Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-16T22:23:07.908Z Has data issue: false hasContentIssue false
  • English
  • Français

A transitive self-polar double-twenty of planes

Part of: Geometry Linear incidence geometry

Published online by Cambridge University Press:  09 April 2009

P. B. Kirkpatrick
P. B. Kirkpatrick
Affiliation:
School of Mathematics and Statistics University of Sydney, NSW 2006, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We demonstrate the existence, in the 5-dimensional projective space over any field J in which 1 + 1 ≠ 0 and −1 is a square, of a non-degenerate double-twenty of planes (ℋ, K) with the property that there is a group of collineations which acts transitively on ℋ ∪ K while each element of the group either maps ℋ onto itself and K onto itself or else swaps ℋ with K. If there is an involutory automorphism of J which swaps the two square roots of −1, then (ℋ, K) is also self-polar (with respect to a unitary polarity). We describe some of the geometry (in both 5-dimensional and 3-dimensional space) associated with the double-twenty (ℋ, K) and its group.

MSC classification

Secondary: 51A45: Incidence structures imbeddable into projective geometries
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 61 , Issue 2 , October 1996 , pp. 249 - 257
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Dembowski, P., Finite geometries, Ergebnisse Math. 44 (Springer, Berlin, 1968).Google Scholar
[2]Hudson, R. W. H. T., Kummer's quartic surface (Cambridge University Press, Cambridge, 1905). Reprinted, 1990.Google Scholar
[3]Kirkpatrick, P., ‘Self-polar double-N's defined by certain paris of normal rational curves’, J. Aust. Math. Soc. 6 (1966), 210–220.Google Scholar
[4]Kirkpatrick, P., ‘A construction for a self-polar double-N associated with a pair of normal rational curves’, J. Aust. Math. Soc. 8 (1968), 415–422.Google Scholar
[5]Room, T. G., The geometry of determinantal loci (Cambridge University Press, Cambridge, 1938).Google Scholar
[6]Room, T. G., ‘Self-polar double configurations, I and II’, J. Aust. Math. Soc. 5 (1965), 65–68, 69–75.Google Scholar
[7]Scott, W. R., Group theory (Prentice-Hall, Englewood Cliffs, 1964).Google Scholar