Aim of the chapter

The aim of this chapter is to show explicitly how certain linear spaces can be embedded in a projective plane. Among such structures are the complements of two lines, of a triangle, of a hyperoval and of a Baer subplane. Here, the notion of a pseudo-complement is crucial. Suppose that we remove a set X of a projective plane P of order n. Then we obtain a linear space P-X having certain parameters (i.e., the number of points, the number of lines, the point- and line-degrees). We call any linear space which has the same parameters as P-X a pseudo-complement of X in P.

We have already encountered the notion of a pseudo-complement, namely the pseudo-complement of one line. This is a linear space with n2 points, n2 + n lines in which any point has degree n + 1 and any line has degree n. We know that this is an affine plane, which is a structure embeddable into a projective plane of order n. Another example may help to clarify the above definition. A pseudo-complement of two lines in a projective plane of order n is a linear space having n2 − n points, n2 + n − 1 lines in which any point has degree n + 1 and any line has degree n − 1 or n. (More precisely, the n − lines form a parallel class.)