Geometry of Derivation, Volume II

This book is concerned mainly with the theory of flocks over skewfields. It begins with discussing what conditions would be required to find a possible way to extend flocks of hyperbolic quadrics and flocks of quadratic cones. This theory completely changes the idea of derivation of an affine plane that contains a derivable net. This volume will give the necessary theory for the reader to understand how to construct examples and become researchers in the field. It shows how to construct four types of determinants, the (i,j)-determinants, which if never zero for the non-zero matrices of the spread will indicate that the first condition for existence of a spread then holds. If applicable, the left unwrapping principle, if this also is valid, will show that a left flock spread is constructed. The book continues the presentation in Geometry of Derivation with Applications, Volume I, and a third volume, Geometry of Derivation, Volume III: Classification of Skewfield Flocks (2026) is also available, both from CRC Press. This is the sixth work in a longstanding series of books on combinatorial geometry by the author, including Subplane Covered Nets, Johnson (2000); Foundations of Translation Planes, Biliotti, Jha, and Johnson (2001); Handbook of Finite Translation Planes, Johnson, Jha, and Biliotti (2007); and Combinatorics of Spreads and Parallelisms, Johnson (2010), all published by CRC Press. Like its predecessors, this book will primarily deal with connections to the theory of derivable nets and translation planes in both the finite and infinite cases. Translation planes over non-commutative skewfields have not traditionally had a significant representation in incidence geometry, and derivable nets over skewfields have only been marginally understood. Both are deeply examined in this volume, while ideas of non-commutative algebra are also described in detail, with all the necessary background given a geometric treatment.