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## Material Information- Title:
- An Involution of period seventeen
- Creator:
- Kenelly, John Willis, 1935- (
*Dissertant*) Hutcherson, W. R. (*Thesis advisor*) Maxfield, J. E. (*Reviewer*) Moore, T. O. (*Reviewer*) Morse, W. P. (*Reviewer*) Barlett, G. R. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1961
- Copyright Date:
- 1961
- Language:
- English
- Physical Description:
- iv, 72, 1 leaves : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Coordinate planes ( jstor )
Coordinate systems ( jstor ) Geometric planes ( jstor ) Graduates ( jstor ) Homography ( jstor ) Hypersurfaces ( jstor ) Mathematics ( jstor ) Plane curves ( jstor ) Tangents ( jstor ) Vertices ( jstor ) Dissertations, Academic -- Mathematics -- UF Geometry, Projective ( lcsh ) Mathematics thesis Ph. D - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- In an extended complex plane with homogeneous coordinates the equations x'1 : x'2 : x'3 = x1: Ex2 : Eax3 define a plane cyclic homography of period p, where p is a prime number greater than two, E a path primitive root of unity and a is an integer greater than unity and less than p. This homography generates an involution of period p. Lucien Godeaux has been the world's leader in studying involutions. Since his paper in 1916 where he used period three, he has published many papers on involutions. Many other authors have contributed to this field. Hutcherson studied involutions of period seven and eleven, Childress studied some of period three, five, and thirteen, Frank studied some of period eleven, and Gormsen studied some of period three, five, and seven. 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 É¸ require detailed study comprising Chapter II, In Chapter III certain projections of É¸ are investigated. A rational surface F, in S11, is exhibited in Chapter IV. Points on this surface are in a one-to-one correspondence with points on the original plane, whereas points of the surface É¸ are in a one-to- seventeen correspondence with points on the plane as well as on the points on surface F. The material in Chapter II was the subject of a joint paper given at the 1960 summer meeting of the American Mathematical Society. The contents of Chapter IV were used in another joint paper given also at this meeting. The reader is referred to the bibliography 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 neighborhoods. Also homogeneous projective coordinates are used exclusively. 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.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 1961.
- Bibliography:
- Bibliography: leaves 69-71.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 022652418 ( AlephBibNum )
13909186 ( OCLC ) ADA4742 ( NOTIS )
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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 one-to-one 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 non-invariant 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) (s-1 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 x-131+ 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 .'3-11 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+ 1-52 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 + 5-22 + 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 + 1-22 + 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 43-X = 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 $1 has a new tangent element (59) X6 k X9 X1 = X2 I X3' 4 X5 = X7 c X10 = 11 r O at the point 0 . 2. Surface (2 Project the surface ~1 from the point 08 (0,0,0,0,0,0,0,1,0,0,0) into the surface 2~ in the space Xg 0 getting (92) X Xq X9X5 g X7 X2X 5 9s X7 X7X1 5g 1 1 11 X1 5- X10= *8 = 0. Two members of the family (52) intersect in 1.92 + 5*42 + 7-12 or 156 fixed points. Thus the order of 2Z is (289 156)/17 or 9. Examine the family of curves which pass through 02 (O, 1, 0), (60) a52 x x62 x3 + al48 xl x52 x131 + a26 x11 x42 X23 + a9 x114 2~ x3 + a59 xl? x2 x95 + al )C17 + a171 +a99 x14 Xx13 0 x ~7 = 0. Note that 02 is an eleven tuple point. Apply the quadratic transformation U twice in succession to obtain (61 z4 (al Z13 + a9 Z Z312 + a26 Z1 25 + a52 z5 + + a99 z 2:0: 51 + a59 21 1,27 z3 + 8171 z1 al48 z9 z27 z13 317 0 5 0 Now apply the transformation V five successive times. The result is (62) z85 (al48 21 + "171 23) + a52 z1 z3,7 + 26 z113 54 z1 + 99 zl4 Z68 314 + ag zl4 27 z55 5927 z51 z5 + al z17 2 9 e 0. One application of V and then six successive applications of U gives (65) z51 ("52 "1 + al48 z$) + 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 + 10-12 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 + 13-12 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 + 5-22 + 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, 1-142 + 1*52 + 1-22 + 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 eighty-nine. 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 eighty-nine 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 one-to-one correspondence between the plane and the surface F, even though there is a one-to-seventeen 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 one-to-seventeen 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) (AB0-A01 (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 d-l + 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 engendree par une homographie de period cing du plan," Mem~oires de la Societe RToyale des Sciences de iege, 3e Series, Tome XVI (1931) pp. 1-25. 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. 49-61. 6. .Geometrie Algebrique. 2 tomes. Liege: Sciences et Lettres, 1948-1949. 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. 1356-1564. 11. "Su leshomoraphes lane cycliques," Memoires de la Societe Royale des Sciences de Liege, eSieTmeX(10) pF. 1-26. 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. 13-14.7 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. 1u3-1U6. 19. and Childress, N. A. "Surf'aces obtained from Involutions Generated by Homographies of Period Three, Five, and Thlirteen," Revista Mathematica y Fisica Teorica, Serie Ai, Vol. 9 (195'7), pp. 41-48. 20. and Gonvsen, S. T. "Maps of Certain Allgebraic Curves Invariant Under Cyclic Involutions of Periods Three, Five, and Seven," Canadian Journal of M~athematics, Vol. rI (1954), pp. 92-98. 21 ,and Kienelly, J. WJ., Jr. "iAn I~nvolution of Period Seventeen Contained on a Rational surfacee in a Space of 11 Dimensions," Notices of American M~athematical Society, Vol. 7 (1960), (Abstract) pp. 479. 21. and Kenelly, J. I., Jr. "'Three Branch Points on a Surface in a pace of Ten ?imensions," Notices of A~merican Mathematical Society, 'Jo1. 7 (1960), (Abstract) cy. -179-480C. 23. M~orelock, James C. "Invariants with Pesnect to Special Protective Transformations." Unpublished form; of Ph.". dissertation, Department of Mlathematics, University of Florida, 1952. 24. Veblen, Oswald, and Young, John W. Protective Geometry. 2 vols. Boston: Ginn and Company, 1910-1C18. 71 25. Winger, R. M. An Introduction to Projective Geometry. Boston: D. C. Heath and C ompany, 1923.--------- 26. Woods, Fredrick S. Higher Geometry. Boston: Ginn and Company, 1922. 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 1956-57 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~ |

