Spread (projective geometry)

A frequently studied problem in discrete geometry is to identify ways in which an object can be covered by other simpler objects such as points, lines, and planes. In projective geometry, a specific instance of this problem that has numerous applications is determining whether, and how, a projective space can be covered by pairwise disjoint subspaces which have the same dimension; such a partition is called a spread. Specifically, a spread of a projective space ${\displaystyle PG(d,K)}$, where ${\displaystyle d\geq 1}$ is an integer and ${\displaystyle K}$ a division ring, is a set of ${\displaystyle r}$-dimensional subspaces, for some ${\displaystyle 0<r<d}$ such that every point of the space lies in exactly one of the elements of the spread.

Spreads are particularly well-studied in projective geometries over finite fields, though some notable results apply to infinite projective geometries as well. In the finite case, the foundational work on spreads appears in André ^[1] and independently in Bruck-Bose ^[2] in connection with the theory of translation planes. In these papers, it is shown that a spread of ${\displaystyle r}$-dimensional subspaces of the finite projective space ${\displaystyle PG(d,q)}$ exists if and only if ${\displaystyle r+1\mid d+1}$. ^[3]

Spreads and translation planes

For all integers ${\displaystyle n\geq 1}$, the projective space ${\displaystyle PG(2n+1,q)}$ always has a spread of ${\displaystyle n}$-dimensional subspaces, and in this section the term spread refers to this specific type of spread; spreads of this form may (and frequently do) occur in infinite projective geometries as well. These spreads are the most widely studied in the literature, due to the fact that every such spread can be used to create a translation plane using the André/Bruck-Bose construction. ^{[1] [2]}

Reguli and regular spreads

Let ${\displaystyle \Sigma }$ be the projective space ${\displaystyle PG(2n+1,K)}$ for ${\displaystyle n\geq 1}$ an integer, and ${\displaystyle K}$ a division ring. A regulus ^[4] ${\displaystyle R}$ in ${\displaystyle \Sigma }$ is a collection of pairwise disjoint ${\displaystyle n}$-dimensional subspaces with the following properties:

1. ${\displaystyle R}$ contains at least 3 elements
2. Every line meeting three elements of ${\displaystyle R}$, called a transversal , meets every element of ${\displaystyle R}$
3. Every point of a transversal to ${\displaystyle R}$ lies on some element of ${\displaystyle R}$

Any three pairwise disjoint ${\displaystyle n}$-dimensional subspaces in ${\displaystyle \Sigma }$ lie in a unique regulus. ^[5] A spread ${\displaystyle S}$ of ${\displaystyle \Sigma }$ is regular if for any three distinct ${\displaystyle n}$-dimensional subspaces of ${\displaystyle S}$, all the members of the unique regulus determined by them are contained in ${\displaystyle S}$. Regular spreads are significant in the theory of translation planes, in that they generate Moufang planes in general, and Desarguesian planes in the finite case when the order of the ambient field is greater than ${\displaystyle 2}$. All spreads of ${\displaystyle PG(2n+1,2)}$ are trivially regular, since a regulus only contains three elements.

Constructing a regular spread

Construction of a regular spread is most easily seen using an algebraic model. Letting ${\displaystyle V}$ be a ${\displaystyle (2n+2)}$-dimensional vector space over a field ${\displaystyle F}$, one can model the ${\displaystyle k}$-dimensional subspaces of ${\displaystyle PG(2n+1,F)}$ using the ${\displaystyle (k+1)}$-dimensional subspaces of ${\displaystyle V}$; this model uses homogeneous coordinates to represent points and hyperplanes. Incidence is defined by intersection, with subspaces intersecting in only the zero vector considered disjoint; in this model, the zero vector of ${\displaystyle V}$ is effectively ignored.

