615
I nternat. J. Math. & Math. Sci.
Vol. 2 NO. 4 (1979) 615-626
DIFFERENTIALS OF THE 2ND KIND ON A PRODUCT SURFACE
J.C. WILSON
Department of Mathematics
Southern Illinois University
Carbondale, Illinois 62901
U.S.A.
(Received January 15, 1979)
ABSTRACT.
This paper deals with the problems of representing an arbitrary
double differential of the second kind, defined on a surface which is the
topological product of two curves, in terms of the products of simple differentials of the second kind on the two curves.
The curves are assumed to
be non-singular and irreducible in a complex projective 2-space.
KEY WORDS AND PHRASES. Algebraic srface, pduct sface, algebraic curv,
double differ of the second kind, simple differentials of the second kind,
Able,an different, genus, extr product, pola crves rest.
T980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
14C50, 52J25.
INTRODUCTION.
Our purpose in this paper is to prove a theorm, stated by Lefschetz,
concerning double differentials of the 2nd kind on an algebraic surface.
will employ the Picard-Lefschetz technique as described in Lefschetz [i].
We
J.C. WILSON
616
This amounts to reducing the initial differential to a new one of the same kind,
but in a more suitable form for our purposes.
A precise statement will be
In all that follows, our ground field, denoted by K, will be
given below.
Also, K(x, y, ...) or K[x, y, ...] will indicate the
the complex numbers.
field of rational functions or ring of polynomials, respectively, in the com-
plex variables x, y,
with coefficients in K.
Suppose C I and C are two distinct, irreducible, non-singular algebraic
2
curves, each in a complex projective 2-space.
the topological product C
4-space.
If C
I
and C
C2,
I
The product surface
,
i.e.,
is then a non-singular surface in projective
affine equations
2 have
respective, then an afflne model, S, of
(, 8)
0 and
(*, 8*)
0,
may be thought of as the surface
of pairs of points (, 8; *, 8*) in complex 4-space.
We will deal pri-
marily with an affine model in this paper. "Our objective is to prove the
"Every double
following:
modulo de, (where
of products
to C
I and
*
A
*
differential of the 2nd kind on
is reducible,
), to a linear combination
is a simple differential on
belongs
of simple differentials of the 2nd kind, where
belongs to C 2 and neither are derived."
See Lefschetz [2].
We point out that the last above statement can also be realized by
In particular, see page 80
using modern techniques of algebraic topology.
of Hodge and Atiyah’s paper "Integrals of the 2
nd
Kind on an Algebraic
Variety" appearing in the Annals of Mathematics, Vol. 62, 1955.
However,
we feel that the classical approach used here has merit in itself since tech-
nique ranks equally with results in algebraic geometry.
SECTION I.
Let R
P( 8
*
8*)
Q(, 8, *, 8*)
be a rational function such that for fixed
and
8
or fixed e* and
8", Q is
.
DIFFERENTIALS ON A PRODUCT SURFACE
not in the ideal generated by
or
[3], i.e., if Q
nomials, then
Q’s
are distinct, irreducible poly-
is of the 2nd kind relative to
2
on S such that
simple differential
tersection of
where the
Qk’
QIQ2
We assume that the double differential
Here we employ the Lefschetz definition
Rde*d is of the 2nd kind on S.
2
a polar curve of
m2’
Qh
provided there exists a
de is regular at the points of in-
2
Such points generally form a curve on S, called
0 with S.
Qh
617
and we designate it by
h"
The simple differential
depends on the choice of h and we say that
is of the 2nd kind on S provided
such an
As an analogy to Lefschetz’s
exists for each polar curve of
2
2"
plane section, we employ the curve C and its copies on S, the latter obtained
2
by varying
"general" C 2.
on C
8
and
First consider the intersection of
I.
h
with such a
A suitable choice of affine co-ordinates will insure that none
of the polar curves pass through a point at infinity on a general C
2.
a non-singular linear transformation will guarantee that both
d
d
8
2
I and
(*)
where d
I
and d
2
are the degrees of
and
,
Also,
and
contain
respectively.
Such co-ordinate changes do not effect the kind of differential involved.
P2(,
8,
Pm (’
2’’ 82")’
with a general C
Suppose
as parameters, we designate by
8
and
Treating
The
2.
el*
and
8,
am*, 8m*)
8i*’s
our general
R
-n
(
8,
i* 8i*)
+
the intersection points of
h
and
8.
