Geometry of elliptic normal curves of degree 6

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.