Let ${\displaystyle F}$ be a field and ${\displaystyle E}$ an ${\displaystyle n}$-dimensional extension field of ${\displaystyle F}$. Consider ${\displaystyle V=E\oplus E}$ as a ${\displaystyle 2n}$-dimensional vector space over ${\displaystyle F}$, which provides a model for the projective space ${\displaystyle PG(2n-1,F)}$ as above. Each element of ${\displaystyle V}$ can be written uniquely as ${\displaystyle (x,y)}$ where ${\displaystyle x,y\in E}$. A regular spread is given by the set of ${\displaystyle n}$-dimensional projective spaces defined by ${\displaystyle J(k)=\{(x,kx):x\in E\}}$, for each ${\displaystyle k\in E}$, together with ${\displaystyle J(\infty )=\{(0,y):y\in E\}}$. ^[6]

Constructing spreads

Spread sets

The construction of a regular spread above is an instance of a more general construction of spreads, which uses the fact that field multiplication is a linear transformation over ${\displaystyle E}$ when considered as a vector space. Since ${\displaystyle E}$ is a finite ${\displaystyle n}$-dimensional extension over ${\displaystyle F}$, a linear transformation from ${\displaystyle E}$ to itself can be represented by an ${\displaystyle n\times n}$ matrix with entries in ${\displaystyle F}$. A spread set is a set ${\displaystyle S}$ of ${\displaystyle n\times n}$ matrices over ${\displaystyle F}$ with the following properties:

• ${\displaystyle S}$ contains the zero matrix and the identity matrix
• For any two distinct matrices ${\displaystyle X}$ and ${\displaystyle Y}$ in ${\displaystyle S}$, ${\displaystyle X-Y}$ is nonsingular
• For each pair of elements ${\displaystyle a,b\in E}$, there is a unique ${\displaystyle X\in S}$ such that ${\displaystyle aX=b}$

In the finite case, where ${\displaystyle E}$ is the field of order ${\displaystyle q^{n}}$ for some prime power ${\displaystyle q}$, the last condition is equivalent to the spread set containing ${\displaystyle q^{n}}$ matrices. Given a spread set ${\displaystyle S}$, one can create a spread as the set of ${\displaystyle n}$-dimensional projective spaces defined by ${\displaystyle J(k)=\{(x,xM):x\in E\}}$, for each ${\displaystyle M\in S}$, together with ${\displaystyle J(\infty )=\{(0,y):y\in E\}}$, ^[2] As a specific example, the following nine matrices represent ${\displaystyle GF(9)}$ as 2 × 2 matrices over ${\displaystyle GF(3)}$ and so provide a spread set of ${\displaystyle AG(2,9)}$. ^[6]

${\displaystyle \left[{\begin{matrix}0&0\\0&0\end{matrix}}\right],\left[{\begin{matrix}1&0\\0&1\end{matrix}}\right],\left[{\begin{matrix}2&0\\0&2\end{matrix}}\right],\left[{\begin{matrix}0&1\\2&0\end{matrix}}\right],\left[{\begin{matrix}1&1\\2&1\end{matrix}}\right],\left[{\begin{matrix}2&1\\2&2\end{matrix}}\right],\left[{\begin{matrix}0&2\\1&0\end{matrix}}\right],\left[{\begin{matrix}1&2\\1&1\end{matrix}}\right],\left[{\begin{matrix}2&2\\2&1\end{matrix}}\right]}$

Another example of a spread set yields the Hall plane of order 9 ^[6]

${\displaystyle \left[{\begin{matrix}0&0\\0&0\end{matrix}}\right],\left[{\begin{matrix}1&0\\0&1\end{matrix}}\right],\left[{\begin{matrix}2&0\\0&2\end{matrix}}\right],\left[{\begin{matrix}1&1\\1&2\end{matrix}}\right],\left[{\begin{matrix}2&2\\2&1\end{matrix}}\right],\left[{\begin{matrix}0&1\\2&0\end{matrix}}\right],\left[{\begin{matrix}0&2\\1&0\end{matrix}}\right],\left[{\begin{matrix}1&2\\2&2\end{matrix}}\right],\left[{\begin{matrix}2&1\\1&1\end{matrix}}\right]}$

Modifying spreads

One common approach to creating new spreads is to start with a regular spread and modify it in some way. The techniques presented here are some of the more elementary examples of this approach.

