AN ABSTRACT OF THE THESIS OF
Daniel Stewart Summers
in
for the degree of Doctor of Philosophy
August 10, 1978
presented on
Mathematic s
Title: ON THE PICARD GROUP OF BRANCHED COVERINGS OF CP2
Signature redacted for privacy.
Abstract approved:
Arnold Kas
The Picard group is investigated for a certain class of nonsingular surfaces: two sheeted branched coverings of CP2. While
I confine the study to double coverings, the results apply to coverings
of any degree. The approach is to describe the Picard group in
terms of properties of the branch curve.
The Picard group of a higher dimensional covering of complex
projective space has a simple description. Let
CPn
IT W
denote
a covering, branched over a nonsingular hypersurface, and let
denote the line bundle of a hyper plane
Picard group of
H
in
CPn. If
n > 3,
[H]
the
is infinite cyclic, and generated by ir*[H].
This is a consequence of the topology of
W,
and therefore is inde-
pendent of the branch locus.
Let
ir W
CP2
denote a double covering, branched over a
nonsingular curve in CP2. The line bundle
Tr4c[H]
can always be
taken as a generator of the Picard group. Additional generators
arise when an irreducible curve in CP2 has reducible inverse
image in the double cover. Some of these curves are in one-to-one
correspondence with divisors on the branch curve possessing special
properties. Necessary and sufficient conditions are given for the
existence of such divisors. More information is obtained by observing that these divisors correspond to square roots on the Jacobian
Variety of the branch curve. Relations between these square roots on
the Jacobian Variety provide interesting linear equivalence relations
between curves on the double cover.
On the Picard Group of Branched Coverings of CP2
by
Daniel Stewart Summers
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
Completed August 1978
Commencement June 1979
APPROVED:
Signature redacted for privacy.
Professor of Mathematics
in charge of major
Signature redacted for privacy.
Chairman of Department of Mathematics
Signature redacted for privacy.
Dean of Graduate School
Date thesis is presented
Typed by Clover Redfern for
August 10, 1978
Daniel Stewart Summers
ACKNOWLEDGMENTS
It gives me great pleasure to express my gratitude to Professor
Arnold Kas for his time, patience and encouraging confidence in this
young mathematician. I am especially thankful for the inspiring
mathematical ability Professor Kas has shared with the students at
Oregon State University.
I am indebted to Professor H. C. Clemens whose interest and
insight has made a lasting contribution to my education as well as this
dissertation.
My special thanks and admiration goes to Professor Kennan T.
Smith for his contribution to a successful completion of my graduate
studies.
Having had the good fortune of being associated with excellent
teachers and committee members, I take this opportunity to thank
Professors Aldo Andreotti, David Butler, Ronald Guenther, John Lee,
Bent Petersen and Don Solmon.
Above all, I offer my devoted thanks to a loving wife Anne, my
greatest collaborator.
To Anne
TABLE OF CONTENTS
Page
Chapter
INTRODUCTION
1
DIVISORS AND LINE BUNDLES
4
SIMPLICIAL HOMOLOGY AND COHOMOLOGY GROUPS
8
ADDITIONAL TOPOLOGICAL AND NUMERICAL
INVARIANTS
THE SHEAF Tr
21
CYw
27
THE PICARD GROUP
35
APPLICATIONS
61
BIBLIOGRAPHY
68
ON THE PICARD GROUP OF BRANCHED COVERINGS OF CPz
INTRODUCTION
In this paper, we study several properties of a certain class of
complex algebraic varieties culminating in an investigation of their
Picard group. These varieties are double coverings of complex
projective n-space CPn branched over a nonsingular hypersurface.
Already, this setting is sufficiently complicated, however, the techniques involved can be expanded to study cyclic, branched coverings
of more general varieties.
The overall theme of this investigation is to exploit the heavy
dependence a covering has to its branch locus. We relate properties
of the covering to properties of the branch locus which are often
simpler or known.
An extra effort is made to maintain a self contained exposition.
Often well known facts are discussed or stated. This also serves to
fix notation.
The first four sections are devoted to computing various topo-
logical and numerical invariants of these varieties. In keeping with
the theme, they are stated in terms of properties of the branch locus.
It is not surprising that a large set of invariants are similar to that
of both
CPn
and the branch locus.
2
These invariants become useful when the investigation of the
Picard group is undertaken in Section 5. For example, if
is a double covering of CPA,
topology of
W,
for
n>3
Tr :W
CPn
it is shown, as a result of the
the Picard group of W is infinite
cyclic.
If
n
2
the situation is complicated. Consequently, most of
Section 5 is devoted to the case when W is a surface. The problem
of describing the Picard group of a particular covering divides
naturally into two problems--generators and relations.
Generators arise when irreducible curves in CP2 have
reducible inverse image in W. Necessary and sufficient conditions
are given for their existence. There are difficulties relating all of
these curves to the branch locus. This forces one to further divide
the problem by considering two types of curves. The two types are
determined by singling out those for which their only singularities are
locally irreducible. The existence of the first type of curves, that is,
irreducible curves with reducible inverse image and which have only
locally irreducible singularities, can be completely characterized by
certain divisors on the branch locus. The remaining curves evade
such a characterization.
The problem of describing the linear equivalence relations
among the inverse image of these curves suffers the same fate. It is
shown that the linear equivalence relations between the curves of the
3
first type can be reduced to linear equivalence relations between
divisors on the branch locus.
There remains a very interesting question as to whether all the
generators and relations can be described in terms of the branch
locus. This problem is discussed throughout Section 5.
The results here, go a long way toward describing the Picard
group and in a few special cases, describe it completely. The techniques developed also provide a useful way of producing coverings
with 'large' Picard group, and have some interesting applications to
the classification of surfaces.
4
1.
DIVISORS AND LINE BUNDLES
We begin by reviewing some definitions and well known facts
about divisors and line bundles.
Let W be a nonsingular complex algebraic variety. A Weil
divisor of
W
is an element of the group
Div W = free abelian group generated by the
irreducible subvarieties of codimension one.
Every meromorphic function
f
on W defines a divisor
(f) = (f) 0
where
set of
denotes the zero set of
(f)0
f.
(f)
-(f)00
f
and
(f)00
denotes the polar
The subgroup
Div
= divisors of meromorphic functions on W,
is called the group of divisors linearly equivalent to zero. The
quotient
Pic W = Div W
Div W
is called the Picard Group of
W.
by
5
There is a sheaf theoretic formulation of the above groups, and
is equivalent when
is nonsingular. Define
W
= sheaf of germs of local holomorphic functions on W.
(y)
= sheaf of germs of local nowhere zero holomorphic
functions on W.
ti.rr = sheaf of germs of local meromorphic functions on W
that are not identically zero on any component of W.
A Cartier divisor is a section of the sheaf
some covering 9,6 = (U.)
tion
{f.}
I
of
dm*
ety
that is, for
W, D can be represented by a collec-
of meromorphic functions f.I
f if. E r(U.rTh U., 01'0. When W
on
U.
I
such that
is nonsingular there is an iso-
morphism
- Div W
r(w,
given by
f
ordX(f)
X
X codim 1
providing a natural equivalence between Weil and Cartier divisors.
There is a short exact sequence of sheaves on W
(1.1)
0
-Dt*
0.
6
From the short exact sequence (1.1) we obtain a long exact
sequence of cohomoiogy groups
o
r(w, g*)
r(w, l'rt* )
r(w, QC)
The last zero is because H1(W,111*)
Thus each divisor
when W is algebraic.
=0
defines a ("6-bundle
D
o.
(5t'*)
5D,
or alternatively
a complex line bundle we shall denote by [D]. The line bundle
is the trivial line bundle if and only if
D
[D]
is a divisor of a mero-
morphic function. Also, every line bundle is the line bundle of a
divisor. In other words
Div W
Pic W - Div
W
r(w,
-- H 1 mey*).
imr (w, yrt,* )
There are advantages to both formulations. The varieties
studied here will be nonsingular, so we need not distinguish between
Weil and Cartier divisors.
Let
V
be another complex algebraic variety and
Tr :V
holomorphic. These groups are natural in the sense, that if we
define similar sheafs on
o
r(v,
6v*)
Tr*i
0 -' r(w, CY*)
V
then the diagram
r(v, -)1*)
TT*
r(w,11t* )
r(v,061)
Hi(v, o'*)
Tri
ir*I
r(w, 00')
Hl(w, 04)
o
0
W
7
commutes. Quite often the symbol
back of divisors
Tr*
:H1(W,
Tr*
I (W,
has multiple uses, e. g. , pull
pull back 01 line bundles
r(v o(1"),
)
etc.
H1 (V, e"="),
Tr*
The correct context will always be
clear, for example, we distinguish between a divisor
responding line bundle
TT* D
or
D
and its cor-
so there is no ambiguity in writing
[D],
Tr*. [D].
