AN IN SOLUTION OF PERIOD SEVENTEEN
By
IrGHN WILLIS KENELLY, JR.
4 DI.EPTATi*.*N 06ESENTED TO THE GRADUATE COUNCIL OF
irHE UNIVERSITY OF FLORIDA
IN irFT*I. Fl.*!i.iLLMENT OF THE REQUIREMENTS FOR THE
DEC BE f OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
January, 1961
ACKNOWLZEDGMENTS
The author expresses his sincere appreciation to
Dr. W. R. Hutcherson, chairman of the supervisory committee, for
his suggesting the problem and counseling before and during the
preparation of this dissertation. Dr. Hutcherson has been a continuing
source of inspiration, and his direction greatly contributed to making
this study possible.
To Dr. J. E. Maxfield, Dr. T. 0. Moore, Dr. W. P. Morse, and
Dr. G. R. Bartlett the author expresses his appreciation for searing
on the supervisory committee and for their assistance in editing this
dissertation.
The typist, Mrs. Th~omas Larrick, was also of great assistance
in the final preparation.
. . . .
811
8. Summary
IV. A RAiTI3ONAL SURFACF F I:
Conclusion .....
FABLE OF CONTENTS
Page
.... ii
. . 1
ACI~O!:tLEDOFC T.. .. .. .. .. .. .
C~rAPTER
I. INTRODUCTION ........
II. A SURFACE $ OBTAINED FROM~ ANI I:MYDLUTIDN~ OF'
PERIOD SEVENJTEEH .
.
. .
and 0'
11
The Imagoe Surface (
Branch Point 0' .
Branch Point O010
Branch Point O'
P
.
. .
" '
. .
oints
.
*

5. Multiplicities of
for Surface .
6. Summary.
III. PROJECTIONS OF THE SUEFACE $
1. Surface S1 ''''**
2. Surface t, ...*
3. Surface 23 '"" "
4. Surface a
5. Surface $5 "
6. Surface $6'
7. Surface @ ...
TABLE OF CONTENTS(Continued)
Page
APPENDIX
I. A METHOD OF FINDING THE ORDER OF A QUINTIC
TANGENT CONE ................... .. 60
II. A METHOD OF INVESTIGATING A FOURTEENTH
ORDER NEIGHBORHOOD .. .. ... .. . 62
III. A METHOD OF DE~MONSTRATINJG THE EXISTENCE
OF Y(X1,Xe, ..., 4).. 66
BIBLIOGRAPHY ................... ........ 69
BIOGRAPHICAL SKETCH ................... ..... 72!
CHAPTER I
INTRODUCTION
In an extended complex plane with homogeneous coordinates
the equations
xl : x2 : x5 = xl : Ex2 : Eax5
define a plane cyclic homography of period p, where p is a prime number
greater than two, E a pth primitive root of unity and a is an integer
greater than unity and less than p. This homography generates an in
volution of period p.
Lucien Godeaux has been the world's leader in studying involu
tions. Since his paper in 1916 (5] where he used period three, he has
published many papers on involutions. Many other authors have contrib
uted to this field. Hutcherson studied involutions of period seven and
eleven (14, 15], Childress studied some of period three, five, and
thirteen [18, 19], Frank studied some of period eleven [3], and Gormsen
studied some of period three, five, and seven (12].
This writer is investigating the mapping of an involution of
period seventeen from a plane onto a surface in a space of ten dimensions
(S10). The three branch points of this surface require detailed study
comprising Chapter II. In Chapter III certain projections of are
investigated. A rational surface F, in S11, is exhibited in Chapter IV.
Points on this surface are in a onetoone correspondence with points
ihe ma1tjrlll in ;:hapter ii uis? rhi subject :f s !::*nt F'per
Lven at the 1960: EnTe~r meeitrtin :f1 ther remerl*.s n rlj'nhematicjl Clt:
[f..The conTt4Tnt.' of *Chapter I!V nere used in jn:othir :Lornt Paper"
Li:n .lso: jt thul meeating (:.1].
Ine reader is referred to the bltllograaphy for introductory
material to this area. One unfamiliar with the usage of terms, symbols,
and techniques of this phase of Algebraic Geometry might not fully
understand certain areas of this dissertation, e.g., first order neigh
borhoods [12]. Also homogeneous projective coordinates are used ex
clusively (15). Since introductory material is plentiful and available
it is usually omitted from most areas and references are mentioned
instead.
As far as the author has been able to determine, most of this
work is original.
CHAPTER II
A SURFACE OBTAINED FROM AN INVOLUTION
OF PERIOD SEVENTEEN
1. The Luage Surface
Consider the homography,
(H) x1 : x2 : x3 = x: Ex2 : E15x5
where E is a primitive seventeenth root of unity. This homography
generates an involution, 117' of period seventeen. A group of 117
is composed of the following seventeen points,
(xl' x2' x5), (Xl, Ex2, E15x3), (Xl, E2 2, E15x3 '
(xl, E3x2, E~11x3)' (xl, E4x2, E x5), (xl, E5x2, E x5 *
(xl, E6x2, E5x5), (xl, E7x2, E x3), (xl, E x2, Ex5 '
(xl, E9x2, E16x5), (xl, E10x2, E14x3), (xl, E11x2, E12x$'
(xlt E12x2, E10x3), (xl, E15x2, E x5), (xl, E14x2, E6x3 '
(xl, E15x2, E4x3), (xl, E16 2, E2 3) .
Now consider the complete noninvariant linear system of order
seventeen in the plane, i.e.,
I ah xi x~ xj 0
+~;r~ a5 x7 x2 x + a x5,~ x, x3 +. a. x4i x5rirlt 10 + al35 x2 x10 x
+ a9 x57 x2 x3 + a7 x4 x2 x31 + al06 x43 x14 + al55 x2 x1~ x
+ al46 xl x xx312 1 x160 x5 0.1)~7_
(4) a~ll X21 x2) + a17 ~210 x6x a40 xC~ x x + a60 x1 x~ x2
+ a74 x6 x5 x8 + a7 x4 xl0 x3 + all? x5 2 x9 + al22 x1 2 x35
+ al45 x1 x2 x3 + al67 x1 2 x10 + 16rln 13n g
(5) ag x4 x5 + a27 ll x22 x5 +, a5 8 x2 x + 6 6 xll
+~~~~~~~ a91 x5 x6 x57 +954 2x2+a8x x x +al5 2
+ al44 xl x23 x5 + al56 xl5 x3 = 0.
(6) a2 x16 X2 + al$ x13 x23 x3 + a53 x10 x2 x5 + a38 x1 x5
+ a62 X1 x2 x5 + a72 x6 x2 x9 + 100 x4 x x5 + all5 x5 2 1
+ al47 X1 X21 x + al 65 x2 xgl e 0.
(7) a25 21 x2 x53 + aB7 x x8 + a55 x18 x5 x3 + ag69 x6 x10 x5
+a90 X5 x2 x3 + 95 x4 xC~+l5 X1x3 x12~ x2 + al56 x12 x7 x3
+ al42 xl x2 x4 + al58 X14. x5 0
(8) aS 216 x5 + al5 x313 x2x2, 4 95 x10 x2 x~ + a64 1 x X6 x34
+ 7 6 x2 x10+ l x4 x8 x5 all x5 x5x1 +2 al"21 x2 1
+ 0 l4x1 xl x + al5x 22=0
(9) al6, x2 x2 + a235 x11 x36 +a59 ~x x x a 51 1 x x x
+ a71 x6 x 3 ag x5 x2 x8 + all2 x5 x11 x3 + al54 x2 2 xj
+al40 xl X2 xl5 + al60 x3 x~ = 0.
9 7 15
+al3 xl x5+a6 x 5.0
+ a83 1 x5 x11x + a8105 x4 2 x5 + al09 x1 x2O x5 + al25 x12 xl5 x5
86 8 5 1
+ al55 xl x23 + al59 X2 x3 0.
(15) a6 21 x21 x3 + a20 xl20 X2 x5 + a455 x9 x a5 g +2 a7x x
+ a75 x6 x2 x5' +1 84 x5 x2x +all6 x53 x2 x~ + al50 x12 x2 xll
+ al57 xl x16X + al64 xll x 0.
(14) a22 xll x6 + a32 0 x2 x63. + a48 x8 x8 x5 + a65 xl x5 x
+ a83 x510 x5 X + al05 xl x5 x13 + "18 10 x5 x5 + al27 x2 x22 x5
+~ ~~ ,15 xl x2 +a57 X2 x135 O.
(15) a5 xl5 x23 + a21 x12 x2 x3 +B a4 x~ x2 x5 + "77 x6 x6 x
+ a82 x5 x2 x11 + a92 x4 xl5 + all8 x3 x2 x6 + al28 x12 x53 xl2
+ alB9 X1 xl5 x5 + al66 210 xB = 0.
(16) a7 xl4 x5 + a24 xll x2 x + a30 xl0 x5 + a50 x18 X27 x2
+ a61 x7 2x 8 5xx l0 4x l9x l x3
+ al50 xl x2 x30O + al55 x2 x56 = 0.
