On the number of real hypersurfaces hypertangent to a given real space curve

Abstract

Let $C$ be a smooth geometrically integral real algebraic curve in projective $n$-space $\mathbb{P}^n$. Let $c$ be its degree and let $g$ be its genus. Let $d$, $s$ and $m$ be nonzero natural integers. Let $\nu$ be the number of real hypersurfaces of degree $d$ that are tangent to at least $s$ real branches of $C$ with order of tangency at least $m$. We show that $\nu$ is finite if $s=g$, $gm=cd$ and the restriction map $H^0(\mathbb{P}^n,\mathcal{O}(d))\rightarrow H^0(C,\mathcal{O}(d))$ is an isomorphism. Moreover, we determine explicitly the value of $\nu$ in that case.