A note on characterizing Hermitian curves via Baer sublines (Q346599)

The author defines Property A as follows. Let \(K\) be a subfield of index 2 of \(F\) and let \(X\) be a set of points of \(\mathrm{PG}(V)\) with \(x_1,x_2\) two distinct points of \(X\) then (1) Any secant line through \(x_1\) or \(x_2\) intersects \(X\) in a Baer-\(K\) subline; (2) \(X \setminus x_1x_2\) is not empty; (3) There exists a point \(y\) in \(\mathrm{PG}(V) \setminus x_1x_2\) such that \(yx\) is a tangent line for every point \(x\) in \(x_1x_2 \cap X.\) The author proves the following result. (1) If \(F\) is a separable quadratic extension of the field \(K\), then the set of points of \(\mathrm{PG}(V)\) satisfying Property A are precisely the \(K\)-Hermitian curves of \(\mathrm{PG}(V).\) (2) If \(F\) is an inseparable quadratic extension of the field \(K\), then the sets of points of \(\mathrm{PG}(V)\) satisfying Property A are precisely the sets of points described by a condition of the form \(\lambda X_0^2 + X_1X_2 \in K\), where \(\lambda\) is in \(F \setminus K\) and \((X_0,X_1,X_2)\) denote the homogeneous coordinates with respect to a fixed reference system.