(17 a9 x2 2 3 + a44 xi x2 x5 + a56 x 10 +2 a 6x2
+ a80 x5 x5;2 + a94 x41 x22 x3 + al20 x3 x2 x5 + al26 x2 x2 $3
+ al41 xl x14 x53 + al68 xi x8 = 0.
Now relate the curves of system (1) projectivel~y to the hyper
planes of S10 by taking the protective transformation
X1
X2
x14 x2x5
5
xll x2 x5
x4
x X2 x
x7 x2x
X6
5x8 x5
X7 8g X
X1 x2 x10 x2 x0x3 xl x5 xll1
10
17
X2
X11
17
x5
Eliminate xl, x2, and x5 in T to obtain a surface in S10
which has for its equations
(1 ) X10 Xg~ X5 16 4 9 5 X3 ]7 X2X 5 ~ 2
4X6 7~ 9 9 X7 7a 11 54 11 1 1 15
This surface is the image of Il7, i.e., a set of 117 corresponds to a
single point of the surface ~. Now investigate the singularities of
the surface 4 at the images of the fixed points of 117. The images
of 01 (1, 0, 0), 02 (0, 1, 0), and 03 (0, 0, 1) are
Or (1,0,0,0,0,0,0,0,0,0,0), Or (0,0,0,0,0,0,0,0,0,1,0), and
1 10
O11 (0,0,0,0,0,0,0,0,0,0,1) respectively.
2. Branch Point 01
This investigation is based on a technique that finds the
projective iuages of successive neighborhoods of the vertices of the
triangle of reference in the plane. This definition of neighborhood
is based on the existence of a quadratic transformation which relates
two planes with homogeneous coordinates in such a manner that a refer
ence triangle vertex in the z coordinate plane is mapped onto its
corresponding reference triangle vertex in the x coordinate plane,
but the x coordinate vertex is mapped to the meaningless point
(0, 0, 0) in the z plane. For example, 01 (1, 0, 0) in the z coordi
nate plane is mapped onto 01 (1, O, 0) in the x coordinate plane,
and 01 (1, 0, 0) in the x plane is mapped onto (0, 0, 0) in the z
coordinate plane. Then the z plane image of the point S(1 + Xa, XB, XY]
from the x plane, will be the image of the first order neighborhood of
01 if the limit is taken as h tends to zero. For a more detailed dis
cussion of neighborhoods the reader is referred to Morelock [25].
Note that members of the family (1) do not in general go
through the vertices of the reference triangle. But if the restric
tion al = 0 is added, a new family (18) will go through.
(1)ag x4 x~ x3 + a26 211 x2~ x5 + a52 x .x6 x + a59 x7 x2 x~
+ 87 x5 x x a x x150 +,3 al55 x2Z x10 + al48 xl x52 xl11
+ al70 x1Z7 + 171 x53 = 0.
Th~e quadratic transformation R and its inverse will relate
P1 (1, 0, 0) in the z coordinate plane to 01 (1, 0, 0) in the x coordi
nate plane and 01 (1, 0, 0) to the meaningless point (0, 0, 0).
(R) xl : x2 : x5 z2 : zlz2 : 22z5
(R)1 z1 : 22 : z5 = x1x2 : x2 : xlx5
Apply the transformation R successively fourteen times to
equation (18) and arrive at
(19) 258 222 111z111 9
+ a2 2~21 zl26 223 + a9 2102 2l27 ,130 + 2~04,~ 52
18 287 4~8 z4 + 1885 ,2145 z~1 13 170 z684 53'
rthis shows thlat the point (z2 z5 0), corresponds to the point ir
the fourteenth order neighborhood of 01 in the direction of xg = 0,
i~e., 0122222222222222 C 012(14) (5, p. 32].
Now apply the transformation, R, fourteen successive times to
the transformation
x2 x3 xq xs
(20)
14 211 4 2 8 6 5 7 9
xl4 x2 x xl x2 x5 xl x2 x3 x1 x2 x5
X6 X7 Xg X9
X5 x2 x5 x4x 10 x2 x0 x5 xl5x1
X10 X11
X17 1~7
The shaplified result is
(21) X2 X3 5 4,
258 221 16 2 204 52 5 119 111 9
zl 25 Z1 z2 z$ Z1 z2 25 Z1 z2 z5
X6 X7 6g
zl87 z48 z4 zl02 zl27 z10 z170 z64 z5
"1 "2 23 1 22 3 Z1 22 5
9 10 X11
z8z45 z11 z258 z 222 z17
1~ 22 5 1 22 2 3
A substitution of z5 = k z2 will allow an all directional
approach to the point (z2 D z3 0 ). This substitution in (21),
after simplification, gives
X2
(22)
258
k z1
X5
2 221 17
k z1 z2
X4
5 204 54
k al z2
X5
9 119 119
k zl z2
6g 7 Xg
k4 3187 z51 k10 z102 zl56 k5 170 z268
k 1 22 k 1 2 Z1
X9
11 85 155
k zl z2
X10
258
Z1
X11
17 258
k z2
As z2 tends to the limit zero, the above gives the equation of
a plane tangent to at 01. It is
X2 = k X10
Xg = X = X5 = Xg = X = Xg Xp Xqq = 0.
Now examine a different quadratic transformation and its
inverse
(s)
(s1
2
z1: : z : 25 x1x3 : xlx2 : x5 '
Apply it seven times to the curve (18), and the simplified result is
(2) ll ("171 z5 + a59 2223 + ag z2+17 317z0
+~ a26 02 z2 ,15 99 z102 z z 6 + 5 85 z6 z30
368 z z45 85 z5 z51 51 z10 z60
+ 87 Z1 22 3 + al48 Z1 2 + alB5 "1 2 0.
This shows that in the seventh order neighborhood of 01, the point on
the curve (18) along the direction x2 = 0 corresponds to the double
point (z2 = z3 = 0).
Repeated application of the transformation S to the transfor
mation (20), yields
(25) X 54X
119 2 102 4 15 85 6 30 119
"1 z2 Z1 z2 zz z1 z2 z$ Z1 z2zz
6g X7 Xg X
68z435 102 531 51l 310 z60 z85 5~ 51
zl6 z2 5 Z1 z2 5 1 2 1 2z
10o X11
17 104 119 2
z2 z3 "1 z5
Substitute z3 = k z2 to allow for an all directional approach
to the point (z2 = z3 0). This simplifies to give
13
X2 3g X4 5g
(26)
z119 kl5 z1102 z17 k50 z85 z54 k all9
6s X7 X
45 6851 16 102 17 60 51 68
k z122 k z1 "2 k al 22
"9 "0 X11
k51 5354 kl04 z119 k2 z11
1 a~ 2 2 k 1
Now let z2 approach the limit zero, and the equation of the
other tangent element to at 01 is
(27) iX5 X4 6 X 7 8 Xg ~ g 10 = 0.
Thnis surface (27) when investigated is seen to be a quadric
cone.d The reader is referred to Gormsen [12] for a different method
for this investigation, i.e., Coble's method.
Hence, the following
Theorem 1: The tangent elements to the surface at the point
01 (1,0,0,0,0,0,0,0,0,0,0) are a plane (25) and a quadric cone (27).
See Appendix I.
3. Branch Point 010
The point 02 (0, 1, 0) corresponds to the point
010 (0,0,0,0,0,0,0,0,0,1,0) in S10 by the transformation T. To study
the tangent elements at this point, examine the system (oo ) of curves
passing through 02, i.e., system (1) with "170 0. We have
(28) ax17 +ax1x2x+a6x11 x2 x5 + a52 x x6 x5
+ a59 xl x2 x93 + a87 x5 x8 x3 + a99 X4 x5 xl0 +155 x2 x10 x5
+ al48 xl x5 x131+ 171 x537 = 0.
Apply the quadratic transformation
(U) xl : x2 : x5 = 2122 : z2 : zlz3
twice in succession to the system (28). We get
(29 z4 (al x5 + a9 z4 z5 + a26 x5 x~ + a52 z2 z5+871z
5) 22 l? 15 zl7 z9 1 17 zl0
+ al35 z3 + a171 Z1 3 + a59 Z1 2 23 + agg zl2 22 3
+ 148 3~1 27 5 ',0
This indicates that 0211' the second order neighborhood point in the
x5 = 0 direction, corresponds to the five tuple point (31 = z5 0) .
Now apply the transformation U twice to the transformation
X1 X2 X3 4q
(30)
xl xl4 x2 x5 X11 2 x2 x8 2 x3
5 9
x1 x2! x
X6
x5 xx3
X
x4 x3 10
Xg
xx10 x3
x9
5 11
xl x2 x3
This gives
11
17
x3
X1
(31)
z5 z34
X2
z4 z54 z3 z
7 9 4 4
z3 Z1 z2z23
z34 5 32 z54 z5
X5
15 1
"1 z2
X
12 17 10
z1 z2 z3
Xs
34 5
22 z3
X
11 17 11
z1 z2 z3
X11
z1 z3
Let z1 E k zg, to allow an approach from all directions to
the image point (zl = z3 = 0),
X1 z "3 4 5s
5 34 4 34 3 34 2 34 13 17 17
k z2 k z2 k z2 k z2 k z2 z3
6g X7 X8 X9 X11
34 12 17 17 34 11 17 17 22 34
k 32 k z2 z3 22 k z2 zB k z3
As z5 approaches the limit zero in (32), the equation of
a tangent element is arrived at
(33) X1 X2 5 X4 X61
~O
X2 g X4 6g X~l
X5 7 'X = X11 0
This tangent surface is verified to be a quintic cone when investigated.
