TITLE: Again: Ample and very ample line bundles
QUESTION [7 upvotes]: I am still on my quest to design a simple example of an ample, globally generated line bundle which is not very ample and also understand the global sections of its smallest, very ample tensor power. Let $\Bbbk$ be some algebraically closed field, Consider 
$$R = \Bbbk[x,y,z]/(yz-x^2)$$
and the projective parabola $X=\operatorname{Proj}(R)$. Let $I=\langle y,z\rangle\subseteq R$ be the homogeneous ideal generated by $y$ and $z$, and consider the ideal sheaf $\newcommand{\cL}{\mathscr{L}}\cL:=I(1)^\sim$, as in, $\cL(X)=I_1=\Bbbk y \oplus \Bbbk z$. Observe that it is globally generated because $y$ and $z$ do not vanish simultaneously on $X$, otherwise $x$ would have to vanish as well. However, of course, the morphism $X\to\newcommand{\P}{\mathbb P}\P^2$ defined by $[x:y:z]\mapsto [y:z]$ is a finite surjection of degree $2$.
Now, there must be some $k\in\mathbb N$ (probably $k=2$) such that $\cL^{\otimes k}$ is very ample. However, I don't understand how it works, in this very concrete example. 

My question is basically: what are the global sections of $\cL^{\otimes k}$ and does $\cL^{\otimes 2}$ contain functions $xz$ and $xy$? 

That would be wonderful, but I can't see how to define either of them on both $X_y$ and $X_z$. More precisely, I would think that
$$\cL^{\otimes 2}(X_y)=\left\{ \frac{fz^2}{y^n} \:\middle\vert\: f\in R_n \right\}$$
but if $xz=fz^2y^{-n}$ then $x=fzy^{-n}$ and this is hard to do when we only have a relation for $x^2$. I must be mistaken somewhere, and this is driving me crazy.

REPLY [3 votes]: $X$ is isomorphic to $\Bbb P^1(\Bbb k)$,by the maps induced by $(u:v) \mapsto (uv:u^2:v^2)$ and $(x:y:z) \mapsto (y:x) = (x:z)$.
$x$ is a global section of $\cL$, because  $x = xy/y \in \cL(X_y)$ and $x=xz/z \in \cL(X_z)$, and those two are equal on $X_y \cap X_z = X_x$ : $xz/z = xyz/yz = xy/y$, so $x \in \cL(X)$. Actually, the pulled back sheaf of $\cL$ to $\Bbb P^1(\Bbb k)$ is $\mathcal O_{\Bbb P^1(\Bbb k)}(2)$, which is ample and also very ample.