DIFFERENTIALS OF THE 2 ND KIND ON A PRODUCT SURFACE

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.


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respective, then an afflne model, S, of 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 following: "Every double differential of the 2nd kind on is reducible, modulo de, (where is a simple differential on ), to a linear combination of products A * of simple differentials of the 2nd kind, where belongs to C I and * belongs to C 2 and neither are derived."See Lefschetz [2].
We point out that the last above statement can also be realized by using modern techniques of algebraic topology.In particular, see page 80 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 not in the ideal generated by or .We assume that the double differential 2 Rde*d is of the 2nd kind on S.Here we employ the Lefschetz definition [3], i.e., if Q QIQ2 Qk' where the Q's are distinct, irreducible poly- nomials, then 2 is of the 2nd kind relative to Qh provided there exists a simple differential on S such that 2 de is regular at the points of in- tersection of Qh 0 with S. Such points generally form a curve on S, called a polar curve of m2' and we designate it by h" The simple differential depends on the choice of h and we say that 2 is of the 2nd kind on S provided such an exists for each polar curve of 2" As an analogy to Lefschetz's plane section, we employ the curve C 2 and its copies on S, the latter obtained by varying and 8 on C I. First consider the intersection of h with such a "general" C 2. 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. Also, a non-singular linear transformation will guarantee that both and Such co-ordinate changes do not effect the kind of differential involved.
The el* and 8i*'s will depend on the choice of and 8. Suppose h is a polar curve of order n.Then, in a neighborhood of P i on our general C2, the following expansion of R is valid: + higher powers of (* .*). 1 We use (e* i* as parameter and all numerators are rational in their variables.

