On regular systems of finite classical polar spaces

Abstract

Let P be a finite classical polar space of rank d. An m-regular system with respect to (k − 1)-dimensional projective spaces of P, 1 ≤ k ≤ d − 1, is a set R of generators of P with the property that every (k − 1)-dimensional projective space of P lies on exactly m generators of R. Regular systems of polar spaces are investigated. Some non-existence results about certain 1-regular systems of polar spaces with low rank are proved and a procedure to obtain m ′ -regular systems from a given m-regular system is described. Finally, three different construction methods of regular systems w.r.t. points of various polar spaces are discussed.