The normal bundle of a rational curve is determined by specific properties of the curve's singularities

Let D denote a rational curve of degree $d_D\ge 4$, with only ordinary singularities and spanning $P^3$. We consider the normal bundle $N_{\phi }$ where φ denotes a generically one-to-one parametrization of D. We study the parameter $d_{0D}$ that controls the splitting of $N_{\phi }$ in terms of line bundles. We find that specific properties of D, related to its singular points, characterize $d_{0D}$. These properties are: the multiplicities and the auxiliary curve associated to the largest multiplicity.