Spreads of 3-space

One can create new spreads by starting with a spread and looking for a switching set, a subset of its elements that can be replaced with an alternate set of pairwise disjoint subspaces of the correct dimension. In ${\displaystyle PG(3,K)}$, a regulus forms a switching set, as the set of transversals of a regulus ${\displaystyle R}$ also form a regulus, called the opposite regulus of ${\displaystyle R}$. Removing the lines of a regulus in a spread and replacing them with the opposite regulus produces a new spread which is often non-isomorphic to the original. This process is a special case of a more general process called derivation or net replacement. ^[7]

Starting with a regular spread of ${\displaystyle PG(3,q)}$ and reversing any regulus produces a spread that yields a Hall plane. In more generality, the process can be applied independently to any collection of reguli in a regular spread, yielding a subregular spread ^[8] ; the resulting translation plane is called a subregular plane. The André planes form a special subclass of subregular planes, of which the Hall planes are the simplest examples, arising by replacing a single regulus in a regular spread.

More complex switching sets have been constructed. Bruen ^[9] has explored the concept of a chain of reguli in a regular spread of ${\displaystyle PG(3,q)}$, ${\displaystyle q}$ odd, namely a set of ${\displaystyle (q+3)/2}$ reguli which pairwise meet in exactly 2 lines, so that every line contained in a regulus of the chain is contained in exactly two distinct reguli of the chain. Bruen constructed an example of a chain in the regular spread of ${\displaystyle PG(3,5)}$, and showed that it could be replaced by taking the union of exactly half of the lines from the opposite regulus of each regulus in the chain. Numerous examples of Bruen chains have appeared in the literature since, and Heden ^[10] has shown that any Bruen chain is replaceable using opposite half-reguli. Chains are known to exist in a regular spread of ${\displaystyle PG(3,q)}$ for all odd prime powers ${\displaystyle q}$ up to 37, except 29, and are known not to exist for ${\displaystyle q\in \{29,41,43,47,49\}}$. ^[11] It is conjectured that no additional Bruen chains exist.

Baker and Ebert ^[12] generalized the concept of a chain to a nest, which is a set of reguli in a regular spread such that every line contained in a regulus of the nest is contained in exactly two distinct reguli of the nest. Unlike a chain, two reguli in a nest are not required to meet in a pair of lines. Unlike chains, a nest in a regular spread need not be replaceable, ^[13] however several infinite families of replaceable nests are known. ^{[14] [15]}

Higher-dimensional spreads

In higher dimensions a regulus cannot be reversed because the transversals do not have the correct dimension. There exist analogs to reguli, called norm surfaces, which can be reversed. ^[16] The higher-dimensional André planes can be obtained from spreads obtained by reversing these norm surfaces, and there also exist analogs of subregular spreads which do not give rise to André planes. ^{[17] [18]}

Geometric techniques

There are several known ways to construct spreads of ${\displaystyle PG(3,q)}$ from other geometrical objects without reference to an initial regular spread. Some well-studied approaches to this are given below.

Flocks of quadratic cones

In ${\displaystyle PG(3,q)}$, a quadratic cone is the union of the set of lines containing a fixed point P (the vertex) and a point on a conic in a plane not passing through P. Since a conic has ${\displaystyle q+1}$ points, a quadratic cone has ${\displaystyle q(q+1)+1}$ points. As with traditional geometric conic sections, a plane of ${\displaystyle PG(3,q)}$ can meet a quadratic cone in either a point, a conic, a line or a line pair. A flock of a quadratic cone is a set of ${\displaystyle q}$ planes whose intersections with the quadratic cone are pairwise disjoint conics. The classic construction of a flock is to pick a line ${\displaystyle m}$ that does not meet the quadratic cone, and take the ${\displaystyle q}$ planes through ${\displaystyle m}$ that do not contain the vertex of the cone; such a flock is called linear.

