NEW ZEALAND JOURNAL OF MATHEMATICS
Volume 44 (2014), 1–5
MAXIMAL PARTIAL LINE SPREADS WITH NEW SIZES IN
P G(4, q)
Sandro Rajola
(Received January 31, 2012)
Abstract. Maximal partial line spreads of P G(4, q) have been investigated by
several authors, but little is known about them. We know very few examples
and the research is still in progress.
In this work we construct a class of maximal partial line spreads of P G(4, q)
with new cardinalities.
1. Introduction
A partial line spread in P G(n, q), finite projective space of dimension n and
order q, is a set of pairwise skew lines. A maximal partial line spread in P G(n, q)
is a partial line spread of this space which cannot be extended to a larger partial
line spread. Throughout this paper, referring to the above line sets, we will omit
the word line and simply call such structures partial spread and maximal partial
spread.
For P G(4, q), simple arithmetical considerations show that a spread, that is a set
of pairwise skew lines covering the space, cannot exist. So spreads are not object
of investigation in P G(4, q).
In P G(r, q), with r greater than four, we know a lot about the spectrum of sizes of
maximal partial spreads (see [4]), while in dimensions three and four the knowledge
of this spectrum is an open problem. So the study of maximal partial spreads in
P G(3, q) and P G(4, q) intrigues many researchers.
Maximal partial spreads in P G(4, q) have been studied by several authors, but little
is known about them and we know only what follows.
By A. Beutelspacher (see [1]) the smallest examples are just the spreads in hyper√ √
planes, of size q 2 +1, and for the interval [q 2 +1, q 2 +q q − q] we know what values
can occur and what the examples are (see also [3]). Beutelspacher also determined
the largest examples, which are of size q 3 + 1.
Besides this, all we have is the density result by J. Eisfeld, L. Storme and P. Sziklai,
given by the interval [q 3 − q + 3, q 3 + 1] (see [2]).
For the previous known results, see also [4] and [5].
In this paper we show a geometric construction of a maximal partial spread in
P G(4, q) of size q 2 + (m + 1)q + 1, for every integer m = 1, . . . , q − 1. By this new
cardinalities follow.
2010 Mathematics Subject Classification 51E23.
Key words and phrases: Maximal partial spreads.
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SANDRO RAJOLA
2. New Examples of Maximal Partial Spreads in P G(4, q)
The aim of this section is to construct examples of maximal partial spreads of
P G(4, q) with new cardinalities.
To do this, let S3 be a hyperplane of P G(4, q), that is a subspace of P G(4, q) of
dimension three.
Let F be a spread of S3 containing a regulus R0 of a hyperbolic quadric I.
This spread does exist, as well known.
Denote by R the regulus of I opposite to R0 . Let r be a line of R, let R1 , R2 , . . . , Rq+1
be the points of r and let r1 , r2 , . . . , rm be m lines of R distinct from r, with
1 ≤ m ≤ q − 1.
Now, let I be the following point set:
I=
m
[
rj .
j=1
Let X be a point of P G(4, q) not in S3 , and π the plane spanned by X and r.
Evidently, we have π ∩ S3 = r.
For every index i = 1, . . . , q + 1, denote by si the line of π through the points X
and Ri and by li the line of R0 through Ri .
Now, for every index i = 1, . . . , q + 1, and with the positions sq+2 = s1 , Rq+2 =
R1 , let F(I, li , si+1 ) be the line set, of size m and depending on I, li and si+1 ,
constructed as follows:
a): each line of F(I, li , si+1 ) meets the sets li ∩ I and si+1 − {Ri+1 , X},
b): distinct lines of F(I, li , si+1 ) meet li ∩I at distinct points and si+1 −{Ri+1 , X}
at distinct points,
c): the lines of F(I, li , si+1 ) cover the set li ∩ I.
Obviously, we have li ∩ si+1 = ∅. By this and by b) it follows that F(I, li , si+1 ) is
a set of mutually skew lines.
By a), b) and c) it follows that there are several ways to construct F(I, li , si+1 ).
Now, for every index i = 1, . . . , q + 1, choose a construction of F(I, li , si+1 ).
Let F 0 be the following line set:
F0 =
q+1
[
F(I, li , si+1 ).
i=1
Let us prove the following theorem.
Theorem 2.1. The line set F 0 is a partial spread of P G(4, q) of size m(q + 1).
Proof. The line set F 0 has size m(q + 1), since each addend of the union F 0 has
size m and distinct addends have no lines in common. Let us prove that the lines
of F 0 are mutually skew. In order to do this, let t1 and t2 be two distinct lines of
F 0.
If t1 and t2 both belong to the same F(I, li , si+1 ), we have t1 ∩ t2 = ∅, as already
said.
So, let us consider the case t1 ∈ F(I, li1 , si1 +1 ), t2 ∈ F(I, li2 , si2 + 1), with i1 6= i2 .
Suppose t1 ∩ t2 6= ∅. By i1 6= i2 , we get li1 6= li2 and si1 +1 6= si2 +1 . Let T1 , T10 ,
MAXIMAL PARTIAL LINE SPREADS
3
T2 and T20 be the following points:
T1 = t1 ∩ li1 ,
T10 = t1 ∩ si1 +1 ,
T2 = t2 ∩ li2 ,
T20 = t2 ∩ si2 +1 .
By the construction of F(I, li , si+1 ) we have T1 6= Ri1 , T10 6= Ri1 +1 , T10 6= X, T2 6=
Ri2 , T20 6= Ri2 +1 and T20 6= X.
The points T1 , T10 , T2 and T20 are all distinct and three by three not collinear, as it
is easy to verify.
Denote by u the line through T1 and T2 and by u0 the line through T10 and T20 .
