Mixed partitions of PG(3,\(q\))
type: Item
instance of: scholarly article
title: Mixed partitions of PG(3,\(q\))
zbMATH Open document ID: 0981.51013
DOI: 10.1007/BF01221060
published in: Journal of Geometry
publication date: 2002-03-18T00:00:00Z
review text: The similarities between finite and continuous geometries are both strong and deep. For instance, in each case, we may ask whether a space can be partitioned into subsets of a certain type. Euclidean 3-space can be partitioned into paraboloids, but not into spheres; in the same way, it may or may not be possible to partition a finite geometry into curves of a specified type. This paper studies mixed partitions, in which a space is partitioned into two types of elements. We may compare to this the division of \(\mathbb{R}^2\) into one point and a family of concentric circles. The author starts with a twisted cubic \(C\) in \(\text{PG}(3,q)\); using results from the literature, he constructs a lifting from a linear collineation of \(\text{PG}(1,q)\) to one of \(\text{PG}(3,q)\) that leaves \(C\) invariant. The lifted collineation generates a group \(G\) acting on \(\text{PG}(3,q)\). Two orbits of \(G\) are lines; and the original cubic \(C\) is of course also an orbit of \(G\). Bootstrapping from the structure of \(C\), the remaining \(q^2-2\) orbits are shown first to be \((q+1)\)-arcs, then specifically to be twisted cubics; this is the main result of the paper. In other results, \(\text{PG}(3,q)\) is partitioned into two lines and \(q^2-1\) conics, and into two lines and \(q-1\) hyperbolic quadrics. An additional partition is given for even \(q\), and for projective spaces of dimension \(n=5,7,9,\dots\). In the latter case, the partition is into normal rational curves and subspaces mutually intersecting in a subspace of dimension \(n-4\). (Considering the empty set as a subspace of dimension \(-1\), the main theorem fits this pattern too.) It is unfortunate that there are several misprints and infelicities in this short paper. Most are easily corrected, a few less so. In the second-last sentence of the proof of Prop. 4.2, the second eigenvalue of \(T^i\) should surely be \((2q+1)i\), not \((2q+1)qi\). In the proof of Prop. 4.3, it seems unnecessary to give a point …
reviewed by: Robert Dawson
Mathematics Subject Classification ID: 51E23
zbMATH DE Number: 1507994
zbMATH Keywords: projective Galois spaces
zbMATH Keywords: mixed partitions
zbMATH Keywords: cubics
author: Antonio Cossidente
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1507994