On Weakly Perspective Subsets of Desarguesian Projective Lines FRANCIS BUEKENHOUT 1. Introduction This paper is mainly motivated by questions arising in higher dimen- sional projective geometry, namely the characterization of so called semi quadrics (see [2]). We recall the axiomatization of a Desarguesian projective line P introduced in [1]. P is a set of at least 3 points together with a mapping A which assigns to each ordered pair of points (p, q) e p2, a set ~ (p, q) of permutations of P whose members are called perspectivities o/ P o] center p and axis q, the mapping A being submitted to: (1) ~i(p, q) is a permutation group fixing p and q, regular on P- {p, q} (2) Every perspectivity a is an automorphism i.e. a74(p,q)a-l= = 71 (a(p), a(q)) for all p, q e P. (3) A (p, q) centralizes A (q, p) for all p, q in P. A subset W of P is called weakly perspective if J W J _> 3 and if for all p, a, b e W, p r a, b the unique element of A(p, p) mapping a onto b, leaves W invariant. If P is identified with K w (oo} where K is a division ring (see [1]) and if W contains 0, 1, oo then we show (Theorem 1) that (i) W- oo is an additive subgroup of K; (ii) W- {o, oo} is closed under the mapping x--.x -1 of K-0; (iii) 1 e W - o o . These properties are shown to be characteristic and (ii) may be replaced by several other conditions. If char K r 2 it is also shown that W is a weakly perspective set containing 0, 1, oo if and only if W - oo is a Jordan division subring of K (Proposi- tion 2). In section 3, we study an ideal W o[ a weakly perspective set U of P which means that W is a proper weakly perspective subset of U and that for all u ~ U and all a, b e W, u ~ a, b, the element of A (u, u) mapping a onto b leaves W invariant. In theorem 2 this situation is characterized in algebraic terms and it is shown that it can occur only if char K = 2. We would like to define a perspective subset of P as a subset S such that (i) S is weakly perspective; (ii) for all p e P-S and all a, b E S, there exists a perspectivity of center p mapping a onto b and leaving S invariant. This definition is inspired from a similar one which works