To study the neighborhood points along the direction xl 0,
examine (28) with the quadratic transformation
1) 2
(V) z1 : z2 : z3 x1x2 : x2x : x5'3
Apply the transformation V five successive times to equation (28);
this gives
See Appendix I.
(54) z85 (al71 Z5 + al48 "1 z5 + al55 z2)+a 1 7
7512~ z2 "9124 2:7 256 11 52 ~ 4 42
+ a59 Z1 22 5 92 5+a62 2 5
4 68 15 8 51 28 5 68 14
+ a99 121 32 3 + a52 21 z2 z5 + a87 Z1 z2 z$ = 0.
This indicates that the fifth order neighborhood point (023(5)) in the
direction xl 0) corresponds to the double point (al = zZ = 0).
Now apply the transformation V repeatedly five times to the
transformation (50), and get
X1 X2 X3 X
1~7 70 Z a 14 17 56 211 54 42 8 51 28
X5 X6 X7 Xg
7 51z29 5 z68 214 4 68 14 2 85
31 z2~1 5 12 2 5 212 z2 z 21 22
"9 X1
8125 8325 2~
To allow for an approach from any direction to the point
(al = zB = 0), let z5 k zl, and obtain
;15 167 8E
k29 z54 z51 kl4 z17 68 kl5 z17 z28 85
9g X11
k e85 k2 z85
As Z1 tends to the limit zero in (36) the other tangent element is
given as
(37) X 2 @XgL. O
X1' "2 5 .g X4 'X 'X5 "67 = 0.
This is seen to be a quadric cone when the order is determined.
Thus,
Theorem 2: The tangent elements to the surface at the point
010 (0,0,0,0,0,0,0,0,0,1,0) are a quintic cone (53) and a quadric
cone (57).
4. Branch Point 011
The point 05 (0, 0, 1) on the plane corresponds under the trans
formation T to the point 0 1 (0,0,0,0,0,0,0,0,0,0,1) To study 011
investigate the system of curves that are members of (1) and have the
restriction al71 0. They are
(58) al 217 +. ax14. x2 x5 a26 x11 X2 x5 + a52 x1 x2 X5
+ 59 x7 x2 x95 + a87 x5 x8 x5 + agg X4 xx50 +1 al5 x.2 0 x5
+ al48 x1 X2 x11 + al 70 x2.0.
Apply the transformation
(M) Xl : x2 : x5 "1"2 : z2z3 : z5
(M) z1 : z2 : zB = x1X5 : X2 : x~x5
eleven times in succession to (58). The result is
(3)187 1717 7 62 119s
(59 z ("170 32 + al48 31) + al z17 "2 + a59 s1 l 2 5
14 140 54 11 109 68 4 51 155
+ ag z1 22 z3 + a26 Z1 z2 "3 + agg "1" z2 z
8 78 102 5 47 156 2 16 170
+ a52 "1" z2 z + a87 Z1 z2 23 + alBB Z1 z2 z$ = 0.
Hence the eleventh order neighborhood point 052(11), in the direction
xl = 0, corresponds to the simple point (al P z2 0).
Now apply the transformation M eleven times to the transformation
X1 X2 X5 4q 5
(40)
17 14 2 11 4 2 8 6 5 7 9
xl xl x2 x5 X1 x2 x 3 xl x2 x3 xl x2 x5
X6 X7 Xg Xp X10
5 8 4 4 5 10 7 10 5 5 11 17
l x2 x5 xl x2 x5 xl x2 x5 xl x2 x5 x2
'1 '~ "3 "1
i. ~II
7a ~11:~,7
31 .; ~1 :, I Zj j
5
7 62 119
z1" z2 z
X6
5 47 136
Z1 z2 zZ
X7
4 31 155
"12 z2 3
Xg
2 16 170
31 Z2! z3
"9
187
z1 zS
187
z2 zz
To allow for an all directional approach to the point
("1 = z2 0), let z2 D k x1, and get
(42) 1
kl7 z187 kl40 zl5z3a4
kl09 z119 z638
k8 Z85 z502
X5
k62! z68 z1319
16 17 170
k "l z$
6
k47 z51 31336
X7
kB1z4z5
31 ,~153
Xp X10
187 187
zS k z5
As z1 approaches the limit zero, we arrive at the equation
of a tangent plane,
(45) X10 k X9
X1=X.Xg X4X; x 5 X6 X7 Xg I0.
Now consider the quadratic transformation
(L) xl : x2 : x5 31z3 : zlz2 : z5
(L)1 z1 : z2 : z$ = x2: x2x3 : x1x5'
Nine successive applications of L to equation (58) give
155 157 17 16 2 156
(44) z5 (al "1 + a59 "2) + a170 Z1 22 + ag zl z2 z5
51 4 119 15 5 156 46 6 102
+ a26 Z1 "2 "3 + a9 zl z2 z5 + a52 Z1 z2 zS
61 8 85 50 35 119, 7:6 z10 z68
+ 87 21 22 zi + al48 "1 2 05+a5 3
This indicates that the point in the ninth order neighborhood
Ln the direction x2 = 0, 051(9), corresponds to the simple point
(al = 32 =0).
Thne transformation (40) under nine successive applications
of L gives
(4)X1 X2' X3 X4
1 315 zl6 z2 316 zB1 2, ~ 211 6 z2 210
X5 6g X7 XB
155 61 8 85 15 5 156 76 10 68
z2 z3 z1 z2 z3 Z1 "223 zS1 22 "3
X9 X10
30 5 119 157 17
"1 z2 z5 z1 z2
Approach the point (zl z2 = 0) from all directions by
substituting z1 k z2,
5
51 54 119
k z2 zz
X7
15 17 156
k z2 z5
X4
k46 51l 2102
76 85 68
k 32 z3
"1
(46) 13
k z5
155
2
X2
16 1716
k z2 z5 3
X6
61 68 85
k z2 zB
"9
kBO z24 19
"10
kl37 zl53
As z2 tends to the limit zero, equations (46) give another
tangent plane,
(47) X1 = k X5
X2 = X5 = X4 6 .g X7 X8 9Q 5 10 = 0.
Note that when the transformation L is applied to equation (58)
and then the transformation M is applied, the result is
(48) z54 (59 32 + a99 Z1 z2 + al48 z2) + al all ~ a357
+ 170 zl1 z25 + a9 z10 2 z5 +26 9z10 31
+ a2 z z1 z5 + 8 7 z2 zj7 + al55 "16 z25 z1 = 0.
X10
25 54
k zl
This indicates that the second order neighborhood point, 0512>
corresponds to the double point (z1 = z2 = 0).
The application of L followed by M to the transformation (40)
gives
X1
(49)
11 8 17
z1 22 zB
X2
10 9 17
"1 zZ z
X5
9~ 10 1~7
X4
z~1 z2 8
5
2 54
z1 z$
9
2 54
32 233
X6
7 12 17
Z1 z2 z
X10
3~1 Zi2
X
a 1 Z 4
X8
6 15 17
"1 z2 Z5
Now approach the point (z1 z2 0) from all directions by
making the substitution z2 = k sl,
X1 X2 X3
(50).
k z7 17 zl7 zl7 klO 17 z17
k 1 23 k1 53 k1 5
k1
1 l7 zl7
"1 53
Xg
2 34
k z5
X5
54
z5
X6 X7 Xp
12 17 17 34 15 17 17
k z1 mi k z5 k z1 zB
I 1 :2 5 :' .4 6 It 18 1 .'311
This surface is demonstrated to be a quadric cone, when investigated.
Thus, the following
Theorem 5: The tangent elements to at the point
O' (OD0,0,0,0,0,0,0,0,001) are two planes (45), (47), and a quadric
11
cone (51) .
5. Multiplicities of Points 01'
010 and 011 for Surface
The surface is of order 17. Two members of the family (18)
intersect at 01, 14*12 + 7*22+ 152 or 51 times [1, p. BO]. Thus, the
system is of degree 289 51 or 238.$ Since the curves (18) are re
lated projectively to the hyperplanes of S10, the multiplicity of the
point 01 on is 51/17 or 3.
Two members of the family (28) intersect at 02 in
1*72 + 522 + 2*52 or 119 fixed points. Since the system intersects
in 289 119 variable points it is of degree 170. Also, the multi
plicity of 010 on is 119/17 or 7.
~Degree is used in the same sense as Godeaux [5].
Two members of the family (58) intersect at Og in
1*62 + 19*12 + 122 + 1*52 or 68 fixed points. The system is of
degree 289 68 or 221, and the point 0' on is of multiplicity
11
68/17 or 4.
