Let $X$ be a projective variety over an algebraically closed field $k$ and $\mathscr L$ a line bundle on $X$. Its section ring,
$$
R(X,\mathscr L) = \bigoplus \_{n=0}^\infty H^0(X,\mathscr L^{\otimes n}),
$$
is a finitely generated graded $H^0(X,\mathscr O\_X)\simeq k$-algebra.


I wonder if the following condition has an established name:


Assume that $\mathscr L$ is very ample and let $\phi:X\hookrightarrow \mathbb P^N$ denote the embedding induced by the global sections of $\mathscr L$. So, in particular the map 
$$
H^0(X,\mathscr O\_{\mathbb P^N}(1))\rightarrow H^0(X,\mathscr L) \tag{$\star$}
$$ 
is surjective. Further assume that 


\begin{equation}
\text{$R(X,\mathscr L)$ is generated in degree $1$,} \tag{$\star\star$}
\end{equation}


that is, by the elements of $H^0(X,\mathscr L)$. In particular, then the embedding $\phi$ is projectively normal, but this is a stronger condition.


I would like to say something like "$\phi$ is *blah*, when this holds", so the question is:  

**Q:** Does this property/condition have an established name in the literature?


If not, I would probably say that "$\phi$ is a *linearly generated embedding* if this condition holds". My rationale for that name is that by $(\star)$ and $(\star\star)$ it follows that $R(X,\mathscr L)$ is the homogenous coordinate ring of $X$ corresponding to the embedding $\phi$
and that $R(X,\mathscr L)$ is generated by the images of linear functions on $\mathbb P^N$.


An ideal answer would give a reference (or more) where this is defined/used, or in absence of a reference would either support the name I am suggesting or argue against it and in that case would suggest an alternative.



Linearly generated embedding?
*This was a comment originally.*


The generation in degree 1 is the same as projective normality at least for normal varieties. 


For simplicity, assume that $X$ is normal.
By Hartshorne, Chapter II, Exercise 5.14, we know that if $X$ is projectively normal (with embedding associated to the complete linear system of $L$), then
$$
H^0(\mathbb{P}^N, O\_{\mathbb{P}^N}(i)) \to H^0(X, L^i)
$$
is surjective for all $i$. Therefore $R(X, L)$ is a quotient of $R(\mathbb{P}^N, O(1)) = S$. But $S$ is generated in degree $1$, and so $R(X,L)$ is generated in degree $1$ as well. 