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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EVYYOTNLL_K6UILZ INGEST_TIME 2011-07-15T21:22:08Z PACKAGE UF00097985_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES PAGE 1 AN INVOLUTION OF PERIOD SEVENTEEN By JOHN WILLIS KENELLY, JR. A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA January, 1961 PAGE 2 UNIVERSITY OF FLORIDA 3 1262 08552 2828 PAGE 3 ACKNCWLEDGMENTS 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 serving on the supervisory committee and for their assistance in editing this dissertation. The typist, Mrs. ITiomas Larrick, was also of great assistance in the final preparation. PAGE 4 TABLE OF CONTENTS Page ACKN01.T:^DGMENTS ii CHAPTER I. INTRODUCTION 1 II. A SURFACE (j) OBTAINED FROM AN Iir/OLUTION OF PERIOD SEVENTEEN 3 1. The Image Surface | 3 Z. Branch Point 0' 6 3, Branch Point 0' 14 4. Branch Point 0' 18 5. Multiplicities of Points 0' 0' and 0' for Surface (jl 24 6, Summary 25 III. PROJECTIONS OF THE SURFACE (j) 27 1. Surface (j)-^ 27 2. Surface 't^ 31 3. Surface \^ 34 4. Surface ip. 26 5. Surface ^^ 38 6. Surface ffg 40 7. Surface (j) 42 8. Summary 44 IV. A R/.TIONAL SUItFACF F IN S 48 Conclusion 59 PAGE 5 TABLE OF CONTENTSÂ— (Continued) III. A METHOD OF DEMONSTRATING THE EXISTENCE 0Fy(x^, x^, ... , x^) 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 66 BIBLIOGRAPHY 69 BIOGRAPHICAL SKETCH 72 PAGE 6 CHAPTER I INTRODUCTION In an extended complex plane with homogeneous coordinates the equations x^ : XX= x^ : Ex^ : e\ 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 involution of period p. Lucien Godeaux has been the world's leader in studying involutions. Since his paper in 1916 [5] where he used period three, he has published many papers on involutions. Many other authors have contributed 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 (S^q). The three branch points of this surface (}) require detailed study comprising Chapter II, In Chapter III certain pro lections of (}) are investigated. A rational surface F, in S^^, is exhibited in Chapter IV. Points on this surface are in a one-to-one correspondence with points PAGE 7 on the original plane, whereas points of the surface (j) are in a oneto-seventeen correspondence with points on the plane as well as \n.th points on surface F. The material in Chapter II was the sub,1ect of a .ioint paper given at the 1960 siBnmer meeting of the American Mathematical Society [22]. The contents of Chapter IV were used in another joint paper given also at this meeting [21]. The reader is referred to the bibliography 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 imderstand certain areas of this dissertation, e.g., first order neighborhoods [12]. Also homogeneous projective coordinates are used exclusively [13] . 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. PAGE 8 CHAPTER II A SURFACE (j) OBTAINED FROM AN INVOLUTION OF PERIOD SEVENTEEN 1. The Image Surface (}) Consider the homography, where E is a prljnitive seventeenth root of unity. This homography generates an ij^volution, I^^, of period seventeen. A group of I is composed of the following seventeen points, (x^, Xg, X3), (x^, Exg, E^\), (x^, e\, e'^\), (x^, e\, E^\), U^, e\, e\), (x^, e5x^, E%), (x^, E^x^, E^x^), (x^, e\, e\), (x^, e^x^, EX3), (x^, E\, E^\), (x^, ElOx^, e14^3), (x^, e^x^, eI^x^), (x^, e1%, e10^3), (x^, e^\, E%), (x3_, e1%, E%), ix^, E^\, E\), (x^, E^\, E^xg). Now consider the complete non-invariant linear system of ordei seventeen in the plane, i.e., 2 ^h 4 ^^ i = PAGE 9 where i + j + k = 17 and h = 1, 2, ..., 171 designates the different coefficients. In this general system there are seventeen systems of curves that are transformed into themselves by the homography H. (1) \^^ ^ % ^^ 4 ^3 -^ ^26 4^ 4 4 * "52 ^? 4 4 * "SS ^ ^2 ^3 * ^87 ^ ^2 ^3 * ^99 ^ ^2 ^ * ^133 ^ ^2 ^3 ^ ^148 ^ ^ ^3 "^ ^170 ^2 "^ ^171 4 = Â°\Zj g^x-i , Xpj Xgj s a-jy xj Xj + a^Q x^" x^ + a^p x^ Xg Xj 7 9 6 4 7 4 11 2 3 6 8 + agg x-^ Xg Xj + a^g x^ Xg Xj + agg x^ x^ x^ + a-j^-^g x-|_ Xg Xg 2 J.4 13 3 8 9 n ^ ^124 ^1 ^2 ^ * ^143 ^1 ^ ^3 * ^169 ^2 ^3 Â°= ^Â• (5) a^Q x^ Xg x| + a^g x^ ^g ^3 + ^54 ^ ^2 ^3 "" ^57 "^ 4 5 7 5 4 2 J.1 3 ^4 2 9 ' + agg x-^ Xg Xj + ag^ x^ Xg x^ + a^^g x^ x^ + a.^^^ ^ ^2 ^ + ^46 "^^2 4* ^154 4 "^3 " Â°(4) a^-j_ x^ Xg + a^^ x^ ^2 ^3 "^ ^40 ^ ^2 ^3 * ^60 ^1 ^ ^' + a^^ x^ Xg Xg + agg x^ x^ Xj + a^^^ x^ x^ x^ + a^^^ x^ x^ + a^ ^_ X. xÂ„ xt + a, ^Â„ xl x}P = 0. 145 n. 2 3 167 2 3 PAGE 10 (5) (6) (7) ^8 ^' 4 ^ ^.v ^' 4 4 ^ %s 4 4 4 * ^67 4 4' . ^91 x^^ 4 4 . ag^ x4 x^ 42 , ,_^^^ ^3 43 ,^ , 3_^^^ ^e ^8 ^ ^. 4' -2 ^ ^13 4' 4 -3 ^ ^33 4' 4 4 ^ ^38 4 4 y Â„7 ..3 , . _^6 Â„2 ,.9 . 4 9 4 ^62 ^ -2 ^3 ^ Â«72 ^ ^2 ^3 ^ \oO ^1 ^2 ^3 ^ 147 ^1 ^2 ^3 -^ ^165 ^2 ^^ = 0. a x^ x^ xlÂ° 115 ^1 ^2 ^3 ^25 4^ -z 4 ^ %n 4 4 ^ ^53 4 4 4 ^ ^69 -i' 4Â° -3 ^ ^90 4 4 4 ^ ^93 4 4' ^ ^110 4 4' 4 ^ ^35 ^ 4 4 ^ \42 4 4 4' ^ ^158 4' -i 0. (8) 3 4' -3 ^ ^5 4' 4 4 ^ ^35 4Â° 4 4 ^ %, 4 4 4 + a X. X xlO. A ^8 Â„5 70 ==1 -z -r * hoz ^i < ^; * \i3 ^1 4 -^^ * h,r -I -i' * ^49 -1 =4Â° -^ =163 4 4' 0. (9) 1? ^ ^6 ^ ^2 " ^ 11 6 9 7 23 ""l ^3 * ^39 ^ x^ X. + a 2 3 51 x^ XX ^x^ 2 ^3 r6 ^9 ^2 .5 Â„4 ^ ^1 ^ -i -^ ^88 -1^ 4 4 ^ ^112 -I 4' -3 ^ ^134 4 4 4 ^140^ xl5 . a Â„13 4 PAGE 11 (10) a^^ x^ ^2 ^3 " ^36 ^ ^Z ^Z * ^66 ^ ^2 ^3 * ^68 ^ 4^ 5 12 4 7 6 3 2 12 2 14 + a^g x^ Xg + a^Q^ x-^ Xg x^ + a^^^ x^ x^ Xg + a^^^ x^ x^ x^ 9 7 4 13 ^151 ^1 ^2 ^3 * ^161 ^2 ^3 0. (11) 15 2 12 4 962 88 ^4 ^1 ^ * ^18 ^1 ^2 ^3 * Â®41 ^1 ^ ^3 "^ ^49 ^1 ^2 ^3 + a x^ x^ x^ + a x^ x^ x^ + a x^ x^Â° x^ * a v^ v^ v^O + a^3 x^ Xg Xj + agg x-^ x^ x^ + a^^^ x^ Xg Xg + a^^^ x^ Xg x^ 16 12 5 n * ^138 ^1 ^3 * ^162 ^2 ^3 " ^Â• (12) a x^^ x^ + a x^^ x^ x^ + a x^ x^ + a v^ v^ ^^ U<; ^12 ^ ^3 * ^34 n. ^2 ^3 * ^46 ^1 ^2 ^65 ^1 ^2 ^3 + ag^ x| xll X3 + a^Q5 y.\ x^ x^ + a^^^ '^ ^2 4^ * ^125 ^? 4^ ^3 " ^153 ^ ""2 ^3 * ^159 ""2 "^3 " Â°* (IS) ag xlS xg X3 + a^Q xl2 ^^ x| . a^g 4 ^2 ^5 ^ ^47 4 4 * ^5 ^^iA^ ^84 444Â°^ ^116 "{44^ ^130 444^ * ^137 ^ 4 "^ ^164 4 ^3 " Â°(^^^ \Z 4' 4 ^ ^32 4Â° -2 4 ^ ^48 4 4 -3 ^ ^63 4 4 -3 + aQ3 x5 xlO x| + a^Q3 x^ x^ x| * a^^^ x^ x^^ ^ a^^^ xf x^^ x^ 0. * ^52 \ 4 4 " ^ 157 \ ^3 PAGE 12 (15) a, 4^ 4 . a^^ ^^ 4 4 . a^^ xj x^ x43 . a^^ xj x^ ^ . ag, x5 X, xll . a,, 4 x^3 ^ ^^^^ ^3 ^8 ^6 ^ ^^^^ ^2 ^3 ^12 * ^139 ^1 4^ ^3 -^ ^166 4Â° ^3 = 0(16) a^ xl4 .3 , 3^^ ^11 ^5 ,^ , ^^^ ^0 ,7 , ,^^ ^8 ,7 ,2 . a,^ xj x^2 x| . ag^ xf x^ x^ . a^^^ ^ ^ ^ , ,^^^ ^ ^1 ,4 ^ ^150 ^4 4Â°-^ ^155 ^2 4^ = 0(17) a^^ xl^ xg 4 . a^^ xf x,^ xf . a,^ xj xlO . ,^^ ^6 ,5 ,6 ^ ^80 ^ 4' ^ ^94 -1 -2' ^3 ^ ^120 4 -I -I -126 ^1 4 4' ^ ^41 ^ 4^ 4 * -168 4 168 ^2 ^3 xÂ° = 0. Now relate the curves of system (l) projectively to the hyperplanes of S^Q by taking the projective transformation (T) Â±. '"'^ Â„ ^^5 ^4 ^5 ^1 ^r. K X' X X" 12 3 12 3 h PAGE 13 Eliminate x-^, x^, and x^ in T to obtain a surface (j) in S-, q which has for its equations i^) hh V9 ^ ^7 ^Al ^5 hi hhl hh = 0. This surface is the Image of I^^, i.e., a set of I^^ corresponds to a single point of the surface (j). Now investigate the singularities of the surface (f) at the images of the fixed points of I The images of 0^ (1, 0, 0), Og (0, 1, 0), and 0^ (O, 0, l) are 0| (1,0,0,0,0,0,0,0,0,0,0), 0|^ (0,0,0,0,0,0,0,0,0,1,0), and 0{i (0,0,0,0,0,0,0,0,0,0,1) respectively. 