There is another short exact sequence of sheaves on W
z
o
ey*
o,
and a long exact sequence of cohomology groups, a portion of which is
1
H (W, Or*)
Hl(W, eV)
The homomorphism
c1
:H1 (W, (30)
its first Chern class c1(),
Each divisor
D
c1
2
H (W, Z)
H2(W, Z)
assigns a line bundle
and has the following interpretation.
on W determined a homology class. Identifying
ech and Singular cohomology, the Poincare dual of this homology
class is
c1[D].
The image of
the Neron-Severi Group of
The dimension of
H1 (W, el/*)
H2 (W, Z)
is called
W.
H1 (W, (91)
is sometimes called the irregu-
larity' of W and denoted by the symbol
Picard Group of W
in
q.
In case q = 0 the
is isomorphic to the Neron-Severi Group of
W.
This shall be the case for the algebraic varieties studied here.
lActually the irregularity is defined by
by Serre Duality dim H (W, y) = dim H(w,
1
0
q = dim H (W,Q 1), but
1
). See Section 3.
8
SIMPLICIAL HOMOLOGY AND COHOMO LOGY GROUPS
2.
In this section the simplicial homology and cohomology groups
are computed for double covers of CPn branched along a nonsingu-
lar hypersurface.
Let
Let
[H]
B
be a nonsingular hyper surface of degree
denote the line bundle of a hyperplane
hypersurface
H
is defined by the zeros of a section
B
the 2d-fold tensor product of [H],
2d
in CPn.
CPA. The
in
s
of [2dH],
and there is a commutative
diagram
\
X2
[2dH]
[dH]
7T
where
X2
CP
is the squaring map. Define W = X2-1 ° s(CPn). It is
clear W is nonsingular and algebraic. The restruction of
W gives a double covering
over,
W
locus and
B
Tr.
We shall call
B
=
Tr-1(B)
W that
the ramification
the branch locus of the covering 7 :W
CPn.
Repeating this construction we have a similar commulative
diagram
t
CPn branched along B. More-
Tr :W
is endowed with a natural involution cr:W
commutes with
TF
9
-
Xd
X2
[dF1]--e[2dH1
71.2d
CPn
where
Xd
is the
power map. As above, this gives a commu-
dth
tative diagram of nonsingular varieties
(2. 1)
W
CP
Here
Tr
along
d
B
and
branched along
is that
IAT°
is a cyclic covering of degree
W
g W°
Tr2d:
WI
CPn
branched
is a cyclic covering of degree 2d
The advantage of studying
B.
d
W°
along with W
can be regarded as a nonsingular hypersurface in
defined by the zeros of the homogeneous polynomial
2d
pn.
z
z )
f = 0 defines B in
-f(z
where
n+1
0'
CPn+1
n
Since algebraic varieties admit simplicial triangulations we
have the following lemma
Lemma (2.2). There are simplicial complexes
ct
with subcomplexes
simplicial maps
s
Tr
and
ki
and
respectively together with
and s:'"
o-
satisfying the
10
following properties.
There are homeomorphisms
CP1I,
(Here the symbol
1336
and lk
B
denotes the space of X,
)
The following diagrams commute
(3)
For every simplex
such that
Let
s
Tr
AE
(A') = A
X there is
a simplex
.
denote the identity map on W. The kernal of the
1
homomorphism
1
w*
-
H (W, Z)
cr
will be denoted by H,(W, Z)+,
F1(W, Z)
that is, the 0--invariant subgroup of
Z).
We use familiar notation found in simplicial homology theory.
For example, the simplicial maps
s
and
s
a-
induce homo-
morphisms of simplicial chain complexes
74
and
morphisms of homology groups
o-*.
The notation
Z)
and
Z),
plicial complexes
H*(W, Z)
and
and
the homo-
being the homology groups of the sim-
and j3c,
H,,(CPn, Z).
Tr
cro,
are used synonymously with
11
Theorem (2.3).
Tr
!
:H*(CPn, Z)
There exists an injective homomorphism
such that the following diagram corn-
H (W' Z)+
mutes
,+
(2.4)
2
H*(W, Z)
H: (CPn, Z)
H
2
F14,(CPn, Z),
where the horizontal maps are multiplication by 2.
Proof. First of all we shall define
on the chain level.
Tr
In this one case we will use the same symbol
Tr
for a homomor-
phism of simplicial chain complexes as well as a homomorphism of
homology.
Define
Tr!
:C q(g,Z)
C (1,Z)
by
.)
where
Tr#Ail = A.
A:
1
(2. 2), and it is clear
homology groups
i
The existence of
.
1
also apparent 7!
boundary map a
(,6+o- A!)
Tr1
#1
is guaranteed by Lemma
is independent of any choice of
A!
is a homomorphism and commutes with the
.
Therefore, we have a homomorphism of
1/4(, Z)
ott, z).
.
It
isI
12
To establish the commuting of diagram (2.4) let
a q-cycle representing [4J] E H(
Then
Z).
(A.'+o-
Tr f(Lp) =
be
1.
')
#
and
Tr !NJ)
Tr#
Ls 1+7
# i
#
0
o-
#
i
)
= 24J.
represent [0] E H(', Z)+. Thus
Conversely, let 0 =
we can write
0 -
cr
for some q+1-chain
0 = 0co
then
and
Tr
Tr
!
#
(0) =
r+cr Al )
i
#
i
= 0 + o-
= 20 + 0(cr
.
cp.
Lett.
= Tr A
#
13
Therefore
Tr
is multiplication by 2.
Tr,
Finally, we show that
q-cycle representing
[4,]
Tr
is injective. Let
E H (X, Z),
(A
where
is a q+1-chain. Suppose
ci9
be a
and suppose
r A I)
Id-c#
892
ya. Each simplex is the
co =
a
sum
ay, =
follows
say
+ o- A
#
Cr Li
#
Ai',
is a q-face of some
is a q-face of either the same simplex
or
or not (recall
belongs to
depending upon whether
It
'ya .
o-
#a'
It I
corresponds to the ramification locus of the covering). Therefore we
can write
21-1 + (v+cr
=
where
8p
0.
Let
-p
cp
v)
p
and note we still have
Now
24i
Tr# 0 1T 141 = Tritac01
= al.(21-1-1-v+
=2
Therefore
Li
0Tr
aTrel
(34
(ri+v)
bounds. This completes the theorem.
No' = Tr
14
Theorem (2. 5).
is a surjection.
F1*(CPn, Z)
Tr*: F1* (W, Z)
Proof. Let [] E H (W, Z) be represented by
cycle
Tr () =
#1
2a.Ai +
71(0 =
1
1
The
can be written
(A!+cr A I)
1
cr.A..
cx.(6!+o- A t)
1
#1
1
A.1 /
E 6C,
Now the chain
=
A
is a cycle and
Trio =
E
Therefore
Tr*
is onto.
By the Poinca/4 duality theorem there is an isomorphism
D:
Z)
H2n-q(W,
Z).
It is defined by the formula
that is, by cap product with the fundamental class
Explicitly, for any 2n-simplex
q-face of
A
and let
Fq(A)
A
let F (A)
p.W
E H2n(W, Z).
denote the front
denote the back q-face of
A.
15
is a q-cocycle representing W E
Now if
represents the fundamental class
p.
W
Hcl(W, Z)
E H2n (W, Z)
q
(A.)>
and
then
F2n-q(L.)].
Using such duality isomorphisms, the composition
D-1
H2n-q(cpn,
CP
H (CPn, Z)
Tr*
----PH
2n-q (W, Z)PH (W, Z)
gives, a priori, a second homomorphism H (CPn, Z)
Theorem (2. 6).
=D
Tr
!
Proof.
W
Tr*o
H (W, Z).
D1
CP
is the unique 2n-cycle representing
If
i
N.
CP n
E H2n(k Z) then
p.
W
E H 2n(ge, Z)
is uniquely represented
1
where again
by
CrliC/A fr
Tr A '
#
i
--- : A
i
.
i 1c=0
Letting
have
be a q-cocycle representing W E Hcl(X, Z) we
16
1
DwTr*[] = (-1)q[
<TT
q
2n-q (k
(o-k,6!)>
#
,
,).1
t
k=0
i
1
= (-1)cl[q(Tr a-# #
1.)>
F2n-q(crk
#
i
k=0
i
1
<,F (.6.)>
q
= (-1)cl[
k
a-#(F
2n-q ( .)!]
1
i k=0
q
= Tr ID
CPn
Therefore
[]
-1
D
= DW ° Tr*
Tr
(A.)> -F2n-q(L.)]
!
as desired.
CP
As for the other coverings in (2.1) we get similar injections
(TT
2d )! = D
(Tr
2d
D
)
cpn and
(Tr
d) ! = D
(d )*
Tr
commutative diagrams
1-1*(W
(Tr2d)*
Z
+
)d
2
2d
H (W
)+
2d
Tr2
, Z)
Z)
D-1
and
17
7F1*(WI, Z)+d
F1*(WI,
(Trd)*
Z)
1-1*(W, Z)
In this context
.
