The Singularities of the 3-secant Curve Associated to a Space Curve

Let C be a curve in P3 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 [GP] one defines closed r-secant varieties to C, r £ N. These varieties are embedded in G, the Grassmannian of lines in P3. Denote by T the 3-secant variety (curve), and assume that the set of 4-secants is finite. Let T be the curve obtained by blowing up the ideal of 4-secants in T. The curve T is in general not in G. We study the local geometry of T at any point whose fibre of the blowingup map is reduced at the point. The multiplicity of 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 T at a singular point. We also give a criterion for whether all the tangential directions are distinct or not.