Showing papers by "Joseph A. Thas published in 2005"

Some Characterizations of Finite Hermitian Veroneseans

Abstract: We characterize the finite Veronesean $${\cal H}_n \, \subseteq \, PG(n(n+2),q)$$ of all Hermitian varieties of PG(n,q2) as the unique representation of PG(n,q2) in PG(d,q), d? n(n+2), where points and lines of PG(n,q2) are represented by points and ovoids of solids, respectively, of PG(d,q), with the only condition that the point set of PG(d,q) corresponding to the point set of PG(n,q2) generates PG(d,q). Using this result for n=2, we show that $${\cal H}_2 \, \subseteq \, PG(8,q)$$ is characterized by the following properties: (1) $$|{\cal H}_2|=q^4+q^2+1$$ ; (2) each hyperplane of PG(8,q) meets $${\cal H}_2$$ in q2+1, q3+1 or q3+q2+1 points; (3) each solid of PG(8,q) having at least q+3 points in common with $${\cal H}_2$$ shares exactly q2+1 points with it.

Trialities and 1-systems of Q + (7, q )

Abstract: The image of a 1-system of Q+ (7, q) under a triality of the D4-geometry, attached to Q+ (7, q), will be investigated. Attention will mainly be paid to the case of a locally hermitian, semiclassical 1-system of a Q(6, q), embedded in Q+(7, q). It is found that its image under a triality is always locally hermitian and semiclassical as well. Moreover, it is a proper 1-system of Q+(7, q) whenever the original 1-system of Q(6, q) is not a spread of some generalized hexagon H(q) on Q(6, q). Finally, some results concerning isomorphisms will be obtained.

On Ferri's characterization of the finite quadric

Abstract: WegeneralizeandcompleteFerri’scharacterizationofthefinitequadricVeronesean V 42 byshowingthat Ferri’s assumptions also characterize the quadric Veroneseans in spaces of even characteristic.© 2004 Elsevier Inc. All rights reserved. MSC: 51E20; 51B99; 51E25Keywords: Veronesean cap; Quadric Veronesean 1. IntroductionLet q be a fixed prime power. For any integer k, denote by PG (k,q) the k-dimensionalprojectivespaceoverthefinite(Galois)field GF (q) of q elements.WechoosecoordinatesinPG ( 2 ,q) and in PG ( 5 ,q) . The Veronesean map maps a point of PG ( 2 ,q) with coordinates (x 0 ,x 1 ,x 2 ) onto the point of PG ( 5 ,q) with coordinates (x 20 ,x 21 ,x 22 ,x 0 x 1 ,x 0 x 2 ,x 1 x 2 ). The quadricVeronesean V 42 is the image of the Veronesean map. The set V 42 is a cap ofPG ( 5 ,q) and has a lot of other nice geometric and combinatorial properties, summarized in[2].Wealsoreferto[2]forcharacterizationsofthiscap,sometimescalleda Veroneseancap .In particular, there exists a characterization of

On Semi-Pseudo-Ovoids

Abstract: In this paper we introduce semi-pseudo-ovoids, as generalizations of the semi-ovals and semi-ovoids. Examples of these objects are particular classes of SPG-reguli and some classes of m-systems of polar spaces. As an application it is proved that the axioms of pseudo-ovoid O(n,2n,q) in PG(4n?1,q) can be considerably weakened and further a useful and elegant characterization of SPG-reguli with the polar property is given.