2. Branch Point 0' This investigation is based on a technique that finds the projective images 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 >diich relates two planes with homogeneous coordinates in such a manner that a reference 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, 0^ (l, 0, O) in the z coordinate plane is mapped onto 0-^ (l, 0, O) in the x coordinate plane. PAGE 14 and 0^ (1, 0, O) in the x plane is mapped onto (O, 0, O) in the z coordinate plane. Then the z plane ijnage of the point S(l + Xa, Xp, \r) from the x plane, will be the linage of the first order neighborhood of 0^ if the lijnit is taken as X tends to zero. For a more detailed discussion of neighborhoods the reader is referred to Morelock [23]. Note that members of the family (l) do not in general go through the vertices of the reference triangle. But if the restriction a^ = is added, a new family (18) will go through. (18) ag xi4 ^Z ^^ ^ ^^^ ^1 ^4 ,2 , ,^^ ^8 ,6 ,3 , ^^^ ^7 ,^ ,9 . ag, 4 4 4 . ag, 4 4 xlO . a,33 xf 4^ ,5 , ,^^^ ^^ ^5 ,11 The quadratic transformation R and its inverse will relate P^ (1, 0, 0) in the z coordinate plane to 0^ (l, 0, O) in the x coordinate plane and 0^ (l, 0, O) to the meaningless point (O, 0, O) . -1 " Â• "2 = ^3 = ^1^2 Â• ^2 Â• ^1^3 ^^^ z, : Z5 : z, = x.Xo : x^ : x.x Apply the transformation R successively fourteen times to equation (18) and arrive at (19) 4^ (aÂ„Â„ .^ . ag .3) . aÂ„, .f^ .1' . a,, ."^ z^" .| * =87 4'' 4' 4 * =148 -!' 4'' 4' * =135 4Â™ 4' 4 0- PAGE 15 10 This shows that the point (zÂ„ = z^ = O), corresponds to the point in the fourteenth order neighborhood of in the direction of Xr, = 0, i.e., 0^22222222222222 " Â°12(14) ^^> P* ^^^Â• Now apply the transformation, R, fourteen successive times to the transformation (20) h PAGE 16 11 A substitution of zÂ„ = k z_ will allow an all directional approach to the point (zp = Zj = O). This substitution in (21), after simplification, gives (22) -"o Â•A-z -^/l A 2 ""3 ^4 ^5 , 238 , 2 221 17 ,3 204 34 , S 119 119 k Z-] k Z-, Zo k z-y Zp k z-, Zp Xg X^ Xg j^4 187 51 10 102 136 ,5 170 68 K Z^ Zg K Z^ Zg K Zj_ Zg ho hi , 11 85 153 238 , 17 238 k z^ zg z-L k zg As Zp tends to the limit zero, the above gives the equation of a plane tangent to (j) at 0' . It is (23) fc = k X^o [X^ = X^ Xg = Xq = X^ = Xg = Xg X^^ = 0. Now examine a different quadratic transformation and its inverse (S) x-| : Xo : Xj = z-| : ZgZj 'Â• z-iZg (S) z^ ! Zg : Zj = x-j^Xg : x^x^ : Xg . PAGE 17 12 Apply it seven times to the curve (18), and the simplified result is (24) 419 (^_^^^ ^2 ^ ^^^ ^^^^ ^ ^^ ^2) ^ ^^^^ ^17 ^104 102 4 15 102 _3 Â„16 ^ Â„ R5 fi 30 * ^26 ^1 ^2 ^3 * ^99 ^1 ^2 ^3 * ^52 ^1 ^2 ^3 + a "^^ z^ z^^ + a 785 ^^ ^31 ^ ^ 51 10 60 ^ + ag^ .^ Zg zg + a^^Q z-L Zg Z3 + a^^^ z^ z^ Z3 0. This shows that in the seventh order neighborhood of 0-, , the point on the curve (18) along the direction x^ = corresponds to the double point (zp Zg = 0) . Repeated application of the transformation S to the transformation (20), yields (25) ^2 PAGE 18 13 (26) Ji_ Â» ^3 ^4 _ h 119 lis kl5 ,102 ,17 ^30 ,85 ,34 j, ^119 ^6 PAGE 19 14 3. Branch Point 0Â« 10 The point 0^ (O, 1, O) corresponds to the point 0{o (0.0,0,0,0,0,0,0,0,1,0) in S^^ by the transformation T. To study the tangent elements at this point, examine the system (oo^) of curves passing through 0^, i.e., system (l) with a-]_^Q 0. We have (28) ai xf . ag x^^ 4 x, . a^^ .^ ^ x| . a^^ xÂ« xÂ« x' * ^59 4 ^Z A * -87 4 4 4 * -99 4 4 4Â° * -133 4 4Â° 4 * ^148 ^1 ^2 ^3 * ^171 x^ = 0. Apply the quadratic transformation (U) x^ : xg : X3 = z^zg : z^ : z^z^ ^1 : Zg : Z3 = x2 : x^x^ : x^x^ twice in succession to the system (28). We get (U)" z^ : z_ : zÂ„ = X? : x^xÂ„ : x. (Â«) zf (a, xf . a, 4 zj . a,g xf x| . a,, ^ z| . ag, ., 4 * -133 4) * -171 4' 4' * -69 4' 4' 4 * -99 4' 4' 4Â° 11 Â„17 Â„11 '148 ^1 ^2 ^3 0. This indicates that 0^^^, the second order neighborhood point in the Xj = direction, corresponds to the five tuple point (z, = zÂ„ O). PAGE 20 15 Now apply the transformation U twice to the transformation (30) A_ . ^^ ^5 X ^5 h^ X7 X 4 -z 4 4 4 4 4 4 4' 4 ^l' ^1 ^1 -Z -3 ^1 ^2 ^3 ^ hi ^5 11 17 Â• n. ^2 ^3 ^3 This gives (31) -Jl ^2 ^3 ^4 ^1 4' 4 4' ^3 " 4 4' 4 Â° Tâ€¢4 h xÂ« X, 5 f6 ^7 Xg ^'4' 4 ^.^T4 ?T? 4' 4 ^ ^1 11 17 11 " 22 17 Â• 1 '2 '3 ^1 ^3 Let z^ k Z3, to allow an approach from all directions to the ijnage point (z^ = zÂ™ = O), PAGE 21 16 (32) h , 5 34 ,4 34 k Zo , 3 34 k Zo 1,2 34 k Zg ,13 17 17 k zg zj hi k z 34 , 12 17 17 k Zo Z-. '-Z ^3 34 , 11 17 17 k Zo Zc ^2 "-i ,22 34 k Zi As Zj approaches the limit zero in (32), the equation of a tangent element is arrived at (33) ^1 ^2 ^3 ^4 ^6 Xg Xg X^ Xg Xg V X g = X^ = Xg = \^ 0. This tangent surface is verified to be a quintic cone when investigated. To study the neighborhood points along the direction x-, = 0, examine (28) with the quadratic transformation (V) Apply the transformation V five successive times to equation (28); this gives I See Appendix I, PAGE 22 17 (34) Zg (^lYx Z3 + ai48 ^^ Z3 + a^sj z^) + a^ z^ Zg ^ ^ Â„7 Â„51 ,29 ^ Â„14 Â„17 Â„56 , 11 54 42 * ^59 ^1 ^2 ^3 * Â®9 ^1 ^2 ^3 * ^26 ^1 ^2 ^3 Â„4 Â„68 Â„15 ^ 8 51 28 5 68 14 Â„ + agg z^ Zg Z3 + agg z^ zg Zg + ag^ z^ Zg Zj = 0. This indicates that the fifth order neighborhood point (Ooj/cO in the direction x, t= 0, corresponds to the double point (z-, =2^ = 0). Now apply the transformation V repeatedly five times to the transformation ( 5C) , and get (35) ^1 ^2 ^3 ^4 17 70 14 17 56 11 34 42 8 51 28 Z-j_ Zg Z-^ Zg Zg Z^ Zg Z5 Z^ Zg Zg Xg Xg X^ Â„7 51 29 5 68 Â„14 4 68 14 2 85 z-L Zg Z3 Z3_ Zg Z3 z-L Zg Z3 z-L Zg ^ ^1 85 85 2 ^1 ^2 ^3 ^2 ^3 To allow for an approach from any direction to the point (z, =. z^ = 0) , let Zr, = k z,, and obtain PAGE 23 18 (36) ^1 PAGE 24 19 ao. x}^ x^ xl + a,Â„ X? x^ xf (38) a^ x^ + ag x^ Xg x^ + agg ^^ ^g ^3 -r ogg a-]_ ^g ^j 4. a v7 y ^9 + a x^ x^ x^ + a x^ x^ x^^ + a x^ x^^ x^ + a^g Xj_ Xg X3 + ag^ x-^ Xg Xg + agg x-j^ Xg X3 + a^^^ x^ Xg x^ ^ ^148 ^ ^2 ^3 * ^170 ^ Â° Â°' Apply the transformation (M) Xl : Xg : X3 z-^Zg : ZgZg : Z3 (M) z-| : Zg : Z3 = x-j^Xj : Xg : XgXg eleven times in succession to (38). The result is /,oN 187 / \ 17 171 7 62 119 (39) Z5 (^170 ^2 * ^148 ^l-* * ^1 ^1 ^2 * ^59 ^1 ^2 ^3 14 140 34 11 109 68 4 31 153 + ag z-L Zg Z3 + agg z^ Zg Zj + agg z-^ Zg Z3 8 78 102 5 47 136 2 16 170 Â„ + agg z-L Zg Z3 + ag7 z^ Zg Zg + 3-^33 z^ Zg Z3 = 0. Hence the eleventh order neighborhood point Osgdi)* i" ''^^e direction x-i = 0, corresponds to the simple point (z-, = Zg Â» O) . Now apply the transformation M eleven times to the transformation (40) ^1 PAGE 25 20 The resxilt is (41) ^1 ^2 ^3 ^4 17 ,171 ^14 140 ,34 ,11 ,109 ,68 ,3 ,78 ,102 ^1 ^2 ^1 ^2 ^3 h ^2 ^3 ^1 ^2 ^3 Xg Xg X^ ,7 62 ,119 ^5 ^47 ,136 4 31 153 2 16 170 Z-L Zg Zg Z^ Zg Z3 Z^ Zg Z3 Z^ Zg Z3 ^ ^10 187 187 ^1 ^3 ^2 ^3 To allow for an all directional approach to the point (z-, " z_ Â« 0), let Zg k z, , anci get (42) h PAGE 26 21 Now consider the quadratic transformation 2 (L) x^ : Xg : Xg = z^Zg : z^z^ : z^ (L)-^ z^ : Zg : Z3 = x^ : x^Xg : x^x^. Nine successive applications of L to equation (38) give ,,^. 