Then, in a neighborhood of P i on
R-n+l (a,
+
We use (e*
el*’ 81")’
the following expansion of R is valid:
C2,
R=
8,
will depend on the choice of
is a polar curve of order n.
h
PI(,
i*
8,
i*
8 i *)
higher powers of (*
.*).
1
as parameter and all numerators are rational in their variables.
618
J.C. WILSON
LEMMA i.
is regular on
PROOF.
There exists a simple differential
h"
m
Define H(, 8, *, 8*)
i=l
R_n(
an expression for
with respect to
8*
8,
and E
Let E
el*’ 8i*)"
We first will obtain
be the resultant of
1
of E and
2 the resultant
1
2 + dlh
on S such that
.,)n.
1
(,
R
lh
Qh
with respect to
8.
and
E2
may be considered as a polynomial in e* with polynomial coefficients in e and
.*’s
as such, has the
For e finite, this implies that
as its only zeroes.
the elementary symmetric functions of the c.*’s are polynomials in a.
.,)nl
(,
i=l
Xh
n
(c, c*), where
Xh
n
For simplicity, set
and *.
is a polynomial in
may be thought of as the projection of
plane.
It is evident that H(, 8, *, 8*) is of order zero on
H(, 8,
i*’ 8i*)
Thus
(*-i*)
i=l
h on the (c,
h" Also,
n
c*)-
m
=1 (i*- .,)n3
R_n(,
8,
H(, 8
R_n(,
8,
i*’ 8i*)
i*’ 8i *)"
Hence
ei*’ 8i *)
m
.,)
(.*
(1.2)
n
j=l
Next we set
V
n
i
n- I
m
R
-n
(
8
(,
i=l
i* 8*)
i
.,)n-i
(1.3)
1
from (1.2), we see that V not only has
Using R
n
-n
h
as a polar curve but can
have poles
i)
at points of intersection of
h
and their occurrence depends on
ii)
iii)
at points where
if
or
8
is such that
is infinite.
with
k’
# h, for then H
k
and 8;
i *=j*
j
# i;
has poles
DIFFERENTIALS ON A PRODUCT SURFACE
2 + d[Vnda]
We can now show that
with n replaced by (n
R
-n
(e, 8 e i *
(e*
i).
8 i *)
(e, 8,
-n
8 i *)
el*’
de*de
ei*)n
(e*
+
d[Vnde] behaves llke
R on
but
h
First
,
1
n
1
+
R
m
1
do*de + d
ei*)n
R
Rde*de
619
I
n
-n
(e
8
8i*)
ei*)n-i
(e*
(1.4)
-n
d
da
ei*)n-i
(e*
i=l
A typical element of the last sum can be written as
(e*
e
Since not both
1.*)n-I
e
and
do
8e
+ 88-n d8
8 vanish,
-n
(n- i)
(e*
8
we can assume
# 0
el ,)n
and get
d8
e de
(n
do*.
Then (1.4) can be written as
R
-n
(e, 8,
(e*
+
el*)
n
m
1
n
8")
i
ei*
1
1
{
el.,)n-i
(e*
i=l
do*de
+
IDR-n
De
DR-n
D8
de
8
i) R-n
(e*-ei*)
n
de* } de
The exterior product then gives
R-n
(e*
e.*)n
1
Thus, in a neighborhood of
replaced by (n- i).
(n
Pi’
i--i (e*
Next, we can
i’
de*de.
+
R-n+2
with the term
we will arrive at the differential
e.*)n
1
d[Vnde] behaves like 2 but with n
define a Vn_ of the same type as
Vn with
1
d[(Vn + Vn_l do] will begin, in some
Rde*de
i) in place of n and Rde*de +
neighborhood of P
R-n
m
de*de-
(e*
el*)
Rde*de +
n-2
Continuing in this manner,
d[Vn + Vn-i
+
+
V2)de],
double differential of the 2nd kind in which the corresponding n is one.
a
620
J. C. WILSON
If we still denote the coefficient of
i
e*
by
ei*
then since
R_l
2
is of
the 2nd kind, its residue relative to any polar curve must be derived, i.e.,
dT
R-I
d-
where T(e, 8,
el*, 8i*)
is a rational function on
See Lefschetz [4].
only of type i), ii), or iii), above.
