗x for x Creg and (ρ,0,0)s (note that it suffices teo
− ∈
show that (ρ,0,0) is in the image of 1 σ∗ because (ρ,0,0)+(ρ,0,0) = (ρ2,0,0)). If at s δ is
− e
as above, choose an integer i such that iδ 1 mod 7. Then
− ≡
(1,i,0) σ∗(1,i,0) = (ρ−δi,0,0) = (ρ,0,0).
−
Hence D = E σ∗E for some divisor E on C. Moreover,
−
(1,1, 1)s σ∗e(1,1, 1)s = (1,0,0)s
− − −
which altogether implies that L M σ∗M−1, where M can be chosen of multidegree as
stated. ≃ ⊗ (cid:3)
Let P denote the Prym variety of f :C C, i.e. the connected component of 0 of the kernel
→
of norm map Nm : JC JC. By definition it is a connected commutative algebraic group.
→ e
Lemma 3.1 implies that P is the variety of line bundles in kerNm of the form M σ∗M−1 with
e ⊗
M of multidegree (0,...,0).
Proposition 3.2. Suppose p (C)= g. Then P is an abelian variety of dimension 6g 6.
a
−
Proof. (As in [4] and [7]). Consider the following diagram of commutative algebraic groups:
(3.1) 0 // T // JC // JN // 0
eNm eNm eNm
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // T // JC // JN // 0
where the vertical arrows are the norm maps and T and T are the groups of classes of divisors
of multidegree (0,...,0) with singular support. Since f∗ is injective on T and Nm f∗ = 7, the
e ◦
norm on T is surjective and
e kerNm|Te ≃ T7 = {points of order 7 in T}.
On the other hand, Lemma 3.1 impliesethat e
ker(Nm :JC JC) P Z/7Z.
→ ≃ ×
e
5
Hence one obtains an exact sequence
(3.2) 0 T P Z/7Z R 0.
7
→ → × → →
Suppose first that C and hence C aere nonsingular. Then C = N and hence T = 0. Since
ker(JN JN) has 7 components, P is an abelian variety.
→ e e e e
Suppose that C and thus also C have s > 0 singular points. Then dimT = dimT = s. Then
e
R is an abelian variety, since f is ramified. We get a surjective homomorphism P R with
e e →
kernel consisting of 7s−1 elements. Hence also P is an abelian variety. Moreover,
e
dimP = dimR = dimJC dimJC.
−
Now, if C has s nodes and N has t connected componeents, then also C has s nodes and N has
t connected components. This implies
e e
dimJC dimJC = p (C) p (C) = 6g 6.
a a
− − −
e e (cid:3)
Let Θ denote the canonical polarization of the generalized Jacobian JC (see [4]). It restricts
to a polarization Σ on the abelian subvariety P. We denote the isogeny P P associated to Σ
e e→
by the same letter.
b
Proposition 3.3. The polarization Σ on P is of type D := (1,...,1,7,...,7) where 7 occurs
g 1 times and 1 occurs 5g 5 times.
− −
Proof. (Following [7, Proposition 2.4]). Consider the isogeny
h :P JN JN.
× →
Clearly ker(h) P[7]. As in [4, Proof of Theorem 3.7],eone sees that ker(h) is isomorphic to the
⊂
group of points a JC[7] such that f∗a P and
∈ ∈
dim ker(h) = dim (JC[7]) 1 = 2g t 1,
F7 F7 − − −
where t = dimT = dimT. Now ker(h) is a maximal isotropic subgroup of the kernel of the
polarizationofP JN,sincethispolarizationisthepullbackunderhoftheprincipalpolarization
× e
of JN. Thisimplies dim (kerΣ JN[7])= 4g 2t 2. Sincedim (JN[7]) = 2g 2t, itfollows
that dim (kerΣ)= 2g F72. Sinc×e kerΣ P[7],−this−gives the asseFr7tion. − (cid:3)
e F7 − ⊂
4. The condition (**)
As in the last section let f : C C be a G-covering of stable curves. Recall that a node
→
z C is either
∈ e
of index 1, i.e Stabz = 1 in which case f−1(f(z)) consists of 7 nodes which are cyclicly
e
• | |
permuted under σ or
of index 7, i.e. Stabz = 7 in which case z is the only preimage of the node f(z) and f
• | |
is totally ramified at both branches of z. Since σ is of order 7, the two branches of z are
not exchanged.
