Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial Articles (Project Euclid)
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The latest articles from Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2019 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Tue, 26 Feb 2019 13:46 ESTTue, 26 Feb 2019 13:46 ESThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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The exterior splash in $\mathrm{PG}(6, q)$: transversals
https://projecteuclid.org/euclid.iig/1551206772
<strong>Susan G. Barwick</strong>, <strong>Wen-Ai Jackson</strong>. <p><strong>Source: </strong>Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial, Volume 17, Number 1, 1--24.</p><p><strong>Abstract:</strong><br/>
Let [math] be an order- [math] -subplane of [math] that is exterior to [math] . Then the exterior splash of [math] is the set of [math] points on [math] that lie on an extended line of [math] . Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry [math] , and hyper-reguli in [math] . We use the Bruck–Bose representation in [math] to investigate the structure of [math] , and the interaction between [math] and its exterior splash. We show that the point set of [math] corresponding to [math] is the intersection of nine quadrics, and that there is a unique tangent plane at each point, namely the intersection of the tangent spaces of the nine quadrics. In [math] , an exterior splash [math] has two sets of cover planes (which are hyper-reguli) and we show that each set has three unique transversal lines in the cubic extension [math] . These transversal lines are used to characterise the carriers and the sublines of [math] .
</p>projecteuclid.org/euclid.iig/1551206772_20190226134618Tue, 26 Feb 2019 13:46 ESTRuled quintic surfaces in $\mathrm{PG}(6,q)$
https://projecteuclid.org/euclid.iig/1551206773
<strong>Susan G. Barwick</strong>. <p><strong>Source: </strong>Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial, Volume 17, Number 1, 25--41.</p><p><strong>Abstract:</strong><br/>
We look at a scroll of [math] that uses a projectivity to rule a conic and a twisted cubic. We show this scroll is a ruled quintic surface [math] , and study its geometric properties. The motivation in studying this scroll lies in its relationship with an [math] -subplane of [math] via the Bruck–Bose representation.
</p>projecteuclid.org/euclid.iig/1551206773_20190226134618Tue, 26 Feb 2019 13:46 ESTA characterization of Clifford parallelism by automorphisms
https://projecteuclid.org/euclid.iig/1551206774
<strong>Rainer Löwen</strong>. <p><strong>Source: </strong>Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial, Volume 17, Number 1, 43--46.</p><p><strong>Abstract:</strong><br/>
Betten and Riesinger have shown that Clifford parallelism on real projective space is the only topological parallelism that is left invariant by a group of dimension at least 5. We improve the bound to 4. Examples of different parallelisms admitting a group of dimension [math] are known, so 3 is the “critical dimension”.
</p>projecteuclid.org/euclid.iig/1551206774_20190226134618Tue, 26 Feb 2019 13:46 ESTGeneralized quadrangles, Laguerre planes and shift planes of odd order
https://projecteuclid.org/euclid.iig/1551206775
<strong>Günter F. Steinke</strong>, <strong>Markus Stroppel</strong>. <p><strong>Source: </strong>Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial, Volume 17, Number 1, 47--52.</p><p><strong>Abstract:</strong><br/>
We characterize the Miquelian Laguerre planes, and thus the classical orthogonal generalized quadrangles [math] , of odd order [math] by the existence of shift groups in affine derivations.
</p>projecteuclid.org/euclid.iig/1551206775_20190226134618Tue, 26 Feb 2019 13:46 ESTA new family of $2$-dimensional Laguerre planes that admit $\mathrm{PSL}_2(\mathbb R) \times\mathbb R$ as a group of automorphisms
https://projecteuclid.org/euclid.iig/1551206776
<strong>Günter F. Steinke</strong>. <p><strong>Source: </strong>Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial, Volume 17, Number 1, 53--75.</p><p><strong>Abstract:</strong><br/>
We construct a new family of [math] -dimensional Laguerre planes that differ from the classical real Laguerre plane only in the circles that meet a given circle in precisely two points. These planes share many properties with but are nonisomorphic to certain semiclassical Laguerre planes pasted along a circle in that they admit [math] -dimensional groups of automorphisms that contain [math] and are of Kleinewillinghöfer type I.G.1.
