Veronesean embeddings of Hermitian unitals
Let \({\mathcal S}=({\mathcal P},{\mathcal L})\) be a point-line geometry. A \textit{Veronesean (projective) embedding} of \({\mathcal S}\) in the projective space \(PG(d,{\mathbf K})\) over a field \(\mathbf K\) is an injective mapping \(\alpha:{\mathcal P}\to\,PG_{0}(d,{\mathbf K})\), where \(PG_{0}(d,{\mathbf K})\) denotes the point set of \(PG(d,{\mathbf K})\), such that the \(\alpha\)-image of any line is a plane oval, and such that the \(\alpha\)-image of \(\mathcal P\) generates \(PG(d,{\mathbf K})\). (It suffices to assume only commutative fields because of the lack of ``classical'' ovals in projective planes over non-commutative skew fields.) Let \(\mathbf L\) be a quadratic Galois extension of \(\mathbf K\), the involution of the corresponding Galois group maps \(x\in{\mathbf L}\) to \(\overline {x}\), and let \(\eta\in{\mathbf L}\setminus{\mathbf K}\) be an arbitrarily chosen element. We consider the mapping \(\beta:PG_0(2,{\mathbf L})\to\,PG_0(8,{\mathbf K})\) with \[ \begin{multlined}\beta(x_0,x_1,x_2)\mapsto (x_0\overline{x}_0,x_1\overline{x}_1,x_2\overline{x}_2; x_0\overline{x}_1+\overline{x}_{0}x_1,x_1\overline{x}_2+\overline{x}_{1}x_2,x_2\overline{x}_0+\overline{x}_{2}x_0;\\ \eta\,x_0\overline{x}_1+\overline{\eta\,x}_{0}x_1, \eta\,x_1\overline{x}_2+\overline{\eta\,x}_{1}x_2, \eta\,x_2\overline{x}_0+\overline{\eta\,x}_{2}x_0).\end{multlined} \] The direct image of \(\beta\) is called the \textit{Hermitian Veronesean} \(\mathcal H\) of \(PG(2,\mathbf L)\) with respect to \(\mathbf L\). Any non-degenerate Hermitian curve \(c\) of \(PG(2,\mathbf L)\) in an obvious way defines a point-line geometry, say \(\mathcal C=(\mathcal P, \mathcal L)\), named the \textit{Hermitian unital over \(\mathbf L/\mathbf K\)} (usually the elements of \(\mathcal L\) are called blocks). The \(\beta\)-image of \(c\) generates a hyperplane \(H\) of \(PG(8,\mathbf K)\) and \({\mathcal H}\cap\,H\) is a Veronesean embedding of \(\mathcal C\) in \(PG(7,\mathbf K)\), one speaks of the \textit{standard Veronesean embedding} of \(\mathcal C\). The authors prove the following main theorem: Let \(\mathcal C=({\mathcal P},{\mathcal L})\) be a Hermitian unital over \({\mathbf L}/{\mathbf K}\). Then every Veronesean embedding of \(\mathcal C\) in \(PG(d,\mathbf F)\), with \(\mathbf F\) any field, and \(d\geq\,7\) is equivalent to the standard Veronesean embedding of \(\mathcal C\). As an application of the main theorem, the authors derive that the subsequent objects coincide: {\parindent6mm \begin{itemize}\item[(1)] the generic hyperplane sections of Hermitian Veronesean in an \(8\)-dimensional projective space, \item[(2)] the Grassmannians of the classical spreads of non-degenerate quadrics of Witt index \(2\) in a \(5\)-dimensional projective space, \item[(3)] the sets of absolute points of trialities of Witt index \(1\). \end{itemize}} As a consequence of this application it is shown: The set of absolute points of a triality without fixed lines, but with absolute points, determines the triality quadric and the triality itself uniquely.
scientific article; zbMATH DE number 5782186
scientific article