We also call a node of C of index i if a preimage (and hence every preimage) under f is a
node of index i. We assume the following condition for the G-covering f : C C of connected
→
stable curves:
e
6 HERBERTLANGEANDANGELAORTEGA
p (C) = g and p (C) = 7g 6;
a a
−
 σ is not the identity on any irreducible component of C;
e
( ) if at a fixed node of σ one local parameter is multiplied by ρi, the other is
∗∗  multiplied by ρ−i, whereρ denotes a fixed 7-th root oef unity;
 P := Pr(f)is an abelian variety.




Underthese assumptionsthe nodesof C areexactly the preimages of thenodes of C. We denote
for i =1 and 7:
e
n := the number of nodes of C of index i, i.e. nodes whose preimage consists of 7 nodes
• i i
of C,
c := the number irreducible components of C whose preimage consists of 7 irreducible
• i e i
components of C,
r := the number of fixed nonsingular points under σ.
• e
Lemma 4.1. The covering satisfies ( ) if and only if r =0 and c = n .
1 1
∗∗
In particular, any covering satisfying ( ) is an admissible G-cover (for the definition see
∗∗
Section 5 below).
Proof. (As in [4] and [7]). Let N respectively N be the normalization of C respectively C. The
covering f : N N is ramified exactly at the points lying over the fixed points of σ : C C.
→ e e →
Hence the Hurwitz formula says
e e e e
p (N) 1 = 7(p (N) 1)+3r+6n .
a a 7
− −
So e
p (C) 1 = p (N) 1+7n +n
a a 1 7
− −
= 7(p (N) 1)+3r+7n +7n .
e ae 1 7
−
Moreover,
p (C) 1= p (N) 1+n +n
a a 1 7
− −
which altogether gives
p (C) 1 = 7(p (C) 1)+3r.
a a
− −
Hence the first condition in ( ) is equivalent to r = 0.
e
∗∗
Now wediscusstheconditionthatP isanabelianvariety. For thisconsideragain thediagram
(3.1). From the surjectivity of the norm maps it follows that P is an abelian variety if and only
if dimT = dimT. Now dimJN = p (N) n 7n +c +7c 1 and thus
a 7 1 7 1
− − −
e edimT =e(n c )+7(n c )+1
7 7 1 1
− −
and e
dimT = (n c )+(n c )+1.
7 7 1 1
− −
Hence dimT = dimT if and only if c = n . (cid:3)
1 1
Let f : Ce C be a G-covering satisfying the condition ( ) with generating automorphism
→ ∗∗
σ. We denote by B the union of the components of C fixed under σ and write
e
C = A Ae B
1 7
∪···∪ ∪
with σ(Ai)= Ai+1 where A8 = A1. e
7
Proposition 4.2. (i) If B = , then C = A A where A can be chosen connected and
1 7 1
∅ ∪···∪
tree-like and #A A = 1 for i = 1,...,7.
i i+1
∩ e
(ii) If B = , then A A = for i = 1,...,7. Each connected component of A is tree-like
i i+1 1
6 ∅ ∩ ∅
and meets B at only one point. Also B is connected.
For the proof we need the following elementary lemma (the analogue of [4, Lemma 5.3] and
[7, Lemma 2.6]) which will be applied to the dual graph of C.
Lemma 4.3. Let Γ be a connected graph with a fixed-poinet free automorphism σ of order 7.
Then there exists a connected subgraph S of γ such that σi(S) σi+1(S) = for i= 0,...,6 and
∩ ∅
6 σi(S) contains every vertex of Γ.
∪i=0
Proof of Proposition 4.2. (As in [4] and [7]). Let Γ denote the dual graph of C. If B = , let
∅
A correspond to the subgraph S of Lemma 4.3. Let v be the number of vertices of S, e the
1
e
number of edges of S and s the numbers of nodes of A which belong to only one component.
1
The equality c = n implies
1 1
v = e+s #A A .
1 2
− ∩
Since 1 v +e 0 and #A A 1 gives s = 0, #A A = 1 and 1 v+e = 0. So A is
1 2 1 2 1
− ≥ ∩ ≥ ∩ −
tree-like. This proves (i).
