On the linear connection index of the algebraic surfaces z^n=f(x,y)

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ON THE LINEAR CONNECTION INDEX OF THE ALGEBRAIC SURFACES z' = f(x, y) By OSCAR ZARISKI DEPARTMENT OF MATHEMATICS, JOHNS HOPKINS UNIVERSITY Communicated May 11, 1929

1. The purpose of this paper is to prove the following theorem: THE3OREM. If n = qa is a power of a prime number and f(x, y) = 0 an irreducible algebraic curve, then the linear connection index R, of the surface z = f(xy) is equal to 0. By this theorem, the above surface is regular and does not possess simple integrals of Picard of the 1st and 2nd kind. The conditon n = qa is essential, as is shown by an example at the end of the paper. The theorem has been proved for n = 2.1 In this case De-Franchis2 has obtained a more expressive result stating that, if the surface Z2 = f(x, y) possesses q simple integrals of the 1st kind (R, = 2q), then the curve f(x, y) = 0 is composed of 2q + 2 or 2q + 1 curves belonging to one and the same pencil. 2. We now proceed to the proof of our theorem. We consider the surface F, given by the equation z =f(x, y),

(1)

where, by hypothesis, the curve f = 0 is irreducible. Let Cx be a generic curve of the pencil C |, cut out on the surface F by the planes x = const., and let p be the genus of Cx. Any one-dimensional circuit I on F is homologous (mod. F) to a linear combination of 2p independent circuits on C;.3 We next proceed to fix a fundamental set of circuits on C; in the following manner: Let m be the order of the branch curve f, and let y, Y2, . . ym be the roots of the equation

f(x, y)

=

0.

If we consider in the y-plane a set of non-intersecting oriented loops, gl, g2, * *, gm, surrounding the points yi, Y2, ..,y*^, respectively, then, when the variable point y describes the loop gi, the n branches zi, z2, . . ., n (Zk = C*k-1 Ziz, = e2ri/n) of the function z are permuted cyclically. To the loop gi there correspond on C; n open overlapping paths starting on different sheets of the n-sheeted Riemann surface of C;. We still denote

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by gi that path which starts on the first sheet and ends on the second sheet. The remaining n - 1 paths may be conveniently denoted by p(gi), 'p (gi),

( (- 0

*.*

where the symbol spo indicates the transformation y' =y, X=

z =Wiz

C; which we now introduce are the following:

The circuits on 7Yi, = g1

X.

gi +

1

Yi,2 = p('Yi,l) = (P(gi)

(i

1, 2,

=

...

-

so(gi + 1)

* *

* Yin = (- (-Yil)

I., m-1).

The (m- 1)n circuits -y,,j are not independent (mod. place the following m - 1 homologies exist: + Yin 0 (mod. C;). ti + 'Yi,2 + These are consequences of the homologies: -

-

-

-

C;).

In the first

(2)

gi + p(gj) + ... + pon - 1(gj) ,- 0 (mod. C-). It is immediately seen that if 6 denotes the H. C. F. of m and n and if 6 $ n, then the point y = o is a branch point of the function z, and that in the neighborhood of y = co the branches of z are distributed in 6 cycles of order n/6. From this there follows immediately: 2p

=

(n

-

1) (m

-

1)

-

(6 - 1).

Hence, in addition to the homologies (2) there must exist further 6 - 1 relations among the circuits -yi. These relations are easily obtained by considering the 6 circuits on Cx which correspond to a circuit in the y-plane surrounding the m points yi, and by observing that they are all homologous to zero. Apparently we thus obtain 6 additional homologies, but it is easily seen that only 6 - 1 of these are independent of the homologies (2). Although the circuits 'Yi j do not form a minimum fundamental set on C-, we prefer for reasons of symmetry not to undertake a further reduction. 3. To each generating relation among the generators gi of the fundamental group G of the curve f = 0, there correspond several homologies (mod. F) among the circuits -yi j. I proved in a recent paper4 that, if the curve f is irreducible, the generators g, are conjugate elements of G. This is all that we require for our present considerations. We have then egi isacoepa +1 = thC

where

g(')

is a

2p (i = 1, . . .ther- 1)c s (3) closed path in the y-plane. To g(i) there correspond on

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Cin (closed or open) overlapping paths starting on different sheets. We still denote by the same symbol g(i) the path which starts on the first sheet, and assume that g(i) ends on the rth sheet. Then g(i) - r 2(gi) - g1 is a circuit ri on C;, i.e., -* *-,o(gl) -

g(i)

=

g1 + rp(gi)

ri +

+ * * * +

r

-2(gl).

If r = 1, then g(i) is itself a circuit F,. From the generating relation (3) we deduce the following homology (mod. F): gi +

ri + gi + q+(gi) +

pr2(gi) + sp- I(go) 1(g1) . . . - P2(gi) (p(gi)

+ p-

-

-

-

(ri),

or

gi + 1

-

ri

p(Fi) + gl,

-

and finally

'yi,l + ri

-

s(pw)

(4)

0.

