Mixed partitions of PG(3,\(q\)) (Q1580997)

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 with coordinates \((x_1,x_2,x_3,x_4)\) the name \(z\) (!), by which it is never referred to again except to set it equal to \(x+y\) (!!) two lines later. Propositions 4.2 and 4.3 require some unravelling. The main point of Prop. 4.2 is that orbits outside the two lines \(l'\) and \(l''\) have length \(q+1\); and the full strength of this is needed in Prop. 4.3. Unfortunately, the author phrases Prop. 4.2 in such a way that, taken literally, it applies only when \(q+1\) is not divisible by 3, in order to make the (ultimately unused) observation that in this case all orbits have this length. The statement of Prop. 4.3 (``Each orbit \dots other than the two lines \dots '') then compounds the confusion by inadvertently implying that the lines \(l'\) and \(l''\) are orbits even when \(3\mid q+1\), which is not the case.