The restricted tangent bundle of a rational curve on a quadric in $\textbf {P}^ 3$

Abstract:

Let ${\psi ^*}{T_{{{\mathbf {P}}^3}}}$ be the pull-back of the tangent bundle to ${{\mathbf {P}}^3}$ via a parametrization $\psi$ of a rational, reduced, irreducible curve $C$ in ${{\mathbf {P}}^3}$ contained in an irreducible quadric surface. Since $C$ is rational, the bundle ${\psi ^*}{T_{{{\mathbf {P}}^3}}}$ splits into the direct sum of three line bundles. In this paper we study the relationship between the degrees of the line bundles of the splitting of ${\psi ^*}{T_{{{\mathbf {P}}^3}}}$ and the geometry of the curve $C$.