The distinct lines u and u0 belong to the same plane, since t1 ∩t2 6= ∅, and therefore
they meet at a point Z.
We get Z ∈ π, since Z ∈ u0 ⊂ π, and Z ∈ S3 , since Z ∈ u ⊂ S3 . Thus we have
Z ∈ π ∩ S3 = r, with Z 6= Ri1 and Z 6= Ri2 . It follows that u and r meet at the
point Z. So the lines li1 and li2 both lie on the plane spanned by the lines u and r.
It follows that li1 and li2 meet: a contradiction, since li1 ∩ li2 = ∅.
The contradiction proves that t1 ∩ t2 = ∅. So the lines of F 0 are mutually skew
and F 0 is a partial spread of P G(4, q).
So the theorem is proved.
For every index i = 1, . . . , q + 1, let αi be the plane spanned by the lines li and
si and let bi be a line of αi through Ri and distinct from li and si . Evidently, we
have αi ∩ π = si , αi ∩ S3 = li , bi ∩ π = bi ∩ S3 = {Ri }.
It is easy to verify that αi1 ∩ αi2 = {X} if i1 6= i2 .
By this and by the fact that X ∈
/ bi for every index i = 1, . . . , q + 1, it follows that
the line set
F 00 = {bi }i=1,...,q+1
is a partial spread of P G(4, q). Obviously we have
|F 00 | = q + 1.
(1)
Now let S be the line set formed by the lines of R distinct from r, r1 , . . . , rm .
Obviously we have
|S| = q − m.
(2)
Let F be the following line set:
F = F 0 ∪ F 00 ∪ S.
Let us prove the following theorem.
Theorem 2.2. The line set F is a partial spread of P G(4, q) of size q(m + 2) + 1.
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SANDRO RAJOLA
Proof. Let us prove that each line of F 0 is skew with each line of F 00 .
In order to do this, let t be a line of F 0 and let bi be a line of F 00 , with 1 ≤ i ≤ q + 1,
such that t ∩ bi 6= ∅.
By
q+1
[
t ∈ F0 =
F(I, li , si+1 )
i=1
it follows that t ∈ F(I, li∗ , si∗ +1 ), with 1 ≤ i∗ ≤ q + 1.
By this we get t ∩ li∗ = T , t ∩ si∗ +1 = T 0 , with T 6= Ri∗ , T 0 6= X, T 0 6= Ri∗ +1 .
By t ∩ bi 6= ∅ we immediately get αi 6= αi∗ and li 6= li∗ .
The line t meets bi , and therefore the plane αi , at a point Y such that Y ∈
/ π and
Y ∈
/ S3 , as it is easy to verify.
So T 0 and Y are two distinct points not in S3 .
Now let Se3 be the 3-dimensional space spanned by the planes π and αi .
By T 0 ∈ Se3 (T 0 ∈ π ⊂ Se3 ) and by Y ∈ Se3 (Y ∈ αi ⊂ Se3 ), it follows that the line t,
which contains the distinct points T 0 and Y , is contained in Se3 . By t ⊂ Se3 we get
T ∈ Se3 . By T ∈ li∗ − {Ri∗ }, T ∈ Se3 and Ri∗ ∈ Se3 (Ri∗ ∈ π ⊂ Se3 ) we get li∗ ⊂ Se3 .
Furthermore, we have li ⊂ Se3 (li ⊂ αi ⊂ Se3 ).
So Se3 contains the two skew lines li∗ and li of S3 and therefore we get Se3 = S3 : a
contradiction, since X ∈ Se3 (X ∈ π ⊂ Se3 ) and X ∈
/ S3 . The contradiction proves
that each line of F 0 is skew with each line of F 00 .
Trivially, each line of F 0 is skew with each line of S, and each line of F 00 is skew
with each line of S, too.
By these conclusions about F 0 , F 00 , S, and by theorem 1 it follows that F is a
partial spread of P G(4, q). Moreover, by theorem 1, by (1) and (2), we get
|F| = q(m + 2) + 1.
So the theorem is proved.
(3)
The line set
L = F − R0
is a partial spread of S3 with
|L| = q 2 − q.
(4)
The line set F is a partial spread of P G(4, q), since theorem 2 holds, and the line
set
L0 = F ∪ L
is a set of mutually skew lines, and therefore a partial spread of P G(4, q), as immediately follows.
Furthemore L0 is maximal, since its lines cover S3 , which is a hyperplane of P G(4, q).
Also, by (3) and (4) we get
|L0 | = q 2 + (m + 1)q + 1.
So L0 is a maximal partial spread of P G(4, q) with size given by (5).
(5)
MAXIMAL PARTIAL LINE SPREADS
5
Thus, for every integer m = 1, . . . , q − 1, there is, in P G(4, q), a maximal partial
spread of size q 2 + (m + 1)q + 1. By this new cardinalities follow.
References
[1] A. Beutelspacher, Blocking sets and partial spreads in finite projective spaces,
Geom. Dedicata 9 (1980), 425-449.
[2] J. Eisfeld, L. Storme and P. Sziklai, On the spectrum of the sizes of maximal
partial line spreads in P G(2n, q), n ≥ 3, Des. Codes Cryptogr. 36 (2005),
101-110.
[3] A. Gács and T. Szőnyi, On maximal partial spreads in P G(n, q), Des. Codes
Cryptogr. 29 (2003), 123-129.
[4] A. Gács and T. Szőnyi, Random constructions and density results, Des. Codes
Cryptogr. 47 (2008), 267-287.
[5] P. Govaerts, Small maximal partial t-spreads, Bull. Belg. Math. Soc. Simon
Stevin 12 (4) (2005), 607-615.
Sandro Rajola
Via Vitaliano Brancati 44,
00144, Roma
Italy
[email protected]