Asume B = and denote
6 ∅
t := #A A ,
1 2
• ∩
m := #A B for i =1,...,7,
i
• ∩
i := # irreducible components of A ,
• A1 1
c := # of connected components of A ,
• A1 1
n := # nodes of A .
• A1 1
Recall that assumption ( ) implies that B does not contain any node which moves under σ.
∗∗
Then
c = i and n = n +r+m.
1 A1 1 A1
For any curve we have n i +c 0 (see [4, Proof of Lemma 5.3]). Thus, if c = n ,
A1 − A1 A1 ≥ 1 1
0= n +t+m i t+m c .
A1 − A1 ≥ − A1
Since C is connected, any connected component of 7 A meets B. But then any connected
∪i=1 i
component of A meets B which implies m c . Hence
e 1 ≥ A1
0 t+m c t 0.
≥ − A1 ≥ ≥
Hence t = 0, m = c and n i +c = 0. So A A = and B is connected. (cid:3)
A1 A1 − A1 A1 i ∩ i+1 ∅
Theorem 4.4. Suppose that f : C C satisfies condition ( ). Then there exist the following
→ ∗∗
isomorphisms of polarized abelian varieties:
e
In case (i), (P,Σ) ker((JA )7 JA ) with the polarization induced by the principal polar-
1 1
≃ →
ization on (JA )7.
1
In case (ii), (P,Σ) ker((JA )7 JA ) Q, where Q is the generalized Prym variety
1 1
≃ → ×
associated to the covering B B/σ.
→
Proof. (As in [4, Theorem 5.4]). In case (i), C is obtained from the disjoint union of 7 copies
of A by fixing 2 smooth points p and q of A and identifying q in the i-th copy with p in the
1 1
e
i+1-th copy of A cyclicly. The curve C = C/G is obtained from A by identifying p and q
1 1
and f : C C is an ´etale covering. Note that JA is an abelian variety, since A is tree-like.
→ e 1 1
e
8 HERBERTLANGEANDANGELAORTEGA
Consider the following diagram
0 // C∗ // JC // (JA )7 // 0
1
Nm eNm m
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // C∗ // JC // JA // 0
1
where m is the addition map. One checks immediately that Nm : C∗ C∗ is an isomorphism.
→
This implies the assertion.
In case (ii) we have
JC (JA )7 JB and P = ker(Nm)0 ker((JA )7 JA ) Q
1 1 1
≃ × ≃ → ×
which immediaetely implies the assertion. (cid:3)
5. The extension of the Prym map to a proper map
Let denote the moduli space of non-trivial ´etale G-covers f : C C of smooth curves
g,7
R →
C of genus g and the moduli space of polarized abelian varieties of dimension 6g 6 with
D
B e −
polarization of type D with D as in Proposition 3.3 and compatible with the G-action. As in
the introduction we denote by
Pr :
g,7 g,7 D
R → B
the corresponding Prym map associating to the covering f the Prym variety Pr(f). In order
to extend this map to a proper map we consider the compactification of consisting of
g,7 g,7
R R
admissible G-coverings of stable curves of genus g introduced in [2].
Let S beafamily of stablecurves of arithmetic genusg. Afamily of admissible G-covers
X →
of over S is a finite morphism such that,
X Z → X
(1) the composition S is a family of stable curves;
Z → X →
(2) every node of a fiber of S maps to a node of the corresponding fiber of S;
Z → X →
(3) is a principal G-bundle away from the nodes;
Z → X
(4) if z is a node of index 7 in a fibre of S and ξ and η are local coordinates of the two
Z →
branches near z, any element of the stabilizer Stab (z) acts as
G
(ξ,η) (ρξ,ρ−1η)
7→
where ρ is a primitive 7-th root of unity.
Inthecase of S = SpecC wejustspeakof anadmissibleG-cover. Inthis case theramification
index at any node z over x equals the order of the stabilizer of z and depends only on x. It is
called the index of the G-cover at x. Since 7 is a prime, the index of a node is either
Z → X
1 or 7. Note that, for any admissible G-cover Z X, the curve Z is stable if and only if and
→
only if X is stable.
