Baer subplanes generated by collineations between pencils of lines (Q861981)

In this paper the authors obtain an interesting results on special Baer subplanes in \(\text{PG}(2,q^{2})\): Let \(\;l_{\infty }\) be a line of \(\text{PG}(2,q^{2})\) and A be an affine Baer subplane of \(\text{PG}(2,q^{2})\backslash l_{\infty }\). Then there exist two distinct point \(A\),\(B\) on \(l_{\infty }\) such that A\(\cup\;l_{\infty } \) is the set of points of intersection of corresponding lines under an \( \alpha _{F}\)-collineation between the pencils of lines with vertices \(A\), \(B\). Moreover A\(\cup\;l_{\infty }\) may be represented by the equation \(x_{3}(x_{1}x_{3}^{q-1}-x_{2}^{q})=0\).