H(W'Z)+2d is the subgroup of
invariant under the action of the cyclic group of order
from the covering
of
d
F-1(Wl, Z)
CPn;
1T 2d-W
and
1-1*(WI, Z)+
F1*(WI, Z)
2d
induced
is the subgroup
invariant under the action of the cyclic group of order
induced from the covering Trd:W' -4' W.
Corollary (2.7). The homomorphisms
Tr*: H*(CPn, Z)
(Tr 2d)* : H*(CPn,
(h1 d)
:
1-1* ( W, Z )
H* (W, Z)
H*(WI, Z)
H* ( W , Z )
are injections.
Theorem (2. 8). The homology groups
H (W', Z)
F1,,,,(W, Z)
and
have no torsion.
Proof. First of all
a nonsingular hypersurface in
Ord),
Z)
Pn+1.
H*(W, Z)
has no torsion because W'
Secondly, as we have noted,
Z)
is
18
is injective. Therefore
Theorem (2. 9).
=
1w'
H (W, Z)+
H (W, Z)
Proof. Recall W' C
T2d
has no torsion.
H (W, Z)
n+
CP1
for all
The automorphism
.
extends to an automorphism
of
n.
q
CPn+1
T
:WI
WI,
by the
formula
t"(zo
z n:z n+1) = (z 0
-z:e2Trii2d
n
0
zn+1
)
Hence we have a commutative diagram
WI C
i
>CPn+1
T
wi c_L.cpr1+1
The corresponding commutative diagram of homology groups is
H (WI, Z)
(CPn+1, Z)
TC1*
H (W1,
i*
H (CP n+1 ,Z)
By the Lefschetz Hyperplane Theorem
q < n. Observe
planes. It follows
is an isomorphism for
is holomorphic and takes hyperplanes to hyper _n+ 1
, Z).
induces the identity on H (CP-
19
-1
Therefore
H
q
= i*
T*
0
,
°
1
* = 1 W*
Theorem (2. 8) there is equality for
fact
(Trd)
q < n which proves
q < n. By Poincarg Duality and
for
(W', Z) = H (W', Z)+2d
q
for
q i n. This together with the
is injective and the involution on W is induced from T
!
implies the theorem is true for
W.
Combining these facts together, we have the following description of the homology groups of double coverings of CPn branched
along a nonsingular hyper surface
(2. 10)
Tr
!
:H q(CPn, Z) '=' H (W, Z),
q
:
Tr
!
Hn(CPn, Z)
:-='
Hn(W,
[H*(W, Z)+]
We can find
dim H (W, Z)
X(W) From the pair
=
n
Z)+,
H*(CPn, Z)
Tr* : H* (W , Z)
if q
is surjective,
2F1*(CPn, Z).
from the Euler-Poincarg characteristic
(W, Bw)
there is long exact sequence of
homology groups from which we conclude
X(W) = X(W, Bw) + X(Bw).
By Lefschetz Duality X(W, Bw) = x(W-Bw). Substituting we have
X(W) = X(W-Bw) + x(B
).
20
Since W - B
is a covering space of CPn - B
x(Mw) =
2x(CPn-B).
Therefore
X(W) = 2X(CPn-B) + x(Bw)
= 2x(CPn-B) + x(B)
= 2X(CPn) - X(B)
= 2(n+1) - X(B)
can also be computed from the highest Chern class of W
X(W)
which we compute in the next section. We conclude this section with
Theorem (2.11). If dim W > 2 then W is simply connected.
Proof. Recall the cyclic, branched covering
(2.1). Let
Trd(x)
Trd:
W'
in
W
x E Bw'. Every closed path in W with endpoints at
is the image of a closed path in W' with endpoints at
x.
Hence the map of pointed spaces
Trd
(AT', x)
(W, Trd(x))
induces a surjection of fundamental groups
(Tr
Again, since
W'
d)*
Tr
1
(W 1)
Tr1(W).
is a nonsingular hypersurface in CPn+1, W' is
simply connected if n > 2. Therefore
Tr1(W) = 0.
21
3. ADDITIONAL TOPOLOGICAL AND NUMERICAL INVARIANTS
Let
be a compact Kahler Manifold.
X
Let
'II
and
"E',3-(
denote the holomorphic cotangent bundle and conjugate holomorphic
cotangent bundle of
X
respectively. Introduce the following
sheaves on X
A'
sheaf of germs of local differentiable sections of the
vector bundle XPT, <. ®
P
sheaf of germs of local holomorphic sections of the
vector bundle XPT*x*
= sheaf of germs of local holomorphic p-forms on X.
Note
0
C2
is by definition the sheaf 0/ , that is, the sheaf of germs
of local holomorphic functions on X.
Some well known results from Hodge Theory are:
The vector spaces
Hq(X,
)
are finite dimensional and
vanish if p+q > n.
(X,
P)
HP(X,
cl).
(Serre Duality) Hq(X, OP)
is dual to
(Hodge Decomposition) Hr(X, C)prq.r Hcl(X, OP)
HP(X,Oq)
is computed by the complex (rAP'*,-5-)
It is convenient to define the numbers
hP' q := dim Hq(X, OP).
example, (2) and (4) imply the useful fact
of OP.
For
22
q = h '0
=
h0 '
1
= 1/2 dim Hi(X, C).
= dim H1(X,
In particular, for the branched double covering
and (2.10) implies
H 1(W,
0)'
=
n > 2. This situation was
if
0
this
CPn,
W
Tr
mentioned in Section 1. We develope it further.
Consider the short exact sequence of sheaves on W
o
where
z
el*
Or
o
is the constant sheaf of integers and
Z
CY*
is the sheaf of
germs of local nowhere zero holomorphic functions on W. There is
a similar short exact sequence of sheaves on CPn. From each we
get long exact sequences of cohomology groups, and from a portion of
these long exact sequences we get a commutative diagram
H1(W,)
H1
7i*
7*
(W,*)
H2(W, Z)
I
H(cpn, 0, .c,
Tr*
1
Hi(CPfl,
I
Tr*
H2(CPn, Z)
)
H2(W, CY)
F12(CPn,
By the above result, this reduces to
(3.1)
0
0 -"'"
Hl(W,&*)
Hz(W, Z)
Tr*
Tr*
H2(W, CY)
I
Hi(CPfl, lY''t.) 1" H2(CP, Z)
I
H2(CPfl,
-.-
23
Lemma (3. 2).
,,,,
:HI (CPn , a * )
Tr*
HI (W,&*)
is an injection.
Proof. This follows from the above diagram and the fact
H2(CPn, Z)
TT*
is injective.
H2(W, Z)
Now we compute the total Chern class of
Tr :W
singular hypersurface of degree
Suppose
where again
branched along a non-
is double covering of CPn,
CPn
W,
2d.
p E Bw. We can choose local coordinates
defined locally by
centered at
... Sn
'
and local coordinates.
with
defined locally by
and such that
13
2.1'
zn = 0;
1 zn
Bw
centered at
Tr
Tr(p) E B
I ''
n
with
= 0;
B
has the local descrip-
tion
,...
Trrg
l.
Let
TW
and
CPn
and
p
the map
T
CP n
.. ' 7n-1, n2
'i'' '
) .
denote the holomorphic tangent bundles of W
respectively. There is a short exact sequence of sheaves
0
where xf
' - " z n-1'7n) = (2-
Ctrw)
(thr),,,
01(Tr*T
)
---` A
0
CPn
is supported on B.W In terms of local coordinates at
24
(thr),,,,
: 0(Tw)p
6(11-*T
)
C Pn
p
is given by
n-
n
a
z.
)
p
1---
il
J
j=1
a. a
3 az.
,..
+ 2a
nnz
azn) p
a
J
There is another exact sequence of sheaves
0 --* 6([Bw.]) ---' a([2Bw]) --'"
where F is concentrated on
0
Bw. In terms of local coordinates,
the first map is defined at each stalk over
(a)p I---*.
(a 7)
np
pE
by
BW
.
This implies the quotient sheaf 3" is isomorphic to ,s,
.
Since
these sheaves are coherent, the Chern character is defined. From
the two exact sequences we get
ch e[2Bw] = ch (Y[Bw] + ch 47
ch( 017r*T
) = ch( eTw) + chi:4-
CPn
Eliminating
ch A we obtain the formula
ch(CYT
)
W
= ch( e r *T
) + ch(MB W ])
CPn
iT
-
ch(ei[2B W ]).
25
The corresponding total Chern class is
)c([B ])c([2B
) = c(rr*T
c(T
CPn
If we let hn denote the dual cohomology class of a hyperplane
in CPn we have
c(T
since
(1+Tr*.h )n+1(1+thr*hn)(1+2thr*h
)
Tr* [d1-11 = [B
)1
by Theorem (2.3) and Lemma (3. 2).