T(de*
m
(V2
lh
+
+
n
+
From the above, it is evident that Rde*de
2
relative to
h
dlh
+ d[ (V 2 +
(e*
i=l
h’ as follows:
Vn)de] behaves like
+
i, so we need to look only at
dei*
T(de*
(.5)
is regular on
but with the corresponding n
m
de)
de
(e*
i=l
then we can immediately show that Rde*de
having poles
If we set
dei*
+
V )de
h
de)
e.*)
Then,
[ i=ml
dT
dT
(e*-
(e*
m
(e*
dei*
de
(e*
i=l
el*)
m
i=l
dT
de
m
(e*
i=l
ei*)dT
(e*
T
e*
ei*
]
de
dede*
de*
+ (e*
T(de*
T
ei*)
de .*)de*
1
(e*
e.*)
1
2
-e.,)2Td(e* e.l*)I
m
e i ,)
fdei*
L de
2
T
(e*
ei
m dei*de I’(e*
d
i=l
a m.,)
dT
de
m
i;1
d
]
Td(e*
e.*)dT
i=l
i=l
ei*)
m
de*-
.dei*
T.(
[---J
+
i;1
(e*
e.*)
2
dede*
d
d’e
de
DIFFERENTIALS ON A PRODUCT SURFACE
m
dr
de
i=l (e*
e.*)
1
dT
de
i=l e*
dei*
m
dede*
T
L" ’d
+
621
2 de*de
i=l (e*
dede*,
.*
1
since the third term in the last sum above is zero and the second and fourth
However,
terms are negatives of one another.
dT
de
m
i=l e*
R_I
m
dde*
i=l e*
ei*
de*de.
el*
This concludes the proof of Lemma i.
For brevity, we will write
lh
Xh
in
From Lemma i, V and U are infinite on the
0 with S, so that in addition to
on any other curve which projects onto
residual intersection by D
LEMMA 2.
D
h
Xh
0.
h’
they are infinite
We will designate this
h.
may be replaced with a similar differential but with
eliminated as a polar curve (or curves).
,
PROOF.
to
lh
(1.5) as
V(e, 8, e*, 8*)de- U(e, 8, e*, 8*)de*,
where V and U are rational on S.
intersection of
lh
(,
For almost any non-singular affine transformation (e, 8, e*, 8*)
*, *),
(, *)-plane
the projections of the transforms of
and D
h
on the
will have at most a finite number of points in common.
Under such a transformation R de*de
R de*de
where we set
h
+ d(Vde
+ d(V
de
Ude*) becomes
U de*),
622
J. C. WILSON
G(, 8, *, 8*)
A, B, m’d G polynomials.
0(, 8)
d
=d
to
Here we have made use of the transformations
(, g, *, g*)
d*.
We can also assume that neither
to the absence of singularities.
to
and
"
be denoted by
LI
I’
with respect to
LI + N
1
(, g, a*, g*)
and ,(*, 8*) to
1"
since
LJ
and
I"
i.e.,
la +
gves
a second resultant, say
LJ 1 N
us, LIa +
NO
are zero, due
nor
Let the resultant of
J
and
with respect
i
Further, the resultant of
G1,
i.e.,
and in the transfoea surface
It follows that
are ero.
in eliminating
lh
can be written as
ih
where
A
and both are polynomials.
in the
h
(4, *)-plane, by
LIA,
,
.LIB
If we call the projection of the transform of
h’
then G
hn(, *)h(,
I
*),
there exist polynomials
n
sxh + th
p().
Using this last equality, we can write
Therefore
I
n
Xh Xh
P ()
h
n
and
s(, *) and t(, *)
such that
G
where
Considering them as polynomials in
are relatively prime polynomials.
with coefficients in
B
h
P () Xh
n
P ()
h
h’
DIFFERENTIALS ON A PRODUCT SURFACE
[Ad
lh
P ()
P
S
+
h P(alh
n
Atdfi- Btd*
P(a)h
n
()h
n
+
P()h
+
tn
Ph p--*
d-
+ Asd- Bsdfi*
P(a)h
dfi*dfi +
It is evident that
t
Bd*]
623
dlh
is regular on the transform of
is true even if we suppress the last term above, i.e.,
we replace
lh
with
p
()h
n
p()h
If
PXh
td- td* dE- Bhd*
lh
and this
h
sd- sd*
we have
n
d*d+ dlh
a differential of the 2nd kind on the transformed surface, regular on
transform and the transform of D is no longer a polar curve of
h
h
lh"
SECTION 2.
The above reduction and replacement can be carried out for all polar
curves
h
of
Returning to the original notation and co-ordlnates, we
2"
would then have a set of
the
h
2+
lh
2+
d(
h
lh
would be regular on
but would have poles for certain values of o and
d(,lh
h
B.
