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.
