6533b82afe1ef96bd128b96e

RESEARCH PRODUCT

A Common Characterization of Finite Projective Spaces and Affine Planes

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Let S be a finite linear space for which there is a non-negative integer s such that for any two disjoint lines L, L' of S and any point p outside L and L' there are exactly s lines through p intersecting the two lines L and L'. We prove that one of the following possibilities occurs: (i) S is a generalized projective space, and if the dimension of S is at least 4, then any line of S has exactly two points. (ii) S is an affine plane, an affine plane with one improper point, or a punctured projective plane. (iii) S is the Fano-quasi -plane.