Discriminants of Convex Curves Are Homeomorphic

For a given real generic curve γ : S1 → Pn let Dγ denote the ruled hypersurface in Pn consisting of all osculating subspaces to γ of codimension 2. In this short note we show that for any two convex real projective curves γ1 : S1 → Pn and γ2 : S1 → Pn the pairs (Pn, Dγ1 ) and (Pn, Dγ2 ) are homeomorphic. §0. Preliminaries and results Definition. A smooth curve γ : S → P is called nondegenerate or locally convex if the local multiplicity of its intersection with any hyperplane does not exceed n, i.e. in local terms γ′(t), ..., γ(t) are linearly independent at every t or its osculating complete flag is well-defined at every point. A curve γ : S → P is called convex if the total multiplicity of its intersection with any hyperplane does not exceed n. The set Conn of all convex curves in P forms 1 connected component of the space NDn of all nondegenerate curves if n is even and 2 connected components (since the osculating frame orients P) if n = 2k+1, see [MSh]. Different results about convex curves show that they have the most simple properties among all curves. In this paper we prove one more result of the same nature. Definition. A curve γ : S → P is called generic if at every point γ(t), t ∈ S one has a well-defined osculating subspace of codimension 2, i.e. in local terms γ′(t), ..., γ(n−1)(t) are linearly independent at every t. Note that any smooth curve γ : S → P can be made generic by a small smooth deformation of the map. The space NDn of all nondegenerate curves is enclosed in the space GENn of all generic curves and consists of several connected components. (The number of connected components in NDn equals 10 for odd n > 3 and equals 3 for even n > 2, see [MSh].) Definition. Given a generic γ : S → P we call by its standard discriminant Dγ ⊂ P the hypersurface consisting of all codimension 2 osculating subspaces to γ. In many cases (algebraic, analytic etc) the assumption of genericity in the definition of discriminant can be omitted. The following proposition answers the question posed by V.Arnold in [Ar2], p.37. 1991 Mathematics Subject Classification. Primary 14H50.