On the saturation sequence of the rational normal curve

TL;DR

This paper investigates the saturation sequence of the rational normal curve's defining ideal, providing bounds on the degrees where the filtration stabilizes, using classical invariant theory techniques.

Contribution

It introduces bounds on the saturation degrees of the ideal's filtration, connecting algebraic geometry with classical invariant theory.

Findings

Established lower bounds on the saturation degrees.

Established upper bounds on the saturation degrees.

Connected saturation sequence properties with classical invariant theory.

Abstract

Let $C \subseteq \P^d$ denote the rational normal curve of order $d$. Its homogeneous defining ideal $I_C \subseteq \QQ[a_0,...,a_d]$ admits an $SL_2$-stable filtration $J_2 \subseteq J_4 \subseteq ... \subseteq I_C$ by sub-ideals such that the saturation of each $J_{2q}$ equals $I_C$. Hence, one can associate to $d$ a sequence of integers $(\alpha_1,\alpha_2,...)$ 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.