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If $L$ is a line bundle on a scheme $X$, what is the ring $A = \oplus \Gamma(X, L^{ \otimes n})$? This ring comes up in an exercise that I am struggling with right now, and I would like some insight into what this ring ... "is." What is the motivation for considering it? Maybe there is some geometric insight that I am missing? How should I think about it?

And also, if $F$ is a quasi-coherent sheaf, $M = \oplus \Gamma(X, F \otimes L^{\otimes n})$ is an $A$ module. Again, what is the geometric meaning of this construction?

My thoughts: If $X$ is projective space, and $L$ a line bundle of positive degree, then $A$ is some veronese subring of the homogeneous coordinate ring. What is $M$ in that case? How does $\tilde{M}$ relate to the sheaf $F$?

(Sorry, I know this is a poorly posed question. I am just getting really confused with the corresponding exercise 13.3H in Ravi's notes - I don't want to ask for a solution to that though. I think I know how a proof should go, but the objects involved are confusing me.)

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I'm not sure this constitutes a complete answer, but at least let me give you some examples and remarks that indicate the importance of this construction (and, hopefully, therefore also a little bit of the geometric intuition behind it).

It's probably useful to play around with it a bit more, beyond the examples I give. What happens for example if $X$ is affine?

Example. Let $\mathscr L$ is a torsion line bundle (say on a smooth projective variety $X$), e.g. $\mathscr L^{\otimes 2} \cong \mathcal O$, but $\mathscr L \not\cong \mathcal O$. Then $\Gamma(X, \mathscr L^{\otimes n})$ is $0$ for all $n$ odd, and $1$-dimensional for $n$ even. This is generated by a nonzero section of $\Gamma(X,\mathscr L^{\otimes 2})$.

Example. If $X$ is a projective scheme and $\mathscr L = \mathcal O(1)$ is some very ample line bundle, then $$A = \bigoplus_{n = 0}^\infty \Gamma(X,\mathscr L^{\otimes n})$$ is the affine coordinate ring of $X$ with respect to the given projective embedding. We know that this ring might depend on the embedding; however, it is always finitely generated (even if $\mathscr L$ is merely ample as opposed to very ample: exercise).

Erratum. The above example is not quite correct, as pointed out by Daniele A. The ring $A$ only equals the affine coordinate ring of $X$ with respect to the given embedding if the embedding is projectively normal. In general, they are isomorphic in large enough degree, hence the ring $A$ is still finitely generated (but not necessarily in degree $1$).

Remark. On the other hand, one can turn this around and ask if we can use this ring to define a projective embedding of $X$, if we have no idea what $X$ is. This is an idea that has proven very useful in the minimal model programme (MMP): if we let $\mathscr L = \omega_{X/k} = \Omega_{X/k}^n$, then we could try to ask whether the natural morphism $$X \to \operatorname{Proj} \bigoplus_{n=0}^\infty \Gamma(X, \omega_{X/k}^{\otimes n})$$ is an isomorphism, or at least a birational map. The right-hand side is called the canonical model of $X$.

Example. If $X = \mathbb P^n$, then this is obviously not true. Indeed $\omega_X = \mathcal O(-n-1)$, none of whose tensor powers has any sections. That is, the right hand side is just a point.

There are examples where the right hand side (if finitely generated) can have any dimension between $0$ and $n$ (the dimension of $X$); this is then the Kodaira dimension of $X$ (although this is not literally the definition of Kodaira dimension).

Remark. On the other hand, it is a celebrated (and very recent!) theorem of Birkar–Cascini–Hacon–McKernan that the canonical ring is finitely generated, whenever $X$ is smooth and projective. This result was obtained independently by Siu using analytic methods in the case where $X$ is of general type. (Don't worry, I don't understand any of the words the two papers use either.)

Reid had already proven (in 1980): if $X$ is smooth, proper, and of general type, under the assumption of finite generation of the canonical ring, the morphism $X \to \operatorname{Proj} \bigoplus \Gamma(X,\omega_{X/k}^{\otimes n})$ is birational.

These two results together more or less solve the minimal model programme for varieties of general type.

Remark. Mark has indicated below that there exist examples of line bundles for which the ring $\bigoplus \Gamma(X,\mathscr L^{\otimes n})$ is not finitely generated. A place to read about this example is Lazarsfeld's Positivity in Algebraic Geometry I, Example 2.3A (p.158).

The idea is to construct a divisor $D$ on a surface $X$ such that the base locus of $mD$ always contains the same curve $C$, but $mD - C$ is base-point free. Then the ring $\bigoplus \Gamma(X,\mathcal O(mD))$ cannot be finitely generated, because otherwise the multiplicity of $C$ in the base locus would go to infinity.

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  • $\begingroup$ Nice answer, but the bit about $2$-torsion seems off to me: you need a generator in degree $0$, but that's it; it's $0$ in odd degrees, you don't need generators there. So the section right is finitely generated. You're right of course that finite generation isn't true for every line bundle, but the examples are a little more elaborate. Note that BCHM proves finite generation of the canonical ring for any smooth variety (including Enriques surfaces, but there it's obvious). What needs general type is the existence of minimal models. $\endgroup$ – Mark Jan 13 '16 at 2:34
  • $\begingroup$ Ah, you're right; I mixed up even and odd. In that case I do not know an example of non-finite generation off the top of my head. $\endgroup$ – Remy Jan 13 '16 at 3:11
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    $\begingroup$ The "usual" example is due to Nagata: you blow up a bunch of points (12?) on an elliptic curve in P^2 that are cleverly chosen with respect to the group law. It's spelled out in Lazarsfeld's book "Positivity in Algebraic Geometry", but I don't have it on me so I can't tell you the precise reference. $\endgroup$ – Mark Jan 13 '16 at 3:14
  • $\begingroup$ Do you mean Hironaka's example? It has a line bundle that does not come from a divisor, maybe that's the joke. (I guess the thing you describe sounds different. I have been unable to locate the example in Lazarsfeld. There is a treatment of Hironaka's example, but that doesn't mention finite generation.) $\endgroup$ – Remy Jan 13 '16 at 3:31
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    $\begingroup$ Just a remark about the case $\mathscr{L}$ very ample: if $X\subseteq \mathbb{P}^r$ is the embedding determined by $\mathscr{L}$ and if by homogeneous coordinate ring of $X$ one means $k[X_0,\dots,X_r]$ modulo the ideal of $X$, then this is not always the same as $\bigoplus_{n=0}^{\infty} H^0(X,\mathscr{L}^{\otimes n})$. For this to be true we need that the embedding is projectively normal, i.e. that all the maps $H^0(\mathbb{P}^r,\mathcal{O}(n)) \to H^0(X,\mathscr{L}^n)$ are surjective. $\endgroup$ – Daniele A Jan 13 '16 at 10:56

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