6. Sammary
The multiplicity of the curves (18), (28), and (58) at the
points infinitely near 01, 02, and 03 respectively have been investigated
and quadratic transformations have been employed to examine the branch
point iuages of these fundamental points. A pictorial diagram of these
multiplicities is given in Figure 1.
1 2 0
02 7 z 2 2 2 2' '1 1 1 1 1 1 1 1 1 1 1 6 05
Figure 1
The tangent elements at 01 constitute a plane and a quadric
cone and the tangent elements at 010 are a quintic cone and a quadric
cone. The branch point 011 is more interesting in that it has two
tangent planes and a quadric cone.
CHAPTER III
PROJECTIONS OF THE SURFACE
1. Surface ~1
The surface projects from the point
010 (0,0,0,0,0,0,0,0,0,1,0) to the surface 1 in the space X10 0.
The equations for the surface 1 are
(61 1 8X5~ 6g 4q 9Qgi 5O X57 X2 5 X2
X7X9 Xg X7 X7 11 X5 X11 1 1 1X5
Two members of the family (28) intersect in 1.72 + 5.22 + 2.52
or 119 fixed points. Thus the order of ~1 is (289 119)/17 or 10
(cf., Chapter II, Figure 1).
Now examine the family of curves (which pass through the
point 02 (0, 1, 0))
(52) a8 5x8x+a5xx6 75 + al48 xl X 51 +~ a 26 x11 x2 x5
+ a9 x4 x5 x30 + a9 xl4, x2 + a59 x7 x2 x~ + al xl7
+ al71 x57= 0.
Notice that the point 02 is a nine tuple point.
Apply the quadratic transformation U twice to the family (52).
The result is
(5) 5 (al z4 + ag "l 3 z5 +26 z 2 z + a52 Z1 zZ + a87 5)
+ 1 0D~ Z zl 11 1 zl zl0 12 z7 z9
+~1 al489 21 21 50 + a99 21 2 5 + 59 3
+ "171 z:1 5~ 0.
Now apply the quadratic transformation V five successive times to (52)
to obtain
54 85 (al48 31 + "171 25) + a8 35 Z6 1+a5az1z7
+ 11 z5 z14 68 114 41 55
"26 Z1 2 541 "99 Z1 2 3 a92
+ a59 57 ,551 z28 1 1 z69 = 0.
Also apply to (52) the quadratic transformation V, then U, and then
V twice in succession. The result is
(55) z68 (a8 z1 + al48 z3 ) + a52 "16 z51 z132 + a26 Z11 z54 z 4
5 3 z47 sl5 6 317 z3,56 10 ,254 ,25
a9 1 28 3 z1 4 2 3 5 z
+ al 321 zi + "171 z4z1 2 4 0
The three previous results indicate that the curves of
system (52) have in common in the neighborhood of 02:
(a) two successive four tuple points 021 and 02113
(b) a four tuple point 025'
(c) four successive simple points 0255, 02355, 023555'
and 0253533'
(d) three successive simple points 0251, 02515, and
025155"
Hence, two curves of system (52) intersect 92 + 3.42 + 7.12
or 156 times at 02. Therefore, the system (52) has degree 289 156
or 153. The sun of the multiplicity of 01 for surface and the
multiplicity of 08 for 1 is 156/17 or 8. But 010 is multiple of
order 7 for ~. Hence 0~ is multiple of order 1 for l'.
Now in a manner similar to the material in Chapter II, apply
the quadratic transformation U twice to the projectivity obtained
from equation (52) and substitute z1 = k z5. As z3 tends to the linit
zero, the result is,
(56) j 1X 3X
X1 X Xg Xq
X2 3g 4 6g
X5 = X7 = X10 ~1 =0.
Hence certain points of 1b, infinitely near 0 situated on (56),
correspond to the points infinitely near 0211. oeta hsi
projection of (33) to the space X10 = 0.
Apply the quadratic transformation V five successive times
to the projectivity and substitute z3 = k al. As sl approaches the
limit zero, one gets the equations
(57) X11 D k Xg
X1 o X2 5g = X4 = X5 = X6' 7 = X10 = 0.
Hence certain points on ~1, infinitely near 08, situated on (57),
correspond to the points infinitely near 025353. ot ta tsi
the projection of (57) to the space X10 c 0.
Now apply to the projiectivity the quadratic transformation V,
then U, and then V two successive times. Substitute z3 o k z1 and take
the limit as zl approaches zero, and obtain
(58) X6 = k Xg
1 = X2 5 43X = X5 = 7 D10' "11 0.
Hence certain points of ~1, infinitely near 08, situated on (58),
correspond to the points infinitely near 025155'
Since (56) and (57) were projections of previous tangent
elements, our new tangent element is the plane that projects (58)
from 08. Hence, the following
Theorem 4: The surface $1 has a new tangent element
(59) X6 k X9
X1 = X2 I X3' 4 X5 = X7 c X10 = 11 r O
at the point 0 .
2. Surface (2
Project the surface ~1 from the point 08 (0,0,0,0,0,0,0,1,0,0,0)
into the surface 2~ in the space Xg 0 getting
(92) X Xq X9X5 g X7 X2X 5
9s X7 X7X1 5g 1 1 11 X1 5
X10= *8 = 0.
Two members of the family (52) intersect in 1.92 + 5*42 + 712
or 156 fixed points. Thus the order of 2Z is (289 156)/17 or 9.
Examine the family of curves which pass through 02 (O, 1, 0),
(60) a52 x x62 x3 + al48 xl x52 x131 + a26 x11 x42 X23
+ a9 x114 2~ x3 + a59 xl? x2 x95 + al )C17 + a171
+a99 x14 Xx13 0
x ~7 = 0.
Note that 02 is an eleven tuple point.
Apply the quadratic transformation U twice in succession
to obtain
(61 z4 (al Z13 + a9 Z Z312 + a26 Z1 25 + a52 z5 +
+ a99 z 2:0: 51 + a59 21 1,27 z3 + 8171 z1
al48 z9 z27 z13
317 0
5 0
Now apply the transformation V five successive times. The result is
(62) z85 (al48 21 + "171 23) + a52 z1 z3,7 + 26 z113 54 z1
+ 99 zl4 Z68 314 + ag zl4 27 z55 5927 z51 z5
+ al z17 2 9 e 0.
One application of V and then six successive applications of U gives
(65) z51 ("52 "1 + al48 z$) + a26 Zl6 ,54 z25 + 99 ,115 ,54 g5
+ ag z531 2l? zZ + a59 2130 z127 m + al 546 z3
+ "171 ,129 2~7 z.~ O .
All this indicates that the curves of (60) have in common
in the neighborhood of Og:
(a) two successive triple points 021 and 0211i
(b) a double point 025'
(c) ten simple points 0255,02535, 02533,3 0253533, 02513
02311, 025111, 0231111' 02511111, n 025111111'
Therefore, two curves of the system (60) intersect
1112 + 2*52 + 122 + 1012 or 155 times at 02. Thus, the system (60)
has degree 289 155 or 136. The sum of the multiplicities of the
points 010 for ~, 08 for 1, and 06 for 2~ must be 155/17 or 9. Hence
06 is multiple of order one for $2.
Now apply the quadratic transformation U twice in succession
to the prcjectivity obtained from (60). Then substitute z1 k z5 and
obsemre that the limit as z3 goes to zero is
(64)Xl 2X
X2 X3 X4X 2 X
X5 I 7 8 9g X 10 D 11 = 0.
Hence to the points infinitely near 021 correspond certain points on
~2 of (64) near 06;. Note that (64) is the projection of (56) to the
space Xg = 0.
Now apply V five successive times to the projectivity and sub
stitute z3 = k zl. As zl tends to the limit zero the result is
(65) rX(1 P X2 X3 4 5q = g X7 Xg ]10 = 0.X1Ck X
Note that (65) is the projection of (57) to the space Xg P 0.
Apply V to the projectivity, and then apply U five successive
times. Substitute z5 = k zl and take the limit as z1 tends to zero;
the result is
(66) )Q = kX4
X1 = X2 .X X5 =X5 7 = Xg = X10 114 0.
Surfaces (64) and (65) are projections of previous tangent
elements. Thus, the additional tangent element to 2E is the plane (66)
as stated below.
Theorem 5: The surface 2Z has a new tangent element
(67) X9 = k X4
(X1 = X2 X5 I X5 X7 X8 = X10 = X11 O
at the point 06.
5. Surface $5
Project the surface ~2 from the point 06; (0,0,0,0,0,1,0,0,0,0,0)
onto the space X6 O to obtain the surface, getting
(4, ) Xq X9X5 g X7 2X5X 2~I:
X7 7~ 11 X11 1~ 11 X5
~cX10 8 6g = 0.
1.112 + 2.32 + 1*22 + 1012
is (289 155)/17 or 8.
pass through 02 (0, 1, 0),
Two members of the family (60) intersect in
or 155 fixed points. Thus, the order of 5
Consider the family of curves which
(68) al8x 5 x1 +a11 1 2 x+a 4 x3 x10 +ax14 x2
+ a59 x7 x2 x53 + al x1~7 + "171 X17 0.
