Hyperbolic flocks and generalizations
Norman Lloyd Johnson · 2023
A hyperbolic quadric H when viewed in an affine form is a classical regulus net within a 4-dimensional vector space V 4 over a finite field https://www.w3.org/1998/Math/MathML" display="inline"> G F ( q ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003324454/24094e3f-974a-4f50-95f4-ded534cb4875/content/math8_1.tif "/> . A flock of H is a covering of H by a set of https://www.w3.org/1998/Math/MathML" display="inline"> q + 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003324454/24094e3f-974a-4f50-95f4-ded534cb4875/content/math8_2.tif "/> mutually disjoint planes in https://www.w3.org/1998/Math/MathML" display="inline"> P G ( 3 , q ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003324454/24094e3f-974a-4f50-95f4-ded534cb4875/content/math8_3.tif "/> . Associated with a hyperbolic flock is a translation plane in V 4 , that admits an affine homology group of order https://www.w3.org/1998/Math/MathML" display="inline"> q − 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003324454/24094e3f-974a-4f50-95f4-ded534cb4875/content/math8_4.tif "/> , one of whose orbits union the axis and coaxis, becomes a regulus net and then all orbits that union the axis and coaxis are regulus nets. The union of these nets define a translation plane; the translation plane of the hyperbolic flock. These two geometries, the hyperbolic flock and the translation plane, are equivalent. There are exactly the following classes; the flocks are the linear flock, where the associated planes of https://www.w3.org/1998/Math/MathML" display="inline"> P G ( 3 , q ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003324454/24094e3f-974a-4f50-95f4-ded534cb4875/content/math8_5.tif "/> share a line, and the Thas flocks, with a few exceptions. The Thas flocks correspond to the regular nearfield planes and the exceptional flocks correspond to certain of the irregular nearfield planes and are due to a number of mathematicians from various different points of view (Bader [ 5 ], Baker-Ebert [ 9 ], Bonisoli [ 17 ], Johnson [ 76 ]). There are three irregular nearfields of orders https://www.w3.org/1998/Math/MathML" display="inline"> 11 2 , 23 2 , 59 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003324454/24094e3f-974a-4f50-95f4-ded534cb4875/content/math8_6.tif "/> . Bader, Bonisoli and Johnson independently determined the same results for all three orders. And, Baker and Ebert showed the same results but for orders https://www.w3.org/1998/Math/MathML" display="inline"> 11 2 , 23 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003324454/24094e3f-974a-4f50-95f4-ded534cb4875/content/math8_7.tif "/> . Each of the authors determined the flocks/translation planes by using essentially different methods. The main point here is that the associated translation planes are all Bol planes; which has been of considerable interest, and there is a complete classification due to Thas and Bader-Lunardon ([ 136 ], [ 8 ]). There are various possible formulations for this classification, depending on whether it is phrased in the associated translation plane or in the hyperbolic flock. Sometimes, names are used, sometimes the name of the algebra coordinatizing the structures is used to describe the structures. For uniformity, here we shall use the name of the coordinate structures for the translation plane version, and the names of the mathematicians finding the flocks in the flock version. Theorem 59 Thas, Bader-Lunardon ( [ 136 ] . [ 8 ] ) Classification of finite hyperbolic flocks/translation planes admitting regulus-inducing homology groups in https://www.w3.org/1998/Math/MathML" display="inline"> P G ( 3 , q ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003324454/24094e3f-974a-4f50-95f4-ded534cb4875/content/math8_8.tif "/> . Plane version: The translation planes are exactly the nearfield planes; based upon the regular nearfields of order https://www.w3.org/1998/Math/MathML" display="inline"> q 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003324454/24094e3f-974a-4f50-95f4-ded534cb4875/content/math8_9.tif "/> and the three irregular nearfields of orders https://www.w3.org/1998/Math/MathML" display="inline"> 11 2 , 23 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003324454/24094e3f-974a-4f50-95f4-ded534cb4875/content/math8_10.tif "/> and https://www.w3.org/1998/Math/MathML" display="inline"> 59 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003324454/24094e3f-974a-4f50-95f4-ded534cb4875/content/math8_11.tif "/> . Flock version: The hyperbolic flocks are exactly the Thas flocks and the flocks of Bader, Baker-Ebert, Bonisoli, Johnson.