153 / s 137 17 16 2 136 (44) Z3 (a^ z-L + a^g z^) + a^^^ z-^ z^ + ag z^ z^ z^ 31 4 119 15 3 136 46 6 102 "^ Â®26 ^1 ^2 ^3 * ^99 ^1 ^2 ^3 "^ ^52 ^1 ^2 ^3 + a z^^ z^ z^S + a z2Â° z^ z^^^ + a JB 10 ^68 ^ q * ^87 ^1 ^2 ^3 * ^148 ^1 ^2 ^3 * ^133 ^1 ^2 ^3 ^' This indicates that the point in the ninth order neighborhood in the direction Xp = 0, ^^i(q), corresponds to the simple point (z^ = zg = 0). The transformation (40) under nine successive applications of L gives (45) ^1 PAGE 27 22 Approach the point (z-^ z^ = O) from all directions by substituting z-j^ k Zg, (46) h PAGE 28 23 This indicates that the second order neighborhood point, Ognp, corresponds to the double point (z-, Â«= Zp = O) . The application of L followed by M to the transformation (40) gives (49) h PAGE 29 24 As Zn approaches the limit zero, a third tangent element is arrived at. It is (51) ^X^.X^Xg.O A " ^2 "^ ^3 = ^4 " ^6 = ^8 " ^0 = 0. This surface is demonstrated to be a quadric cone, when investigated. Thus, the following Theorem 5 ; The tangent elements to (j) at the point 0' (0,0,0,0,0,0,0,0,0,0,1) are two planes (43), (47), and a quadric cone (5l) . 5. Multiplicities of Points 0' 0' , and 0' for Surface (j) The surface ({> is of order 17. Two members of the family (18) intersect at 0^, 14-1^ + 7-2^ + 1-3^ or 51 times [l, p. 30]. Thus, the system is of degree 289 51 or 238.* Since the curves (18) are related protectively to the hyperplanes of S , the multiplicity of the point 0' on (J" is 51/17 or 3. Two members of the family (28) intersect at Op in 1Â«7 + 5'2 + 2*5 or 119 fixed points. Since the system intersects in 289 119 variable points it is of degree 170. Also, the multiplicity of Oj on (j) is 119/17 or 7. 'Degree is used in the same sense as Godeaux [5] PAGE 30 25 Two members of the family (38) intersect at O5 in 1-6^ + 19-1^ + 1-2^ + 1-3^ or 68 fixed points. The system is of degree 289 68 or 221, and the point 0' on (j) is of multiplicity 68/17 or 4. 5. Summary The multiplicity of the curves (18), (28), and (38) at the points infinitely near 0, , 0_, and 0^ respectively have been investigated and quadratic transformations have been employed to examine the branch point images of these fundamental points. A pictorial diagram of these multiplicities is given in Figixre 1. '2 7 2 2 2 2 2 111111111116 ^3 Figure 1 PAGE 31 26 The tangent elements at 0Â« constitute a plane and a quadric cone and the tangent elements at 0' are a quintic cone and a quadric cone. The branch point 0' is more interesting in that it has two tangent planes and a quadric cone. PAGE 32 CHAPTER III PROJECTIONS OF THE SURFACE (j) 1. Surface (j). The surface (f projects from the point 0|q (0,0,0,0,0,0,0,0,0,1,0) to the surface (J)^ in the space X = 0. The equations for the surface (j)-, are = 0, ((tl) ^8^5 ^6 \ ^8-^5 ^3 -^7 "^2^5 ^2 X^Xg Xg X^ X^X^^ Xg X^^ X^X^^ X^Xg Two members of the family (28) intersect in 1.7^ + 5-2^ + 2-5^ or 119 fixed points. Thus the order of ({)-]_ is (289 119)/l7 or 10 (cf.. Chapter II, Figure l) . Now examine the family of curves (which pass through the point Og (0, 1, 0)) .5 ^8 ^4 ,6 ^3 .5 ^11 Al ^4 ^2 (52) ag^ xÂ£ Xx| + a^g x^ xx+ a^^g x^ x^ x^ + a^g x^x^ x4 3 -_10 14 2 7 Q J.7 + agg x^ Xg x^ + ag x^ Xg Xg + a^g x^ Xg Xg + a^ x^ + a^^^ x^'7 = 0. Notice that the point 0Â„ is a nine tuple point. 27 PAGE 33 28 Apply the quadratic transformation U twice to the family ( 52) . The resvilt is (53) z^^ (a^ zj -Iag z-^ Zj + a^g z^ z| + a^^ z^ Z3 + ag^ z^) 10 17 11 11 17 10 12 17 9 + ^148 ^1 ^2 ^3 * ^S9 ^1 ^2 ^3 * ^59 ^1 ^2 ^3 + a z^l z^'^ Now apply the quadratic transfonnation V five successive times to (52) to obtain (54) zf (a^^3 z^ . a^,^ Z3) . ag^ z^ zf ,13 , ,^^ ,8 ,51 ,27 . Â„ Â„11 Â„34 _41 . Â„ Â„4 Â„68 Â„14 . Â„ Â„14 _17 Â„55 "" ^26 \ ^2 ^3 * ^99 \ ^2 ^3 * ^9 \ ^2 ^3 . a Â„7 Â„51 28 17 69 ^ q * Â®59 ^1 ^2 ^3 * ^1 ^1 ^3 ^ Also apply to (52) the quadratic transformation V, then U, and then V twice in succession. The result is (55) Zg (ag^ z^ + a-j^^g z^ ) + a^^ z-|_ z^ Zj 4a^g z^ Zg Zg . Â„ Â„5 _47 _13 . Â„ Â„16 Â„17 Â„36 . Â„ Â„10 Â„34 Â„25 * ^99 \ \ H * ^9 \ h ^3 * ^59 \ \ ^3 . a _21 _48 . . Â„4 Â„51 Â„14 ^ q + a^ z^ Zg + a^^^ z^ Zg z^ = U. PAGE 34 29 The three previous results indicate that the curves of system (52) have in common in the neighborhood of : (a) two successive four tuple points , and j (b) a four tuple point j CO (c) four successive simple points 0,33, 0^333, 0^3335, ^^ Â°253353^ (d) three successive simple points 0^3^, 0^^^^, and Â°23133Hence, two curves of system (52) intersect 9^ + 3.4^ + y.i^ or 136 tmes at 0^. Therefore, the system (52) has degree 289 136 or 153. The sum of the multiplicity of OJ^^ for surface (j) and the multiplicity of 0> for ^^ is 136/17 or 8. But Oj^^ is multiple of order 7 for (}). Hence 0^ is multiple of order 1 for ({) . 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 z^ = k Z3. As z^ tends to the limit zero, the result is, (56) h h h h\ 4 ^2 ^3 ^4 ^6 0, X5 = X, = Xo = X^o = Y ^ 0. Hence certain points of (j)^, infinitely near 0^, situated on (56), correspond to the poij^ts infinitely near 0^^^. Note that this is a projection of (33) to the space X,q = 0. PAGE 35 30 Apply the quadratic transformation V five successive times to the projectivity and substitute Zg k z-, . As z^ approaches the limit zero, one gets the equations (57) Ihl"^ h X^ o Xg o Xj = X^ = Xg = Xg = X^ = X^Q 0. Hence certain points on (t)^ , infinitely near 0' situated on (57), i o correspond to the points infinitely near '^oZZZZZ' ^*^'^Â® that this is the projection of (57) to the space X,^ = 0. Now apply to the pro.iectivity the quadratic transformation V, then U, and then V two successive times. Substitute z, Â« k z, and take the limit as z-, approaches zero, and obtain (58) [^6 = ^ ^ [X^ = Xg = Xj = X^ = Xg = X^ = X^Q X^^ = 0. Hence certain points of ({)-,, infinitely near Oi, situated on (58), correspond to the points infinitely near ^27>17>Z' Since (56) and (57) were projections of previous tangent elements, our new tangent element is the plane that projects (58) from 0'.. Hence, the following o Theorem 4 ; The surface (j), has a new tangent element (59) fXg = k Xg X^ = X^ X3 = X^ o Xg = X^ . ^10 = ^11 = at the point 0^. PAGE 36 31 2 . Surface Project the surface (})^ from the point 0Â» (0,0,0,0,0,0,0,1,0,0,0) into the surface ^^ in the space X = 0, getting (*p) I Xg X^ XgX^ X3 x^ x^x^ x| Xg X^ X^X3_^ Xg X^^ l^X^^ X^Xg X^Q = Xg = 0. Two members of the family (52) intersect in 1-8^ + 3-4^ + v-i^ or 136 fixed points. Thus the order of (j)^ is (288 136) /17 or 9. Examine the family of curves which pass through Op (O, 1, O) , (60) a,^ xf x6 x| . a,,3 x, x^ xll . a^^ xll ^ x| . a,g xf x| ^^ + ag xl4 x2 X3 + agg xj x^ x| + a^ xl7 ^ a^^^ x^^ = 0. Note that 0^ is an eleven tuple point. Apply the quadratic transformation U twice in succession to obtain (61) zf (a, 4 . ag 4 ., . a,3 z, ^ . a,, z|) * a^.g .f .1' .11 ao 17 Â„io ,11 Â„17 Â„8 ,20 Â„17 " ^88 ^1 h ^3 ^ ^59 ^r ^2^3 ^ ^171 \ ^3 ' Â°Now apply the transformation V five successive times. The result is (Â«^' 4' (-148 h ^ =171 -5) * Â«5e -? 4' 4' * -Z6 4' -r -f * Â«99 ^ 4' 4' * =9 4^ 4' 4' * 'S9 'I 4^ 4"" . Â„ Â„17 ^69 n + a, z, Zp " U, PAGE 37 52 One application of V and then six successive applications of U gives (63) =F( 3co Zt + a 148 ^3 '26 ^1 16 ^34 ,2 , Â„15 Â„34 Â„3 Igg Z-L Zg Z3 ^ ^9 ^f 4^ 4 * ^59 ^r ^2 -3 2Â° "^^ zf . a^ z46 ,6 + ^171 zf zl^ z6 . 