On the rank index of projective curves of almost minimal degree

In this article, we investigate the rank index of projective curves $\mathcal{C}\subset {\mathbb{P}}^r$ of degree $r+1$ when $\mathcal{C}={\pi }_p(\tilde{\mathcal{C}})$ for the standard rational normal curve $\tilde{\mathcal{C}}\subset {\mathbb{P}}^{r+1}$ and a point $p\in {\mathbb{P}}^{r+1}\setminus {\tilde{\mathcal{C}}}^3$. 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 $\le k$. Our results show that the rank index of $\mathcal{C}$ 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 $p\in {\tilde{\mathcal{C}}}^3\setminus {\tilde{\mathcal{C}}}^2$.