Let's say I have a complex projective variety $X\subseteq\mathbb P^n$ with homogeneous coordinate ring $S=\bigoplus_{d\ge 0} S_d$. The localization by some homogeneous $f\in S$ (of nonzero degree) yields a graded ring $S_f=:R=\bigoplus_{d\in\mathbb Z} R_d$. Now, instead of looking at the spectrum of the degree zero part, I could look at $R_{\ge 0}=\bigoplus_{d\ge 0} R_d$. I have an inclusion of graded $\mathbb C$-algebras $S\hookrightarrow R_{\ge 0}$ and both are positively graded, so I may very well consider the morphism $$Y:=\mathrm{Proj}(R_{\ge 0})\xrightarrow{\quad\textstyle\pi\quad}\mathrm{Proj}(S)=X.$$ Since $\mathrm{Spec}(R_0)=X_f$, the variety $Y$ is projective over $X_f$. However, what is the nature of $\pi$? Is it surjective? I am sure this must have interested someone before, but I cannot find anything in the 'standard' literature on algebraic geometry.

More generally, is there any deeper geometric meaning to taking the positive part of a graded ring?

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