´╗┐

Spreads in PG(3,q) admitting several homology groups of order q+1


Abstract


The set of translation planes with spreads in PG(3,q) admitting at least three homology groups with distinct axes of order q+1 is completely determined. Apart from the Desarguesian and Hall planes of order q<sup>2</sup>, the only possible plane is the Heimbeck plane of order 7<sup>2</sup> admitting several quaternion homology groups of order 8. A classification is also given of all translation planes with spreads in PG(3,q) that admit at least three distinct homology groups of order q+1. Recent results conneting translation planes with spreads in PG(3,q) admitting cyclic affine homology groups of order q+1 with conical flocks spreads provide the background for applications showing how the associated collineation groups are interrelated.

DOI Code: 10.1285/i15900932v24n2p9

Keywords: Homology groups; Translation planes; Spreads; Flocks of quadratic cones; Hyperbolic fibrations

Classification: 51E23; 51A40

Full Text: PDF


Creative Commons License
This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.