Let us write
in the form
W2
where
such that
A(o, [3 o*,
S(o, 6, o*, B*)G(o)
A, B, and G are polynomials and A/B
is regular except possibly for
points of infinity.
LEMMA 3.
where
W
2
Ado*do
BG
can be reduced to the form
(O,
*,
IB(a)
B*) do*do
is a polynomial.
PROOF.
We begin by replacing the affine co-ordinates o* and
projective co-ordinates
o0*, i*’
and
o2"
in both
where B is to be regarded as a polynomial in
* with
(o*, 8*) and B(o, 8, o*, B*)
e0*’ l*’ 2"
with coefficients
J.C. WILSON
624
in the field
form
A
Y+
By Kapferer’s Theorem [5], A is representable in the
K(e, 8).
ZB, Y and Z polynomials
s0* al*
in
.
Thus, A
polynomial in
Y+
_
This is true since
and B is a finite simple point
ZB and on the surface, A
s0* el* e2*
a2*"
and B with a multiplicity at
must vanish at the simultaneous zeros of
least as great as B and any such zero of
of
and
with coefficients in
ZB, so that A/B
K(e, 8).
Z, a
Returning to
affine co-ordinates, we see that the ratio A/B is a polynomial in e*,
8
with rational coefficients in e and
so that we can write
F(, 8, *, 8*)
D(e, 8)
B
where F and D are polynomials.
D and (, 8), i.e., E
8*
y+
Let E be the resultant with respect to 8 of
6D.
Therefore,
F
D
E
D
F
y
E-
and on the surf ace
D
E
s0 is a zero of order r of E, then F must be divisible by (
If
A/B has no poles for
since
(, 8, *, 8*) on S.
finite.
2
0 )r
Thus, F/E is a polynomial
Finally, we can write
W2
and W
E
nd*d
(2.1)
G(e)
has poles only at the zeroes of G and possibly e
at our final reduced form for
2’
i.e.,
polar curves on S are C 2 or its copies.
kind may, by subtraction of a suitable
(/G)de*d (modulo
.
We have arrived
dl),
whose only
Any double differential of the 2nd
dl,
be reduced to one of the form
(2.1).
THEOREM.
Any double differential of the 2nd kind on
can be reduced,
DIFFERENTIALS ON A PRODUCT SURFACE
modulo
dl,
to the form
cij i
rials of the 2nd kind on C
PROOF
1
and C
We begin by writing
j*
A
2 and no
2
0
i
or
i
in the form
-
and m.* are simple differen3
3
*
is derived
(2.1), i.e.,
de*de.
2
If
where
625
is of degree d in 8", we can write
c’s
where the
Cd[,
8,
+
+
e*](8*)d + Cd_l[e,
Cl[a,
are polynomials.
polynomial in *.
8, *]8" +
8, e*](8*)
c0[a,
d-i
8, *],
Also, each of the c’s can be written as a
Let the degree of c in e* be d
i
i.
Then
each c
can be
i
written
c
i
b
d
i
+ b i,d.-i (u*) di-i +
(e*)
i,d.1
1
where the b’s are polynomials in
and
8.
+
bi,le*
+
b
It is then clear that
can be
written as a finite sum of terms of the form
(8*) q
where P is a polynomial and p and q are positive integers.
P( 8) (*)P(8*) q
G()
Consider
(2.2)
d*de.
This is a double differential of the 2nd kind on S and can be written as
(0*) P(8*) qdo*]
(2.3)
a()
a product of two Abelian differentials, the first on C
the second on C
I.
2 and
Each of them must be of the 2nd kind on their respective curves since,
J.C. WILSON
626
Let the
if not, the double differential would have non-zero residues on S.
genus of C I be gl ad that of C 2 be g2
d9 2g
dgl, d2,
respectively.
Also, let
dl
d2
d
U2g I
be bases for differentials of the 2nd kind on C
and
I and C 2,
2
Then
2g
(,)P(8,)qd,--
2
i
kdk
and
2g
Pd_
G
where the
c’s are
constant.
i
I
cid i’
Thus,
P(s, 8) (e*) p (8*)qd*d
G()
CikdVkd
where
Cik
cik.
Since each term in
w/G da*de can be so written, the
theorem follows.
REFERENCES
i.
Lefschetz, Solomon. Algebraic Geometry, Princeton University Press,
1953, p. 196-210.
2.
Ibid, p. 221.
3.
Ibid, p. 202.
4.
Ibid, p. 206.
5.
Van der Waerden, B. L. Einfuehrung in die Algebraische Geometrle,
Springer, Berlin, 1939, p. 209.