On Geometry and Combinatorics of Finite Classical Polar Spaces

TL;DR

This paper explores the geometry and combinatorics of finite classical polar spaces, addressing open problems and their connections to combinatorial objects, with contributions to the understanding of their structure and related combinatorial designs.

Contribution

It provides new insights into the geometric and combinatorial properties of finite classical polar spaces, including results on spreads, ovoids, and their relations to combinatorial structures.

Findings

Results on existence of spreads and ovoids in polar spaces

New bounds on partial spreads and ovoids

Connections established between polar spaces and combinatorial objects

Abstract

Polar spaces over finite fields are fundamental in combinatorial geometry. The concept of polar space was firstly introduced by F. Veldkamp who gave a system of 10 axioms in the spirit of Universal Algebra. Later the axioms were simplified by J. Tits, who introduced the concept of subspaces. Later on, from the point of view of incidence geometry, axioms of polar spaces were also given by F. Buekenhout and E. Shult in 1974. The reader can find the three systems of axioms of polar spaces in Appendix A. Examples of polar spaces are the so called Finite classical polar spaces, i.e. incidence structures arising from quadrics, symplectic spaces and Hermitian varieties, which are in correspondance with reflexive sesquilinear forms. It is still an open problem to show whether or not classical polar spaces are the only example of finite polar spaces. Nowadays, some research problems related to finite classical polar space are: existence of spreads and ovoids; existence of regular systems and $m$-ovoids; upper or lower bounds on partial spreads and partial ovoids. Moreover, polar spaces are in relation with combinatorial objects as regular graphs, block designs and association schemes. In this Ph.D. Thesis we investigate the geometry of finite classical polar spaces, giving contributions to the above problems. The thesis is organized as follows. Part I is more focused on the geometric aspects of polar spaces, while in Part II some combinatorial objects are introduced such as regular graphs, association schemes and combinatorial designs. Finally Appendix B, C and D are dedicated to give more details on, respectively, maximal curves, linear codes and combinatorial designs, giving useful results and definitions.