Sextactic and type-9 points on the Fermat cubic and associated objects (Q6639804)

Recall that for any arbitrary smooth plane curve \(C\) of degree \(d\geq 3\) a point \(p \in C\) that is not an inflection point is called sextactic if there exists an irreducible hyperosculating conic \(D\) at \(p\), i.e. the intersection index of curves \(C,D\) at \(p\) is equal to \(i_{p}(C,D)\geq 6\). A point \(p\) on a curve \(C\) is called of type \(9\) if there exists an irreducible plane cubic curve which intersects \(C\) at \(p\) with multiplicity \(9\). In the paper under review the authors find exactly \(72\) points of type \(9\) to the Fermat cubic curve \(F : x^{3} + y^{3} +z^{3}=0\) and they study the line arrangements associated with these points of type \(9\) and the sextactic points to \(F\). Moreover, the authors study the geometry of conic arrangements associated with the sextactic points and points of type \(9\). Among others, they show that for any type \(9\) point \(p \in F\) there are exacly \(9\) conics which intersect \(F\) with multiplicity \(3\) at \(p\) and another type \(9\) point \(p'\). All these \(324\) conics intersect with multiplicity \(9\) in \(72\) so-called shadow points off the Fermat cubic and \(1944\) ordinary double points.