-

Transforming (4) by sp, sp2, . . . np - 1 we obtain the following system of n(m - 1) homologies among the n(m- 1) circuits -y1,j: yi,l +

ri

-

p(Fi)

-

o,.0 + i(ri) - ri 'Yin (i=1, 2, ... .,Mm-1).

0, 'Yi,2 + p(Fi) - p2(rP)

.

n

-

0

(5)

In (5) the r's are linear combinations of the 'y s. We prove in the next sections that if n is a power of a prime number, then the determinant A of the coefficients of the system (5) is different from zero. Hence, if n= q, then for any of the circuits y,j:

0,

Ayi,j and this proves our theorem. 4. Let m-1

ri

n

E E

bs't eS' m-1

Then, since P(-Ys,n) = Ys,, we have p(FO) ' must be replaced by b). We set a't= bi) b"

n

i where b1 'vt,

s=1 t=

bSi)

(6)

for any value of s and t, except for s = i and t = 1, in which case we set at,)= bY) - b + 1. (6a)

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The index i being fixed, the first of the homologies (5) becomes m-1

n

a,

a"0.: ]r

(7)

sol t=ll

The remaining n - 1 homologies can be obtained from (7) by applying successively the substitution

(a~i ai)

a())

ai) . ai . ., a d ,

...

.(a(')

1,1,

a"i)-,

.., a( _ l.)

to the coefficients. Hence the matrix of the coefficients a") of the system (5) is made up of (m -1)2 circulants A ") (s, i = 1, 2, . . ., m - 1), where a(i) a(i) a() (i) a(z) a(is n a3,1 . . s,n-1 A"?) .

afi) s,

=-H1 ani) 11

.

s,

a(i) .

s

a"' II symbolically in the following form:

We may write our matrix

A

a(i) ..

A (1)

All) (2)

=

A (1) M1

A(2)

A(2)

.

A(m-l) A(m-l)

(

. . .

l

We observe that it follows from (6) and (6a) that n

ai) tall

=

(8)

b ,i

where &s,; = 0, if s $ i, and 6jj = 1, i.e., the sum of the elements of a row (or of a column) in each matrix A(') is equal to 0, except the main diagonal matrices A (i) in which this sum is 1. In the following section we deduce a formula for the value of the determinant A - |i) : If we put f'() = a, + a")x + . + ()Xn 1 (9)

and if we form the determinant

fA1)(x) f(x) =- f21(x) W .~

fA() A(2)(X)

.. . . .

f

..

f(

m-1

.

f ) I(

.

2) l

(9a)

then A

=

where 1, W, W2, . .

f(1)f(Co1)f(w2) . . . f(C."-1), *, ,n-1 are all the nth roots of unity.

(10)

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The formula (10) extends to determinants made up of circulants the known formula for the evaluation of a circulant. Using the formula (10) and taking into account the relations (8) we qa, then A $ 0. In fact, it follows from can easily prove that, if n (8) and (9) that Af (1) = and hence, from (9a), that (11) f(1) = 1. If, however, A = 0, then for some value k, 1 < k < n - 1, we will have

f(wO)

(IIa)

0.

=

Let Wk be a primitive vth root of unity (n = 0 (mod. v)), and let s.(X)

0

=

be the irreducible equation with integer coefficients which Wk satisfies. By (Ila) we have (12) f(x) = 0 (mod. Vo(x)). It can be easily shown that if v =

qo is a power of a prime number q, then

pVP(l)

=

(13)

q.

In fact, relation (13) holds for j3 = 1, for Vq (X) =

Xq -1 + Xq-2 +

We prove now that if (13) holds for any for ,3. In fact, we have

Vq6(X)pq

-(X) W

.

.

.q(X)

= x2$

+

+1

x

f3' less than f3,

+

x2$

+

v

then it holds also +

X

+ 1.

Hence,

Pqp(I). q'

q

,

and consequently

Vqu(l) = q. If n that

=

qa, then v

=

qf,3

<

f(l)

Q. E. D.

a, and hence from (12) it follows by (13) 0 (mod. q),

which contradicts (11). Thus the hypothesis A = 0 leads to a contradiction, and hence A $ 0.

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5. To prove the formula (10) we proceed as follows. We consider the matrix 11 d

=

1 n-i

...

1

* *

W.2

1

Wi

1

WI

(02

1

co,n-1

W2n-1.

2

2

2

.

*

n Xn -1

.

and we adopt the following symbolic notation:

mr-1 d O ... o o d ..o

D=

0 0 ... d

for the determinant of order n(m - 1), where d stands for the matrix introduced above and each 0 for the zero-matrix of order n. If we now form the product AD, it is easily seen that the corresponding determinant can be written symbolically in the form:

d(l)' d(l) AD

=

d(2)

d(2)

1

2

d~mi- )

..d(l)

d(2)md~m-l)

-1)

d~7'

(14)

*** m-1

where each element d(i) stands for the following matrix:

Ai) (w1)

As () d (i)-

W f() (1) f(i) (1) C°o1 lf(i) (w)

Ai)(Cn

. ..

f(i) (1)

n

. . .