],
Next we relate the Canonical Bundle of W to the canonical
bundle of
line bundle
CPn. By definition, the canonical bundle of W
Xn'T)..
defines a divisor
w
If
is a meromorphic section of
co
on W and [K] =
K
XnT*
a meromorphic n-form on CPn. We can pull
morphic n-form
1,
Tr (p )
. . .
7r*cp
on
z
1"n
z
centered at
, zn)dz1A...A dzn
Tr* cp = 22" f(2.. .
n
go
2
.
n
)dz 1A.
.
Ad'n
Therefore we have the following formula of divisors
Kw = Tr*K
CP
n
+B
W
.
be
back to a mero-
EB
.
Let
then
In terms of local coordinates
W.
centered at p E Bw and
49 = f(z 1,
.
C2n
is the
.
26
Therefore
(3. 3)
[Kw] = Tr*[-(n+1)H] (8) Tr*[dH]
= 7*[(d-n- 1)H].
and
(3.4)
hn' ° = dim Fl°(W, On) = dim AW, aTr*[(d-n- 1)H]).
27
THE SHEAF Tr
* aw
4.
We continue our investigation of double coverings
with nonsingular branch locus
down functor
Tr*
in
B
Tr
CPn
:w
CPn. We can apply the push
to the structure sheaf
0.W of W and obtain
the sheaf Tr
* aw on CPn.
*
01w
is locally free.
xE
CPn
and suppose
Lemma (4. 1).
Proof. Let
Tr
Then there is a small neighborhood
branch locus
B.
such that
is evenly covered by
U
rw, IT * Crw)
Now suppose
= r (Tr
xE
with local coordinates
U' 01w) =
and
IT
Tr- 1(x)
series
rw, 0,
cPn
zl' .. zn centered at
BW
x
CP n).
x
and local coordinates
such that
of
)4* r(u, Ce
B. We can choose a neighborhood
U
such that
Zi,
of
B
... ,Z"n
is defined locally by 'In = 0,
is given locally by
(2' 1 ,
Any
U
Therefore
Tr- '(U).
-1
defined locally by zn = 0,
centered at
does not belong to the
x
-
. .
.
,
n)
h'
f E r(u, Tr* (..9'w ) = r (Tr1 U , aw)
2
(T1' .
.
.
'
n- 1 ' a'n) .
can be expressed as a power
x
is
28
CO
a (21)2'.e
n
f(2") =
=0
where
'2' =
(2'1'
. .
, 21n-1).
Let
00
a
fo(Z) =
2.e
)-1)
(
n
1=0
00
fl() =
4.12
a2.e +1(z )zn .
.Q=0
It follows
f(21) = f (11,12) + n f1
0
n
ZZ)
= f0(z) + n -f 1(z).
The
F(Tr-1U,
which proves
Let
I
Tr
01w
6w)
r(u,
63e
CPn
)st Znr(U,
is locally free.
denote the trivial line bundle on
Theorem (4. 2).
),
CPn
Tr* &147..--=- (5V(I
means Whitney sum of line bundles.
[-d111),
CPn.
where the direct sum
29
Proof. Since
is locally free, it is the sheaf of germs
Tr. 1;,'w
of local holomorphic sections of a vector bundle
The involution
of
E,
of
P.
9.A,=
W defines a vector bundle involution
o-:W
(U.)1
11=
such that
E
E
cr
is the direct sum of the eigensubspaces
and each fibre of E
To see that
of rank two.
E
is a Whitney sum of line bundles let
be a covering of CPn by coordinate neighborhoods
{g.}.
is defined by the 6/1.-cocycle
given by a collection
{6-\.}
Locally
CT
is
of holomorphic functions
U. X C
U. X C2
1
1
satisfying
A2
=1
U.XCZ
i
A
= g. .o.
1
13
13
onUU.
1
There is a holomorphic change of coordinates, after possibly
choosing a smaller open covering of
A
i
is a diagonal matrix consisting of the eigenvalues of
is sheet interchanging the eigenvalues are
,
cisely, there is a collection {Ti.N
t 1=1
T. e F(U., GL(2, C))
1.
such that on each fibre
CPn,
and
1
such that
1
T.&
T.1 = A =
1 1
1
01
{.1
-
Since
-1. More pre-
of local sections
1
(1.
o-
30
-
for all x E U., 1 < i < N. Let g:.13 =
U.
1
,
1
1
of
then at each point
T.g..T.1
we have
U.
Ag
jA.- 1 = AT igiiT;lA
-1
=
cr
1
1
13
j
-1
T.3
-1
= T.g..T j
1
g:j
This implies
61* ED
is a diagonal matrix for each
Therefore, the structure group of
X E U. (Th U..
to
g:.(x)
(1*.
This is equivalent to E
E
can be reduced
being the Whitney sum of
two line bundles.
Let
and
Ll
L2
consider their pull back to
denote these two line bundles. We shall
W. The new local coordinates are simi-
lar to those we used above:
1,
z1,
such
U.
that=
0
defines
... ,
in Tr-1(U.)
in U.
,z
Bw
in
Tr
-1(U.),
Z
n
defines
=0
and
1.
Tr(zz
) = (z 1 .. ... z n-1' (z n
1'
)
).
B in
31
Recall from the proof of Lemma (4.1) that in these coordinates the
local sections in
F(Tr
1U, 0'
have the form
)
f(zi)
Hence the
Ti f (zi).
n 1
1 = {g!
-cocycle
Tr
+
TT}
Therefore, one of these line bundles, say
pulled back to
W;
has the form
is trivial when
L 1,
and the other is the dual to the line bundle of the
divisorB W' that is,
Tr*L2
= [-B] = Tr*[-d1-1].
Since
Tr*
is
injective this proves the theorem.
Corollary (4.3).
Proof.
dim Ho(W, O'Tr*[mF1])
The functor
7,1/4
=
(m+n )
= H0 (CP
n
bk.
Tr 0 Tr*[mH]).
By the projection formula
The refore
(m-d+n).
does not effect cohomology so
H 0(W, .0/ Tr*[mH])
Tr
+
eTr*[mH] = Oi[mH] CS) Tr, 8(w
32
Tr,
9'(I
[mH]
(Yrr*[mH]
431 e[(rn-d)H].
It follows
dim HO(CPn,
Let f
IT*
0/Tr4c[mli]) = (m+n) + m-d+n)
be the homogeneous polynomial of degree
defines the branch locus
B.
that
be the covering of
Let CU= (U.)n1
1 1=
.
by affine open sets U.1 = {(z
0
CP
2d
-z
n
)1
z.
0}.
The CU-cocycle
defines the line bundle [mH] on CPn. If we regard
31
is a
z./z.
31
31 as a function on ir-1(U.r-N U.) then {(z./z.)"}
{(z./z.)na}
3
1
Tr
-.1
a-cocycle representing the line bundle
There are holomorphic functions
h. (z /z.,
and
s
h.13
/h.
(z. /z.)d
3
since
1
root on Tr-1(U.)
we shall call
If m
Ni f
z
0
1
.quare
Let
n
Tr* [rriFI]
on
h.
/z.) = f(z 0 /z i'
z
1
f(z /z
z
0
N/ f
a form of degree
n
Tr' (U) such that
/z.)
n
/z.)
has a holomorphic
denote the collection
d
on
{h.},
and
W.
is a positive integer then by the above discussion homo-
geneous forms of degree m define sections of
Corollary (4.4). Every section of
homogeneous form of the type
Tr*[m1-1]
Tr*[mH].
is defined by a
33
if m > d
RN/ f + Q
if m < d,
where R and Q are homogeneous polynomials in
degree
CPn
of
m-d and m respectively.
Proof. As R and
vary we get a subspace of sections of
Q
dimension
(m+n)
(m-d+n)
AW,(Yrr*[mH]).
So by Corollary (4.3) this subspace must be all of
Corollary (4.3) also provides a way of computing the topological
index of W when n = 2. The quadratic form, induced from the
intersection pairing on H2(W, R),
and
p-
has
p+
negative eigenvalues. The index
positive eigenvalues
T(W)
is defined to be
the signature of this quadratic form
,
T(W) = p
+
-p
.
Using the highest Chern class of W one computes
dim H2 (W,R) = p+ + p- = 2 + (2d-1)(2d-2).
By the Hodge index theorem [Hirzebruch; 5]
p+
=
2h20 + 1.
34
dim H()(W, eYTr*[(d-3)H]), so by Corollary (4.3)
By (3.4), h2'
2 0 = 1 (d-1)(d-2). Combining, we arrive at the formula
h
T(W) = 2 -2d2.
