On the saturation sequence of the rational normal curve

Let $C\subseteq {\mathrm{\P }}^d$ denote the rational normal curve of order $d$. Its homogeneous defining ideal $I_C\subseteq \text{QQ}[a_0,\mathrm{.}\mathrm{.}\mathrm{.},a_d]$ admits an $S{L}_2$-stable filtration $J_2\subseteq J_4\subseteq \mathrm{.}\mathrm{.}\mathrm{.}\subseteq I_C$ by sub-ideals such that the saturation of each $J_{2q}$ equals $I_C$. Hence, one can associate to $d$ a sequence of integers $({\alpha }_1,{\alpha }_2,\mathrm{.}\mathrm{.}\mathrm{.})$ which encodes the degrees in which the successive inclusions in this filtration become trivial. In this paper we establish several lower and upper bounds on the ${\alpha }_q$, using \emph{inter alia} the methods of classical invariant theory.