Repeated application of the transformations U
that members of the family (68) have in common at 02:
(a) one triple point 021'
(b) one double point 0211'
(c) thirteen simple points 025 025, .,
025 0255 0255 and 02553
and V will show
0213(8), 025'
Rence, tw~o members of the system (68) intersect in
1*122 + 1*5 + 1*2 + 15*1 or 170 fixed points at 02. Therefore,
the system (68) has degree 289 170 or 119. This indicates that the
sum of the multiplicities of 010 for ), 08 for 61'O6; for ~2, and
0~ for 53 is 170/17 or 10. Thus, 04 is of multiplicity one for 5'.
In a manner very much like those used before, apply the quadratic
transformations U and V repeatedly to the projectivity and observe that:
(a) certain points near 0~ on 53 situated on
(69) X X1X5 = 0
X~ ,(m 1 5, 6, 7, 8, 9, 10, 11)
correspond to the points infinitely near 0211'
(b) certain other neighborhood points near 04 on $3
situated on
(70) X11 = k Xg
IXm = (m a 1, 2, 5, 5, 6, 7, 8, 10)
correspond to the points infinitely near 0233555'
(c) and similarly, other points on $5 situated on
(71) j ~Xm = 0,X X(m 1, 2, 5, 6, 7, 8, 10, 11)
correspond to the points infinitely near 0213(8)'
Mate that (69) is the projection of (64) to the space X6 O
and that (70) is the projection of (65) to the space X6 = 0.
Thec new tangeint elmentii i the~ plane projectingj (711 fr:.n the
potnt. j Henrce, the fo:llowirng
TIheorem 6: The surface 113 has a new tangent element
(72) X3 = k Xg
=i '0, (m = 1, 2, 5, 6, 7, 8, 10, 11)
at the point 04'
4. Surface ~4
Project the surface $5 from the point 0~ (0,0,0,1,0,0,0,0,0,0,0)
to the space X4 O to obtain the surface
(44> 9g ~ 5 X7 X2X 5 2
X10 ~8 6 4g = q 0.
Two members of the family (68) intersect in 1.122 + 1.52 + 1.2
+ 1312 or 170 fixed points at 02 (0, 1, 0). Thus the order of 4q is
(289 170)/17 or 7.
Examine the family of curves whiich pass through 02,
(75) a26 x1 x2 xiZ + a09 4~ x23 x503 + 89l4 x2 53 + a59 X X2 x9
+ al x~7 + 171 x57 =0.
Successive applications of the transformation U and V indicate that
the members of the family (75) have in common at 02:
(a) one four tuple point 025'
(b) three double points 021, 0211, and 0251'
(c) seven stuple points 02515, 02515, 02311' 025111'
0211,0511,and 0251(6).
Thus, two members of the system (73) meet at 02 in
1.152 + 1*42 + 522 + 7*12 or 204 fixed points, and the system has
degree 289 204 or 85. This indicates that the sum of the orders of
010 for ~, 0' for 1, 06 for ~2, 04 for $5, and 0' for 4q is 204/17 or 12.
Consequently, O~ for ~4 is multiple of order two.
As before, apply the quadratic transformations to the projec
tivity, then make the necessary substitution of z1 k z3 or z 5 k zl,
and take the limit as z5 or zl tends to oero respectively. This gives:
(a) certain points near Of on d4situated on
(74) XX2 13=
( = 0, (m =4, 5, 6, 7, 8, 10, 11)
correspond to the points infinitely near 0211'
(b) similarly, certain points on ~4 situated on
(75) X5 =k X7
(Im 0, (m = 1, 2, 4, 5, 6, 8, 10, 11)
correspond to the points infinitely near 025111111'
(c) and other points on ~4 situated on
(76) X1
=, 0, (m D 1, 2, 5, 4, 5, 6, 8, 10)
correspond to the points infinitely near 025153'
The curve (74) is the projection of (69) to the space X4 0.
Hence, the following
Theorem 7: The surface d4 has the two new tangent elements
i =m 0,X X(m = 2, 4, 5, 6, 8, 10, 11)
Xm = (m = 2, 3, 4, 5, 6, 8, 10)
at the point 0 .
Project the
to the space Xg = 0
s6) I
5. Surface ~5
surface ~4 from the point 0~ (0,0,0,0,0,0,0,0,1, 0,0)
to obtain the surface
4x i
5s X11 X1 11
X10 8g X6 4q 9g O.
Two members of family (73) intersected in 204 fixed points
at 02. Thusi the order of 5g is (289 204)/17 or 5.
Applications of the transformations U and V indicate that
the family
(79) a99 x4 x3 %0 + a xl4 x.2 x3 + 59 x7 x2 xB3 + al 217
+ 171 x53 0,
has a group of multiple points at 02. These multiple points are:
(a) one triple point 0233
(b) one double point 021;
(c) twelve simple points 0231, 02513, 025153 0211'
0215, 02155, ..., 0215(7) and 0215(8)'
Hence, two members of the system (79) have in common at 02,
1142 + 1*52 + 122 + 12*12 or 221 fixed points, and the degree of the
system is 289 221 or 68. Delete from 221/17 the sum of the orders
of 010 08, 06 04, and 09. Thus, the point 05 is of multiplicity one
for 65'
Applications of U and V to the projectivity give:
(a) the points on ~5 situated on
(80) X1 = k X2
Im = 0, (m 4, 5, 6, 7, 8, 9, 10, 11)
correspond to the points infinitely near 0211;
(b) the points on ~5 situated on
(81) 1 E~n 0,X1 X(m 1, 2, 4, 5, 6, 8, 9, 10)
correspond to the points infinitely near 023155'
(c) the points on ~5 situated on
(82) Xm~= O ,x k (m 1, 4, 5, 6, 8, 9, 10, 11)
correspond to the points infinitely near 0213(8).
Note that the surface (80) is the projection of (74) to the
space X9 = 0, and that (81) is the projection of (76) to the same space.
Hence, the following
Theorem 8: The surface ~5 has a tangent new element
(83) X2 ., k X7
[Xm (m 1, 4, 5, 6, 8, 9, 10, 11)
at the point 05'
6. Surface $6
surface ~5 from the point 05 (0,0,1,0,0,0,0,0,0,0,0)
to obtain the surface
X711 X1X11 X1X
X0 8X=X6'Xq 4 g 3.0.
Project the
to the space X5 0
(4 )
'Rwo members of the family (79) intersected in 221 variable points at 02.
Thus, the order of 16 is (289 221)/17 or 4.
Now use the transformations U and V to find the multiple points
at 02 of the family
(84) ag xc14 x2~ x5 + a59 xl x2 x5 + al 17 + 17 x1 o.
The multiple points are:
(a) seven double points 025, 0251' ... 025(5), and 025(6)'
(b) two stuple points 021and 0211'
Consequently, two members of the system (84) have in comnmn at
02, 1*152 + 7.22 + 2*12 or 255 fixed points, and the degree of the
system is 289' 255 or 54. Since the sum of the orders of O' ,0', O'
10' 8' 6
04, Of and 0' is 15, the value 255/17, or 15, iaplies that the multi
plicity of 0; for the surface (6 is 2.
Applications of U and V to the projectivity give:
(a) the points on ~6 situated on
Xm O)1 = 0 (m 5 4 5 6 8 9 1 0 1 1 )
correspond to the points infinitely near 02113
(b) the points on ~6 situated on
X X2 11' O
X= O, (m= 1, 5, 4, 6, 8, 9, 10)
correspond to the points infinitely near 023(6).
(85)
The surface (85) is the projection of the surface (80) to the
space X5 = 0. Hence, the following
Theorem 9: The surface ~6 has a new tangent element
(87) X2 X2X11 O
SXm~ (m c 1, 3, 4, 6, 8, 9, 10)
at the point 07'
7. Surface 7
The surface ~6 projects from the point Q; (0,0,0,0,0,0,1,0,0,0,0)
to the space X7 O to a new surface
( 7) 2 X2 11= O
X 10 =X X6 4 X X = X7 .0.
Two members of the family (84) intersected in 255 fixed points at 02'
Thus, the order of ~7 is (289 255)/17 or 2.
The transformations U and V establish the multiple points
at 02 for the family
(88) a59 x1 x2 x93 + al x1~7 + "171 x37 O.
These are sixteen simple points 0 1, 0215, 02135, '..., 0213(8), 025'
0231, 02311, ... 0231(5), and 0231(6). Thus, two members of the
system (88) have in common at Op 1*162 + 16*12 or 272 fixed points, and
the degree of the system is 289 272 or 17. The multiplicities of 010'
08, 06, O O O 07, and 02 total to 272/17 or 16. Therefore 02
is a simple point for 7.
Apply U and V to the projectivity and observe that:
(a) the points on ~7 situated on
(89) X5 k ll
xm = 0, (m = 1, 5, 4, 6, 7, 8, 9, 10)
correspond to the points infinitely near 025111111'
(b) the points on situatedd on
(90)~u 0,~,X X(m = 5, 4, 6, 7, 8, 9, 10, 11)
correspond to the points infinitely near 0213(8)*
The surface (89) is the projection of (86) to the space
X = 0. Hence, the following
Theorem 101 The surface $7 has a new tangent element
X1 = k X5
(1 = 0, (m c 3, 4, 6, 7, 8, 9, 103, 11)
at the point 0
8. Summary
A sequence of projected surfacor was described and the
tangent elements investigated. The orders of these surfaces were
arrived at and the multiplicities of the points calculated.