W

- lf(i)

(co. 1) -

1n-1fj(So- D

Let us now consider the Laplace expansion of the determinant (14) in terms of the elements of the columns of order 1 + k, 1 + k + n, 1 + k + 2n, ..., I + k + (m-2)n (O < k < n)-e1). We denote by Bk the matrix composed of these columns. Any determinant of order m-1 extracted from the matrix Bk has the following form: C

jf(l) Yi) (Wk) (Wk) 72f(i2) (CL) )2f(i2) (Wk)

wrm-lf(zr-1)i(ok)

r

km-lfi2(m 1(Wk)

*

.

.

C)k fm-1l(Wk)

(Cik~fm. . .

w

l

,~ (15)

Ikml mm L 1) ('Wk)

where il. i2, . . ., imn- are numbers of the series 1, 2, . . ., m- 1, ii < i2

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< im_ , and ji, j2, .. ., jm-1 are numbers of the series 0, 1, 2, . . .. Every determinant (15), in which two of the indices i are equal, vanishes. Hence in the non-vanishing determinants we must have ii = 1, Ai = 2, ... ., im = m - 1, and so, by (9a), the value of such a determinant is equal to <

n - 1.

f (Wk) *fiWk

i

+

Jm-l

Thus all non-vanishing determinants (15) have a common factor f(wk). If we then consider the matrix Ck of as many rows and columns as Bk ((m-1) n rows, m -1 columns): Wk

0 0

...

0 0

co-lk

0

. ..

0

o

1

...

o

W0

.

0 0

o

CWk

...

0

o

o

...

1

o

o

. ..

W~k

o

0

. ..

1

Ck =

...

Wk1

We see by mere inspection that every determinant (15) extracted from the matrix Bk is equal to the corresponding determinant extracted from the matrix Ck multiplied by the factor f(wk). Consequently the value of the determinant (14) remains unaltered if we replace in it the elements ofthecolumnsoforderk + ,k + 1 +n, ...,k + 1 + (m - 2)n by the elements of the columns of the matrix Ck, provided we multiply the resulting determinant by the factor f(Wk). If this is done successively for k = O 1, ., n - 1, the determinant (14) is reduced to the following form: AD = f(l) f(w') f(wO2) ... f(ct..) D', where the new determinant D' is easily seen to be nothing else than the determinant D. Hence D' = D, and consequently A = f(l) fw1) f(CO2) . . . f(On-1),

which proves the formula (10). 6. We want to show in this section by an actual example that the con-

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dition n = qa, under which we proved our theorem, is essential. Let us consider the surface F given by the following equation: (16) Z6 = $p2(X, y) + {,3(x, Y), = 0 represent a cubic curve and a conic, respectively. where sp = 0 and 0/Here the branch curve of the function z is an irreducible sextic. However, we show that F possesses a simple integral of Picard of the 1st kind. In fact, if we consider the following pencil of surfaces:

4,(X, y) - t Z2 = 0,

(17)

where t is a parameter, by eliminating a, between (16) and (17) we have (18) the equation z6(1 - t3) - p2(x, y) = 0 But the surface (18) breaks up into two surfaces $°

+ Zax/i7C3 = 0,

$-

Z

;

= O.

and this shows that the generic curve C of the pencil C cut out on F by the surfaces of the pencil (17) breaks up into two curves D = D1, D = D2. When C varies in the linear pencil C |, the curve D varies in an elliptic pencil D . In fact, the elements of the pencil D are in one-to-one correspondence with the points of the elliptic curve u2 = 1-t3. (19) The abelian integral of the 1st kind /t U = I-

I

I

attached to the curve (19), evidently becomes a simple integral of Picard of the 1st kind attached to the surface F, when we put t _ 6U =e> , z2

z3

Hence the surface (16) is irregular, R, > 0, and-we observe incidentallyit can be easily shown that R1 = 2q 2. 1 H. Lacaze, "Sur la connexion lineaire de quelques surfaces algebriques," Ann. FacultA Sciences l' Universitg Toulouse, 2nd series, 3, 1901, pp. 151-215. 2 De-Franchis, "I piani doppi dotati di due o piA differenziali totali di prima specie," Rendiconti dell' Accademia Lincei, 5th series, 13, pp. 688-695 (1904). 8 See E. Picard and G. Simart, Thiorie des fonctions algibriques de deux variables independantes, 1, Chapt. IV, sec. 11, p. 86 (Paris, Gauthier-Villars, 1897); S. Lefschetz, L'Analysis Situs el la giometrie algetbrique, p. 33 (Paris, Gauthiers-Villars, 1924). 4 "On the problem of existence of algebraic functions of two variables possessing a given branch curve," Am. J. Math., 51, No. 2, April, 1929.