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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.

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Sándor, a quick (dumb) question. Is there an easy example where this is a stronger condition than projectively normal (lets assume that $X$ is a normal variety itself)? In particular, if $X$ is projectively normal, then I thought all the $$H^0(\mathbb{P}^N, O_{\mathbb{P}^N}(i)) \to H^0(X, L(i))$$ maps are surjective (Hartshorne, II, Ex 5.14)? Doesn't that imply that $R(X, L)$ is generated in degree $1$, since $R({\mathbb{P}^N}, O(1))$ is generated in degree $1$ and we now have a surjection between the two algebras? Or is the point you want a name for non-normal schemes/varieties? – Karl Schwede Apr 6 '13 at 11:00
Karl, you are perfectly right! The situation I have is somewhat more complicated and in order to make it reasonable I simplified the situation to a point that it lost the juice. I will have to think about the more general situation, but this is certainly a good point. Thanks! Why don't you post this as an answer, so I can accept it and with that close the question? – Sándor Kovács Apr 6 '13 at 14:45
up vote 4 down vote accepted

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.

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How about "linearly/projectively normal and normally generated" or "linearly/projectively normal and $N_0$"?

Mumford in "Varieties defined by quadratic equations" defines $X$ to be normally generated if the section ring is generated in degree $1$. Green and Lazarsfeld in "On the projective normality of complete linear series on an algebraic curve" call the same condition $N_0$.

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Thanks Piotr, I will check these out. Your suggestion sounds reasonable based on these references, but it's a bit long. Thanks, anyway. – Sándor Kovács Apr 6 '13 at 5:12
Also, I would like a name for the embedding, not for the section ring being generated in degree $1$. I understand that this is a tiny difference, but still... – Sándor Kovács Apr 6 '13 at 5:14

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