As shown in [2] or [1, Chapter 16], the moduli space of admissible G-covers stable of
g,7
R
curves ofgenus g is anaturalcompactification of . Clearly thecoverings satisfying condition
g,7
R
( ) are admissible and form an open subspace of .
g,7 g,7
∗∗ R R
e
Theorem 5.1. The map Pr : extends to a proper map Pr : .
g,7 g,7 D g,7 g,7 D
R → B R → B
Proof. The proof is the same as the proof of [7, Theorem 2.8] just rfeplacineg 3-fold covers by
7-fold covers. So we will omit it. (cid:3)
9
6. Generic finiteness of Pr
2,7
From now on we consider only the case g = 2, i.e. of G-covers of curves of genus 2. So
dim = dim = 3 and is the moduli space of polarized abelian varieties of type
2,7 2 D
R M B
(1,1,1,1,1,7) with G-action which is also of dimension 3. Let [f : C C] be a general
2,7
→ ∈ R
point and let the covering f be given by the 7-division point η JC.
∈ e
Lemma 6.1. (i) The cotangent space of at the point Pr ([f : C C]) is identified
D 2,7 D
B → ∈ B
with the vector space 3 H0(ω ηi) H0(ω η7−i) .
i=1 C ⊗ ⊗ C ⊗ e
(ii) The codifferentiLal of t(cid:0)he map Pr2,7 : 2,7 D at th(cid:1)e point (f,η) is given by the sum of
R → B
the multiplication maps
3
H0(ω ηi) H0(ω η7−i) H0(ω2).
C ⊗ ⊗ C ⊗ −→ C
Mi=1 (cid:0) (cid:1)
Proof. (i): Considerthecomposedmap Pr2,7 π . Thecotangent spaceoftheimage
g,7 D D
R −→ B −→ A
of [f :C C] in is by definition the cotangent at the Prym variety P of f. It is well known
D
→ A
that the cotangent space T∗ at 0 is
e P,0
6
(6.1) TP∗,0 = H0(C,ωCe)− = H0(C,ωC ⊗ηi).
Mi=1
e
According to [8] the cotangent space of at the point P can be identified with the second
D
A
symmetric product of H0(C,ωe)−. This gives
C
6 e 3
(6.2) T∗ = S2H0(ω ηi) H0(ω ηi) H0(ω η7−i)
AD,P C ⊗ ⊕ C ⊗ ⊗ C ⊗
Mi=1 Mi=1 (cid:0) (cid:1)
Since the map π : is finite onto its image and the group G acts on the cotangent
D D
B → A
space of at the point, we conclude that this space can be identified with a 3-dimensional
D
B
G-subspace of the G-space T∗ which is defined over the rationals. But there is only one such
AD,P
subspace, namely 3 H0(ω ηi) H0(ω η7−i) . This gives (i).
i=1 C ⊗ ⊗ C ⊗
(ii): Itis well knLown t(cid:0)hat thecotangent space of 2,7(cid:1)at a point(C,η) withoutautomorphism
R
is given by H0(ω2) and the codifferential of Pr : at (C,η) by the natural map
C 2,7 R2,7 → AD
S2(H0(C,ωe)−) H0(ω2). The assertion follows immediately from Lemma 6.1 (i) and
equations (6C.1) an−d→(6.2). C (cid:3)
e
Theorem 6.2. The map Pr : is surjective and hence of finite degree.
2,7 2,7 D
R → B
Proof. Since the extensionfPr eis proper according to Theorem 5.1, it suffices to show that
2,7
the map Pr is generically finite. Now Pr is generically finite as soon as its differential at
2,7 2,7
f
the generic point [f : C C] is injective. Let f be given by the 7-division point η.
2,7
→ ∈ R
According to Lemma 6.1, the codifferential of Pr at [f :C C] is given by (the sum of) the
e 2,7 →
multiplication of sections
e
µ : 3 H0(C,ω ηj) H0(C,ω η7−j) H0(C,ω2).
C,η ⊕j=1 C ⊗ ⊗ C ⊗ −→ C
Since isirreducibleand isopenanddensein , itsufficestoshowthatthemapµ
2,7 2,7 2,7 X,η
R R R
is surjective at a point (X,η) in the compactification , even if Pr is not defined at (X,η).
e R2,7 2,7
So if µ is surjective at this point, it will be surjective at a general point of . Moreover,
X,η 2,7
R
it suffices to show that µ is injective, since both sides of the map are of dimension 3.
