Secant rank and syzygies of projections of elliptic normal curves

TL;DR

This paper investigates the syzygies of projections of elliptic normal curves, establishing bounds for the Green--Lazarsfeld index based on secant rank, and characterizes these bounds for general points.

Contribution

It provides sharp bounds for the Green--Lazarsfeld index of projected elliptic curves in terms of secant rank, using geometric realizations via elliptic ruled surface scrolls.

Findings

Bounds for the Green--Lazarsfeld index in terms of secant rank.

Equality of index and bounds for general points.

Explicit formula for the index of general projections.

Abstract

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 $\mathrm{index}(C_q) \le s-3$, and equality holds for a general point $q$ of $C^s$. In particular, $\mathrm{index}(C_q) = \lceil \frac{d}{2} \rceil - 3$ 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.