J. Geom. 83 (2005) 1 – 4 0047–2468/05/020001 – 4 © Birkh¨ auser Verlag, Basel, 2005 DOI 10.1007/s00022-005-0004-2 Projective planar spaces Paola Biondi ∗ Abstract. A local condition on a planar space is given which is sufficient for its points, lines and planes to be the points, the lines and some subspaces of a projective space. Mathematics Subject Classification (2000): 51A05. Key words: Projective spaces, planar spaces. 1. Introduction Many local or global geometric conditions on various incidence structures have been inves- tigated in order to prove that they are sufficient, sometimes also necessary, for the given structures to be projective spaces (see, e.g., [2, 3, 4, 6]); here we study this problem in the context of planar spaces. Note that the planar spaces are the d -dimensional linear spaces when d = 3 [1]. A linear space is a pair L =(S , L), where S is a set of points and L is a set of subsets of S , which are called lines, such that any line contains at least two points and any two distinct points lie on exactly one line. A subspace of a linear space L =(S , L) is a subset of S containing the line through any two of its distinct points. It is immediate that the intersection of subspaces is a subspace, too. If X ⊆ S , the subspace generated by X is the intersection of all subspaces of L containing X (since S itself contains X, such a subspace always exists). We write 〈X〉 for the subspace generated by X. If X ={a 1 ,a 2 ,...,a n } or X = X ′ ∪{a} (a 1 ,a 2 ,...,a n ,a are points and X ′ is a subset of S ), we also write 〈X〉=〈a 1 ,a 2 ,...,a n 〉 or 〈X〉=〈X ′ ,a〉, respectively. So, for any two distinct points a and b, 〈a,b〉 denotes the line through a and b. A planar space is a triple S =(S , L, P ), where L =(S , L) is a linear space and P is a set of proper subspaces of L, called planes, such that any three non-collinear points (i.e., points not on a common line) lie in a unique plane and every plane contains three non-collinear points. ∗ Work supported by the National Research Project “Structure geometriche, Combinatoria e loro applicazioni” of the italian M.U.R.S.T. 1