0. All this indicates that the curves of (60) have in common in the neighborhood of Og: (a) two successive triple points Op, and Op-,-,; (b) a double point O-gj (c) ten simple points 0^35,02333, 0^3333, 0^33333, 0^31, Â°2311' Â°23111' Â°231111' Â°2311111> ^^ Â°23111111Therefore, two curves of the system (60) intersect 1-11 + 2-32 + 1-22 + io.i2 or 153 tijnes at 0^. Thus, the system (60) has degree 289 153 or 136. The sum of the multiplicities of the points O'q for (j), 0^ for ^^, and 0^ for ^^ must be 153/17 or 9. Hence 0' is multiple of order one for (])Â„. Now apply the quadratic transformation U twice in succession to the prcjectivity obtained from (60). Then substitute z, , k z, and -1o observe that the limit as Zg goes to zero is (64) X-L Xg X3 ^2 ^3 ^4 " 0, ^ = ^8 = ^9 = ^0 = ^1 = 0- PAGE 38 53 Hence to the points infinitely near correspond certain points on '^Z Â°^ ^^'^^ "Â®^^ ^6' ^Â°^Â® ^^^^ ^^^^ ^^ ^^Â® projection of (56) to the space X = 0. o Now apply V five successive times to the projectivity and substitute Zg = k z^. As z tends to the limit zero the result is (65) fx^^ . k Xg Note that (65) is the projection of (57) to the space Xg = 0. Apply V to the projectivity, and then apply U five successive times. Substitute Zj = k z-]_ and take the limit as z-, tends to zeroj the result is (66) j^-^\ W h h h h h " ho " hi ''' Surfaces (64) and (65) are projections of previous tangent elements. Thus, the additional tangent element to <^^ is the plane (66) as stated below. Theorem 5 ; The surface (}) has a new tangent element (67) I X^ k X^ 19 4 1 X^ =. Xg = X3 = Xg Â» X^ = Xg = X^Q = X^^ at the point 0' D PAGE 39 34 3. Surface (j)Â„ Project the surface d from the point 0' (0,0,0,0,0,1,0,0,0,0,0) 2 6 onto the space XÂ„ = to obtain the surface, getting V PAGE 40 35 0Â« for (|) is 170/17 or 10. Thus, 0Â» is of multiplicity one for (j) . ^ ^ Q 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 (i)Â„ situated on 4 3 (69) |X^ X^Xj = _^ =0, (m = 5, 6, 7, 8, 9, 10, ll) correspond to the points infinitely near ; (b) certain other neighborhood points near 0' on (j)-. situated on (70) [X^^ = k Xg _X =0, (m 1, Z, 3, 5, 6, 7, 8, lO) correspond to the points infinitely near 0_Â„Â„Â„Â„Â„; (c) and similarly, other points on (j) situated on (71) fXj = k Xg X^ 0, (m = 1, 2, 5, 6, 7, 8, 10, ll) correspond to the points infinitely near ^^-,-z.(a\' Note that (69) is the projection of (64) to the space X = and that (70) is the projection of (65) to the space X = 0. PAGE 41 36 The new tangent element is the plane projecting (71) from the point 0'. Hence, the follovdng Theorem 6: The surface 6 has a new tangent element 3 (72) AÂ«7 = K Aq \= 0, (m 1, 2, 5, 6, 7, 8, 10, ll) at the point 0' , 4. Surface Project the surface ^ from the point 0' (0,0,0,1,0,0,0,0,0,0,0) to the space X. Â« to obtain the surface (^ XgXg Xg X^ XgXg Xg ^7^1 ^5 ^1 ^^1 ^^5 ^0-^8 = ^6 = ^4 = Â°0, Two members of the family (68) intersect in 1-12^ + 1Â«3^ + 1.2^ + 13-1^ or 170 fixed points at (O, 1, O) . Thus the order of (j) is (289 170) /17 or 7. Examine the family of curves which pass through 0Â„, Al ^.4 ^2 A ^3 UO A4 ^2 (73) a^g x^XX+ agg x^ xx+ a^ xjx^ x^ + a^^ x^ x^ x 7 ^ ^9 3 ^ "i ^^ -^ ^71 ""â€¢ " Â°- PAGE 42 S7 Successive applications of the transformation U and V indicate that the members of the family (73) have in common at : (a) one four tuple point 0Â„Â„; (b) three double points "^onT* ^"d Â°?'^t ' (c) seven sJinple points 0^^^^, 0^^^^^, 0^^^^, 0^3^^, Â°231111' Â°2311111' ^"^ Â°231(6)' Thus, two members of the system (73) meet at 0in 1-13^ + 1-4^ + 3-2^ + 7-1^ or 204 fixed points, and the system has degree 289 204 or 85. This indicates that the sum of the orders of 0' for (}), 0' for (j) , 0' for f^, 0' for (j) . and 0' for (f. is 204/17 or 12. XU o lb ic4 o 8 4 Consequently, 0^ for (j) is multiple of order two. As before, apply the quadratic transformations to the projectivity, then make the necessary substitution of z, = k zÂ„ or z, = k z, , and take the limit as zÂ„ or z, tends to zero respectively. This gives: (a) certain points near 0' on (ji situated on y 4 (74) f X? X X =0 2 13 X^ = 0, (m = 4, 5, 6, 7, 8, 10, ll) correspond to the points infinitely near j (b) similarly, certain points on (J) situated on (75) U3 k X^ X^ = 0, (m = 1, 2, 4, 5, 6, 8, 10, ll) correspond to the points infinitely near Opsii-i-ii-i > PAGE 43 (c) and other points on (j). situated on (76) X,, = k X^ \ = 0, (m o 1, Z, 3, 4, 5, 6, 8, lO) correspond to the points infinitely near 23133' The curve (74) is the projection of (69) to the space X^ = 0. Hence, the following Theorem 7 ; The surface (ji^ has the two new tangent elements (77) X^ = k X^ 0, (m = 1, Z, 4, 5, 6, 8, 10, ll) and (78) X^, . k X^ at the point 0' . .\ = 0, (m = 1, 2, 3, 4, 5, 6, 8, lO) 5. Surface Project the surface (|)^ from the point 0^ (0,0,0,0,0,0,0,0,1,0,0) to the space Xg = to obtain the surface Â«}).) X3 X^ XgXg Xg ^5 ^1 ^1^1 ^^5 0, ^ ^10 = -^8 ^6 " Â•'^4 " ^9 Â°- PAGE 44 39 Two members of family (73) intersected in 204 fixed points at 0^. Thus the order of (Ji is (289 204) /l7 or 5. Applications of the transformations U and V indicate that the family (79) agg x^ 4 40 . ag ^4 ,2 ,^ , ,^^ ^7 ,^ ,9 , ,^ ^7 * ^171 ^3 ' Â°' has a group of multiple points at 0^. These multiple points are; (a) one triple point Op,; (b) one double point 0_, j (c) /twelve smple points 0^3^, Ogg^^, 0^^^^^, 0^^, Â°213* Â°2133' Â•Â•Â•Â» Â°213(7) ^''^ Â°213(8) ' Hence, two members of the system (79) have in common at , 1-14 + 1-5^ + 1-2^ + 12-1^ or 221 fixed points, and the degree of the system is 289 221 or 68. Delete from 221/17 the sum of the orders Â°^ Â°io' %' Â°6' Â°4' ^^ %^^^^' ^^^ PÂ°^^ % is of multiplicity one for (|)g. Applications of U and V to the projectivity give: (a) the points on (})situated on (80) fX^ = k Xg X^ Â» 0, (m = 4, 5, 6, 7, 8, 9, 10, ll) correspond to the points infinitely near O^,,; PAGE 45 (b) the points on ^c situated on 40 (81) X^^ o k X^ X^ =0, (m 1, 2, 4, 5, 6, 8, 9, lO) correspond to the points infinitely near '^2Z1ZZ> (c) the points on (p^ situated on (82) Xg = k X^ I^ =0, (m 1, 4, 5, 6, 8, 9, 10, ll) correspond to the points infinitely near ^oiZiQ)' Note that the surface (80) is the projection of (74) to the space Xg = 0, and that (8l) is the projection of (76) to the same space. Hence, the following Theorem 8 : The surface (|)jhas a tangent new element (83) Xg k X,7 X^ 0, (m 1, 4, 5, 6, 8, 9, 10, ll) at the point 01. 