An alternate method of computing the index of
W
is by Hirzebruch's
index theorem [Hirzebruch; 51 using the Chern classes of W which
we have computed:
T(W) =
12
(c1-2c2),
or by the G-signature theorem given in [Hirzebruch; 6].
Additional properties of branched coverings of CPn which
includes a useful characterization of the Hodge filtration of W can
be found in [C. H. Clemens; 2].
35
5.
THE PICARD GROUP
In this section we investigate the Picard group of W where
Tr
W
branched over a smooth
is a double covering of CPn,
CPn
hypersurface
B
of degree
2d. From diagram (3.1) we have an
injection
(5.1)
Hl(W,
HZ(W, Z)
Cygt)
and by (2.10)
H2(W, Z)
This implies
Pic W
Z
if
n> 3.
Z and is generated by
Tr*[--1]
if
n > 3.
Therefore, for topological reasons, the Picard group of these
branched double covers have a simple description if dim W > 3.
On the other hand, we have found that the rank of
large when W
Pic W
H2 (W, Z)
can be
is a surface. There is of course no guarantee that
is large on this basis alone.
The approach taken here is to describe the Picard group of W
in terms of properties of the branch curve
because specifying a branch curve
covering
Tr
:W
CP,
B
B.
This is a natural idea
completely determines the
therein, one expects special properties of
the surface W to reflect themselves in special of the branch curve
B. We have seen this is so for the topology of
W,
and because of
36
(5. 1) some of the topological properties of the covering
CP2
Tr : W
will be useful here.
Fix such a double covering
and involution
B
with a branch locus
Let us agree to write the group opera-
W.
o- :W
CP2,
Tr :W
tion of line bundles additively, thus if L and L'
L + L'
bundles, we write
are two line
instead of L 03) L'. Suppose
line bundle on W such that
L
is a
By (5. 1) we can regard L
cr* L = L.
as a cohomology class. By Theorem (2.3) the homomorphism
o
Tr*
!
is
Tr*
Since
H2(W'
Z)+
H2(W'
H2(W, Z)+,
H2 (W, Z)+
Tr
L E H2(W, Z)
is multiplication by
Z)+
2.
Dual to this
which is also multiplication by 2.
we have
2L = Tr*
o
Tr L
Tr*[mli.]
for some integer m. Using the intersection pairing on H2(W, Z)
4L
=
(Tr*[mH])2 = 2[mFI]
=
2m2
Therefore m is divisible by 2. This implies
because there is no torsion in
H2(W, Z).
L = Tr*[(m/2)H]
Thus we have shown
Lemma (5.2). Every o--invariant line bundle on W
is the
pull back of a line bundle on CP2.
If
L
is an arbitrary line bundle on W we get a cr-invariant
line bundle by forming the sum
L + o-* L.
Therefore, character izing
37
the Picard group of W divides into two problems: (1) Describing
the irreducible curves in CP
image in W,
which have a reducible inverse
and (2) finding the linear equivalence relations
among the components of their pre-images.
Generators. By Lemma (3.2) and the above discussion we can
always take
Tr*[1-1]
to be a generator of Pic W. We will character-
ize the existence of additional generators.
Let
C
be an irreducible curve in CP2.
Lemma (5.3). A necessary condition that
-1
Tr
(C)
is
reducible is that the local intersection
(C
Proof. Suppose
xEB
B)x = 0 mod 2
for all x E B.
Tr*(C) = A + A*
where
0-*(A) = A*.
and set y = Tr-1(x). Then
(C
B)
= (Tr*C B
)
Wy
= (A BW)y + (A* B W) y
= 2(A BW)y
since
(A B W) y
= (cr*A
0-*B
W) y = (A* B W) y
.
Let
38
j:B
Let
CP2
denote the inclusion of
in
B
CP2. If
C
is a curve in CP2 (not necessarily irreducible) that satisfies
(C
B)x = 0 mod
2
for all
x
B we can define the following effec-
E
tive divisor
1
D=
2
XEB
Thus
2D = j*C.
On the other hand, an effective divisor
ZD
j*mH
(-I
D
which satisfies
denotes linear equivalence of divisors) implies the
existence of a curve in CP2
satisfying the necessary conditions of
Lemma (5. 3 ).
There is some classical language that is useful here, called
residuation [Basset; 1].
D on
Definition (5.4). A divisor
residual, written res
[ID] = 0
if
divisor of degree m in CP2.
we say
D
is residual to D'
D
If
D
if
B
is said to have zero
is the restriction of a
and
D'
are divisors on
res [D+DI] = 0.
We state some basic properties of divisors on
language of residuation:
(a)
If
res [D] = 0 and resn[D1 = 0 then
resm+n[D+DI] = 0.
B
using the
39
(b)
If
res [D] = 0 and
is an integer then reskm[kD] = 0.
k
The subscript m in the definition, referring to the degree of the
plane divisor, is not always important. So in some cases m
suppressed. Thus
res[D] = 0
simply means
is
is cut out on
D
B
by a divisor of CP2.
We now turn to sufficient conditions. Let
Cm
be an
irreducible curve of degree m in CP2 such that
(C
m B) x = 0 mod 2
for all
x
E
/ 1(C
2
m
The sum
B.
B)
x
x
XEB
can be regarded as either a divisor DB on
on
C
B
or a divisor
D
.
Recall the construction of the covering W branched along
\
B.
X2
(5. 5)
[CM]
Tr
[2d1-1]
s
CP2
where
W=
s
-1
X2
is a section of
[2d1-1]
which defines
B
and
s(CP2).
Letting
i:C
CP2
denote the inclusion of
we can restrict diagram (5.5) to
C
111
C
in
CP2rn
40
x2
i*[2dF-1]
Cm
Theorem (5. 6).
[D
rr*(C
is reducible if and only if
)
=
Proof. If [DC] = i*[dH]
then there is a homogeneous
rn
polynomial of degree
vanishes on
d
which when restricted to
CP2
in
such that
i*[dEl]
= (constant). s
X2 °
rr*(C
)
Conversely, if
s': C
i*[dFl]
is reducible.
rr*(C
such that
)
= A + A*,
sI(C
)
then there is a section
A. Now A defines a divisor
on the zero section of iqdH]. By identifying
section of
C
and
Tn.
and nowhere else. Hence, this polynomial
D
defines a section s' :C
Therefore,
C
iqd1-1],
with the zero
C
the line bundle of this divisor is clearly equal to
i*[dFl].
rn
We can use this theorem to provide sufficient conditions for the
existence of such curves. We state the conditions in terms of
divisors on the branch curve
B.
Let
obtained by restricting a hyperplane
HB
H
in
denote the divisor on B
CP
to
B.
41
Let D be a divisor on
Theorem (5. 7).
B
of degree md
such that
resm[2D] = 0
((m-d)H
(D) >
(dHB-D) >
for
C
0.
).
((d-m)HB).
Then there is a curve
component of
res[D]
but
C
of degree at most m such that each
has reducible inverse image. We abbreviate
(D)
dim FI°(B, CY[D]).
Proof.
Conditions (i) and (ii) imply that
linearly equivalent to
D,
D,
or a divisor
is effective and there is a curve
degree m satisfying (1) D =
2
(C B) x
of
C
(2) not all com-
x
XEB
ponents of
C
are multiple curves.
Condition (iii) says there is a polynomial
that vanishes on
CP
on
C.
CI'
D
and that
Q
of degree
1
D1 =
2
C,
say
(C-C1 B)x x and apply the
xEB
theorem to D1. This is why the conclusion asserts the curve
has degree at most m.
The existence of the polynomial
where
i:C -4" CP2
in
does not vanish identically
Q
If however Q vanishes on some component of
then we define
d
Q
implies
[D] = igc[d1-1]
is an inclusion. Therefore by Theorem (5.6)
42
each component of the curve
reducible inverse image in
(of degree at most m) has
C
W.
There are some interesting observations from Theorem (5. 6)
and Theorem (5.7). First of all, if the conditions of Theorem (5. 6)
are satisfied for a curve
C,
then
the only irreducible curves in CP
deg C < 2d. This does not mean
that have reducible inverse
image in W must have degree at most
There is a way of
2d.
characterizing all irreducible curves in CP2 that have reducible
inverse image.
Let
g
be a homogeneous polynomial of degree m that
defines an irreducible curve
such that
CP2
in
C
reducible. Write Tr*C = A + A*. Note that
Now A meets
A*
defines a section
defines
s :C
meets
C
iqd1--1]
= md.
B.
If
A A* = md then
as in Theorem (5. 6)
(A*
-s). Hence m = deg C < 2d.
In general A -A* > md,
on
= A* B
A
in at least md points (counting multiplicities),
that is, over the points where
A
is
Tr*C
so we define the following divisor
C:
E=
1
(A A*)
yE TT
Tr(y)
(C - B
The divisor is nonzero if and only if
A
meets
A*
away from the
E
43
ramification locus
defines a section of
B
.
