Generators of a graded algebra defining bundle over elliptic curve

Question

I have a question about a statement from P. Wagreich's paper "Elliptic Singularities of Surfaces" (page 425):

We consider an elliptic curve $X$ and a line bundle (=invertible sheaf) $L$ on $X$.

Then, following EGA one finds that $V=Spec(\oplus_n \Gamma(X,L^{\otimes -n}))$ (it's a question of convention; often author's assign instead the scheme $Spec(\oplus_n \Gamma(X,L^{\otimes n}))$).

My first question is what does the author mean by the bundle $L \to X$?

I thought that using GAGA-principles in order to "identify" invertible sheaves with line bundles the corresponding line bundle $B_L$ to $L$ over $X$ is nothing but $Spec(\oplus_n \Gamma(X,L^{\otimes -n}))$ so $V$. See for example here

But on the other hand $V$ is introduced by the author as already the resolution of the "bundle" $B_L \to X$. This confuses me. So if $V$ is already the resolution what is the associated bundle $B_L \to X$ to $L$?

Second question: I don't understand how to verify that the graded algebra $\oplus_n \Gamma(X,L^{\otimes -n})$ is generated by $1 \in \Gamma(X,L^{-1}, \wp \in \Gamma(X,L^{-2}), \wp ' \in \Gamma(X,L^{-3})$

Honestly, I have no idea. Does it arise from a general principle for line bundles over elliptic curves?

Remark: I have already asked the same question here: https://math.stackexchange.com/questions/3303909/generators-of-a-graded-algebra

Answer 1

To answer your second question: it follows from the Riemann-Roch theorem. Since $\deg(L)=-1$ we have $\Gamma(X,L^m)=0$ for $m\geq0$. Therefore, by RR on $X$, $\dim \Gamma(X,L^{-m}) = 1 + \deg(L^{-m}) - g(X) = m$. Now, in order to have $\dim \Gamma(X,L^{-m}) = m$ we see that $V=\bigoplus_{m\geq0} \Gamma(X,L^{-m})$ must be generated by at least $x\in\Gamma(X,L^{-1})$, $y\in\Gamma(X,L^{-2})$, $z\in\Gamma(X,L^{-3})$, where $$\Gamma(X,L^{-1}) = \langle x\rangle, \quad \Gamma(X,L^{-2}) = \langle x^2,y\rangle, \quad \Gamma(X,L^{-3}) = \langle x^3,xy,z\rangle, \quad \Gamma(X,L^{-4}) = \langle x^4,x^2y,y^2,xz\rangle, \quad \text{etc.}$$ We have $\dim \Gamma(X,L^{-6})=6$, but we find seven monomials $x^6,x^4y,x^2y^2,y^3,x^3z,xyz,z^2\in \Gamma(X,L^{-6})$ so a relation must hold between them. (In the paper you cite, $x,y,z$ are taken to be $1,\wp,\wp'$ and the relation is $z^2 - 4y^3 + g_2x^4y + g_3x^6=0$.) Since the Hilbert series of $V$ is $1 + \sum_{m\geq1} mt^m = \frac{1-t^6}{(1-t)(1-t^2)(1-t^3)}$ we expect that these three generators with this relation in degree 6 generate the whole ring, as they do.