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Systems of Lines Amd Planes on the Quadric Systems of lines and planes on the quadric hypersurface in spaces of four and five dimensions Item Type text; Thesis-Reproduction (electronic) Authors Fish, Anne, 1924- Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 01/10/2021 14:55:47 Link to Item http://hdl.handle.net/10150/319188 SYSTEMS OF LINES■AMD PLANES ON THE QUADRIC HYPERSURFACE SPACES OF FOUR AND- FIVE DIMENSIONS - by Anne Fish A TLes is submitted to the faculty of the Department of Mathematics in partial fullfilment of the requirements for the degree of Master of Science in the Graduate College University of Arizona 1948 Approved: Direeto^ of Thesis t Date & R A& TABLE OF CONTENTS I? O (TU. O "fc X OH a a © © a © o © © © © o © o © © ©ooo oool Chapter I Geometry of M-Dirnensiona 1 Projective Space © » 2 II Ruled Surfaces in Three-Dimensional Projective Space © © © © © © © © © © © © 10 III Systems of Lines on a Quadric Surface in Three-Dimensional Projective Space © © « © 13 IV Systems of Lines on a Quadric Hypersurface in Four-Dimensional Projective Space « © © © 23 V Systems of Planes on a Quadric Hypersurface in a Projective Space of Five Dimensions © © 34 Bibliography © © © © © © © © © © © © © © © © © © © s © © © 4V 194189 INTRODUCTION It has long been common knowledge that an infinity of straight lines lie entirely on every quadric surface in ordinary three-dimensional space and that- these .lines possess interesting and important properties„ This thesis is an investigation of the analagous prob­ lems for quadric hyper surf aces* in spaces of four and five dimensions * A system of lines is shown to exist on every quadric hypersurface in four-dimensional space, The proper­ ties of this system of lines differ markedly from those of corresponding systems in ordinary space„ In five-dimensional space an infinity of planes are shown to lie entirely on each quadric hypersurface e ' None of these results is new* The credit for their CD original discovery goes, to the Italian geometers, CK Veronese and Co SegreHowever, the results are scattered and the preparation of this thesis involved their rediscovery by the present writer» Algebraic geometry is used throughout„ Dro Edwin J . Purcell, Professor of Mathematics at the University of Arizona, has been ofgreat assistance in the preparation of this thesis* ■ - * Co Segre, Encykiopadie der lathematisehen ffissenschaften. Band III, 2. Tell, p. 851= - 2 - CHAPTER I GEOMETRY OF N-DIMENSIONAL PROJECTIVE SPACE In a projective flat space of n dimensions (n is any positive integer), let a point be defined by the n ratios of n + 1 coordinates : These coordinates may be any complex numbers, not all zero, A hyperplane lying in n-dlmensional space is defined by the n +1 equations f>Xi" A ly i + A2y i'1' A3y l v t1 ~ 1»2 » • • .,n +1), 1 2 3 in which P is a factor of proportionality, y^, y^, y^, ..., and y^ are the coordinates of n given points, and A-^, Ag, A , a n are homogeneous parameters. If the parameters are eliminated, there results a single linear equation in x^ (1=1,2,...,n + 1). Thus, a hyperplane is a flat space of (n-1) dimensions, and is determined by any n points of n-dimensional space. The symbol will represent a hyper­ plane; the prime indicates that this is a flat space and the subscript denotes the dimension of the space. - 3 - A variety (k is any positive integer) is defined by the n +1 equations a 1y ^ ^ 2y^ + A-^y^ +• • • i » ( ^*.*,n ■** 1)» 1 2 3 in which P is a factor of proportionality, y^, y^, yit n—k + 1 and y^ are the coordinates of n-k + 1 given points, and A -j_, A g t A -i-1 are homogeneous parameters . If the parameters are eliminated, there result k linear equac­ tions in ( i = 1,2, •. • , n 1). Therefore, an is deter­ mined by any n-k + 1 points of n-dimensional space, and is the intersection of k hyperplanes. A straight line is defined to be a one-dimensional extent of points whose coordinates, satisfy the n 1 equat­ ions pxi“ Aiyi+ A 2zi» (1 “ 1»2,... ,n + 1) in which P is a factor of proportionality, A ^ and A 2 are homogeneous parameters, and y^ and are the coordinates of two fixed points in n-dimensional space. Thus, any two points of n-dimensional space determine a straight line. If the parameters, A^ and Ag, are