Regular functions on line bundle minus zero-section Let $X$ be a non-singular complex algebraic variety (say quasi-projective if necessary) and $L$ a line bundle on $X$. Let
$$Z:=L-\{\text{zero-section}\}.$$
What is the relationship between regular functions on $Z$ and the ring
$$R:=\bigoplus_kH^0(X,L^k)?$$
The sentence "relating $R$ to the regular functions on $Z$'' is used in [1] and I'm not sure what it means.
[1] P. Heinzner, A. Huckleberry, Kählerian structures on symplectic reductions. Complex analysis and algebraic geometry, 225–253, de Gruyter, Berlin, 2000.
 A: The relation between the two is that they agree, but actually more can be said. The variety $Z$ in your notation is naturally a $\mathbb{G}_m$ torsor over $X$, so that there is a free action of the multiplicative group on the total space $Z$ of this torsor. The equation $\Gamma(Z, \mathcal{O}_Z) \cong \oplus_k H^0(X,L^k)$ is then an isomorphism of graded rings ,where the right hand side is graded by $-k$ and the left hand side by the grading induced from the $\mathbb{G}_m$-action. Explicitly, the $k$-th piece on the left hand side consist of functions $f$ on $Z$ such that $f(tx) = t^k f(x)$ for $t \in \mathbb{G}_m$. 
To see why this isomorphism holds, one only has to identify fucntions on $Z$ which are homogeneous of degree $k$ with $L^{-k}$. For $k = 1$ this follows from the more or less tautological fact that linear functionals (i.e. homogeneous functions of degree 1) along the fibers of $L$ agree with sections of $L^* = L^{-1}$. The rest of the $k$-s are similar, just note that unctions which are homogeneous of degree $k$ along the fibers form the sections of a line bundle, and that locally each such function is a product of $k$ functions of degree 1.         
