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Let $X$ be a projective variety over $\mathbf C$ and assume it has all the good hypotheses one can wish for, and let $ \mathcal{L} $ be a line bundle on $ X $. One can consider the complete linear system associated to $ \mathcal{L} $, namely the projective space $$ \mathbb{P}(H^0(X, \mathcal{L})), $$ whose points correspond to effective divisors linearly equivalent to $ \mathcal{L} $. This defines a rational map $$ \varphi_{\mathcal{L}}: X \dashrightarrow \mathbb{P}(H^0(X, \mathcal{L})^*) $$ given by evaluating a basis of global sections of $ \mathcal{L} $ at each point of $ X $.

On the other hand, one can consider the section ring of $\mathcal{L} $, $$ R(X, \mathcal{L}) := \bigoplus_{n \geq 0} H^0(X, \mathcal{L}^{\otimes n}), $$ which is a graded ring. Taking its Proj yields a scheme $$ \text{Proj}(R(X, \mathcal{L})), $$ and $ \mathcal{L} $ induces a natural (possibly rational) map $$ \psi_{\mathcal{L}}: X \dashrightarrow \text{Proj}(R(X, \mathcal{L})). $$

My question is: What is the precise relationship between the rational map $ \varphi_{\mathcal{L}} $ defined by the complete linear system and the map $ \psi_{\mathcal{L}} $ to $ \text{Proj}(R(X, \mathcal{L})) $?

In particular:

  1. When do these maps agree or factor through each other?
  2. How does the structure of the section ring (e.g., generation in degree 1) influence this relationship?
  3. Can one view the linear system as defining the degree-1 part of $ \text{Proj}(R(X, \mathcal{L})) $? If so, how precisely?

Any clarification or references would be appreciated!

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  • $\begingroup$ for non experts: Include an example where the maps $\phi_L, \psi_L$ are not "the same map". $\endgroup$ Commented May 31 at 16:09
  • $\begingroup$ ..for projective space they define the "same" map. $\endgroup$ Commented May 31 at 16:31
  • $\begingroup$ Clearly there are cases they don't agree, but $\phi$ probably factors through $\psi$. You probably already noted this but if $\mathcal{L}$ is effective this can be seen by noting that the graded ring has generators in some bounded degree, then take sections of $H^0(X, \mathcal{L}^{\otimes n})$ so that the first bunch are living in $H^0(X, \mathcal{L})$. Probably this generalizes, I can't see it though $\endgroup$ Commented May 31 at 17:28
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    $\begingroup$ The first morphism is toward ${\rm Proj}({\rm Sym}^\bullet(R_1))$. So the first one factors through the second one via ${\rm Proj}(\cdot)$ applied to the natural map of graded ring ${\rm Sym}^\bullet(R_1)\to R_\bullet$ This last map is surjective if $R_\bullet$ is generated by $R_1$, and then the same map induces a closed immersion. $\endgroup$ Commented May 31 at 18:56
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    $\begingroup$ Also, if the line bundle is ample, then the second morphism is an isomorphism, provided the line bundle is generated by its global sections. See stacks.math.columbia.edu/tag/01Q1 $\endgroup$ Commented May 31 at 19:10

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The word you want is “projectively normal”. When L is very ample, the section ring is the integral closure of the homogeneous coordinate ring. There is a Hartshorne exercise about this in Chapter 2 somewhere, possibly exercise II.5.14.

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