2005
DOI: 10.1515/advg.2005.5.2.279
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The embedding of (0, 2)-geometries and semipartial geometries in AG(n, q)

Abstract: A ð0; aÞ-geometry, a d 3, fully embedded in AGðn; qÞ is always a linear representation [2]. In [4] the ð0; 2Þ-geometries fully embedded in AGð3; qÞ are classified up to linear representations. Here we extend this result to full embeddings in AGðn; qÞ. As a corollary we classify the semipartial geometries with a > 1 fully embedded in AGðn; qÞ, but not as a linear representation.

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Cited by 7 publications
(3 citation statements)
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“…Hence we classify the (0, 2)-geometries embedded in AG (3, q). We use this classification in [5] to prove by an induction argument that if S is a (0, 2)-geometry embedded in AG(n, q), q = 2, then S is a linear representation of a set K ∞ in π ∞ , or S = I(n, q, e), or n = 2 and the lines of S together with π ∞ form a dual hyperoval, or n = 3 and S = A(O ∞ ), or n = 4 and S = TQ(4, q). As a corollary it then follows that if S is a semipartial geometry with α > 1 embedded in AG(n, q), q = 2, then S is a linear representation of a set K ∞ in ∞ , n = 2 and the lines of S together with π ∞ form a dual hyperoval, or n = 4 and S = TQ(4, q).…”
Section: Proposition 22 ([4]mentioning
confidence: 99%
“…Hence we classify the (0, 2)-geometries embedded in AG (3, q). We use this classification in [5] to prove by an induction argument that if S is a (0, 2)-geometry embedded in AG(n, q), q = 2, then S is a linear representation of a set K ∞ in π ∞ , or S = I(n, q, e), or n = 2 and the lines of S together with π ∞ form a dual hyperoval, or n = 3 and S = A(O ∞ ), or n = 4 and S = TQ(4, q). As a corollary it then follows that if S is a semipartial geometry with α > 1 embedded in AG(n, q), q = 2, then S is a linear representation of a set K ∞ in ∞ , n = 2 and the lines of S together with π ∞ form a dual hyperoval, or n = 4 and S = TQ(4, q).…”
Section: Proposition 22 ([4]mentioning
confidence: 99%
“…So the results in [7], in this paper and in [6] add up to the complete classification of (0, 2)-geometries embedded in AG(3, q), q = 2 h , h > 1, which have at least one plane of type IV (those without planes of type IV were already classified in Theorem 1.3). This classification for AG (3, q) is then used in [8] to obtain the complete classification of (0, 2)-geometries embedded in AG(n, q), q = 2 h , h > 1, which have at least one plane of type IV. As a corollary of the classification of (0, 2)-geometries embedded in AG(n, q), q = 2 h , h > 1, and of Theorem 1.3 we obtain the following result.…”
Section: Theorem 13 (De Clerck and Delanotementioning
confidence: 99%
“…As a corollary of the classification of (0, 2)-geometries embedded in AG(n, q), q = 2 h , h > 1, and of Theorem 1.3 we obtain the following result. [8]). Let S be a semipartial geometry spg(s, t, , ), > 1, embedded in AG(n, q), q > 2.…”
Section: Theorem 13 (De Clerck and Delanotementioning
confidence: 99%