A rational sextic associated with a Desargues configuration
In the complex projective plane \({\mathcal P}= \text{PG}_2 \mathbb{C}\), consider a general Desargues configuration \({\mathcal D}\) (defined by two perspective triangles such that none of the 10 lines contains more than three of the ten points of \({\mathcal D}\)). There is a unique orthogonal polarity \(\delta\) of \({\mathcal P}\) mapping each of the given perspective triangles onto the other one (and hence inducing a duality of \({\mathcal D}\)). The absolute points of \(\delta\) form a conic \(C\). The projective group \(\Lambda= \text{PGL}_3 \mathbb{C}\) is sharply transitive on quadrangles. For each point \(u\) outside \({\mathcal D}\) (not on a line of \({\mathcal D}\)) there is a unique projective collineation \(\lambda_u\in \Lambda\) fixing \(u\) and mapping the first triangle of \({\mathcal D}\) onto the second one. In general, the fixed points of \(\lambda_u\) form a triangle. The author determines the set \(U\) of those points \(u\) for which \(\lambda_u\) has no more than two fixed points. He proves that \(U\) consists of two components, one being the conic \(C\), the other a rational curve \(S\) of degree 6 which has a node at each point of \({\mathcal D}\). In suitable coordinates with respect to \({\mathcal D}\), a rational parametric representation of \(S\) is given; it has been found using MACSYMA. In the last two sections, the author discusses questions of reality and special Desargues configurations (with 1, 2, or 3 points of \({\mathcal D}\) belonging to \(C\)).
scientific article; zbMATH DE number 663757