A Characterization of Projective Spaces by a Set of Planes |
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Authors: | Alexander Kreuzer |
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Institution: | (1) Mathematisches Seminar, Universität Hamburg, Bundesstr. 55, 20146 Hamburg, Germany |
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Abstract: | This note deals with the following question: How many planes of a linear space (P, $\mathfrak{L}$ ) must be known as projective planes to ensure that (P, $\mathfrak{L}$ ) is a projective space? The following answer is given: If for any subset M of a linear space (P, $\mathfrak{L}$ ) the restriction (M, $\mathfrak{L}$ )(M)) is locally complete, and if for every plane E of (M, $\mathfrak{L}$ (M)) the plane $\bar E$ generated by E is a projective plane, then (P, $\mathfrak{L}$ ) is a projective space (cf. 5.6). Or more generally: If for any subset M of P the restriction (M, $\mathfrak{L}$ (M)) is locally complete, and if for any two distinct coplanar lines G1, G2 ∈ $\mathfrak{L}$ (M) the lines $\bar G_1 ,\bar G_2 \varepsilon \mathfrak{L}$ generated by G1, G2 have a nonempty intersection and $\overline {G_1 \cup {\text{ }}G_2 }$ satisfies the exchange condition, then (P, $\mathfrak{L}$ ) is a generalized projective space. |
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Keywords: | projective spaces embeddings planes of linear spaces |
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