Fisher and Thas ^[19] show how to construct a spread of ${\displaystyle PG(3,q)}$ from a flock of a quadratic cone using the Klein correspondence, and show that the resulting spread is regular if and only if the initial flock is linear. Many infinite families of flocks of quadratic cones are known, as are numerous sporadic examples. ^[20]

Every spread arising from a flock of a quadratic cone is the union of ${\displaystyle q}$ reguli which all meet in a fixed line ${\displaystyle m}$. Much like with a regular spread, any of these reguli can be replaced with its opposite to create several potentially new spreads. ^[21]

Hyperbolic fibrations

In ${\displaystyle PG(3,q)}$ a hyperbolic fibration is a partition of the space into ${\displaystyle q-1}$ pairwise disjoint hyperbolic quadrics and two lines disjoint from all of the quadrics and each other. Since a hyperbolic quadric consists of the points covered by a regulus and its opposite, a hyperbolic fibration yields ${\displaystyle 2^{q-1}}$ different spreads.

All spreads yielding André planes, including the regular spread, are obtainable from a hyperbolic fibration (specifically an algebraic pencil generated by any two of the quadrics), as articulated by André. ^[1] Using nest replacement, Ebert ^[22] found a family of spreads in which a hyperbolic fibration was identified. Baker, et al. ^[23] provide an explicit example of a construction of a hyperbolic fibration. A much more robust source of hyperbolic fibrations was identified by Baker, et al., ^[24] where the authors developed a correspondence between flocks of quadratic cones and hyperbolic fibrations; interestingly, the spreads generated by a flock of a quadratic cone are not generally isomorphic to the spreads generated from the corresponding hyperbolic fibration.

Subgeometry partitions

Hirschfeld and Thas ^[25] note that for any odd integer ${\displaystyle n\geq 3}$, a partition of ${\displaystyle PG(n-1,q^{2})}$ into subgeometries isomorphic to ${\displaystyle PG(n-1,q)}$ gives rise to a spread of ${\displaystyle PG(2n-1,q)}$, where each subgeometry of the partition corresponds to a regulus of the new spread.

The "classical" subgeometry partitions of ${\displaystyle PG(n-1,q^{2})}$ can be generated using suborbits of a Singer cycle, but this simply generates a regular spread. ^[26] Yff ^[27] published the non-classical subgeometry partition, namely a partition of ${\displaystyle PG(2,9)}$ into 7 copies of ${\displaystyle PG(2,3)}$, that admit a cyclic group permuting the subplanes. Baker, et al. ^[28] provide several infinite families of partitions of ${\displaystyle PG(2,q^{2})}$ into subplanes, with the same cyclic group action.

Partial spreads

A partial spread of a projective space ${\displaystyle PG(d,K)}$ is a set of pairwise disjoint ${\displaystyle r}$-dimensional subspaces in the space; hence a spread is just a partial spread where every point of the space is covered. A partial spread is called complete or maximal if there is no larger partial spread that contains it; equivalently, there is no ${\displaystyle r}$-dimensional subspace disjoint from all members of the partial spread. As with spreads, the most well-studied case is partial spreads of lines of the finite projective space ${\displaystyle PG(3,q)}$, where a full spread has size ${\displaystyle q^{2}+1}$. Mesner ^[29] showed that any partial spread of lines in ${\displaystyle PG(3,q)}$ with size greater than ${\displaystyle q^{2}-{\sqrt {q}}}$ cannot be complete; indeed, it must be a subset of a unique spread. For a lower bound, Bruen ^[30] showed that a complete partial spread of lines in ${\displaystyle PG(3,q)}$ with size at most ${\displaystyle q+{\sqrt {q}}}$ lines cannot be complete; there will necessarily be a line that can be added to a partial spread of this size. Bruen also provides examples of complete partial spreads of lines in ${\displaystyle PG(3,q)}$ with sizes ${\displaystyle q^{2}-q+1}$ and ${\displaystyle q^{2}-q+2}$ for all ${\displaystyle q>2}$.

