SPG systems and semipartial geometries (Q2736575)

A semipartial geometry (SPG) is a point-line incidence structure \({\mathcal S}=(P,B,I)\) with each point on \(t+1\) lines, each line on \(s+1\) points, no two lines meeting in two points. There are two other parameters: if two points are not collinear, there are \(\mu\) \((\mu>0)\) points collinear with both of them. If a point \(x\) and a line \(L\) are not incident, then there are either 0 or \(\alpha\) \((\alpha >0)\) points \(x_i\) and either 0 or \(\alpha\) lines \(L_i\), respectively, such that \(xIL_i Ix_iIL\).NEWLINENEWLINENEWLINESemipartial geometries were introduced in 1978 by J. Debroey and J. A. Thas and include the well-known partial geometries, partial quadrangles and generalized quadrangles. For a major survey of the subject see the article by \textit{F. De Clerck} and \textit{H. Van Maldeghem} in the Handbook of Incidence Geometry: Buildings and Foundations, 433-475 (1995; Zbl 0823.51010).NEWLINENEWLINENEWLINEThe paper under review starts with a new investigation of necessary and sufficient conditions for a set of subspaces of \(PG(n,q)\) to be an SPG regulus, a structure that provides the ingredients for a construction of SPG. Then this notion is modified to the concept of SPG system, a special set of totally singular subspaces of a nonsingular (or nearly nonsingular) polar space. A new construction of SPG is given using an SPG system. Many examples are given, some of which turn out to be new. In particular, there arises a new family of SPG with parameters \(s=q^n-1\), \(t=q^{n+1}\), \(\alpha= 2q^{n-1}\), \(\mu=2 q^n(q^n-1)\), with either \(q\) any prime power and \(n=2\), or \(q=2^h\), \(h\geq 1\), and \(n\geq 3\).NEWLINENEWLINENEWLINEThis article is extremely well-written and is a very significant contribution to the study of semipartial geometries.