Projective Convexity in P Implies Grassmann Convexity

In this note we introduce the notion of Grassmann convexity analogous to the wellknown notion of convexity for curves in real projective spaces. We show that the curve in G2,4 osculating to a convex closed curve in P is Grassmann-convex. This proves that the tangent developable (i.e. the hypersurface formed by all tangents) of any convex curve in P has the ‘degree’ equal to 4. Here by ‘degree’ of a real projective hypersurface we mean the maximal total multiplicity of its intersection with a line. §1. Preliminaries and results Let us first recall some basic definitions. Definition. A smooth closed curve γ : S → P is called locally convex if the local multiplicity of intersection of γ with any hyperplane H ⊂ P at any of their intersection points does not exceed n = dimP and globally convex or simply convex if the above condition holds for the global multiplicity, i.e for the sum of local multiplicities. Local convexity of γ is a simple requirement equivalent to the nondegeneracy of the osculating Frenet n-frame of γ, i.e. to the linear independence of γ(t), ..., γ(t) for any t ∈ S. Global convexity is a rather nontrivial property studied under different names (e.g. disconjugacy of linear ordinary differential equations) since the beginning of the century. (There exists a vast literature on convexity and the classical achievements are well summarized in [4]. For one of the most recent developments see e.g. [1].) Let Gk,m denote the usual Grassmannian of k-dimensional real planes in R m (or equivalently, (k − 1)-dimensional planes in P). Definition. Given an (m − k)-plane L ⊂ R we call by the Grassmann hyperplane HL ⊂ Gk,m associated to L the set of all k-dimensional subspaces in R m nontransversal to L. Remark. Grassmann hyperplanes is a well-known classical concept in Schubert calculus, see e.g. [3]. More exactly, HL coincides with the union of all Schubert cells of positive codimension in Gk,m constructed using some complete flag containing L as a subspace. The complement to each HL is the open Schubert cell isomorphic to the standard affine chart in Gk,m. If n−k ≥ k then each HL considered in some other standard affine chart is isomorphic 1991 Mathematics Subject Classification. Primary 14H50.