Spreads of classical polar spaces

The classical polar spaces are all embedded in some projective space ${\displaystyle PG(d,K)}$ as the set of totally isotropic subspaces of a sesquilinear or quadratic form on the vector space underlying the projective space. A particularly interesting class of partial spreads of ${\displaystyle PG(d,K)}$ are those that consist strictly of maximal subspaces of a classical polar space embedded in the projective space. Such partial spreads that cover all of the points of the polar space are called spreads of the polar space.

From the perspective of the theory of translation planes, the symplectic polar space is of particular interest, as its set of points are all of the points in ${\displaystyle PG(2n+1,K)}$, and its maximal subspaces are of dimension ${\displaystyle n}$. Hence a spread of the symplectic polar space is also a spread of the entire projective space, and can be used as noted above to create a translation plane. Several examples of symplectic spreads are known; see Ball, et al. ^[31]

References

1. André, Johannes (1954), "Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe", Mathematische Zeitschrift , 60: 156–186, doi:10.1007/BF01187370, ISSN   0025-5874, MR   0063056, S2CID   123661471
2. Bruck, R. H.; Bose, R. C. (1964), "The Construction of Translation Planes from Projective Spaces" (PDF), Journal of Algebra, 1: 85–102, doi:10.1016/0021-8693(64)90010-9
3. This is ultimately a consequence of the fact that a finite field of order ${\displaystyle p^{d+1}}$ has a subfield of order ${\displaystyle p^{r+1}}$ if and only if ${\displaystyle r+1\mid d+1}$.
4. This notion generalizes that of a classical regulus, which is one of the two families of ruling lines on a hyperboloid of one sheet in 3-dimensional space
5. Bruck, R. H.; Bose, R. C. (1966), "Linear Representations of Projective Planes in Projective Spaces" (PDF), Journal of Algebra, 4: 117–172, doi:10.1016/0021-8693(66)90054-8 , page 163
6. Moorhouse, Eric (2007), Incidence Geometry (PDF), archived from the original (PDF) on 2013-10-29
7. Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007), Handbook of Finite Translation Planes, Chapman&Hall/CRC, ISBN   978-1-58488-605-1 , page 49
8. Bruck, R.H. (1969), R.C.Bose and T.A. Dowling (ed.), "Construction Problems of finite projective planes", Combinatorial Mathematics and Its Applications, Univ. of North Carolina Press, pp. 426–514
9. Bruen, A.A. (1978). "Inversive geometry and some translation planes, I". Geometriae Dedicata. 7: 81–98. doi:10.1007/BF00181353. S2CID   122632353.
10. Heden, O. (1995). "On Bruen chains" (PDF). Discrete Mathematics. 146 (1–3): 69–96. doi: 10.1016/0012-365X(94)00058-0 .
11. Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007). "Nests". Handbook of Finite Translation Planes. Boca Raton, FL: Chapman & Hall/CRC. ISBN   978-1-58488-605-1.
12. Baker, R. D.; Ebert, G. L. (1988). "Nests of size ${\displaystyle q-1}$ and another family of translation planes". Journal of the London Mathematical Society. 38 (2): 341–355. doi:10.1112/jlms/s2-38.2.341.
13. Ebert, G. L. (1988). "Some nonreplaceable nests". Combinatorics '88. Research Lecture Notes in Mathematics, Mediterranean, Rende. 1: 353–372.
14. Baker, R. D.; Ebert, G. L. (1988). "A new class of translation planes". Annals of Discrete Mathematics. Amsterdam: North-Holland. 37: 7–20. doi:10.1016/S0167-5060(08)70220-6. ISBN   9780444703699.