The following figures are an outline of the multiple points
that the generating curves have at the 02 vertex of the triangle of
reference in the plane.
1
51
02. L2f
Figure 2
05
. 3 02 q ITE i
Figure 5
0"1
Figure 5
"1
Figure 4
S01
05
Figure 6
01
02 /` ~ 03
14 3
Figure 7
02
15
o z
i7 \\o
Figure 9
o3 o
Figure 8
Tangent Element
to
Surface at Point
S17 010 7 jX1 23 5q 4 X6
10 X1 X5 X4 X6 X ,
and
Xm (m = 5,7,9,1156,)
and
Xm 0, (m = 1,2,5,4,5,6710 1)
01 10 0 1 X6 = k Xo
and
SXm = (m 1,2,3,45,7,,10,11)
93 O' 1 Xg = k Xg
and
= 0, (m = 1,2,5,5,7,8,10,11)
The following chart lists the various results of this chapter
and some infonnation on from Chapter II.
Order Point MuBltiplicity
of on of
Surface Surface ;Surface Point
47 0 o 2 X5 = k Xg
and
xm o, (m 1,2,4,s,6,8,10,11)
X11 .k X
and
= 0O, (m = 1,2,5,4,5,6,8,10)
5a 5 0 1 ~ X2 =k X7
and
X= 0, (m = 1,4,5,6,8,9,10,11)
~6 4 0 ~ 2 X X2 11' =
and
X= 0, (m = 1,3,4,6,8,9,10)
~7 2 02 1 I k
and
0O, (m~ = ,4,6,7,8,9,10,11)
The author realizes that the results obtained in this chapter
only begin to identify the information that is obtainable about the
projected surfaces. Further study will undoubtedly yield many other
fascinating truths, illuminating the facts of this chapter.
Tangent Element
to
Surface at Point
Order
of
Surface
Point
on
ISurface
Multiplicity.
of
Point
Surface
CIIAPTER IV
A RATIONAL SURFACE F IN S11
To a certain plane curve shown below, of order seventeen, and
which is not invariant under H corresponds on a curve of order two
hundred eightynine. This curve is cut out on by a seventeenth
order hypersurface. Furthermore, the coefficients of the equations
of the latter surface are functions of the coefficients of the equa
tion of the plane curve considered.
In order to see this, consider the plane curve of order
seventeen,
(92) 91 El c ijk x x2 x5 = 0,
where
i + j+ k =7.
Apply H sixteen times in succession to (92); this gives
9 = c Ew(n) cij x xj x = 0,
n ik 2 3
where
i + j+k 17
n = 2, 3, ..., 17,
and w(n) is the remainder when (n 1)(j + 15k) is divided by 17.
The curve,
(95) el 62 3 .. 817 E 0,
corresponds to a curve C on ~, where C is in birational correspondence
with each of the curves 9m = 0 (m = 1, 2, ..., 17). That is, to a point
of C corresponds seventeen points of the plane with one of the seventeen
points on each of the seventeen curves considered.
The curve (93) meets a curve of (1) in two hundred eightynine
groups of Il7. This implies that the hyperplane related to (1) inter
sects C in 289 points. Hence, C is of order 289.
Let us vary 91 in a continuous manner in its plane until its
equation becomes equal to (1). The corresponding C varies on and
reduces to the section of cp by the hyperplane,
(94) al X1 + a9 X2 + a26 X3 + a52 4q + a59 5g + a87 6g + a99 X7
+ al33 Xg + al48 "9 + "170 X10 + al1 X11 0,
counted seventeen times. That is, the section of is made by the
reducible hypersurface of order 17,
(95) (al X1 + a9 X2 + a26 X3 + ... + "171 X11 17 = 0.
Itnis implies that the curves C are cut out on by seventeenth order
hypersurfaces.
Now, consider 91 r 0 varying in the plane and becoming
equation (2). The curve (93) becomes
(96) (g(x1, x2, x3 ~17~ 0 ,
and the curve C becomes a curve A counted seventeen times. Consequently,
A must be cut out on 4 by a hypersurface of order seventeen.
By simplifying (96) and applying T one arrives at the following equation
for the hypersurface
(9,7) yl(X1, X21 ... X11) a (g(x1, x2 ,x3 )17 al7 12 ~1
+ al6 a29 X11 X2 Xg 1 + ... + al69 Xo X1 o 0
The fact that the xi's ( i c 1, 2, 3) group together into factors
of I. (i = 1, ... 11) can be demonstrated by solving certain equations
relating the exponents of xi (i = 1, 21, 5) obtained from possible powers
of terms of g (Xl, x2, x3) to the exponents of xi (i = 1, 2, 5)
obtained from possible factors of X. (i = 1, ... 11).
Take a surface F in S11 whose equations are:
X12= 1(Ei, X2' 11)
(F) 0/ 8o1 X5~ 6 X %X5 X3 X7 X2X5
X4X6 X79 X, X7 X X11 X5 X11 X1 11 X1X5
Now the author demonstrates that F is a rational surface.
To do this, a projective correspondence is set up between the plane
and F using the following transformation T
(T ), 1 2, X
17 x14 2 11 x4 2 8 6 5 7 9
Xl x x2 x5 x x2 x3 11 X2 x3 xl x2 x3
X6 X7 X K X10
x5 x8, x5 x 3 lO x 10 ~Ox53 xl x2 1 x27
11 X12 1.
x~17 g~x1, x2, x3
X3
This transformation orders to each point of the plane a unique point
of P. It needs to be shown that the converse is true. A development
of T'1 will show this and thus show that F is a rational surface.
The first of the following ten equations comes directly from T .
The others are derived with successive multiplications by X4 5g 6 and
applications of T'.
(98) g(22, X2, x3)~ 12
(99) al24 4q X5 6 x~ x2 x134, 169 X4 X5 6g x8 x932
+ a42 3g 4 5g x X2 x83 + a75 X4 6g 212 x3
10 7 6 4 7
+ a96 4q 6 $ x1 x2 + al43 K4 6s 8 xl X2 x3
4 11 29 6
+ all9 X3 X5 Xg xl 2 x3 + al7 X2 4q 9g x X2 x52
+ a29 X2 Xq XQ x7 x9 x3 +a8X X 1 3x
= X4 5g 6 12'
(100) al24 ~ 5 6~ 11 x~ x22 x6 + al69 X2 5g X2 11 x7 x92 x3
+ a75 1 ~ X26 Xg x2 x2 x14 + a96 X1 4 X26 % x8 x
+ a43 2, 6 12 3+a72 5 11 xx1x3
+ a29 3~ 2 X~ xl x2 + 5 0 x64x3
+ a42 X2 X2 2g 1 x4 x11x + all9 X2a 5 X2~ x x2 x3
P X2 X52 X6 X12'
(101) a2 3 5s X3 X11 x2 x32 14 + l923 X5 56I X11 x 93
+ a75 1~ X3 5~ 9 11 9 x2 x36 + a96 1 3~X X53 5 11 Xl~1 xl5
+ al43 X3 2~ X5 2:9 xl x~ x3 + al7 2~ X3 ~ ]10 IlX11 x6 x2
+ a29 3~ 5~ 5 2g x1 4 xllX + a58 3g X45 3 X X10 x10 x2
+ a42 e X,3 X53 X6 X8 x3 X2 x8 + all9 X2 X5 X6 X~ x12 x53
P~ x1 x 6 12
(102) al24 X2 4 X5 6g 21 x1 x2 x6 + al69 X2 4 X5 X6 nl 1 x7 X x
+ a75 1 X2 4 1x2x 1
4 4 64 7
+ a96 1 X4 X5 6g Xg Ill xl X2 x3
+l43 94 ,2 6 8 9 1 5+aX 01 4 xl x
+a29 X5 X5 9 10 x3 x6 + a58 X2 g X4 5 0xlx5x
+ a42 2 X3 X4 4~ X8 x8~ x9 + all9 5 X5 5q X6 8S g X10 X2
4 ,4 5 612
(105) al4X 5562 2 143 + al69 X2 5 95 6~ X1 x8, X93
+12 a7~54 1g 56 91 21 x9 2x
5 7 ~X X i L1 5 ~ 1~25
+ a96 1 4~ 5 6~ 8g 9g X11 Xl 5
+ al45 1 2 X6 9~ 11 x4 x:11 x32
+ al? i 5~ 5~ 8 10o 11 7 x2~ x37
+ a29 X3 5C X9 X10 xl x13 x53 + a58 X2 5g 4 5 9Q 20 X16 x2 x'3
+a42 X2 93 X5 X55 8 x5 x6 X8 + all9S 3~ 4 5 X61 X8 X4, 2 %10 x2
P~ X5 X5 X6 2.