eliminated, there result n-1 linear equations in x^ (l=l,2,...,n+l). These n -1 linear equations define an ^n-(n-l)“ Si» l*e*» a line. - 4 - A plane Is defined to be a two-dimensional extent of points whose coordinates satisfy the n + 1 equations Pxi= Alyi+ A2Zi* A3w i’ (1=1,2, . ..,n + 1), in which P is a factor of proportionality, Ag, and A-j are homogeneous parameters, and y^, , and w^ are the coordinates of three fixed points in n-dinensional space. A plane, therefore, is defined by any three points. If the parameters, A ^ , Ag, and A ^ , are eliminated, there result n -2 linear equations in x^ (i= 1,2,...,n v 1). These n -2 equations define an sn-(n-2)~ ®2» or a plane. Consider, now, the Intersection of these flat spaces. In a space of n dimensions, let an and an intersect. An is defined by n-m homogeneous linear equations in n ♦ 1 variables, x^ ; and is defined by n-r homogeneous linear equations in n + 1 variables, x^. A point common to the flat spaces, and 3^, must satisfy 2n-r-m homogeneous linear equations in n + 1 variables, x^. These equations have a solution if and only if 2n-r-mn +-1, or if 2n-r-m1=n, since £ homogeneous equations in q. variables have a solution if and only if*p<q.* Consider the relationship 2n-r-m = n; * N. B . Conkwright, Introduction to the Theory of Equations (Ginn and Company, 1941), p. 144. - 5 - let this be written as n-r-m< C, or m 4- r-n > 0. If m 4- r-n = a, then the 2n-r-m equations define an S^, for, 2n-r-m=n-a and n-a equations define an . This relationship gives a con­ venient method for determining the dimension of the intersec­ tions of flat spaces. A variety of order (or degree) two and dimension (n-l) o is represented by By order two, it is meant that a straight line meets the variety in two points or lies entire­ ly on it. A quadric hypersurface is a variety of order two and dimension (n-l). It is defined to be the locus of points whose coordinates satisfy the equation 4- 2a x x + n 4- l,n 4- 1 n 4- 1 12 1 2 (1) 2al,n +l xlxn +1 where a ^ (i = l,2,...,n+l; J = 1,2, ... ,n 4-1) is any complex constant, and the relationship a ~ ajj_ exists. It is desirable that this equation of a quadric hypersurface be written in a simpler form; this may be achieved by a change of coordinates, as follows: the points of the n-dimensional space which is being considered are located with reference to a hypertetrahedron. This hypertetrahedron has n 4- 1 ver­ tices* (C,0, ...,l), (0,0, ...,1,G), ..., (0,1,0,«.«,0), and - 6 - (1,0, No n -%-l of these vertices lie in the same hyperplane • The hypertetrahedron therefore consists of n 4-1 vertices and n -t-1 hyperplanes which do not have a common intersection• A point in n-dimensional space is fixed with reference to its distances from these n +1 hyperplanes. If a point lies at a distance p-^ from the hyperplane 0, P2 from Xg=*0, ..., pn from xn = 0, and Pn+1 from x^ ^ = 0, then the coordinates of the point are determined by the ratios 1 = kp1:kp2:kp5 :kPn>1, where k is a factor of proportionality. The point (l,l,...,l), or the unit point, and the n 4-1 vertices may be fixed at will, and all other points are thus determined. To each point in this space of n dimensions, there is associated a particular flat space with respect to a given quadric hypersurface; the dimension of the flat space is the same as that of the quadric hypersurface • The point is termed a "pole" and the corresponding flat space is called the "polar" of the given point with respect to the given quadric hypersurface. The polar of a point, Y, with respect to the quadric hypersurface defined by equation (1) is de­ fined to be the locus of points whose coordinates satisfy the n 4-1 equations 0ui = a11y14- a12y2 4- ••• + aljn+1yri^ 1, (I - 1, 2 n 4-1 ); (aik= akl) -7 - 0 is a factor of proportionality. A self-polar hypertetrahedron^ is defined to be such that each vertex is the pole of the opposite hyperplane. In particular, if the hypertetrahedron of reference be chosen as a self-polar hypertetrahedron with respect to the quadric hypersurface, (1), then equation (1) reduces to allXl* a22X2 + * • * + a n * l,n + lXL l ~ C - Upon equating /aT^x^ to x^ ( i = 1,2, ... ,n + 1) and then dropping the primes, the equation is in its simplest form: x?1 2 + xl3 x2n -t- 11= 0.
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