15. Baker, R. D.; Ebert, G. L. (1996). "Filling the nest gaps". Finite Fields and Applications. 2 (1): 45–61.
16. Bruck, R.H. (1973). "Circle geometry in higher dimensions. II". Geometriae Dedicata. 2 (2). doi:10.1007/BF00147854. ISSN   0046-5755. S2CID   189889878.
17. Dover, Jeremy (1998). "Subregular Spreads of PG(2n+1,q)". Finite Fields and Their Applications. 4 (4): 362–380. doi: 10.1006/ffta.1998.0222 .
18. Culbert, Craig; Ebert, Gary (2005). "Circle geometry and three-dimensional subregular translation planes". Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial. 1 (1): 3–18. doi: 10.2140/iig.2005.1.3 . ISSN   1781-6475.
19. Fisher, J. Chris; Thas, Joseph A. (1979). "Flocks in PG(3,q)". Mathematische Zeitschrift. 169 (1): 1–11. doi:10.1007/BF01214908. ISSN   0025-5874. S2CID   121796426.
20. Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007). "Infinite Classes of Flocks". Handbook of Finite Translation Planes. Boca Raton, FL: Chapman & Hall/CRC. ISBN   978-1-58488-605-1.
21. A specific example of this phenomenon is illustrated in Dover, Jeremy M. (2019-02-27). "A genealogy of the translation planes of order 25". arXiv: 1902.07838 [math.CO]. to relate two translation planes of order 25 found by computer search in Czerwinski, Terry; Oakden, David (1992). "The translation planes of order twenty-five". Journal of Combinatorial Theory, Series A. 59 (2): 193–217. doi:10.1016/0097-3165(92)90065-3.
22. Ebert, G. L. (1989). "Spreads Admitting Regular Elliptic Covers". European Journal of Combinatorics. 10 (4): 319–330. doi:10.1016/S0195-6698(89)80004-6.
23. Baker, R. D.; Dover, J. M.; Ebert, G. L.; Wantz, K. L. (1999). "Hyperbolic Fibrations of PG(3,q)". European Journal of Combinatorics. 20 (1): 1–16. doi: 10.1006/eujc.1998.0249 .
24. Baker, R. D.; Ebert, G. L.; Penttila, Tim (2005). "Hyperbolic Fibrations and q-Clans". Designs, Codes and Cryptography. 34 (2–3): 295–305. doi:10.1007/s10623-004-4861-8. ISSN   0925-1022. S2CID   21853272.
25. Hirschfeld, J. W. P.; Thas, J. A. (1991). General Galois geometries. London: Oxford University Press. p. 206. ISBN   978-1-4471-6790-7. OCLC   936691484.
26. Bruck, R. H. (1960). "Quadratic extensions of cyclic planes". In Bellman, Richard; Hall, Marshall (eds.). Combinatorial Analysis. Proceedings of Symposia in Applied Mathematics. Vol. 10. Providence, Rhode Island: American Mathematical Society. pp. 15–44. doi:10.1090/psapm/010. ISBN   978-0-8218-1310-2.
27. Yff, Peter (1977). "On subplane partitions of a finite projective plane". Journal of Combinatorial Theory, Series A. 22 (1): 118–122. doi:10.1016/0097-3165(77)90072-3.
28. Baker, Ronald D.; Dover, Jeremy M.; Ebert, Gary L.; Wantz, Kenneth L. (2000). "Baer subgeometry partitions". Journal of Geometry. 67 (1–2): 23–34. doi:10.1007/BF01220294. ISSN   0047-2468. S2CID   121116940.
29. Mesner, Dale M. (1967). "Sets of Disjoint Lines in PG(3, q)". Canadian Journal of Mathematics. 19: 273–280. doi:10.4153/CJM-1967-019-5. ISSN   0008-414X. S2CID   123550829.
30. Bruen, A. (1971). "Partial Spreads and Replaceable Nets". Canadian Journal of Mathematics. 23 (3): 381–391. doi: 10.4153/CJM-1971-039-x . ISSN   0008-414X. S2CID   124356288.
31. Ball, Simeon; Bamberg, John; Lavrauw, Michel; Penttila, Tim (May 2004). "Symplectic Spreads". Designs, Codes and Cryptography. 32 (1–3): 9–14. doi:10.1023/B:DESI.0000029209.24742.89. ISSN   0925-1022. S2CID   8228870.