Characterization of nongeneric curve models

Characterize the curves C_0 in the projective space P^{n-1} that are nongeneric in the sense that the linear system governing the structure functions of an isotrivial cone structure of equation type admits nontrivial solutions, and hence that the rigidity conclusion of Theorem 1 fails.

Background

Theorem 1 establishes that an open dense subset of curves in P^{n-1}, for n>3, models only flat isotrivial cone structures arising from scalar ordinary differential equations. In the proof, nonflat possibilities are associated with nontrivial solutions of a linear system for the frame structure functions, modulo conformal rescaling.

The remark explains that the existence of such a nontrivial solution imposes a polynomial differential relation on the parametrizing functions of C_0. Although an example of the resulting condition is given in dimension four in terms of the osculating cone or tangential variety, a complete characterization of all exceptional, nongeneric curves is left unresolved.