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Let $X$ be a variety over $\mathbb{C}$. Let $D$ be an effective divisor on $X$. I heard there is a natural rational map $X\dashrightarrow Proj (R)$ where $R=\oplus_{n=0}^\infty H^0(X,nD)$.

My question:

  1. How is this map defined?

  2. When is this a birational map?

  3. Is $Proj (R)$ the image of the morphism defined by the linear system $|D|$ as in Hartshorne?

Thank you for the help!!

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  1. The map is defined by sending $x \in X$ to the homogeneous ideal of sections vanishing at $x$. (Note that it could be the case that every nonconstant section vanishes at $x$, in which case the ideal is the irrelevant ideal, and the map is not defined at $x$.)

  2. The map is birational precisely when $D$ is big. Depending on your definition of big, that is a tautology, but there are other characterisations of big divisors that make this more informative. See Chapter 2 of Positivity in Algebraic Geometry by Lazarsfeld.

  3. $ \operatorname{Proj } (R)$ need not be the image of the map (not morphism in general) given by $\mid D \mid$, because $R$ need not be generated in degree 1. For example if $D$ is ample but not very ample, then the morphism defined by $\mid D \mid$ will be some finite morphism from $X$ whose image is unlikely to be $X$ itself, whereas $\operatorname{Proj}(R)$ will be isomorphic to $X$. The simplest example is to take $D$ to be an effective divisor of degree 2 on an elliptic curve.

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  • $\begingroup$ Thanks for the great answer. One follow up: in the case when $R$ is generated in degree $1$, we do have $Proj(R)$ is the image, right? ( I believe $R$ can be considered as a quotient by homogeneous ideal of $k[x_1,\ldots,x_n]$ where $n= h^0(X,D)$. $\endgroup$
    – Rust Z
    Commented May 15, 2017 at 2:46

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