6. Surface Proiect the surface (j) from the point 0' (0,0,1,0,0,0,0,0,0,0,0) to the space X, Â» to obtain the surface ((l)g) I X, XgXg X^ ^11 ^1^11 Vs 0, . '^o " Â•'^s " '^e " -^4 " -^ X3 = 0. PAGE 46 41 Two members of the family (79) intersected in 221 variable points at 0_, Thus, the order of ^^ is (289 22l)/l7 or 4. Now use the transformations U and V to find the multiple points at Og of the family (84) ag 44 ^ x^ ^ a^^ ^7 ^^ ^9 ^ 3^ ^17 ^ 3^^^ ^7 . q. The multiple points are: (a) seven double points 0^^, Og^^, ... , ^^.ZiS)' ^^'^ Â°23(6)* (b) two simple points 0Â„ and Op^, . Consequently, two members of the system (84) have in common at Og, 1.15^ + 7-2^ + 2-1^ or 255 fixed points, and the degree of the system is 289 255 or 34. Since the sum of the orders of 0' , 0Â», 0' 10' 8* 6' 0Â« 0^, and 0' is 13, the value 255/17, or 15, implies that the multiplicity of 0' for the surface ^. is 2. I b Applications of U and V to the projectivity give: (a) the points on ^c situated on (85) [x^ = k Xg X^ =0, (m Â» 3, 4, 5, 6, 8, 9, 10, ll) correspond to the points infinitely near Op-, , ; (b) the points on (pÂ„ situated on (36) ( 1^ XgX-L^ = \= 0, (m = 1, 3, 4, 6, 8, 9, lO) correspond to the points infinitely near Op^/g-v. PAGE 47 42 The surface (85) is the projection of the surface (80) to the space X = 0. Hence, the following Theorem 9 ; The surface (})g has a new tangent element (87) at the point 0'. f 2 ^5 hhi \ = 0, (m = 1, 3, 4, 6, 8, 9, 10) 7. Surface (Ji The surface ^^ projects from the point 0^ (0,0,0,0,0,0,1,0,0,0,0) to the space XÂ„ = to a new surface i^n) 2 ^5 ^2^11 hz Xg Â» X^ = Xc X3 = X^ 0. Two members of the family (84) intersected in 255 fixed points at 0Â„. Thus, the order of (j)^ is (289 255) /l7 or 2. The transformations U and V establish the multiple points at Og for the family (88) a^g x^ Xg x| + a^ x^^ + a 171 ^3 ^Â• T^ese are sixteen simple points 0^^, 02^3, 0^^^^, ... , 0^^3(3^, 0^3, Â°231Â» Â°2311' *Â•Â• ' Â°231(5)' ^^^ Â°23l(6)* ^^^Sj "two members of the system (88) have in common at 0^, 1-16^ + 16-1^ or 272 fixed points, and the degree of the system is 289 272 or 17. The multiplicities of 0' PAGE 48 43 Â°fi' Â°Â«' Â°^ Â°c> Â°7Â» Â°^' ^^^ Â°o ^o""^^! ^Â° 272/17 or 16. Therefore 0' is a simple point for (j)Â„. Apply U and V to the projectivity and observe that: (a) the points on ({>Â„ situated on (89) f Xg . k X^^ X^ 0, (m = 1, 3, 4, 6, 7, 8, 9, lO) correspond to the points infinitely near ; (b) the points on ({) situated on (90) f X^ = k Xg X^ = 0, (m = 3, 4, 6, 7, 8, 9, 10, ll) correspond to the points infinitely near Oo-it/'q\. The surface (89) is the projection of (86) to the space I= 0. Hence, the following Theorem 10 : The surface ({) has a new tangent element X^ = kX5 XjÂ„ = 0, (ra 3, 4, 6, 7, 8, 9, 10, ll) at the point 0' PAGE 49 44 8 . Summary A sequence of projected surface; 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 Op vertex of the triangle of reference in the plane. 5/ 5/ 07/ ^. Figure 2 / \ / \ / \ .".4' \ 0. .l-JJ ^ ^^ 0, <',Â•! OilO " 3 'J 211:1 Figure 4 Figure 5 PAGE 50 45 15 H Figure 6 Ot 0, /U~L IH i Figure 7 \ 0. 15 1 H \ Figure 8 Figure 9 PAGE 51 46 The following chart lists the various results of this chapter and some information on (j) from Chapter II. ; Order | Point j Multiplicity of . on j of Surface , Surface ^ Surface ! Point Tangent Element to Surface at Point ^1 -^2 ^3 ^4 h h h h h \ and \ 0, (m = 5,7,9,11) ^9 ^8^1 Â° and X 0, (m = 1,2,3,4,5,6,7) X = kXp and \ = 0, (m 1,2,3,4,5,7,10,11) XÂ„ = k X, and \' 0, (m = 1,2,3,5,7,8,10,11) X3 . k Xg and X^ = 0, (m = 1,2,5,6,7,8,10,11) PAGE 52 47 j Order j Point : Multiplicity j of i on of Surface | Surface | Surface Point Tangent Element to Surface at Point i '^ X3 = kXg and \ 0, (m Â» 1,2,4,5,6,8,10,11) X-Li , k X7 and ; \ = 0, (m = 1,2,3,4,5,6,8,10) *5 PAGE 53 CHAPTER IV A RATIONAL SURFACE F IN ll To a certain plane curve shovm below, of order seventeen, and which is not invariant under H corresponds on t a curve of order two hundred eighty-nine. This curve is cut out on ({^ by a seventeenth order hypersurf ace . Furthemore, the coefficients of the equations of the latter surface are functions of the coefficients of the equation of the plane curve considered. In order to see this, consider the plane curve of order seventeen, (92) 0, . Z c, ., x^^ x^ x^ = 0, where i + .1 + k = 17. Apply H sixteen times in succession to (S2); this gives 3 = Z E^(") c. , xi xJ x^ = 0, n ijk 1. 2 3 * where i + ,i + k 17 n = 2, 3, ..., 17, and w(n) is the remainder when (n l)(,i + 15k) is divided by 17. The curve, (93) e^e^e^ ... e^^ = o, corresponds to a curve C on ^, where C is in birational correspondence with each of the curves 9^ Â» (m = 1, 2, ..., 17). That is, to a point 48 PAGE 54 49 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 (l) in two hundred eighty-nine groups of I-j^^. This mplies that the hyperplane related to (l) intersects C in 289 points. Hence, C is of order 289. Let us vary 9-^ in a continuous manner in its plane until its equation becomes equal to (l). The corresponding C varies on (j) and reduces to the section of (j) by the hyperplane, (94) a^ X, . a, Xg . a^g X3 . a^^ X^ . a^g X^ . ag, Xg . a^g X, * ^133 ^8 * ^148 ^ ^ ^170 ^10 "^ ^171 ^1 = Â°> counted seventeen times. That is, the section of (j) is made by the reducible hyper surf ace of order 17, (95) i-^h-sh* ^zeh* '^ -17lhl^^' 0. This implies that the curves C are cut out on (j) by seventeenth order hyper surfaces. Now, consider 6^ Â« varying in the plane and becoming equation (2). The curve (93) becomes ^^^^ (g(x^, x^, x^))^' = 0, and the curve C becomes a curve A counted seventeen times. Consequently, A must be cut out on (j) by a hypersurface of order seventeen. PAGE 55 50 By simplifjong (S6) and applying T one arrives at the foUowing equation for the hyper surf ace (87) Vi/(X^, X^, ... , X^^) , (g(x^, .^, x,))" . .11 xj= X^^ The fact that the x^'s ( i = 1, 2, 3) group together into factors of X_^ (i = 1, ... , 11) can be demonstrated by solving certain equations relating the exponents of x^ (i = 1, 2, 3) obtained from possible powers of terms of g (x^, x^, x^) to the exponents of x. (i = 1, 2, 3) obtained from possible factors of X. (i = 1, ... ii) . Take a surface F in S whose equations are: av ^^=^(\,\, ... ,:i^^) (F) hoh hh h h ^-^ r^S h ^7 ^2^5 ^2 hh ^7^9 ^ ^7 ^7^1 ^5 ^1 hhl hh Now the author demonstrates that F is a rational surface. To do this, a pro.iective correspondence is set up between the plane and F using the following transformation T . PAGE 56 51 (t') ''l PAGE 57 52 (100) 'l24 ^4 ^5 ^6 ^1 ^i ^2 ^3 * ^16S ^4 ^5 ^6 ^1 ^ 4 ^3 2 2^2 14 -, ^2 ^2 ^ 8 9 * ^5 \ ^4 ^6 ^ ^ ^2 ^3 * ^96 ^1 ^4 ^6 ^ ^2 ^3 * ^43 ^4 4 ^8 ^9 4^ 4 * ^^^ ^? ^^^ ^ ^"^ ^^ 17 2 4 5 ^^ * a Y^ Y^ ^2 10 ^7 ^ ^ Y y2 y2 Y 6 4 7 * ^29 ^3 ^4 h\ ^2 * ^58 ^3 ^4 ^5 ^0 ^ ^2 ^3 ^ ^42 ^2 ^4 4 h 4 4^ 4 ^ ^119 4 4 4 4 4 4 '^44 4\z(101) '124 ^2 ^4 ^5 ^6 ^1 ^ ^2 ""S "^ ^169 ^2 ^4 ^5 ^6 ^1 ^2 '^3 .3 ,,3 9 2 6 75 13 3 h h h h hi ^ -2 -3 ^ ^96 h K h ^6 ^1 ^ -2 -3 + a X^ X^ X^ X x'^ yÂ° V * ^ Y^ Y^ Y^ Y Y 6 4 7 * ^143 ^4 ^5 ^6 ^9 ''l "^2 ^3 * ^7 ^2 ^4 ^5 ^0 ^1 ^1 ^2 ^3 4a X^ X^ X X^ x^ x^^ x^ 4. Â« Y Y^ Y^ Y Y 10 "7 4agg Xj X^ Xg Xg x^ Xg X3 + a^Q Xg X^ X^ Xg X^^ x^ x^ + a X X^ Y^ Y Y v^ v^ ^8 ^ a y2 y3 y^ y^ 12 5 + a^2 Xg X^ Xg Xg Xq x^ Xg X3 + a^^g X^ X^ Xg Xg x^ X3 ^444hz' PAGE 58 (102) 53 ^4 ^2 9 2 6 124 ^2 ^4 ^5 ^6 ^11 ^1 ^2 ^3 * ^169 ^2 ^4 ^5 ^6 4l ^1 ^ 2 ^3 A ^4 + ^75 \ Xg X^ Xg Xg X^ x^ x^ X3 14 96 6 4 7 h h h h h hi ^ ^2 ""s X X x; X Xp 12 5 ^143 -4 -5 -6 -8 ^ ^^ ^3 * ^17 4 4 4 ho hi 4 4^ 4 r3 v4 ^3 .3 6 .13 3 * ^Z9 ^l ^4 K \ ho ^ -I -I * =58 h h ^: 4 h ho ^ -f ^ 2 3 A ^4 ^2 8 9 3 ^4 10 7 * =42 h h ^4 ^i ^; -I -3 * -lis h K n h K ^ -i 444 = (^44 4 ^2(103) .2 ^5 Y*^ Y^ Y v^ v*^ < 14 ^2 ^ 5 2 fl o "124 ^2 ^4 ^5 ^6 ^1 ^ ^2 ^3 * ^69 ^2 4 ^5 4 hi 4 ^3 * ^5 ^ ^2 ^ 4 ^ 4l 4 4 ^3 + age ^ ^4 X5 Xg Xq Xg X^^ x^ X3 ^ \43 ^ ^ 4 4hhl4 4' 4 " ^7 4 4 4 ^8 ^0 ^1 ^ 4 ^3 ^ "29 4 ^ 4 ^ ^0 -1 -2' -3 ^ ^58 ^2 ^3 ^ 4 ^ 4o ^l' 4 4 ^ "42 ^2 ^3 ^4 4 4 ^' 4 ^3 ^ "119 ^3 4 4 4 4 ^9 ^'^ 10 7 ^2 pxf X 5 ^5 4 "5 ^6 ^2' PAGE 59 (105) 54 rZ ^6 ^6 ^3 9 2 6 2 ^6 ^6 ^3 7 9 (104) a^^^ X^ XÂ° X3 XÂ° Xj^ xj x^ x^ . a^^g X^ XÂ° X^ X^ X^\ x{ x^^ X3 * ^5 ^1 ^2 ^4 ^6 ^9 \l \ ^2 ^3 .aÂ„ X^X^X X^X X^ x^^x^^x^ 96 14569 11 12 3 \43 ^ < 4 4 4 \l 4Â° 4 * Â»17 4 4 4 '^ 4 =^1 4 12 5 + aÂ„Â„ x5 X? X? X 6 4 7 2 v6 ^6 8 9 29 3 ^4 ^5 ^9 ho -L -2 ^3 ^ ^58 ^2 ^3 ^4 ^5 ^9 ^0 ^2 4 * ^42 ^2 \ h h ^ ^2 "^3 * ^119 ^2 ^3 ^ ^5 ^ ^ ^ ^ ^2 ^ P 4 4 4 hz3 y7 y y^ Â„3 2 14 ^124 ^2 \ h h \l ^ ^2 ^3 " ^169 "2 "4 "5 "6 XÂ„ X. X. XÂ„ X^^ x^ x^ ^75 h 4 < 4 ^ 4i 4 4 4 ^ ^96 4 ^I ^5 4 4 4i 4Â° 4 + a^^5 X-j^ Xj X^ Xg Xg Xg X-^^ x-^ x^ x^ + a X^ x"^ x'^ X X^ X ^2 ^^ ^^ X . y3 t'7 y? y3 7 12 5 + ^17 ^2 \ Xg Xg X^Q X^ x^ x^ X3 + a^g X3 X^ ^^ X^^ X^ x^ X3 ^ ^58 ^2 4 4 4 ^' 2 ^ 4 11 2 2 7 7 4 7 9 9 ^0 ^ ^2 ^3 -^ ^42 ^2 ^4 ^5 ^8 ^9 ^ ^2 ^3 '119 h h h h h h \ \ ^2 ""3 e4 4 4^^. PAGE 60 55 (106) a^2^ Xg X^ Xg Xg X^^ x^ x^ x^ + a^gg X^ X^ X^ Xg X^-|_ x^ Xg x^ r3 ^8 .3 2 '75 h h h ^6 ^9 ^ "^ "^2 "^3 14 X. X? X? X?. X? X? '86 ^ 4 ^5 ^6 ^9 ^ "2 "3 rZ ^8 ^3 12 5 ^43 h h h h h h hi "^ ""3 a. X^ X^ X^ X^ X.