If it is nonzero, neither
nor
A*
i*[dI-11.
Suppose the branch curve
polynomial
A
B
is defined by a homogeneous
of degree 2d, and let R be a homogeneous poly-
f
so that R
nomial of sufficiently large degree, say
on the divisor
E.
vanished
We will use Theorem (5. 6) to obtain a relation
between the polynomials
g, f and
R. The idea is to form a new
double covering such that the degree of
C
is less than the degree
of branch locus.
There is commutative diagram of holomorphic maps between
the total spaces of line bundles
(5.8)
[dH]
X2
[(c11-/ )H]
I
1 X2
AZ
[2dE1]->
[2(d+f )H]
The map X2
[cu-]]
has been defined in Section 1. The map
[(d+f )1-]]
is given locally by a collection
1Z.
U. X C
U. X C
by
(z,R(z0/z
where U. = {(z 0
n
)1z.
0}.
mutes with projection to CP2.
zn/z.)-t)
It is clear diagram (5.8) also corn-
44
A-0
We can form a double covering 'if: W
the reducible curve t of degree
CP2
branched along
defined by the zeros of
2(f +d)
R f. This covering is different from the other coverings we have
considered in that W is a singular algebraic variety, however, it
possesses many of the properties that the nonsingular double coverings have. In particular, Theorem (5. 6) is applicable.
If
is as above, then
C
reducible in
A,-1
C) = Tr
C.
R(Tr
Therefore
is
Ti*C
Write i'f*C = A + A*. Note that we can form the
W.
divisor
1
D
",
(C B)
x.
x
xE B
We can apply Theorem (5. 6) and conclude there is a homogeneous
polynomial
Let
Q
F
of degree
dig
Q meets
B.
CP2
that vanishes on
be the unique effective divisor on
is defined by Q = 0;
D+F
in
Since
that is,
such that
is the other points where
F
res2(d+1 )[2(D+F)] =
B
D.
0
and
resm[2D] = 0
it
follows
res2(d+i )-m[2F] = 0. Therefore, there is a homogeneous
polynomial h of degree 2(d+f )-m such that h = 0 defines 2F
on
B.
Lemma (5. 9). We have
plying
h
and
Q
gh = R2f - Q2,
after possibly multi-
by some appropriate complex numbers.
45
Proof. Since
also defines
2(D+F)
gh
defines
0
on
2(D+F)
on
/1.
and
Qz = 0
it follows some linear combination of
13
the two vanished identically on
B.
Since
'13/
is defined by R2f = 0
and
deg gh = deg Qz = deg R2f
we get the desired equation.
Theorem (5.10). If
that
)
is an irreducible curve in CP2 such
C
is reducible and
singularities away from
deg C < 2d,
B,
C
has only locally irreducible
then one of the following holds.
and neither of the two components of Tr-1(C)
is linearly equivalent to a divisor pulled back from CP2.
deg C
2d,
and both components of
equivalent to
Tr-1(C)
is linearly
7* dH.
Proof. By Lemma (5.9) there is an equation relating
(5.11)
and g
gh = R2f - Q2
Examination of (5.11) implies if
singularities away from
(5.12)
so
f
deg g < 2d.
B
then
gh
C
deg R
f-
Q2
has only locally irreducible
0.
Hence
46
If
deg C = 2d,
and (5. 12) becomes
deg h = 0
then
g = f - Q2
are defined by zeros of
It follows the two components of Tr-1(C)
"if-Q
and
Therefore by Corollary (4.4) if deg C = 2d
NI f + Q.
then each component of Tr-1(C)
is linearly equivalent to
Tr*dH.
This proves (ii).
Next assume m = deg C < 2d. In this case,
conditions of Theorem (5. 6) because
d
that vanished on
is a polynomial of degree
Q
1
D=
(CB)x
satisfies the
C
x.
As pointed out above,
xE B
when this condition is satisfied, the two components of
meet only on the ramification locus
in
B
W.
-1
Tr
(C)
Thus
A A* = md
(5. 13)
2m2
=
[Tr*C]2
=
(A+A*)2
=
2A2
- 2A A*
Assume, for a contradiction, that either
A
or
A*
is
linearly equivalent to the pull back of a divisor on CP2. Now the
rank of the subgroup of cr-invariant line bundles on W
Hence mA
nA*,
but A2
=
A*2
and both A and
effective, so we can conclude A -I A*,
A2
= A A*.
A*
are
which in turn implies
But this together with (5. 13) forces
we have assumed is less than 2d.
is one.
deg C = 2d
which
47
We can say more about the singular set of these curves.
Corollary (5. 14). Suppose
Theorem (5. 10), then
that is
C
satisfies the hypothesis of
C
is nonsingular at each point of B cm C,
is totally tangent to the branch locus B.
C
is a homogeneous polynomial in CP2
Proof. If F
denote (8F/8z0' 'BF/az
aFiaz2). Thus
let
vF
F
is nonsingular if vF
B
and
never vanishes.
Again let f(zo z1, z2) = 0
define
in
C
define
By Lemma (5.9) we have an equation
CP2.
gh = f
Applying
V
g(zo'z1,z2)
- Q2
we have
gvh + hvg = vf - 2QvQ.
At each point of
B rm C
this becomes
hvg = vf
because
since
B
Q=g=0
on
B r--N C.
Therefore,
Vg
0
on
B rm C
is nonsingular.
When we relax the assumption that a curve
irreducible singularities away from
B
C
has only locally
there is no bound on the
degree of C. We shall investigate this situation, but first we discuss
48
another interesting observation of Theorem (5.6) and Theorem (5.7).
Suppose
is a divisor on the branch curve
D
that
B
There is a
satisfies conditions (1), (ii) and (iii) of Theorem (5.7).
certain symmetry to these conditions, because the divisor
'15=
D
dHB -
also satisfies conditions (i), (ii), and (iii). Therefore,
to the divisor
-1
m such that
"..,
curve
there corresponds a curve
D
TT
is reducible and to
(C)
e...,
D there corresponds a
-1 ".,
of degree at most 2d-m such that
C
of degree at most
C
(C)
Tr
is reducible.
We make the following definition.
Definition (5.15). Suppose
and
C
such that each irreducible component of
inverse image in
D=
2
(C B) x
W. We say
x
and
xEB
divisors on
CI
C
are curves in CP2
and
C'
is residual to C'
C
1
D' =
2
(C' B)x x
has reducible
if the divisors
are residual
XEB
B,
that is
res[D+DI] = 0.
According to this definition
C
and C".., are residual curves.
This observation and language will be useful when relations are discussed.
We return to the general case of describing the irreducible
curves in CP2
such that their inverse image is reducible. It may
happen that a totally tangent curve of arbitrarily large degree may
possess reducible singularities (e.g. double points) that forces
49
-1
Tr
(C)
to be reducible.
><D
(5. 16)
>CD
3
LTr
even
odd
Topologically, the singularity creates a loop in
C
which 'winds
round' the branch curve an odd number of times (see diagram (5. 16)).
More precisely, suppose
C
is irreducible,
C
Tr*C = A + A*,
and
does not satisfy the conditions of theorem (5. 6). As we have
pointed out, when this occurs A meets
fication locus
(A A*)
by 2,
0-(y)
I0
BW'
If
(A A*)
0
A*
for
y
so the additional contribution to
away from the ramiB
then
A A*
is divisible
so we can write
(5.17)
A A* = md + 2X
Theorem (5. 18). Let
(and hence
A*)
.
be as above and suppose that
C
is not linearly equivalent to the pull back of a
divisor on CP2. Then
X>
1
(m2 -2md).
A
50
Proof. By the Nakai-Moisizon criterion [Hartshorne; 4], the
ramification locus
Bw
is ample. Clearly (A-A*) Bw = 0,
so
by the Hodge index theorem
0 > (A-A*)2
=
2A2
- 2A .A*,
with equality if and only if A -I A* (since
(5.13)
A2
=
By
H1 (W, icy) = 0).
m2 - A A*. Combining this with (5. 17) we get
>
4
(m2-2md).
We have given necessary and sufficient conditions for the
Pic W. The previous theorem
existence of additional generators of
enables us to divide them into two types. Those arising from curves
satisfying the conditions of Theorem (5.6) constitutes the first type.
By Theorem (5.7) these relate nicely to the branch curve. The other
type arises from factors of polynomials of the form R2f
-
Q2,
and
we may assume they have degree greater than the degree of the branch
locus. Relating the second type of curves to the branch locus may
require more than simple conditions similar to those in Theorem (5.7).
Relations. Suppose
C
and
C
are irreducible plane curves
satisfying the conditions of Theorem (5.6). Then C
and
to reducible divisors in W. We can write Tr*(C) = A + A*
Tr*C = A + A*.
a' lift
and
There is of course no natural way to distinguish
A
51
from
likewise for
except that A* = o-*A,
A*
and
A
A*.
