The singularities of the $3$-secant curve associated to a space curve

Abstract:

Let $C$ be a curve in ${P^3}$ over an algebraically closed field of characteristic zero. We assume that $C$ is nonsingular and contains no plane component except possibly an irreducible conic. In [${\mathbf {GP}}$] one defines closed $r$-secant varieties to $C$, $r \in N$. These varieties are embedded in $G$, the Grassmannian of lines in ${P^3}$. Denote by $T$ the $3$-secant variety (curve), and assume that the set of $4$-secants is finite. Let $\tilde T$ be the curve obtained by blowing up the ideal of $4$-secants in $T$. The curve $\tilde T$ is in general not in $G$. We study the local geometry of $\tilde T$ at any point whose fibre of the blowing-up map is reduced at the point. The multiplicity of $\tilde T$ at such a point is determined in terms of the local geometry of $C$ at certain chosen secant points. Furthermore we give a geometrical interpretation of the tangential directions of $\tilde T$ at a singular point. We also give a criterion for whether all the tangential directions are distinct or not.