^ XL x.^ x^ x' \7 -^2 ^4 ^5 ^8 ^0 ^1 ^ ""2 ^ * ^28 -^ ^3 -^ -^ ^0 -^1 ^1 ^ ^3 "^ ^58 -^3 -^4 -^5 -^8 -^9 Ho ^1 ^2 "*" ^42 ^2 ^^3 ^4 "^5 ^ ^9 ^ ^2 ^ * ^119 ^ ^3 -^4 ^ ^6 ''^ ^ ^1 ^2 ^3 " ^ ^4 ^ ^ ^12' (107) A ^9 .9 ^4 14 "124 ^2 ^4 ^5 ^6 \l "^ ""2 ^3' "^ ^ Y T^ Y Y"^ Y -v -v 169 ^2 ^4 ^5 ^6 ^11 ^2 ^3 300 4 926 + a^5 X-L Xg X^ X^ Xg X^-,_ x^ Xg Xg agg \ X^ Xg Xg X^ X^^ x^ Xg Xg + a^^g X^ X^ X^ X^ Xg X^^ x^ x 3 9 9 2 2 10 7 + a^^ Xg X^ Xg Xg Xg X^Q X^^ x^ Xg 1 X X^ X^ X^ X X? X x^ x^ x*^ ^29 ^ ^3 ^4 ^5 ^8 ^0 11 n ^2 ^3 40Q 22 4 11 2 ^ ^58 -^3 ^ ^ ^8 ^ ^10 ^1 ^2 ^3 3 9 9 4 xÂ„ x: X" x^ xÂ„ x,^ xr xX x 42 2 4 5 3 6 9 ^^10 '^l ^2 ^Z ,9 ^9 ^2 ^4 13 3 Q . Q ..Q + a^g X^ Xg X^ Xg Xg Xg Xg x^ Xg x^ = rx^ X^ Xg X^g PAGE 61 56 Note that the previous ten equations are linear in Thus, Cramer's rule can be easily used to solve for these expressions. The ten equations give (108) 4 4 4 . _Â£v^ . where A and A. (i Â« 1, 2, ... , 10) are the determinants that are functions of the ten constant coefficients and of X. (i = 1,2,3,4,5,6,8,9,10,11). Now from T' one obtains (109) x^^ pXi-L. This gives with (108) the following .6 Â„4 (110) PAGE 62 57 Next, combine (llO) and (113) to get (114) ^2 'l''5'^\ 4 "'"Ihi The transformation T' yields (115) xj x^ x| = PXj. Also, the ten equations give (116) X2 X3 = These two equations combine to give (117) PAGE 63 58 These two equations combine to give 8 (la) ., ^>h From the transformation T' obtain (122) x^ x^ X3 = pX^ The ten equations give (123) =^ ==2 ^S 4 The above two equations combine to give x^ A XÂ„ (124) x| ^8^2 * Now (121) and (124) combine to give ^2 (125) ^2 ^ ^8 ^ x^ 2 ' ^8 ^9 ^12 Finally combine (II8) and (125) to give the inverse of the transformation T' . It is 7 4 2 2 A 2 S 4 4 "a '9 h ^Iz PAGE 64 59 Hence, there is a one-to-one correspondence between the plane and the surface F, even though there is a one-to-seventeen correspondence between the surface (j) and the plane and also between (j) and F. Conclusion Using an homography, an involution of period seventeen was generated; and certain siirfaces obtained from this involution were investigated. A family of plane curves invariant under this involution vras projected, by means of the transformation T, to the hyperplanes of a space of ten dimensions (S.,^). From this a surface (|), with points on it in a one-to-seventeen 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 projections of this surface (p. The surfaces an-ived at by successive projections were ({). (i Â« 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 X^ Â„ proportional to the function g(x^ , Xp, xÂ„) , a projectivity T', mapping the points of the plane onto the points of a surface F in S.^ , , 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. PAGE 65 APPENDIX I A METHOD OF FINDING THE ORDER OF A QUINTIC TANGENT GONE There are various techniques of determining the order of a surface. The particular method illustrated here employs the definition from V7oods [26, p. 390]. Examine as an example the surface (33). 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) X^ = X^ =. Xg = X^^ = (127) x| X^Xj (128) X^ X^X^ = (129) X"^ X X =0 4 3 6 (130) X^ X^Xg = (131) 2 A^ X^ = (i = 1, 2, ... , 11) (132) Z B. Xj = (,j = 1, 2, ... , 11). 60 PAGE 66 61 The combination of equations (126), (127), (l3l), and (132) will give after simplification (133) (A^B^Q A^^B^) X| . (A^B^^ A^^B^ X^X^ . (A^B^^ A^^B^) X^ * ^h\0 ho\^ hh ^ ( VlO ^10^6^ ^3^6 Now substitute equation (128) to eliminate X and then use (129) to remove XÂ„, (154) (A^B^^ A^^B^) X^ . (A^B^Q A^^B^) l^ * ^VlO " ^0^3^ 44 ^ VlO \oV Vl ^ ^'6\0 hoV ^6 * (^8^10 ^10^8^ 4h = Â°Now employ (130) to arrive at (155) (A^B^Â„ A^^B^) X= . (A^B^g A^Â„B^) X^ . (A^B^^ A^^B^) X^ * ^Uho *ioV 44 * (^^6=10 ^lo^e) =^64 Note that the solution of the fifth degree equation (135) indicates that the tangent element is a quintic cone. PAGE 67 APPENDIX II A METHOD OF INVESTIGATING A FOURTEENTH ORDER NEIGHBORHOOD In the investigation of involutions that involve large values of p such that E^ = 1, there arises the problem of applying a quadratic transformation repetitively to a large equation. For example, in the study of ' , 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 z, zi z, \inder R goes into z-, "'"'*' ^ z^ "*" z,. The a^ _^ term stops any factoring of zÂ„'s in the simplification. Thus for any given term the k, or the exponent of z , remains constant under applications of R. The a^^ term allows only one z to be factored out in the simplification. Hence, for a given term the z^ exponent will increase by the constant value k 1 under each application. Now the a,Â„-, term allows only z-, "*" "^ ^ ~ to be factored out. This final result after simplification is 2i + j-i-j-k + 17 .i + k-1 k i-k + 17 i + k-1 k ^1 ^2 "5 ^'^ ^1 "2 ^3For a given term z-, increases by the constant 17 k and Zp increases by the constant k 1 for each application. This pattern develops only after one complete application of R. 62 PAGE 68 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). PAGE 69 64 ^A o 1142 863 7 9 5 84 39x^^x2x3 a.gxf x^x| a^, x^X2X3 a,, x^ x^ x^ ag, x, x^ X3 14 2 1 11 4 2 6 3 7 1 S 5 8 4 30 1 26 3 2 22 6 3 15 7 S 18 9 4 46 1 41 4 2 36 8 3 23 15 9 31 12 5 62 1 56 5 2 50 10 3 31 23 9 44 15 4 78 1 71 6 2 64 12 3 39 31 9 57 18 4 04 1 86 7 2 78 14 3 47 39 9 70 21 4 110 1 101 8 2 92 16 3 55 47 9 83 24 4 126 1 116 9 2 106 18 3 63 55 9 96 27 4 142 1 131 10 2 120 20 3 71 63 9 109 30 4 158 1 146 11 2 134 22 3 79 71 9 122 33 4 174 1 161 12 2 148 24 3 87 79 9 135 36 4 190 1 176 13 2 162 26 3 95 87 9 148 39 4 206 1 191 14 2 176 28 3 103 95 9 161 42 4 222 1 206 15 2 190 30 3 111 103 9 174 45 4 238 1 221 16 2 204 32 3 119 111 9 187 48 4 PAGE 70 65 4 3 10 2 10 5 5 11 17 99 \ ^2 ^3 ^133 ^1 ^2 ^3 ^148 ^1 ^2 ^3 ^170 4 17 ^171 ^3 4 3 10 2 10 5 11 10 10 14 12 5 18 19 10 26 16 5 25 28 10 38 20 5 1 5 11 17 17 7 13 11 17 14 14 17 13 23 11 34 13 30 17 19 33 11 51 12 46 17 32 37 10 50 24 5 25 43 11 68 11 62 17 39 46 10 62 28 5 31 53 11 85 10 78 17 46 55 10 74 32 5 37 63 11 102 9 94 17 53 64 10 86 36 5 43 73 11 119 8 110 17 60 73 10 98 40 5 67 82 10 110 44 5 49 83 11 136 7 126 17 55 93 11 153 6 142 17 74 91 10 122 48 5 61 103 11 170 5 158 17 81 100 10 134 52 5 67 113 11 187 4 174 17 88 109 10 146 56 5 73 123 11 204 3 190 17 95 118 10 