Nevertheless, we can specify linear equivalence relations between the
divisors
and A*. In some cases, they are determined
A, A*, A
by linear equivalence relations between the divisors
1
D=
1
B)x x and
(C
=
xEB
and
C
Proof. Suppose
Q
The polynomial
2D
+
2D".'
C
be as above. If
/o'
Tr*dH.
defines
g=0
defines the branch curve
polynomial
B.
then, after possibly relabeling, we have
=0
A+
f=0
on
x
xE B
Theorem (5.19). Let
resd [DC +DHC
2
', B)
(C
of degree
g
h
B.
C,
defines
h=0
resd[Dc+De] = 0
If
that vanished precisely on
d
of degree
2d
and
C
there is a
D
C
+
C
vanished precisely on
hence after possibly multiplying
and
h
Q by
appropriate constatns we have
gh = f -
Q2
Using this, we can label the components of
that if Tr*(C) = A + A* then
A
is defined by
Also, we can arrange it so Tr*(C) = A + A*
h = N/ f + Q = 0.
Thus
A+A
Tr*C
g
and Z
in such a way
\If+ Q = 0.
is defined by
is defined in W by the zeros of
52
f + Q.
But
NI if + Q
defines a section of
Therefore
7*[d1-1].
A + A -1 Tr*dH.
According to Definition (5. 15),
nJ
C
and
C
are residual
curves. We have proven there is a relation between the inverse
image of residual curves. As an example, suppose
a two sheeted covering of CP2
Tr :W
CP2
is
branched along a smooth quartic
curve B4. A quartic always admits a bitangent
L.
L must
satisfy the conditions of Theorem (5. 6) because it is simply connected. This implies there is a plain conic passing through the points
p1
and
pz
where
L
is tangent to
through six other points of
curve
C
totally tangent to
is a residual curve to
Tr*(L) = A + A*
and
B4
B4.
say p3,
54
This conic also passes
p8. There is a cubic
at the points
p3,
p8
and
C
L. Now Theorem (5. 19) implies we can write
Tr*C = B + B*
A+B
As pointed out above, a curve
and there is a relation
Tr*2H
C of degree m that satisfies
the conditions of Theorem (5.6) has a residual curve of degree at
most
2d-m. This implies we need only consider such curves of
degree up to half the degree of the branch curve.
To view the problem of describing all linear equivalence relations in
W,
let
Bw
W be the inclusion of the ramification
53
locus in W. The restriction of line bundles on W to
B
gives
a homomorphism
p.* : Pic W --' Pic Bw .
is the identity on
Since the involution o-:W
W
of line bundles of the form
L - cr*L
p.*.
Pic W
{L
the subgroup
is contained in the kernel of
Because of this, we shall mean by
(5. 20)
B
the homomorphism
11*
Pic B
L}
The precise form of the kernel of
p.*
.
is unknown. As a
starting point, I make the following conjecture and then investigate to
what extent it is true.
Conjecture. The homomorphism
la*
in (5.20) is injective.
A convenient way of looking at this is to return to the use of
divisors. We can choose a set of divisors
{A1,
, AN}
of W so
that the line bundles
[A1],
,
[AN]
and
Tr*[H]
form a basis of Pic W. We can also choose Al,
, AN to be very
ample. There is a positive integer mi assigned to each A.
because
54
Tr*m.H
- A.
A+
I
for
1<
i < N.
Thus every divisor
divisor of the form
Ai
in W is linearly equivalent to a
a.A. - miT*H;
I
and since
1
we may replace the coefficients
+ m:frItH
-
D
'2AL-12
by
a.
1.
So we write
II
EA. - mTr*H mod{E-T*E}
D
1.
where
is either
E
Suppose
D
r,
-o-*D.
but
1
or
0.
is a divisor and [D] E ker IL*. We claim
D
To see this, observe
r must be zero because
fore line bundles in the kernel of
for some integer
D + cr*D -i2 ri-r*H
0
[I*
versely, a skew invariant line bundle
-z (D+cr*D)Tr*FI = Zr.
There-
are skew invariant. ConL
satisfies
41* L = 11* ( L - o-*L ) = 0
and so determines a point of order
2
in
Pic0
B
.
To show
is an injection, one must show that skew invariant line bundles, not
of the form
L - IT* L,
restricts to nontrivial half-periods in
Pic0 Bw.
Define a skew invariant divisor to be a divisor
[D] = -0-*[D]. If
D
D
such that
is a skew invariant divisor then clearly
55
By replacing D by its linear equivalent representative
D Tr*H = 0.
it follows
E .A. - mir*H modulo{E-o-*E}
11 Tr*H - mr*H ir*H
EA
0 = D-1r*F1 =
-
/71.
E
2M..
1
I.
This motivates the following definition. Define the group
1
1.
Provided at least one of the
m.
I.,
a.m. = 0 mod 2}
1
I.
are odd, we have an exact sequence
ZN
o
2ZN
where the first map is
aN) I-' /(al'
Z
2Z
ii
aA. I-- (al' ... , aN) and the second
airni Note that )4
/241
is isomorphic to
i
Pic W
skew
{L-cr*L}
Pic W
cr*. {L-cr*L}
(5.21)
the skew invariant subgroup of the involution
Pic W
We have a homomorphism.
{L-cr*L}
2
PicoB
56
given by
/
A.
)1c
Its image is contained in the half-periods of
Pic0 B
.
At this point, we mention a topological analogue of (5.21).
Recall that H2(W' Z)+ denotes the 0--invariant subgroup of
H2(W' Z). It is useful to define the skew invariant classes as elements
of
H2(W'
Z)+
Z2
Now there is a way of assigning to each skew invariant class an
element of Hi(B, Z2),
which topologically can be identified with the
points of order
PicoB. Interestingly, the point of order
2
in
will be zero if and only if the class is of the form t - cr*
2
.
This is done as follows. Let {D}
be the pencil in
S SEUre
CP2 spanned by the branch locus D 0 = B and the divisor D00=2C,
where
is a nonsingular curve of degree
C
transversally. If
C
d
that meets
is sufficiently general the pencil has irre-
ducible singular fibres Dt,
consisting of only one ordinary
, Dt
l
r
node. Associated to the singular fibres are so called vanishing cycles
Nil
in
B
that generate
H1 (B
"Z)
where each of the vanishing
57
Ei, ... , Er
cycles bound 2-cells
CP2.
in
Corresponding to this pencil, there is a pencil
W,
50 = Bw and
spanned by
actually parameterized by a two sheeted covering
branched over
0
and
---''
CP],
The covering is defined by
co
is a Lefschetz pencil.
{-DI }
SE CP
1
Proof. The ramification locus
the form
s
and
NI f
polynomial
Q
1r-1C
1
is defined by the zeros of
B
is defined by the zeros of a homogeneous
(regarded as a form on W). If (so, s1) E CP
1
-s = s N./f - s0Q are members of this
2
2
Dt = tif - t0Q2 where (t0 : t1) = (s0 :s ) Now
and
= s Nil* + s Q
pencil lying over
0
D
1
1
bs has one ordinary node if and only if
Tr- 1( C )
CP1
(s 0' s2).
Lemma (5.22).
1Y
in
2
(s0 : s1)
then
,.....r.,1
S E....r-
This pencil is
1(C).
= Tr -
-61c
{ffs}
is nonsingular, we conclude
Dt
{ffs}
does. Also, since
1
is a Lefschetz
,...1-'.,,
s Et.....
pencil.
This pencil has singular fibres
5s
s
1
lying over
t DD .
tr
-s 1/
''
The 2-cells
r r r'
5- sr
E1,
.
Z
ri
lift to 2-cells
58
with the same vanishing cycles
B
and
(using the isomorphism between
{N.}
The skew invariant classes can be represented by
B).
r.-r!).
I
I
cycles of the form
The homomorphism Tr* : H2(W, Z)
is onto by
F-12(CP2, Z)
Theorem (2.5), and H2(W, Z)+ maps isomorphically onto
by Theorem (2.3). This implies the homomorphism
2H2(CP2, Z)
Tr*.
H2(W,
Z)
2
Cit)
H2(W'
Z)
H2(CP, Z2)
Z2
is a surjection.
Using the pencil
{Ds}
s Ear
1
'
C. H. Clemens [3] has proven
there is an exact sequence
(5. 23)
0
a H2(W'
Hl(BW' Z2)
where the injection a is given by
jep
2 jett.
/24,
>
Tr
Z2
H (W Z)+
2
Returning to the group
Z)
H2
(CP2 ' Z2 )
0
'
{Yi} I
{r-r}.
we clearly have an injection
H2(W, Z)
2
Z2
+
H (W, Z)
This homomorphism is given by first forming the line bundle
associated to a representative divisor in
,
then assigning to this
59
line bundle its first Chern class in
H2(W, Z).
