The intersection of certain quadrics

L. M. Brown
1938 Proceedings of the Edinburgh Mathematical Society  
We investigate in this paper a certain special family of quadric varieties, that is of F|_ 1 's in [R]. Now among the more important properties of a quadric in [R] is that it possesses a system or systems of " generators," i.e., the quadric may be taken as the locus of certain families of subspaces, the behaviour of these depending on the parity of R. If i? is even, a quadric Ffn-i in [2n] contains a single family of [n-l]'s, so it seems likely that in discussing special families of quadrics in
more » ... lies of quadrics in [2n] an important type will be obtained by constraining the quadric to pass through a number of [n -l]'s. The freedom of quadrics in [2n] is n(2n-\-3), and since the postulation of an [n -1] for quadrics is \n(n + 1)> the freedom of a quadric in [2n] which is to contain k assigned [n -l]'s in general position is n (2n + 3) -\hn (n + 1). In order therefore that the freedom should not be negative we must have k ^ 4 + 2/(n +1), and it follows that (except for the trivial case where n = 1) the maximum number of [n -l]'s which may be assigned to a quadric is four. We therefore will discuss here the nature of the family of quadrics obtained, subject only to the condition of possessing four assigned [n -l]'s in general position. By the reasoning given above, such a family has freedom n, and the quadrics will thus have a common intersection consisting of a y n _! of order 2 n+1 . It is found that this F n _ x (which we shall call the base of the family) is highly degenerate, and we shall limit ourselves to an investigation of the component parts of the base and of their relations to one another. The establishment of the base occupies § 1 to § 4. In § 1 we define \n varieties, of known orders and method of generation, which must belong to the base. We then show in § 2 that these varieties must lie in eight certain [2n -3]'s. We calculate the orders of the varieties
doi:10.1017/s0013091500008403 fatcat:6ndwy6yceneq7pf3m7lcokqiyy