Inversive Planes of Even Order
نویسنده
چکیده
1. Results. An inversive plane is an incidence structure of points and circles satisfying the following axioms: I. Three distinct points are connected by exactly one circle. II. If P, Q are two points and c a circle through P but not Q, then there is exactly one circle c' through P and Q such that cC\c'~ [P]. III. There are at least two circles. Every circle has at least three points. For any point P of the inversive plane 3 , the points 5̂ P and the circles through P form an affine plane 31 (P). If 3 is finite, all these affine planes have the same order (number of points per line); this integer is also termed the order of 3 . An inversive plane of order n consists of n + l points and n(n + l) circles; every circle contains n + 1 points, and any two points are connected by n + 1 circles. Let ^ be a projective space of dimension d>\ (we shall only be concerned with d = 2, 3, and we do not assume the theorem of Desargues if d = 2). A point set ® in $ is called an ovoid if I'. Any straight line of ty meets S in at most two points; IV. For any P£(5, the union of all lines x with xP\Ë= {P} is a hyperplane. (This is called the tangent hyperplane to S in P.) It is straightforward to prove that the points and the nontrivial plane sections of an ovoid in a three-dimensional projective space form an inversive plane. The purpose of the present note is the announcement, and an outline of proof, of the following partial converse:
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تاریخ انتشار 2007