Tubes of even order and flat ?.C 2 geometries

Cameron, Peter J‎.; Ghinelli, Dina

doi:10.1007/bf01266318

Cited by 4 publications

(6 citation statements)

References 7 publications

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“…We denote it SY , after Shult and Yanushka [30] Shult and Yanushka [30] described SY as an affine expansion (although they did not define affine expansions explicitly in their paper); they also noted that all boxes of SY on firm quadrangles of SY are closed, but they did not seem to be aware that SY is a quotient of H (12,3). This fact is noted in Pasechnik [25].…”

Section: Therefore Proposition 69 Modulo Strong Isomorphism the Nmentioning

confidence: 96%

“…(3,2). The following C 2 .c geometry is the smallest member of a family of flat C 2 .A f -geometries constructed in [22] (see also [12]). Given a three-dimensional subspace S of PG(4, 2) and a spread L in it, choose a line L ∈ L. We take as 'quads' the 3-subspaces of PG (4,2) containing L but distinct from S; as 'lines' we take the points of PG(4, 2) not belonging to S; as 'points' we take the planes X of PG(4, 2) with X ∩ S a line of L different from L. All 'points' are declared to be incident to all 'quads' whereas the incidence relation between 'lines' and 'points' or 'quads' is inherited from PG (4,2).…”

Section: Flat C 2 C-geometriesmentioning

confidence: 99%

“…REMARK. It is well known that the points, pairs, triples and quadruples of points of the Steiner system S (12,6,5) together with the hexads of S(12, 6, 5) form a 3-point extension S of AG (2,3), namely a matroid with diagram and orders as follows:…”

Section: Therefore Proposition 69 Modulo Strong Isomorphism the Nmentioning

confidence: 99%

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Parallelism and Cubes inC2.c-Geometries

Pasini

1998

European Journal of Combinatorics

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We characterize C 2 .c-geometries that are truncations of almost-thin C n -geometries and C 2 .c-geometries covered by truncated almost-thin buildings of type C n . Then we show how to profit from those characterizations in the investigation of a number of special cases. The proof of our main theorem is a rearrangement of the proof of a theorem by Brouwer on rectagraphs. A generalization of Brouwer's theorem is also given.

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Section: Therefore Proposition 69 Modulo Strong Isomorphism the Nmentioning

confidence: 96%

Section: Flat C 2 C-geometriesmentioning

confidence: 99%

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Parallelism and Cubes inC2.c-Geometries

Pasini

1998

European Journal of Combinatorics

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“…(Perhaps it coincides with the ( q ϩ 1) / 2-fold cover ofq described in [3] . ) [22] (see also [5]) , with O a classical hyperoval of PG (2 , q ) , q any power of 2 . That extension is flat ; namely , all of its points are incident with all its planes (whence ( LL ) fails to hold in it) .…”

Section: Examplementioning

confidence: 99%

A New Family of Extended Generalized Quadrangles

Fra

Ṗasechnik²,

Pasini

1997

European Journal of Combinatorics

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For every hyperoval O of PG (2 , q ) ( q even) , we construct an extended generalized quadrangle with point-residues isomorphic to the generalized quadrangle T * 2 ( O ) of order ( q Ϫ 1 , q ϩ 1) . These extended generalized quadrangles are flag-transitive only when q ϭ 2 or 4 . When q ϭ 2 we obtain a thin-lined polar space with four planes on every line . When q ϭ 4 we obtain one of the geometries discovered by Yoshiara [28] . That geometry is produced in [28] as a quotient of another one , which is simply connected , constructed in [28] by amalgamation of parabolics . In this paper we also give a 'topological' construction of that simply connected geometry .÷ 1997 Academic Press Limited 1 . I NTRODUCTION An extended generalized quadrangle (also called c . C 2 geometry ) is a (connected) geometry belonging to the following Beukenhout diagram :The orders s and t are assumed to be finite . Given a finite generalized quadrangle Q and an extended generalized quadrangle ⌫ , if all point-residues of ⌫ are isomorphic to Q then we say that ⌫ is an extension of Q .In the context of extended generalized quadrangles , the following property is equivalent to the Intersection Property ([21 , Lemma 7 . 25]) :( LL ) no two distinct lines are incident with the same points . (We warn the reader that some authors use the name 'extended generalized quadrangle' only for c .C 2 geometries satisfying ( LL ) . This convection is adopted in [6] , for instance . )We follow the notation of [24 , chapter 3] for finite generalized quadrangles , but we take the liberty of writing T * 2 ( O ) q instead of T * 2 ( O ) , in order to remind ourselves of the order q of the projective plane to which the hyperoval O belongs . (We recall that , given a hyperoval O of the plane at infinity of AG (3 , q ) , q even , the generalized quadrangle T * 2 ( O ) q consists of the points of AG (3 , q ) and of the lines of AG (3 , q ) , with the point at infinity belonging to O . The orders of T * 2 ( O ) q are q Ϫ 1 and q ϩ 1 .More properties of T * 2 ( O ) q will be recalled in Section 1 . 2 . 1 . ) In Section 1 . 1 we briefly survey the known examples of extended generalized quadrangles .In Section 2 , for every hyperoval O of PG (2 , q ) ( q ϭ 2 n , n any positive integer) we construct an extension ⌫ q of T * 2 ( O ) q satisfying ( LL ) . Our construction is entirely geometric . As we will explain in Section 2 . 5 , it imitates an idea of [17] and [19] , which goes back to Buekenhout and Hubaut [2] .As there are dif ferent types of hyperovals in PG (2 , q ) when q Ͼ 4 , the isomorphism type of ⌫ q depends on the type of O . But it also depends on the choice of a line l ϱ of † In memory of Giuseppe Tallini . 155

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“…This paper deals with a generalization of combinatorial structures living in a threedimensional projective space over a Galois field called tubes which were introduced in [1] in connection with a construction of a class of geometries.…”

Section: Introductionmentioning

confidence: 99%

Tubes in three–dimensional projective spaces

Ballico

Cossidente

2007

J. geom.

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A partial tube in PG(3, q) is a pair T = {L, L}, where {L} ∪ L is a collection of mutually disjoint lines of PG(3, q) with the property that for each plane π of PG(3, q) through L, the intersection of π with the lines of L is an arc. Here, we generalize the notion of partial tube allowing the ground field to be any algebraically closed field. To a generalized partial tube we will associate an irreducible surface of degree d in P 3 K providing upper bounds on d. (2000): 51E20. Mathematics Subject Classification

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Tubes of even order and flat ?.C 2 geometries

Cited by 4 publications

References 7 publications

Parallelism and Cubes inC2.c-Geometries

Parallelism and Cubes inC2.c-Geometries

A New Family of Extended Generalized Quadrangles

Tubes in three–dimensional projective spaces

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