158 60 5 79 133 11 221 2 206 17 102 127 10 170 64 5 85 143 11 238 1 222 17 PAGE 71 APPENDIX III A METHOD OF DEMONSTRATING THE EXISTENCE OF\j;(x^, x^, ... , x^) The following is one way of justifying that the terms of 17 (g(x,, Xp, xÂ„)) can be expressed in terms of the X.'s (i Â« 1^ ... , 11) of the transformation T. Suppose that the a-iy, apo* Â•Â• > *'t rq "terms are raised to the powers b-. , b^, ... , b,^^ respectively and one factors out Tf^where 17 i 1, 2, ... , 11 and j = d-^, dg, ... , d^^. Then(g(x-|^, Xg, x^)) will be expressible as products of X.'s if all possible combinations of integral values of b. = 0, 1, ... , 17, and Z b . = 17 are such that the equations (136), (137), and (138) have integral solutions of d.'s where 2 d. = 17. Equations (136), (137), and (138) are the conditions that cause the exponents of the x^ , x , and x to be the same in the powers and the factors. (136) 12 b-^^ + 10 bg + 9 bg + 7 b^ + 6 bg + 4 bg + 5 b^ + 2 bg + bo 17 d^ + 14 dg + 11 dg + 8 d^ + 7 dg + 5 dg + 4 d^ + 2 dg + dQ. (137) 7 bg + 2 bj + 9 b^ + 4 bg + 11 bg + 6 by + bg + 13 bg + 8 b^Q 2 dg + 4 dj + 6 d^ + dg + 8 dg + 3 dy + 10 dg + 5 do + 17 d^^. 66 PAGE 72 67 (138) 5 b^ + 6 bg + b^ + 7 bg + 2 bg + 8 b^ + 14 bg + 3 bg + 9 b^Q dÂ„ + 2 d, + 3 d^ + 9 d^ + 4 d+ 10 d_ + 5 do + 11 d_ + 17 d 23456 78 9 Now use Z b. = 17 and Z d. = 17 to eliminate bg and d-i lin equations (136), (137), and (138). The results are 11" (139) 34 + 10 b]_ + 8 bg + 7 bj + 5 b^ + 4 bg + 2 bg + by bg 2 b 10 17 d-,_ + 14 dg + 11 dj + 8 d^ + 7 dg + 5 dg + 4 dy + 2 dg + dg, (140) 17 b-]^ + 6 bg + bj + 8 b^ + 3 bg + 10 bg + 5 by + 12 bg + 7 b 10 2 d^ + 4 dÂ„ + 6 d. + dc + 8 d_ + 3 dÂ„ + 10 do + 5 d_ + 17 d^^, 2 o 45 6 7 o 9 iU (141) 51 + 9 b^ + 14 bg + 8 bg + 13 b^ + 7 bg + 12 bg + 6 by + 11 be + 5 b^Q = 17 d-|_ + 16 dg + 15 dg + 14 d^ + 8 dg + 13 dg + 7 dy + 12 dg + 6 d^ + 17 d^Q. Observe that the above three equations are not independent, i.e., equations (139) and (140) added together give equation (l4l) . Consequently, if (139) and (140) are satisfied then (l4l) will automatically be satisfied. To exhibit that (139) and (140) have a common solution assume that the b , b , b , b , b , b are each equated respectively to d,, d^y d^, dÂ„j dp, dp. The problem now is reduced to showing that PAGE 73 68 equations (142) and (143) have a common solution. (142) 34 + 10 b-L bg 2 b-^Q 17 d^ + 14 d^ + 11 dj, (143) 17 b^ + 12 bg + 7 b^Q = 2 dg + 4 dg + 17 d^^. Now eliminate bg in the above two equations to obtain (144) 25 + 7 b^ b^Q = 12 d^ + 10 d^ + 8 dj + d-^Q. The author has examined individually all the possible variations of b^ and b-j^^ 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. PAGE 74 BIBLIOGRAPHY 1. Childress, N. A. "Surfaces Obtained from Involutions Generated by Homographies of Period Three, Five, and Thirteen," Unpublished form of Ph.D. dissertation. Department of Mathematics, University of Florida, 1954, 2. Dessart, J. "3ur les surfaces representant 1' involution engendree par une homographie de period cinq du plan," Memoires de la Societe Royale des Sciences de Liege , 3Â® Series, Tome XVT (,1931) , pp. 1-25. 3. Frank, Stanley. "Certain Cyclic Involutory Mappings of Hyperspace Surfaces." Unpublished form of Ph.D. dissertation, Department of Mathematics, University of Florida, 1960. 4. Godeaux, Lucien. C ours de Geometrie Superieure . Fascicule II. Liege: Libr.airie Bourguignon, 1937. 5. . "Etude elementaire sur 1' homographie plane de periode trois et sur une surface cubique," Nouvelles Annales de Mathematiques , 4^ Serie, Tome XVI (1916), pp. 49-61. 6. . Geometrie Algebrique . 2 tomes. Liege: Sciences et Lettres, 1948-1949. 7. . Introduction a la Geometrie Pro-'ective Hvnerspatiale . Liege: Librairie Bourguignon, 19 39, 6 . . Introduction a la Geometrie Superieure . 2^ edition. Liege: G. Thone, 1946. G. Thone, 1952. Memoire sur les Surfaces Multiples, Liege: 10. . "Recherches sur les involutions rycliques anpartenant a une surface algebrique," Bulletin de 1' .'-dademie Royale de Belginue (Classes de Sciences), 5Â® Series, Tome X .11 (1931}, pp. 1D56-1364. 11. . "Sur les homographies plane cycliques," Memoires de la Societe Royale des Sciences de Liege , 3Â® Serie, Tome XV (1950) , pp. 1-26. 12. Gormsen, Svend T. "Maps of Certain Algebraic Curves Invariant under Cyclic Involutions of Period Three, Five and Seven." Unpublished form of Ph.D. dissertation, Department of Mathematics, University of Florida, 1953. 69 PAGE 75 70 13. Graustein, V/illiam C, Introduction to Higher Geometry . New York: The Macmillan Company, 1946. 14. Hutcherson, V. K. ";. Cyclic Involution of '^rder Seven," Bulletin of the American Mathematical Society , Vol. 40 (1934), pp. 1^;3-151. 15. . "A Cyclic Involution of Period Eleven," Canadian Journal of Mathematics . Vol. IH (l95l), pp. 155-158. 16. . "Maps of Certain Cyclic Involutions on Two Dimensional Carriers, " Bulletin of the /unerican Mathematical Society , Vol. 37 (1931), pp. 759-765. 17. . "Invariant Curves of Order Tdght," Revista Mathematica y Fisica Teorica , Serie A, Vol. 9 (1952) , pp. 13-14. 18. , et Childress, N. A. ":.tude d'une involution cyclique de periode cinq," Bulletin de l'.n.daderfiie Foyale de Belgique (Classes de Sciences), 6 Serie, Tome XL (1954), pp. 103-106. 19. , and Childress, N. A. "Surfaces obtained from Involutions Generated by Homographies of Period Three, Five, and Thirteen," Revista Mathematica y Fisica Teorica , Serie A, Vol. 9 (1957), pp. 41-48. 20. , and Gormsen, S. T. "Maps of Certain Algebraic Curves Invariant Under Cyclic Involutions of Periods Three, Five, and Seven," Canadian Journal of Mathematics , Vol. 'Tl (1954), pp. 92-98. 21. , and Kenelly, J. VJ. , Jr. "An Involution of Period Seventeen Contained on a Rational Surface in a Space of 11 Dimensions," Notices of American Mathematical Society , Vol. 7 (i960), (Abstract) pp. 479. ' ~~^ 22. , and Kenelly, J. VJ. , Jr. "Three Branch Points on a Surface in a Space of Ten Dimensions," Notices of American Mathematical Society , Vol. 7 (1960), (Abstract) rr. 479-480. 23. Morelock, James C. "Invariants with Respect to Special Projective Transformations." Unpublished form of Ph.D. dissertation. Department of Mathematics, University of Florida, 1952. 24. Veblen, Oswald, and Young, John VJ. Projective Geometry . 2 vols. Boston: Ginn and Company, 1910-1918. PAGE 76 71 25. Winger, R. M. An Introduction to Projective Geome try. Boston: D. C. Heath and Company, 1923. 26. Woods, Fredrick S. Higher Geometry . Boston: Ginn and Company, 1922. PAGE 77 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 High 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. This degree was conferred with honors at the following commencement exercises on May 25, 1957. He entered the University of Mississippi in the fall of 1S56 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 1956-57 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 Charraaine Ruth Voss of Covington, Louisiana, in 1956. She is also a graduate of Southeastern Louisiana College. 72 PAGE 79 This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved 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/Soiences Dean, Graduate School SUPERVISORY COMMITTEE Chairm^ i I A. y^/y^ ./)-<.-^^ ^d 77?.^^^ finT^j'^C PAGE 80 i, d S7 7 |