Lemma (5. 24). Using the duality between
Z)
H2(W'
H2(W'
a.A
If
and
H2(W,'Z)d-
1
1
satisfies
II
ce.Af
It follows
Tr
® Z2'
TL,.
a.m.
= 0 mod 2
I
then
Tr*4mH.
sends the homology class of
homology class of a curve of degree
in
H2(W Z)
is contained in the kernel of
/24
the image of
Proof.
Z)
Z2
2m
iAi
in
CP2,
to the
which is zero
H2(CP2' Z2).
Thus, there is a topological way of assigning to each skew
invariant line bundle a point of order two in
Pic0 B
if and only if the line bundle is of the form
L-o-*L.
which is zero
It is an open question whether the homomorphism (5.21) is dual
to the homomorphism
a.
We can summarize the entire problem via the homomorphism
in (5.2 1). The problem of finding generators reduces to finding
half-periods in the image of (5. 21), and the problem of describing
60
relations reduces to describing the kernel of (5.21). The results
given here deliver a partial answer to each, their greatest asset
being their constructive nature.
61
APPLICATIONS
6.
The direct application of the results in Section 5 depends of
course on how much is known about the branch curve. To illustrate
how the geometry of the branch locus relates to curves on the surface
W we will apply these results to a double covering branched along a
smooth quartic curve. In this case, much is known about the branch
curve.
Also, some properties of the branch locus translate to intrinsic
properties of the surface. For example, we will give a criterion for
a double covering branched over a nonsingular sextic to be elliptic.
Before giving these examples, we introduce some additional structure
of Riemann surfaces.
Let
C
be a Riemann surface of genus
Jacobian variety of
C
g > 2. We define the
to be
1
H (C,C1) -= group of line bundles on C
Hl(C, Z) of degree zero.
J(C)
Choose a basis
. .
CA)
of Abelian differentials for
O.)
1'
C
and
consider the complex torus obtained by forming the quotient of
by the lattice
A
consisting of the vectors
(k)
1'
'
wg
E H1 (C Z).
Cg
62
After choosing a base point b0 in C, Abels theorem and the
Riemann-Roch theorem implies every line bundle in J(C) is the
line bundle of a divisor of the form P1 +
g bO Using this
P
representation of line bundles, we get an isomorphism J(C) Cg /A
given by
P.1.0)
[p1+
+pg-g bo]
The usefulness of J(C)
l'
b
i=1
P.
b0
g
.
includes an imbedding
J(C)
given by
rP
=(
(A)1,...,
bo
which extends to the
r th
b0
on
C(r)
wo.)
5
symmetric product
=p.
where
rP
by
iP(Pi)
can be identified with effective divisors of degree
C.
J(C)
also has the structure of a principally polarized Abelian
variety whose polarization explicitly defines the Riemann theta
63
function
C.
of
B
(with respect to
Its zero locus defines a periodic divisor on
which defines a divisor 8 in
A)
VC),
Cg
called
the theta divisor.
The concluding examples will depend upon some facts pertaining
to Riemann's solution to the Jacobi inversion problem, which relates
properties of the theta divisor to special effective divisors on C. A
detailed treatment can be found in [Lewittes; 7].
These facts center around the existence
of a pointin
b0
called a Riemann
depending only upon the base point bo,
J(C),
constant. For a given
If
8,
e
that
e
eE
J(C)
we have
then there is a unique divisor
D E C(g)
such
C(g-1)
such
.
I11( D)
Ljo
If
eE8
that
there is an effective divisor
e = 4J(D)
and f(D)
+16b
DE
equals the multiplicity
0
of
(3)
6 at e.
-2Xb0
where
=
eE6
By (2) and (3) if
effective divisor
D
K
is a canonical divisor on
is a half-period in VC),
of degree
g-1
such that
the basis of periods, the half-periods in
of
4E
then there is an
2D
is a canonical
Such divisors are called theta characteristics. Relative to
divisor.
where
C.
E
and
5
are
1Xg
J(C)
are given by
e = ro
vectors whose entries are either
0
1/2. A theta characteristic is even or odd depending upon whether
is even or odd. Some additional results found in [7] are
64
The multiplicity of 0
at an even (resp. odd) half-period
is even (resp. odd).
All the odd half-periods lie on 0
2g-1(2g - 1)
Let
Tr
:W
B4
CP2
and there are
of them.
CP2
be a nonsingular quartic curve in
a double covering branched along
B4.
and
We shall con-
sider the class of curves that satisfy the conditions of Theorem (5.6)
that lead to additional generators of Pic W.
First we make a reduction. According to the remarks which
follow Theorem (5.7) we need not consider cubic curves because they
are residual to lines. Every conic that satisfies the required conditions also has a residual curve. In fact, in this case, a one dimensional family of residual conics that contains a degenerate one,
consisting of two lines. So again we have reduced the problem to
studying totally tangent lines to
Bitangents of
B4
B4.
are in one-to-one correspondence with
effective square roots of a canonical divisor, i. e. theta character is-
tics. A result of Lewittes [7] states a Riemann surface of genus
3
has an even theta characteristic if and only if it is hyperelliptic. A
plane quartic is never hyperelliptic, so we conclude
B4
has only
odd theta characteristics. By property (5) above, there are
22(23-1) = 28
odd theta characteristics; and therefore 28 bitangents
65
of
The 28 bitangents lift to 56 exceptional curves
B4.
E.
1" E
28'
in
E*
28
E*
1'
W.
Let us consider the linear equivalence relations among these
exceptional curves. We have the obvious
E. + El<
1
1
1<i
-J E. + E*
3
< 28
3
giving 27 relations. Next suppose that among the 28 half-periods
28
{e.}.
corresponding to the 28 odd theta characteristics, we have a
1=1
relation
e.
11
in
+ e. + e.13 + e.
=0
12
Then by property (2) we have
J(C).
4
)
k= 1
Thus
D.
11
+ D.
12
+ D.
13
+ D.
14
= 0.
4
Lk
- canonical divisor of
B4
2H.
4
are residual curves,
By Theorem (5.19) the bitangents
11
14
and therefore give rise to a relation between E.
11
.
14
11
14
A computation shows there are precisely 19 such independent relations
28
between {e.}.1=1 .
Therefore we have
66
p = rank Pic W > 56 - 27 - 19 =
But since W
is rational, it is well known
p -= 8.
Therefore we
have given an explicit description of the entire Picard group of
For a second application, let
CP2
Tr :W
covering branched along a nonsingular sextic
B6'
W.
be a double
W
is a Kummer
surface.
Proposition (6. 1).
If
has an even theta characteristic,
B6
then W is an elliptic surface.
Proof. An even theta characteristic corresponds to halfcanonical divisor
D
such that
(D)
is even, so in particular
i(D) > 2. By Theorem (5.7) there is a totally tangent cubic curve
such that
write
Tr*(C)
is reducible. C has a residual curve C. We can
Tr*C = A + A*
and
ir*ei = A + A*;
we have a relation A + A
and by Theorem (5. 19)
Using the intersection pairing on
Tr*3H.
W and formulas (5.13) we have
(A)2
=
2
(Tr*3H)2 - A A*
=9-9
= 0,
and similarly
(A*)2 = 0,
(Av)2
=0
and
(Z*)2 = 0.
Next,
67
18 = (Tr*3H)2 - (A+Z)2
=
" + (A)
(A)2 + 2A
2
= 2A
so
A -A = 9. Next,
18 = (ir*3 H)2 = (A+A*) Tr*3H
= 2A Tr*3H
and
9 = A Tr*3H
=A
=
(Av44)
A -A*
A A'
= 9 + A Z*
Therefore A A* = 0.
These intersection numbers imply (1) A is linearly equivalent
to
A*
because
(A -./t6ii* )2
Hodge index theorem,
implies
(2)
A
A
=0
and
'X*
ftd
CP1.
A*
because
= 0.
H(W,d)
1
have arithmetic genus equal to one. There-
fore, the linear system spanned by
4 :W
so by th.e
is numerically equivalent to A*. This
is linearly equivalent to
Both A and
fibration
(A-Z*) Tr*H = 0
A
and
Z*
defines an elliptic
68
BIBLIOGRAPHY
A. B. Basset, A Treatise on the Geometry of Surfaces, Deighton
Bell and Company, Cambridge, 1910.
C. H. Clemens, The General Theory of Double Solids (to appear).
C. H. Clemens, On Double Coverings of Simply Connected
Algebraic Surfaces (to appear).
R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York,
1977.
F. Hirzebruch, Topological Methods in Algebraic Geometry,
Third Enlarged Edition, Springer-Verlag, New York, 1966.
F. Hirzebruch, The Signature of Ramified Coverings, in Global
Analysis, pp. 254-265, University of Tokyo Press, Tokyo, 1969.
J. Lewittes, Rieman.n Surfaces and the Theta Function, Acta Math.
111 (1964), 37-61.