A frequently studied problem in finite 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.
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.
Construction of a regular spread is most easily seen using an algebraic model.
; this model uses homogeneous coordinates to represent points and hyperplanes.
[6] 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
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.
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.
Bruen[9] has explored the concept of a chain of reguli in a regular spread of
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
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.
Unlike a chain, two reguli in a nest are not required to meet in a pair of lines.
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.
from other geometrical objects without reference to an initial regular spread.
, 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
planes whose intersections with the quadratic cone are pairwise disjoint conics.
that do not contain the vertex of the cone; such a flock is called 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
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.
Baker, et al.[28] provide several infinite families of partitions of
For a lower bound, Bruen[30] showed that a complete partial spread of lines in
Bruen also provides examples of complete partial spreads of lines in
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.