On regular parallelisms in PG(3,q)

✍ Scribed by G. Lunardon

• Publisher: Elsevier Science
• Year: 1984
• Tongue: English
• Weight: 868 KB
• Volume: 51
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(84)90003-7

✦ Synopsis

Inaction A prrrallelism of S3 = PG(3., 4) is a set 9 of 4*+ 4 + 1 spreads such that, if $I and sz are two distinct spreads of 9, then & and s2 do not have a common line. If all spreads of 9 are regular we say that g is a regular parallelism of SJ. In [3] Bruck studied a amstruction of a projective plane of order 4* +4 using a set of parallelisms of S3. When 4>2 the only known parallelisms are tbme of Denniston [8] and Beutelspacher [l]; these parallelisms have one regular spread and the other 4*+4 subregular of index 1. Thus, with these parallelisms we cannot construct a set of parallelisms with the properties required by Bruclk in [3_l, because when 4 > 2 there is no parallelism of S, which contains two regular spreads.

In this paper we prove that IFor 4 odd there are no regular parallelisms of S3. 2.1. A partial t-spread of S, == PG(d, 4) is a set 9' of t-dimensional subspaces of S, such that any point of S, is incident at most with one element of 3'. The set 9' is a t-spread if each point of S, is incident with exactly one element of 9. Following Dembowski [7, p. 2201 a t-regulus in S2t+l is a partial t-spread $R having 4 +l elements with the properties that if a line meets three distinct elements of 9R in a point, then this line meets each member of 9; such a line is called a tranmersai line of St. A 1-regulus is one ruling of a nondegenerated hyperbolic quadric in S,. A t-spread of S 2t+l is said to be regdar if for any three distinct elements of 9 the t-regulus determined by these three t-dimensicqal subspaces is contained in 3'. 2.2. Let S, be embedded as a subgeometry in ST =PG(3,4*). Let cr be the unique involutorial collineation of Sg whilch f&s every point of Ss.