Geometry of elliptic normal curves of degree 6

TL;DR

This paper investigates the geometry of elliptic normal curves of degree 6 in projective 5-space, determining their defining equations, properties under projection, and the algebraic structure of their secant varieties.

Contribution

It explicitly computes the equations of quadrics through the curve and describes the generators of the ideal of the secant variety, providing new insights into their algebraic and geometric properties.

Findings

The space of quadrics through the curve is characterized.

Generators of the secant variety's ideal are explicitly identified.

Projections of the curve exhibit specific normality and ideal generation properties.

Abstract

In our work we focus on the geometry of elliptic normal curves of degree 6 embedded in $\mathbb{P}^5$. We determine the space of quadric hypersurfaces through an elliptic normal curve of degree 6 and find the explicit equations of generators of $I(\text{Sec}(C_6))$. We study the images $C_p$ and $C_{pq}$ of a sextic $C_6$ under the projection from a general point $P \in \mathbb{P}^5$ and a general line $\overline{PQ} \subset \mathbb{P}^5$. In particular, we show that $C_p$ is $k$-normal for all $k \geq 2$ and $I(C_p)$ is generated by three homogeneous polynomials of degree 2 and two homogeneous polynomials of degree 3. We then show that $C_{pq}$ is $k$-normal for all $k \geq 3$ and $I(C_{pq})$ is generated by two homogeneous polynomials of degree 3 and three homogeneous polynomials of degree 4.