All cubics are self-isotomic
In [J. Geom. 105, No. 2, 243--271 (2014; Zbl 1301.51029)], the authors show that every planar cubic \(\mathcal J\) is self-isogonal with respect to some triangle on \(\mathcal J\). Here, the authors show that \(\mathcal J\) is also self-isotomic. Their proof uses the self-isogonal transform of \(\mathcal J\) but does not fit into the general framework to study Cremona transformations that fix \(\mathcal J\), as developed in above paper. This singular triangle of the self-isotomic transform can be constructed as follows: Denote by \(G\) a pole of the line at infinity \(L_\infty\) with respect to \(\mathcal J\) and by \(P\), \(Q\), \(R\) the intersection points of \(\mathcal J\) and \(L_\infty\). There exists an isogonal transform \(\mu\) with \(\mu(G) = G\), \(\mu(\mathcal J) = \mathcal J\) and a conic \(\mathcal K\) through \(\mu(P)\), \(\mu(Q)\), and \(\mu(R)\) such that \(G\), \(L_\infty\) are dual with respect to \(\mathcal K\). The residual intersection points \(A\), \(B\), \(C\) of \(\mathcal K\) with \(\mathcal J\) determine the singular triangle of an isotomic transform that fixes \(\mathcal J\). The authors do not explicitly mention this but it seems that their results are true in projective planes over algebraically closed fields as otherwise the pole \(G\) does not need to exist.
scientific article; zbMATH DE number 6706780