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Let $X$ be a projective variety over a field $k$, and $\mathcal L$ a vector bundle on $X$, i.e. a locally free $\mathcal O_X$-module of finite rank. For each $n\geq 0$, $\text{Sym}^n \mathcal L$ is a vector bundle on $X$, hence $H^0(X, \text{Sym}^n \mathcal L)$ is a finite-dimensional $k$-vector space. Consider the graded $k$-algebra

$$(\text{Sym } \mathcal L) (X) := \bigoplus_{n=0}^\infty H^0(X, \text{Sym}^n \mathcal L).$$

What kind of conditions on $X$ and $\mathcal L$ ensure that this algebra is finitely-generated? Is it always the case?

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The ring you describe is the ring of global sections of the total space of the vector bundle $\mathcal{L}$, i.e. $\mathbf{Spec}(\mathrm{Sym}^\bullet(\mathcal{L}))$. (This notation is the "global Spec" construction for a sheaf of $\mathcal{O}_X$-algebras.)

In general, this does not need to be a finitely-generated $k$-algebra (see this shocking counterexample, for a rank-2 vector bundle over an elliptic curve: http://math.stanford.edu/~vakil/files/nonfg.pdf).

It is true for (very?) ample line bundles, since then it is the homogeneous coordinate ring of $X$ under the corresponding embedding. (Slight white lie here -- it's the integral closure of the homogeneous coordinate ring.)

It is true for arbitrary line bundles on toric varieties apparently: http://arxiv.org/pdf/alg-geom/9608034.pdf.

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    $\begingroup$ Very interesting answer! $\endgroup$ – Georges Elencwajg Jan 24 '15 at 9:28
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    $\begingroup$ Thanks Jake, resourceful as always! :) $\endgroup$ – Bruno Joyal Feb 3 '15 at 3:51

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