Degree $5$ embeddings of genus 1 curves, plucker embedding of a Grassmannian... Here is something Ravi Vakil says after 19.9F in his notes. He is talking about embeddings of a genus 1 curve $C$ by complete linear series associated to the divisors $O(nP)$.
"The beautiful structure doesn’t stop with degree 4, but it gets more complicated. For example, the degree 5 embedding is not a complete intersection (of hypersurfaces),but is the complete intersection of G(2,5) under its Plucker with five hyperplanes (or perhaps better, a codimension 5 linear space)."
2 questions about this:


*

*The degree 5 embedding is a map of $C$ into $\mathbb{P}^4$. The plucker embedding of $G(2,5)$ lives in $\mathbb{P}^9$. So I am not sure what he means when he says the degree 5 embedding "is" the intersection of the Plucker embedding with the planes. Currently I would be willing to believe a statement along the lines of: if $X$ is the Plucker embedding of the Grassmannian $G(2,5)$, then the intersection of $X$ with 5 generic hyperplanes gives a smooth genus one curve. I guess there is something I am missing.

*Does anyone know a good reference for this? I am fascinated by this image. (I know about the papers by Tom Fischer that deal with the description using Pfaffians, but it doesn't seem to be obviously connected to the description with the plucker embedding of the Grassmannian. Besides, I am still confused about that description, as hopefully I communicated in point 1.)
Thanks!
 A: Let $W,V$ be $5$ dimensional vector spaces. Let $f\colon \wedge^2W\to V$ be a linear map. 
Consider the following diagram:
$$\require{AMScd}
\begin{CD}
E @>{i}>> \mathbb{P}(V)\\
@VVV @VVV \\
\mathrm{Gr}(2,W) @>{\mathrm{Plu}}>> \mathbb{P}(\wedge^2W)
\end{CD}$$
To show $E$ is an elliptic curve, we calculate its canonical bundle by adjunction formula:
$$K_E=K_{\mathrm{Gr}(2,W)}|_E+\det(N_{E/\mathrm{Gr}(2,W)}|_E).$$
Note that $$N_{E/\mathrm{Gr}(2,W)}|_E=i^*N_{\mathbb{P}(V)/\mathbb{P}(\wedge^2W)}=i^*\mathcal{O}(1)^{\oplus5}$$
Also note that $$K_{\mathrm{Gr}(2,W)}=\mathrm{Plu}^*(\mathcal{O}(-\dim W))=\mathrm{Plu}^*(\mathcal{O}(-5))$$
So $K_E=0$, $E$ is an elliptic curve. 
(For the relation with Pfaffian: Pick basis $\{w_1,\dots, w_5\}$ of $W$, we can write $f=a_{ij}\cdot w_i\wedge w_j$, where $a_{ij}=-a_{ji}\in V.$ So we view $a_{ij}$ as functions on $\mathbb{P}(V)$. 
The Pfaffian of all the $4\times 4$ minors in $(a_{ij})$ vanishes
 exactly when $\mathrm{rk}(a_{ij})=2$. In this case, vanishing locus of Pffafian coincides with fiber product with Grassmannian.)
