Secant rank and syzygies of projections of elliptic normal curves

We study the syzygies of projections of elliptic normal curves. Let $C\subset {\mathbb{P}}^{d-1}$ be an elliptic normal curve of degree $d\ge 5$, and let $C_q$ denote the projection of $C$ from a point $q$. We obtain sharp bounds for the Green–Lazarsfeld index of $C_q$ in terms of the secant rank of $q$. More precisely, if $q\in C^s\setminus C^2$, where $C^s$ is the $s$-th secant variety of $C$, then $\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(C_q)\le s-3$, and equality holds for a general point $q$ of $C^s$. In particular, $\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(C_q)=\lceil \frac{d}{2}\rceil -3$ for a general point $q$ in ${\mathbb{P}}^{d-1}$. The proof realizes projected elliptic curves as hyperplane sections of elliptic ruled surface scrolls and exploits the known syzygetic properties of these scrolls.