Very Strange Projective Curves

Projective Curves Such That a General Point of the Ambient Projective Is Contained in a Unique Osculating Hyperplane of the Curve

“…Example 1. We point out how to use [3] and [1] to construct all X with ♥ (or ♠) for which there are n − 1 linearly independent points P 1 , . .…”

“…Fix integer m ≥ 0, s ≥ 1 and e > 0. The construction in [1] gives all strange curve X ⊂ P n with P 1 as their strange point, with Y as their image by the linear projection ℓ and such that ℓ|(X \ {P 1 } has separable degree s and inseparable degree p e . Among these curves the ones with ♠ are the one for which at each step the separable degree, s, is 1 (you need to start a strange plane curve with separable degree 1).…”

“…We recall that there is a construction of all strange plane curves ( [3] for n = 2, [1] when n > 2); in the case n = 2 it involves the multiplicity µ ≥ 0 of X at the strange point, o, the separable degree s of the the rational map τ induced on the normalization of X from the linear projection from o and the inseparable degree p e of τ (we have deg(X) = µ + sp e ). We do not know a way to construct all curve with ♥ or ♠, but we have a way to construct two classes of such curves (see Examples 1 and 2).…”