Lines having contact four with a projective hypersurface (Q1111633)

Let \(X\subset {\mathbb{P}}^{n+1}({\mathbb{C}})\) be a projective hypersurface and \(p\in X\). The third contact cone of X at p, \(C_ p^ 3\), is the set of all lines in \({\mathbb{P}}^{n+1}\) having contact \(\geq 4\) with X at p. If dim(X)\(\geq 3\) then the map \(p\mapsto (projective\quad moduli\quad of\quad C_ p^ 3)\) usually is a local immersion [answering a conjecture of \textit{P. Griffiths} and \textit{J. Harris}, in Ann. Sci. Éc. Norm. Supér., IV. Sér. 12, 355-452 (1979; Zbl 0426.14019)], and one can prove a rigidity theorem: X is determined by the projective moduli of its \(C_ p^ 3\)'s and certain fourth order invariants. This immersion property may fail e.g. if X is a homogeneous space. We study this case also.