(104) al24 i X46 X5 6~ 31 xC x2 x6 + al69 2 6 5~ 3 sX L1 x:~7 x 3
+ a5 1X2 6 66 X, X1 x2 x2 X~14
+ a96 6X X5 X6 X~ X21 X4X xllx
+ a43 6 X566) X6 $ 11x10 x2 +a 7 al X26 X6 6r X8~ O 11%2 X53
+ a29 5~ 6 6~ 20 xLg r6 x2 x3 + a584 X2 5, X46 6~ 20 I x X8 x
+ 82 2 ,6 X6 X x3 1n xL 2 x3 + all9 X2 g ) 5 6 6~ 8 g X9 xl x13
=~ x X6 X6 X6 2
(105) al24 32 X X5 6 51x2 x2 x1 + 69 3gX 4 5g 6s 51 x8 x9
+ a75 Xl X2 4~ X76 g 31~ "2 2 x +32 a96 '2 "4 x5 x6 21 @0 x2
+ al43 X1 5g 4 25 6 2g 1 1 X6 2 x3
+ a7 2 ^4 X5 Xs X120 X11 x5 x26 x8 + a296 3 4 X5 X50 11 1t x5
+ 58 Xl 2 X 4~ X57 9 o x xll x3 + 1 a42 X2 X5 8 974 xlxx
+ all9 X1 5g 6 X5 6~ X8 X9 xl x2 xl
= PX4 X5 X6 12'
(106) al24 8 X3 5 ^6^1sX 41 x9 xZ x6 + al69 X~ [ Xg X5 X8 1 Xl x2 x
96 ,5 8 5I 8 21 xQ XL :
9~ ~ o6 x I X i x9
S19 X1 3g X~ X~ X~ x9 xl 1 x X X
96 5 5 18 X xO x91 x5 + al43 5 X9 11y ~5 88 x12: x3"
29~ ~~ 2 a 8 X 1
"119 X1 5X X4 X56 ^8 ^9 xl 21 x3 "1
:i,:ts tl~st. triz F''~;i:u~ ~Ftl e.~lljt.iC'II~ gri. iln~jr in
r "r.
jr,,l P ~? l.t.. ths tra",:l 1~111..~. r;l.
Ti?~l, ~_rimcr'i rulf sjn ?e ss~il:. u;cj t~~ ~ci:a iir r.lie~e ErF~rf~sl:~n~.
The ten equations give
x6 X2 x5 1
(108)
where CI and di (i 1, 2, ... 10) are the determinants that are
functions of the ten constant coefficients and of
Xi (i = 1,2,5,4,5, 6,8,9,10,11). Now from T' one obtains
xf P 11*
(109)
This gives with (108)
(110)
the following
x6 x2 g X12
x10 A Ill
In like manner the ten equations give
x12 x5 = (6 11
(111)
and Tr yields
(112)
Now combine the above
(115)
two equations,
x5 61 X12
57
Next, combine (110) and (113) to get
(114) x2 62 5 X3
x4 AS 2 11
The transformation Tr yields
(115) x7 xx9 =P5
Also, the ten equations give
8 9 10 12o X
(116) x2 x3 A~
These two equations combine to give
(117)15
Now combine (114) and (117) to give
x2 61 bA X5 52
(118)
X1 65 d10 4 21
The transformation T' also yields
(119) xl xx11
and the ten equations give
(120) xl" x13 x 1
These two equations combine to give
x3 9~
(121)
x8 93 12
From the transformation T' obtain
2 10 5
(122) xl X2 x5 8 'g
The ten equations give
2 14 e 8 X12
(123) xl x2 x3 "
The above two equations combine to give
(124) x2 8~a
x9 a8 X12
X3
Now (121) and (124) combine to give
(125) x2 2 Xg Xg
x3 2
5s A 9 X12
Finally combine (118) and (125) to give the inverse of the
transformation T'. It is
(q,1 1 x 2
x52
642 7 9X
Hence, there is a onetoone correspondence between the plane
and the surface F, even though there is a onetoseventeen correspond
ence between the surface and the plane and also between and F.
Conclusion
Using an homography, an involution of period seventeen was
generated; and certain surfaces obtained from this involution were
investigated. A family of plane curves invariant under this involu
tion was projected, by means of the transformation T, to the hyper
planes of a space of ten dimensions (S10). From this a surface Q, with
points on it in a onetoseventeen correspondence to the points of the
plane, was arrived at. Then a study of the tangent elements at three
branch points was carried out.
The next section of the study constituted a series of projec
tions of this surface 4. The surfaces arrived at by successive pro
jections were ~i (i c 1, 2, ... 7). Then certain tangent elements
at selected points on the various surfaces were exhibited.
By adding to the transformation T used previously, an additional
coordinate X12 proportional to the function g(x1, x2, x3)' a projectivity
Tr, mapping the points of the plane onto the points of a surface F in S11'
is established. Now each point of the plane is mapped onto a point on
the surface F. By exhibiting the inverse of T' each point of F is mapped
onto a point on the plane. Hence, the surface F is rational.
APPENDIX I
A METHOD OF FINDINGS THE ORDER
OF A QUINTIC TANGENT CONE
There are various techniques of determining the order of a
surface. The particular method illustrated here employs the definition
from Woods [26, p. 590].
Examine as an example the surface (35). The equations of this
surface combined with those of two general hyperplanes will give an
homogeneous equation in two homogeneous variables. The degree of this
final equation will be the order of the original surface.
Solve simultaneously the equations:
(126) X5 X7 = 10 11 =
(127) X X1X5 =
(128) X3 2 4 = 0
(129) X X3X6 
(130) X2 4Xg =O
(151) AX=Oi=1,2.. 11
(152) Z B. X. = 0 (j = 1, 2, ... 11).
The combination of equations (126), (127), (151), and (152)
will give after simplification
(153) (A1B10 A10Bl) X + (A2B10 A10B) X2 3+ (A5B10 A10B5) 2
+ (A4B10 A10B4) X3X4+ 63g10 A10B6) 5 6
+ (AB10 A0B ) X3Xs" O .
Now substitute equation (128) to eliminate X2 and then use (129)
to remove X3,
(154) (AB0A01 (21 1B 56 + A5B10 A10B3) X2X6
+(A4B10 A10B4) X4X~ 6 6810 A0B6) 6
+ (A 310 A10B8 X6Xg 8 .
Now employ (150) to arrive at
(155) (A1B10 A10 1) 65 + (A 310 A10B2 X6 8 + (A5B10 A10B5) X5X8
+ (A4B10 A10B4) X25X8 63B10 10B6) 6 X8
+ (A 810 A10B ) X58 = 0.
Note that the solution of the fifth degree equation (135)
indicates that the tangent element is a quintic cone.
APPENDIX II
A METHOD OF INVESTIGATING
A FOURTEENTH ORDER NEIGHBORHOOD
In the investigation of involutions that involve large values
of p such that EP = 1, there arises the problem of applying a quadratic
transformation repetitively to a large equation. For example, in the
study of 01, a quadratic transformation R had to be applied fourteen
times to a seventeenth degree equation, cf., Chapter II.
The problem is not quite as difficult as the reader might first
expect. The use of homogeneous coordinates makes the computation
slightly less involved.
The following is a description of how a pattern develops.
Observe that a term zl z z3 under R goes into zli+ z + z3.
The a170 term stops any factoring of z5's in the simplification.
Thus for any given term the k, or the exponent of z3, remains constant
under applications of R. The ao term allows only one z2 to be factored
out in the simplification. Hence, for a given term the z2 exponent
will increase by the constant value k 1 under each application.
Nowthea17 tem alow ony i + j + k 17 to be factored out.
This final result after simplification is
2i + j i j k + 17 j + k 1 k i k + 17 i + k 1 k
z1 z2 zZ or z1 z2 zZ"
For a given term z1 increases by the constant 17 k and z2 increases
by the constant k 1 for each application. This pattern develops
only after one complete application of R.
63
The above explanation is not meant to be a proof that a similar
constant increase pattern develops under certain types of quadratic
transformations applied to any homogeneous equation, even though a
related theorem might conceivably be constructed. The explanation
is included here simply because it happened in all the applications
of this paper and it was a considerable time saver.
The following chart of numbers is included as a display of the
exponents of the z's under fourteen applications of the transformation
R to equation (18).
