Geometry of Derivation, Volume II by Johnson, Norman L.;

Short description:

This book is concerned mainly with the theory of flocks over skewfields and gives the necessary theory for the reader to understand how to construct examples and become researchers in the field of combinatorial geometry. 

Long description:

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 Derivations 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.

Table of Contents:

Part 1: When Quasifibrations become Spreads  1. Quasifibrations  2. Unwrapping  3. Twisted Extensions  4. Semifield Planes from Cyclic Algebras  5. Proper Quasifibrations of Dimension 2  Part 2: Skewfield Flocks-A Window  6. The Main Points and Ideas  Part 3: Foundations of Flock Theory  7. Building the Foundation  8. Generic and Non-Generic Flocks  Part 4: Framework for Flock Theory  9. Setting up Flock and Spread Connections  Part 5: Left σ−A Flocks and Spreads  10. Left σ − A-Hyperbolic Flocks  11. Left σ − A-Conical Flocks; 1st and 2nd Main Theorems  12. The Lower Left Form Theory  13. The 1st General Theorem of Flocks over Skewfields  Part 6: Right τ − A∗ Flocks and Spreads  14. τ − A∗ Hyperbolic Flocks  15. Right τ − A∗ Hyperbolic 1st and 2nd Main Theorems  16. τ − A∗ Right Conical Flocks  17. The Right Upper Form Theory  18. Four “Easy” Problems  Part 7: The Kaleidoscope of Derivable Nets  19. The Conical and Hyperbolic Isomorphism Questions  Part 8: Apps of the Kaleidoscope  20. Resolution and Return-Flock Spreads  21. A Class of Linear 1 − Cc−Conical Flocks  Part 9: Double Covers  22. The Left Generic Elation Double Nets  Part 10: The Group of Conical Flock Spreads  23. Why Semifields?  24. Omnibus Theorem  Part 11: Quaternion Division Ring Variations  25. 1-A Left Conical Spreads  26. Why Unwrapping?  Part 12: Left Pseudo-Regulus-Inducing Homology Groups and Transposition  27. Inversing-Right Hyperbolic Spreads