PROJECTIVE CURVES SUCH THAT A GENERAL POINT OF THE AMBIENT PROJECTIVE IS CONTAINED IN A UNIQUE OSCULATING HYPERPLANE OF THE CURVE

We study the non-degenerate integral curves X ⊂ Pn such that a general point of Pn is contained in a unique osculating hyperplane of X (they are a generalization of the strange curves to the case n > 2). AMS Subject Classification: 14N05, 1499


Introduction
Let X ⊂ P n , n ≥ 2, be an integral and non-degenerate curve defined over an algebraically closed field K.We ask when the natural map from the variety of (n − 1)-osculating spaces of X to P n has separable degree 1, i.e. for a general O ∈ P n there is a unique osculating hyperplane of X containing O. We call ♠ such a property.Set p := char(K).It is easy to check that X does not exists if p = 0 or if p > deg(X) (Proposition 4 and Remark 1).In positive characteristic we need to choose a definition of osculating variety and osculating space.We use [4], but essentially only use the existence of a non-empty open subset U ⊂ X reg and an integer x ≥ n such that for each P ∈ U there is a unique hyperplane O(X, P, n − 1) ⊂ P n with order of contact x with U at P , while all other hyperplanes have order of contact < x with X at P .This shows an ambiguity of the definition of ♠: unique hyperplane or unique P ∈ U such that O ∈ O(X, P, n − 1)?We take the second one as the definition of ♠, but usually it is easy to modify the proofs to adapt to the other interpretation (we call ♥ the alternative interpretation).We prove the following results.
Proposition 1. X has property ♥ if and only if there is a codimension 2 linear subspace M ⊂ P n such that O(X, P, n − 1) ⊃ M for a general P ∈ U .X has ♠ if and only if it has ♥ and for a general P ∈ U the hyperplane O(X, P, n − 1) is not an osculating hyperplane of X at another point of U .
(a) X has property ♥ if and only it is a strange curve.(b) X has property ♠ if and only if it is a strange curve and the linear projection from its strange point has separable degree 1.
In particular by a theorem of Lluis ([5]) the case p = 2 and X a smooth conic is the only case with n = 2, ♥ and X smooth (it also has ♠).
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).

The Proofs
Proposition 3. If X has ♠, then it is rational, i.e. its normalization has genus 0.
Proof.Let f : C → X be the normalization map.Write P n = P(V ∨ ) with V an (n+1)-dimensional vector space.Let x be the degree of the intersection at a general P ∈ X of the osculating hyperplane O(X, P, n − 1) and ℘ := P (X, x − 1, f * (O X (1))) the bundle of principal parts of order x − 1 of the line bundle f * (O X (1)).The composition C → X ֒→ P n induces a map W ⊗ O C → ℘ whose image is a rank n vector bundle E on C and a surjective map u ′ : P(E) → P n with separable degree one.The projection P(E) shows that C has genus 0.
Proof of Proposition 1: First assume that M exists.For any O ∈ P n there is a unique hyperplane containing both M and O. Since X is non-degenerate, we get that P n is the image of the (n − 1)-osculating variety of X. Hence X has ♥.Now assume that X has ♥.Fix a general O ∈ P n and take any P ∈ U with O ∈ O(X, P, n − 1).Let S ⊂ X be the set of all Q ∈ U with O(X, Q, n − 1) = O(X, P, n − 1).S is a finite set.Fix any e ∈ U \ S and set M := O(X, P, n − 1) ∩ O(X, e, n − 1).Since e / ∈ S, we have O(X, e, n − 1) = O(X, P, n − 1) and M is a hyperplane of O(X, P, n − 1) not containing O. Since . Therefore X has ♥ with M as associated codimension two linear subspace.The statement concerning ♠ follows from the one concerning ♥.Let X ⊂ P n an integral non-degenerate curve.Fix an open subset U ⊆ X reg on which the osculating sequence in the sense of [4] is the general one for X.Fox each P ∈ U and each integer t ∈ {1, . . ., n − 1} let O(X, P, t) be the t-dimensional linear osculating subspace to X at P .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 , . . ., P t ∈ P n such that for each t = 1, . . ., n − 1 a general t-osculating space of X contains the linear span of {P 1 , . . ., P t }; call ♣ this property.Note that the construction will not depend, up to a projective equivalence, from the choice of the ordered (n − 1)-ple of linearly independent points.If n = 2, then we use Proposition 2 and the construction of strange plane curves done in [3].Now assume n > 2 and that the construction is done in P n−1 .Let ℓ : P n \ {P 1 } → P n−1 be the linear projection from P n .Set O i := ℓ(P 1 ), 2 ≤ i ≤ n − 1.The point O 2 , . . ., O n−1 are linearly independent.Take Y ⊂ P n−1 with ♣ with respect to the points O 2 , . . ., O n−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).The curve discovered by J. Rathmann ([6, Example 1.2]) has ♣ (see [2,Example 1] for the explicit equations of Y ).
Example 2. If n > 2 there are non-strange curves with ♠.Here there is a way to construct them.Fix a codimension two linear subspace M ⊂ P n .Choose a system of homogeneous coordinates x 0 , . . ., x n such that For general f 1 , . . ., f n−1 the scheme X has dimension one and hence it is a locally complete intersection and deg(X) = p e 1 +•••+e n−1 .For general f 1 , . . ., f n−1 we have X ∩ M = ∅ (e.g. if e i = e for all i it is sufficient that the (n − 1) × (n − 1) matrix with c 1/e i,j as entries is invertible).The linear projection ℓ : P n \ M → P 1 from M induces a generically bijective X red → P 1 .Therefore to check that X is an integral curve it is sufficient to check that it is smooth at a general P ∈ X. Fix P ∈ X reg and let H be the hyperplane spanned by P and M .We have (H ∩ X) red = {P }, because M ∩ X = ∅ and ℓ|X has separable degree one.Hence H is the osculating hyperplane to X at P .

Proof of Proposition 2 :
Part (a) is the case n = 2 of Proposition 1. Part (b) follows from part (a).

Proposition 4 .Remark 1 .
In characteristic zero there is no curve with ♥.Proof.If n = 2, just use Proposition 2 and that in characteristic zero no curve, except lines, have a strange point.Now assume n > 2 and take M as in Proposition 1.We have x = n − 1 by [4, Theorem 15].Fix a general Q ∈ M and call ℓ : P n \ {Q} → P n−1 and set Y := ℓ(X).Y is a non-degenerate curve whose general osculating hyperplane has order of contact n + 1 with the curve at the contact point, contradicting [4, Theorem 15].Assume p > 0. The proof of Proposition 4 and [4, Theorem 15] gives deg(X) ≥ p for each curve X with ♥.