A new characterization of projections of quadrics in finite projective spaces of even characteristic |
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Authors: | F De Clerck N De Feyter |
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Institution: | aDepartment of Mathematics, Ghent University, Krijgslaan 281 - S22, B-9000 Gent, Belgium |
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Abstract: | We will classify, up to linear representations, all geometries fully embedded in an affine space with the property that for every antiflag {p,L} of the geometry there are either 0, α, or q lines through p intersecting L. An example of such a geometry with α=2 is the following well known geometry . Let Qn+1 be a nonsingular quadric in a finite projective space , n≥3, q even. We project Qn+1 from a point r∉Qn+1, distinct from its nucleus if n+1 is even, on a hyperplane not through r. This yields a partial linear space whose points are the points p of , such that the line 〈p,r〉 is a secant to Qn+1, and whose lines are the lines of which contain q such points. This geometry is fully embedded in an affine subspace of and satisfies the antiflag property mentioned. As a result of our classification theorem we will give a new characterization theorem of this geometry. |
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Keywords: | Affine partial linear spaces Projections of quadrics Antiflag types |
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