8 6 5
a52 xl x2 x3
7 9
a59 X1 X2 x3
7 1 9
15 7 9
23 15 9
51 25 9
39 31 9
47 59 9
55 47 9
65 55 9
71 65 9
79 71 9
87 79 9
95 87 9
105 95 9
111 105 9
119 111 9
58 4
a87 xl X2 X5
14 2 11 4 2
ag xl X2 x3 a26 X1 x2 x3
2 1
0 1
0. 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
0 1
O 1
0 1
0 1
O 1
4 2
3 2
4 2
5 2
6 2
7 2
8 2
9 2
10 2
11 2
12 2
13 2
14 2
15 2
1 6 2
8 4
9 4
12 5
15 4
18 4
2L 4
24 4
27 4
50 4
3B 4
56 4
59 4
42 4
45 4
4 8 4
4 5 10 2 10 5 5 11
ag xl x2 x5 al55 "1 x2 x5 al48 xl x2 x5
17 17
"170 x2 "171 x3
4 3 10
L1 10 10
L8 19 10
!5 28 10
52 37 10
59 46 10
16 55 10
53 64 10
30 73 10
j7 82 10
74 91 10
31 100 10
38 109 10
!5 118 10
)2 127 10
2 10 5
.4 12 5
!6 16 5
j8 20 5
,0 24 5
;2 28 5
'4 32 5
16 36 5
;8 40 5
.0 44 5
12 48 5
i4 52 5
~6 56 5
;8 60 5
'0 64 5
1 5 11
7 15 11
15 23 11
19 SB 11
25 45 11
51 55 11
37 63 11
45 75 11
49 85 11
55 93 11
61 105 11
67 115 11
75 125 11
79 135 11
85 145 11
17 0
14 0
15 0
12 0
11 0
10 0
9 0
8 0
7 0
6 0
5 0
4 0
S0
2 0
1 0
0 17
14 17
30 17
46 17
62 17
78 17
94 17
110 17
126 17
142 17
158 17
174 17
190 17
206 17
222 17
AFFEDIX III
0~ L~al** 1.+*v[u ~ i3T E EI."' l'
The~ foLlowirng ie cne way, of urtifyi,g rng ht the terms of;
r the ?trnsfo.rmatio n T.
juppjSe that the jl'5 ,,. '... "i. ?Sr tms re rslped to thr
i 1, =, ... 11 san J = al, j2' ... JU 11.6X1 Thngx,22 x5)
will be expressible as products of X.'s if all possible combinations of
integral values of b. c. 0, 1, ... 17, and C bj 17 are such that the
equations (136), (157), and (138) have integral solutions of d.'s where
Z d. = 17. Equations (136), (137), and (158) are the conditions that
cause the exponents of the %1, x2, and x5 to be the same in the powers
and the factors.
(136) 12 bl + 10 b2 + 9 bg + 7 b4 + 6 b5 + 4 b6 + 3 b7 + 2 b8 + b9
17 dl + 14 d2 + 11 d5 + 8 d4 + 7 d5 + 5 d6 + 4 d7 + 2 d8 + dg.
(137) 7 b2 + 2 b3 + 9 b4 + 4 b5 + 11 b6 + 6 b7 + b8 + 15 bg + 8 bl0
P 2 d2 + 4 dg + 6 d4 + d5 + 8 d6 + 5 d7 + 10 d8 + 5 dg + 17 dlO'
(158) 5 b1 + 6 b3 + b4 + 7 b5 + 2 b6 + 8 b7 + 14 bg + 5 b9 + 9 bl0
= d2 + 2 dg + 3 d4 + 9 d5 + 4 d6 + 10 d7 + 5 d8 + 11 d9 + 17 611'
Now use C bi = 17 and Z di = 17 to eliminate bg and d11in
equations (156), (157), and (158). The results are
(139) 34 + 10 b1 + 8 b2 + 7 bg + 5 b4 + 4 b5 + 2 b6 + b7 b9 2 blO
= 17 dl + 14 d2 +11 d5 + 8 d4 + 7 d5 + 5 d6 + 4 dl + 2 d~ + dg,
(140) 17 b1 + 6 b2 + b3 + 8 b4 + 5 b5 + 10 b6 + 5 b7 + 12 bg + 7 bl0
= 2 d2 + 4 d3 + 6 d4 + d5 + 8 d6 + 5 d7 + 10 dg + 5 dg + 17 dl0'
(141) 51 + 9 bl + 14 b2 + 8 b3 + 13 b4 + 7 b5 + 12 b6 + 6 b7 + 11 bg
+ 5 610 = 17 dl + 16 d2 +15 d2 + 14 d4 + 8 d5 + 13 d6 + 7 d7
+ 12 d8 + 6 dg + 17 dlO'
Observe that the above three equations are not independent,
i.e., equations (159) and (140) added together give equation (141).
Consequently, if (159) and (140) are satisfied then (141) will
automatically be satisfied.
To exhibit that (139) and (140) have a common solution assume
that the b b b4, b5, b6, b7 are each equated respectively to
d4, d5, d6, d7' d d The problem now is reduced to showing that
equations (142) and (143) have a common solution.
(142) 54 + 10 b1 bg 2 blO = 17 dl + 14 d2 + 11 d3'
(143) 17 b1 + 12 bc + 7 610 = 2 d2 + 4 di + 17 dlO'
Now eliminate bg in the above two equations to obtain
(144) 25 + 7 bl bl0 = 12 dl + 10 d2 + 8 d3 + dl0'
The author has examined individually all the possible variations
of b1 and bl0 in equation (144). They are considerably too many to be
listed here.
The above analysis is included to explain to the reader how the
number of test situations are substantially reduced.
BIBLIOGRAPHY
1. Childress, N. A. "Surfaces Obtained frDE InVcrlutiOnS
Generated by Hiomographies of Period Three, F~ive, and Thirteen. "
Unpublished. form of Ph.D. dissertation, Department of Mathematics,
University of Florida, 1954.
2. Dessart, J. ".;ur les surfaces represen~tant 1'involution
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Societe RToyale des Sciences de iege, 3e Series, Tome XVI (1931)
pp. 125.
3. Frank, Stanley. "Certain Cyclic Involutory Mappings of
Hyperspace Surfaces." Unpublished form of Ph.D. dissertation, Depart
ment of M3athematics, University of Florida, 1960,
4. Godeaux, Lucien. Cours de Geometrie Superieure.
Fascicule II. Liege: Librairie Bourguignon, 1957.
5.. "Etude elementaire sur 1'homographie plane
de period troisetsur une surface cubique," Nouve~lles Annales de
Mathematiques, 4e~ Serie, Tome X~VI (1916), pp. 4961.
6. .Geometrie Algebrique. 2 tomes. Liege:
Sciences et Lettres, 19481949.
7. .Introduction a la Geometrie Prosective Hyner
spatiale. Liege: Librairie Bourguignon, 1959.
8. introduction a la Geometrie Superieure.
2e edition. Liege: G. Thane, 1946.
9. Memoire sur les Surfaces Mlultin7les. Liege:
G. Thone, 1952.
10. "Recherches sur les involutions eycliqjues appar
tenant a une suface algebrique," Bulletin de 1'ltdademie Royale de
Belgrique (Classes de Sciences), 5eSeisToeX1 11)
pp. 13561564.
11. "Su leshomoraphes lane cycliques," Memoires
de la Societe Royale des Sciences de Liege, eSieTmeX(10)
pF. 126.
1.Gormsen, Svend T. "ME~aps of Certain Algebraic Curves Invar
iant under Cyclic Involutions of Period Three, Five and Seven."
Unpublished form of Ph.D. diissertation, Department of M~athenatics,
University of Florida, 1955.
Vol. 57 (1951), pp. 759;765
17. "Invariant Curves of Order :ight," Pevista
Mlathema.tica yFisi.ca Teorica, Serie A, Vol. 9 (1952), pp. 1314.7
18. et Childress, N. A. "l~tude d'une involution
cyclique de period cing Bulletin de 1'Adademie Foyale de Belgique
(Classes de Sciences), 6 Serie, Tome XL (1954), pp. 1u31U6.
19. and Childress, N. A. "Surf'aces obtained from
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Thlirteen," Revista Mathematica y Fisica Teorica, Serie Ai, Vol. 9
(195'7), pp. 4148.
20. and Gonvsen, S. T. "Maps of Certain Allgebraic
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Period Seventeen Contained on a Rational surfacee in a Space of 11
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71
25. Winger, R. M. An Introduction to Projective Geometry.
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BIOGRAPHICAL SKETCH
John Willis Kenelly, Jr., was born at Bogalusa, Louisiana,
on November 22, 1935. He attended the public schools of the City of
Bogalusa and was graduated from Bogalusa RLigh School in May, 1953.
Immediately thereafter he entered Southeastern Louisiana College,
Hammond, Louisiana, and completed the requirements for the degree of
Bachelor of Science with a major in mathematics in August, 1956. Ti
degree was conferred with honors at the following commencement exercises
on May 25, 1957. He entered the University of Mississippi in the fall
of 1956 and received the degree of Master of Science with a major in
mathematics on August 18, 1957. Subsequently he entered the University
of Florida in the fall of 1957 and has pursued graduate studies since
that time, with the exception of the summer of 1960.
During his studies at the University of Florida the author has
been a graduate assistant and later an instructor of mathematics.
At the University of Mississippi he was a graduate fellow during the
school year 195657 and a visiting Associate Professor for the summer
term of 1960.
The author is an active member of the Mathematical Association
of America and the American Mathematical Society.
The author married Charmaine Ruth Voss of Covington, Louisiana,
in 1956. She is also a graduate of Southeastern Louisiana College.
This dissertation was prepared under the direction of the
chairman of the candidate's supervisory committee and has been approve?
by all members of the committee. It was submitted to the Dean of the
College of Arts and Sciences and to the Graduate Council and was
approved as partial fulfillment of the requirements for the degree
of Doctor of Philosophy.
January 28, 1961
Dean, College of Arts and/tylences
Dean, Graduate School
SUPERVISORY COMMITTEE:
Cha~
