On the geometry of Baer subplanes in \(PG(2,4)\) (Q1587911)

The author recalls basic facts on hyperovals, Baer subplanes of \(PG(2,4)\), unitals and the Gewirtz graph. Then he defines certain sets of Baer subplanes in \(PG(2,4)\) called butterflies. These butterflies are used to determine parameters of the graph \(\Gamma\) on \(\mathbb{C}\) which is an \(L_3(4)\)-orbit of Baer subplanes in \(PG(2,4)\). It is shown that \(\Gamma\) is the collinearity graph of the geometry (101) of Buekenhout with the diagram \[ \underset {1}\circ\diagrtripleBar\underset {2}\circ\overset{\vtop{=2em\centerline{1079}\vglue 5pt}}\diagrtriplebar\underset {1}\circ. \] Moreover, the author proves that, if \(L_3(4)\) or \(L_3(4):2_1\) are automorphism groups, this geometry is determined up to isomorphism by its diagram.