</p>projecteuclid.org/euclid.iig/1551206776_20190226134618Tue, 26 Feb 2019 13:46 ESTRegular pseudo-hyperovals and regular pseudo-ovals in even characteristichttps://projecteuclid.org/euclid.iig/1559700152<strong>Joseph A. Thas</strong>. <p><strong>Source: </strong>Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial, Volume 17, Number 2, 77--84.</p><p><strong>Abstract:</strong><br/>
S. Rottey and G. Van de Voorde characterized regular pseudo-ovals of [math] , [math] , [math] and [math] prime. Here an alternative proof is given and slightly stronger results are obtained.
</p>projecteuclid.org/euclid.iig/1559700152_20190604220235Tue, 04 Jun 2019 22:02 EDTConics in Baer subplaneshttps://projecteuclid.org/euclid.iig/1559700153<strong>Susan G. Barwick</strong>, <strong>Wen-Ai Jackson</strong>, <strong>Peter Wild</strong>. <p><strong>Source: </strong>Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial, Volume 17, Number 2, 85--107.</p><p><strong>Abstract:</strong><br/>
This article studies conics and subconics of [math] and their representation in the André/Bruck–Bose setting in [math] . In particular, we investigate their relationship with the transversal lines of the regular spread. The main result is to show that a conic in a tangent Baer subplane of [math] corresponds in [math] to a normal rational curve that meets the transversal lines of the regular spread. Conversely, every 3- and 4-dimensional normal rational curve in [math] that meets the transversal lines of the regular spread corresponds to a conic in a tangent Baer subplane of [math] .
</p>projecteuclid.org/euclid.iig/1559700153_20190604220235Tue, 04 Jun 2019 22:02 EDTOn triples of ideal chambers in $A_2$-buildingshttps://projecteuclid.org/euclid.iig/1559700154<strong>Anne Parreau</strong>. <p><strong>Source: </strong>Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial, Volume 17, Number 2, 109--140.</p><p><strong>Abstract:</strong><br/>
We investigate the geometry in a real Euclidean building [math] of type [math] of some simple configurations in the associated projective plane at infinity [math] , seen as ideal configurations in [math] , and relate it with the projective invariants (from the cross ratio on [math] ). In particular we establish a geometric classification of generic triples of ideal chambers of [math] and relate it with the triple ratio of triples of flags.
</p>projecteuclid.org/euclid.iig/1559700154_20190604220235Tue, 04 Jun 2019 22:02 EDTOpposition diagrams for automorphisms of small spherical buildingshttps://projecteuclid.org/euclid.iig/1559700155<strong>James Parkinson</strong>, <strong>Hendrik Van Maldeghem</strong>. <p><strong>Source: </strong>Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial, Volume 17, Number 2, 141--188.</p><p><strong>Abstract:</strong><br/>
An automorphism [math] of a spherical building [math] is called capped if it satisfies the following property: if there exist both type [math] and [math] simplices of [math] mapped onto opposite simplices by [math] then there exists a type [math] simplex of [math] mapped onto an opposite simplex by [math] . In previous work we showed that if [math] is a thick irreducible spherical building of rank at least [math] with no Fano plane residues then every automorphism of [math] is capped. In the present work we consider the spherical buildings with Fano plane residues (the small buildings ). We show that uncapped automorphisms exist in these buildings and develop an enhanced notion of “opposition diagrams” to capture the structure of these automorphisms. Moreover we provide applications to the theory of “domesticity” in spherical buildings, including the complete classification of domestic automorphisms of small buildings of types [math] and [math] .
</p>projecteuclid.org/euclid.iig/1559700155_20190604220235Tue, 04 Jun 2019 22:02 EDT