On the rank index of projective curves of almost minimal degree

TL;DR

This paper studies the rank index of certain projective curves of almost minimal degree, establishing bounds and exact values based on the position of the projection point, contributing to the understanding of their algebraic properties.

Contribution

It determines the rank index of these curves is at most 4 and exactly 3 when the projection point is a coordinate point, providing new insights into their algebraic structure.

Findings

Rank index of the curves is at most 4.

Rank index equals 3 when the projection point is a coordinate point.

Analysis of the case where the projection point lies on the third secant variety.

Abstract

In this article, we investigate the rank index of projective curves $\mathscr{C} \subset \mathbb{P}^r$ of degree $r+1$ when $\mathscr{C} = \pi_p (\tilde{\mathscr{C}})$ for the standard rational normal curve $\tilde{\mathscr{C}} \subset \mathbb{P}^{r+1}$ and a point $p \in \mathbb{P}^{r+1} \setminus \tilde{\mathscr{C}}^3$. Here, the rank index of a closed subscheme $X \subset \mathbb{P}^r$ is defined to be the least integer $k$ such that its homogeneous ideal can be generated by quadratic polynomials of rank $\leq k$. Our results show that the rank index of $\mathscr{C}$ is at most $4$, and it is exactly equal to $3$ when the projection center $p$ is a coordinate point of $\mathbb{P}^{r+1}$. We also investigate the case where $p \in \tilde{\mathscr{C}}^3 \setminus \tilde{\mathscr{C}}^2$.