A geometric construction of partial geometries with a Hermitian point graph (Q1864583)

A partial geometry \(\text{ pg}(s,t,\alpha)\) is a point-line incidence structure in which two distinct lines are incident with at most one point, each line is incident with \(s+1\) points, each point is incident with \(t+1\) lines, and for each non-incident point-line pair there are \(\alpha\) lines incident with the point and intersecting the line. In [\textit{R. Mathon}, Geom. Dedicata 73, 11-19 (1998; Zbl 0927.51014)] a class of partial geometries \(\text{ pg}(q-1,\frac{q^2-1}{2},\frac{q-1}{2})\), \(q=3^{2h}\), was constructed algebraically. In the present paper, the author gives a geometric construction for these partial geometries. This geometric construction also works if \(q\) is an odd power of 3, giving rise to new partial geometries. The point graphs of the partial geometries which arise from the construction are the Hermitian graphs which were first defined in [\textit{D. E. Taylor}, Some topics in the theory of finite groups, Ph.D. Thesis, Univ. Oxford (1971)].