Two-strange projective curves

Abstract

The notion of second order strange projective curve in P n , n м 3, is introduced. A non-degenerate curve C in P n , n м 3, is said to be a second order strange curve if it admits osculating conics and such that all its osculating conics pass through a common point called 2-strange point. An existence-construction result is provided and some examples are furnished. Introduction.Let K be an algebraically closed field of characteristic p > 0. A strange curve in the n-dimensional projective space P n is a non-linear irreducible projective curve, with the property that all its tangent lines at simple points pass through a given point called center or strange point.Such curves were studied by E. Lluis in [9], showing that all strange curves are singular, except for conics in characteristic two, see also [11] and [6], [2], [1], for further developments.In this paper, we present a generalization of strange curves. Precisely, we introduce the notion of second order strange curve in P n .A non-degenerate curve C in P n , n м 3, is said to be a second order strange curve (or 2-strange curve) if it admits osculating conics and such that all its osculating conics pass through a common point (2-strange point).We stress that this is a characteristic p phenomenon not possible in characteristic zero and as far as we know, this is the stronger and more general non-linear generalization of the notion of strange curve. We are able not only to prove the existence of such curves, but also to construct them explicitly. Our curves are contained in cones of P n with vertex a point P and basis a curve in a hyperplane H of P n , not containing P. Choosing any curve D with a birational model in H, we find a curve C in the cone over D which is birational to D.We recall some preliminary notions. Let C be a non-degenerate irreducible curve in P n . Then for a general point P of C, there is a complete flag of linear subspacesis called the (Hasse) sequence of invariants of C , [12], [8], [14], [7], [3].In Section 2 we discuss the notion of generic osculating conic to a curve at a point P, giving conditions on uniqueness.