J.C. WILSON
LEMMA i.There exists a simple differential lh on S such that 2 + dlh is regular on h" m PROOF.Define H(, 8, *, 8*) R (, .,)n.We first will obtain 1 i=l an expression for R_n( 8, el*' 8i*)" Let E 1 be the resultant of Qh and with respect to 8* and E 2 the resultant of E 1 and with respect to 8. E 2 may be considered as a polynomial in e* with polynomial coefficients in e and as such, has the .*'sas its only zeroes.For e finite, this implies that the elementary symmetric functions of the c.*'s are polynomials in a.Thus (, .,)nl is a polynomial in and *.For simplicity, set (*-i*) n i=l i=l n n Xh (c, c*), where Xh may be thought of as the projection of h on the (c, c*)- plane.It is evident that H(, 8, *, 8*) is of order zero on h" Also, m H(, 8, i*' 8i*) =1 (i*-.,)n3R_n(, 8, i*' 8i 2), we see that V not only has h as a polar curve but can -n n have poles i) at points of intersection of h with k' k # h, for then H has poles and their occurrence depends on and 8; *=j* j # i; ii) at points where is such that i iii) if or 8 is infinite.Since not both e and 8 vanish, we can assume 8 # 0 and get d8 Then (1.4) can be written as Thus, in a neighborhood of Pi' Rde*de + d[Vnde] behaves like 2 but with n replaced by (n-i).Next, we can define a Vn_ 1 of the same type as Vn with (n i) in place of n and Rde*de + d[(Vn + V n_l do] will begin, in some R-n+2 neighborhood of P i' with the term n-2 Continuing in this manner, (e* el*) we will arrive at the differential Rde*de + d[Vn + Vn-i + + V2)de], a double differential of the 2nd kind in which the corresponding n is one.i If we still denote the coefficient of e* ei* by R_l then since 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 h having poles only of type i), ii), or iii), above.See Lefschetz [4].If  then we can immediately show that Rde*de + dlh is regular on h' as follows: From the above, it is evident that Rde*de + d[ (V 2 + + Vn)de] behaves like 2 relative to h but with the corresponding n i, so we need to look only at dei* This concludes the proof of Lemma i.
For brevity, we will write lh in (1.5) as lh V(e, 8, e*, 8*)de-U(e, 8, e*, 8*)de*, where V and U are rational on S. From Lemma i, V and U are infinite on the intersection of Xh 0 with S, so that in addition to h' they are infinite on any other curve which projects onto Xh 0. We will designate this residual intersection by D h.LEMMA 2. lh may be replaced with a similar differential but with D h eliminated as a polar curve (or curves).
PROOF.For almost any non-singular affine transformation (e, 8, e*, 8*) to (, , *, *), the projections of the transforms of h and D h on the (, *)-plane will have at most a finite number of points in common.
Under such a transformation R de*de + d(Vde Ude*) becomes where we set R de*de + d(V de U de*), G(, 8, *, 8*) A, B, m'd G polynomials.Here we have made use of the transformations 0(, 8) to (, g, *, g*) and ,(*, 8*) to (, g, a*, g*) in eliminating d =d d*.We can also assume that neither nor are zero, due to the absence of singularities.Let the resultant of and with respect to " be denoted by I' i.e., la + J I" Further, the resultant of i and with respect to gves a second resultant, say G1, i.e., LI + N 1" us, LIa + LJ 1 N and in the transfoea surface LI 1 since LJ and NO are ero.The above reduction and replacement can be carried out for all polar curves h of 2" Returning to the original notation and co-ordlnates, we would then have a set of lh such that 2 + d( lh would be regular on where A, B, and G are polynomials and A/B is regular except possibly for points of infinity.
Ado*do LEMMA 3. W 2 BG where is a polynomial.can be reduced to the form (O, IB *, B*) do*do (a) PROOF.We begin by replacing the affine co-ordinates o* and * with projective co-ordinates o0*, i*' and o2" in both (o*, 8*) and B(o, 8, o*, B*) where B is to be regarded as a polynomial in e0*' l*' 2" with coefficients J.C. WILSON in the field K(e, 8).By Kapferer's Theorem [5], A is representable in the form Y + ZB, Y and Z polynomials in s0* al* and a2*" This is true since A must vanish at the simultaneous zeros of and B with a multiplicity at least as great as B and any such zero of and B is a finite simple point of .Thus, A Y + ZB and on the surface, A ZB, so that A/B Z, a polynomial in s0* el* e2* with coefficients in K(e, 8).Returning to affine co-ordinates, we see that the ratio A/B is a polynomial in e*, 8* with rational coefficients in e and 8 so that we can write and W 2 has poles only at the zeroes of G and possibly e . We have arrived at our final reduced form for 2' i.e., (/G)de*d (modulo dl), whose only polar curves on S are C 2 or its copies.Any double differential of the 2nd kind may, by subtraction of a suitable dl, be reduced to one of the form where P is a polynomial and p and q are positive integers.Consider P( 8) (*)P(8*) q G() d*de. (2.2) This is a double differential of the 2nd kind on S and can be written as (0*) P(8*) qdo*] a() 3) a product of two Abelian differentials, the first on C 2 and the second on C I.
Each of them must be of the 2nd kind on their respective curves since, J.C. WILSON if not, the double differential would have non-zero residues on S. Let the d genus of C I be gl ad that of C 2 be g2 Also, let dl 2 d and U2g I dgl, d2, d9 be bases for differentials of the 2nd kind on C I and C 2, ei*)n-i A typical element of the last sum can be written as -n d8 (ni) -n (e* e 1.*)n-I 8e do + 88 (e* el ,)n do*.
de I'(e* ei*)dT -. , ) 2 T d ( e * e.l* )term in the last sum above is zero and the second and fourth terms are negatives of one another.However,

h
the h but would have poles for certain values of o and B. Let us write 2 + d(,lh in the form h W2 A(o, [3 o*, S(o, 6, o*, B*)G(o)

3 PROOF
2nd kind on C 1 and C 2 and no or -We begin by writing 2 in the form (2.1), i.e., 2 de*de.If is of degree d in 8", we can write Cd[, 8, e*](8*)d + Cd_l[e, 8, e*](8*) + + Cl[a, 8, *]8" + c0[a, 8, *], d-i where the c's are polynomials.Also, each of the c's can be written as a polynomial in *.Let the degree of c i in e* be d i.Then each c can be le* + b where the b's are polynomials in and 8.It is then clear that can be written as a finite sum of terms of the form (8*) q the c's are constant.Thus, where Cik cik.
(e*) (8*)qd*dCikdVkdSince each term in w/G da*de can be so written, the DIFFERENTIALS ON A PRODUCT SURFACEWe can now show that 2 + d[Vnda] Rde*de + d[Vnde] behaves llke R on h but with n replaced by (n i).First It follows that lh can be written as It is evident that dfi*dfi + dlh is regular on the transform of h and this is true even if we suppress the last term above, i.e., differential of the 2nd kind on the transformed surface, regular on h transform and the transform of D h is no longer a polar curve of lh" a