Linear spaces of affine type (Q1191616)

Let \(({\mathcal P},{\mathcal L})\) be a finite linear space, that is a finite set \(\mathcal P\) whose elements we call points, \(\mathcal L\) a family of parts on \(\mathcal P\), whose elements we call lines, such that any line has at least two points, two distinct points are contained in just a line and \(| {\mathcal L}|\geq 2\). Let \(n+1\) be the maximum number of lines on a point; then the integer \(n\) is called the order of \(({\mathcal P},{\mathcal L})\). The number of lines passing through a point \(P\) is called the degree of \(P\). In this paper finite linear spaces \(({\mathcal P},{\mathcal L})\) of order \(n\) which satisfy the condition \(|{\mathcal L}|=|{\mathcal P}|+n\) are investigated. Some classes of these spaces are characterized and a complete classification of those with constant point degree is obtained.