Discriminants of convex curves are homeomorphic

Abstract

For a given real generic curve $\ga: S^1\to \Bbb {RP}^n$ let $D_\ga$ denote the ruled hypersurface in $\Bbb {RP}^n$ consisting of all osculating subspaces to $\ga$ of codimension 2. A curve $\ga: S^1\to \Bbb {RP}^n$ is called convex if the total number of its intersection points (counted with multiplicities) with any hyperplane in $\Bbb {RP}^n$ does not exceed $n$. In this short note we show that for any two convex real projective curves $\ga_1:S^1\to\Bbb {RP}^n$ and $\ga_2:S^1\to\Bbb {RP}^n$ the pairs $(\Bbb {RP}^n,D_{\ga_1})$ and $(\Bbb {RP}^n,D_{\ga_2})$ are homeomorphic answering a question posed by V.Arnold.