X,η
10 HERBERTLANGEANDANGELAORTEGA
Consider the curve
X = Y Z,
∪
the union of two rational curves intersecting in 3 points q ,q ,q which we can assume to be
1 2 3
[1,0],[0,1],[1,1] respectively. A line bundle η = (η ,η ) on X of degree 0 is uniquely deter-
X Y Z
mined by the gluing of the fiber over the nodes ·ci , given by the multiplication by a
OY|qi → OZ|qi
non-zero constant c . We may assume c = 1 and since η7 we have c7 = c7 = 1. Notice
i 3 X ≃ OX 1 2
that ω = (1) and ω = (1) and the restrictions η ,η are trivial line bundles.
X|Y OY X|Z OZ Y Z
From the exact sequence
0 ( 2) ω ηi βi (1) 0
→ OZ − → X ⊗ X → OY →
follows that h0(X,ω ηi )= 1 for i = 1,...,6. Moreover, since H0(Z, ( 2)) = 0, the map
X ⊗ X OZ −
β induces an inclusion H0(X,ω ηi ) ֒ H0(Y, (1)) for i = 1,...,6.
i X ⊗ X → OY
Therefore,tostudytheinjectivityofthemapµ ,itisenoughtocheckwhethertheprojection
X,η
of 3 H0(X,ω ηi ) H0(X,ω η7−i) to H0(Y, (1)) H0(Y, (1)) is contained in
⊕i=1 X ⊗ X ⊗ X ⊗ X OY ⊗ OY
the kernel of the multiplication map
H0(Y, (1)) H0(Y, (1)) H0(Y, (2)).
Y Y Y
O ⊗ O −→ O
We claim that the line bundleω = (ω ,ω ) is uniquely determined and one can choose the
X |Y |Z
gluing c at the nodes q to be the multiplication by the same constant. To see this, first notice
i i
that, since (X,ω ) is a limit linear series of canonical line bundles,the nodes of X are necessary
X
Weierstrass points of X. Let s H0(X,ω ( 2q )) be a section giving a trivialization of ω
3 X 3 Y
∈ −
and ω away from q . For i = 1,2, we have (1) s−31 s3 (1) , which implies that
Z 3 OY |qi → OX|qi → OZ |qi
c = c . Similarly, by using a section in H0(X,ω ( 2q )) one shows that c = c .
1 2 X 2 1 3
−
A section of ω ηi (1) for i = 1,2,3 is of the form f (x,y) = a x+b y, with a ,b
X|Y ⊗ Y ≃ OY i i i i i
constants. Suppose that the sections f are in the image of the inclusion
i
H0(X,ω ηi )֒ H0(Y, (1)).
X ⊗ X → OY
By evaluating the section at the points q and using the gluing conditions one gets a = ci 1
i i 1 −
andb = ci 1. Oneobtainsasimilarconditionfortheimageofthesections ofH0(X,ω η7−i)
i 2− X⊗ X
in H0(Y, (1)). Set j = 7 i. By multiplying the corresponding sections of ω ηi and
OY − X ⊗ X
j
ω η we have that an element in the image of µ is of the form
X ⊗ X X,η
(2 ci cj)x2+(2 ci cj)y2 (2 cj ci +cjci ci cj +cicj)xy.
− 1− 1 − 2− 2 − − 1− 2 1 2− 1− 2 1 2
Hence, after taking the sum of such sections for i = 1,2,3 we conclude that there is a non-trivial
element in the kernel of µ if and only if there is a non-trivial solution for the linear system
X,η
Ax= 0 with
2 c c6 2 c c6 c6c 2 c c6
− 1− 1 − 1 2− 1 2 − 2− 2
A =  2 c2 c5 2 c2c5 c5c2 2 c2 c5 
− 1− 1 − 1 2− 1 2 − 2− 2
2 c3 c4 2 c3c4 c4c3 2 c3 c4
 − 1− 1 − 1 2− 1 2 − 2− 2 
Clearly, if c =1 for some i or c = c the determinant of A vanishes. We compute
i 1 2
1
detA= c6(c3 c5)+c5(c6 c3)+c4(c2 c )+c3(c5 c6)+c2(c c4)+c (c4 c2).
7 1 2 − 2 1 2 − 2 1 2− 2 1 2− 2 1 2 − 2 1 2 − 2
Supposethat c = 1and c = ck for some 2 k 6. Then astraightforward computation shows
i 6 2 1 ≤ ≤
that detA = 0 if and only if k = 3 or k = 5.
6