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This is a rather basic question that I have never really thought about until now, when I was forced to think about a sheafy version. This condition is needed for example to define the relative Proj construction, and I don't understand what exactly about it is making something 'projective.' Here's an example result who's proof I understand algebraically, but not geometrically. I would like to explain geometrically what this proof is about.

Consider the well-known result that if $f: Z \to X$ is a projective, birational map between quasiprojective varieties, there is a coherent sheaf of ideals $\mathscr{I}$ on $X$ so that $Z \simeq \tilde{X}$, the blow up with respect to $I$. This is proven in Hartshorne and the proof sketch is as follows.

Since $f$ is projective we have a map $i: Z \hookrightarrow \mathbb{P}^n_X$ and an invertible sheaf $\mathscr{L} = i^\ast \mathcal{O}_X(1)$ on $Z$. We define $\mathscr{S} = \bigoplus_{d \geq 0} \mathscr{L}^d$. Since $\mathscr{L}$ is coherent, each power is and the sum is quasicoherent. The condition to make $Z$ the Proj of something is that $\mathscr{S}$ needs to be generated in degree $1$. This isn't the case, so we work affinely and find a related algebra corresponding to a degree $D$ Veronese embedding so that after composing $i$ above with the Veronese embedding, the condition holds.

Some more technicalities follow in this argument concerning whether or not we have a coherent sheaf of ideals, and arranging this to be the case, but the primary issue has already arisen.

What is this Veronese embedding doing, geometrically? I understand how it takes degree $D$ things to hyperplanes - this isn't quite what I'm looking for and is a little bit on-the-nose. Stated another way, what is composing with this embedding fixing about this object in projective space that is detected about being generated in degree $1$, that isn't there in general? Can we see visually/geometrically what fails in the Proj construction when this condition isn't met?

I apologize if this question is still not clear, and I worry that more rambling is only making it worse, so I'll stop here and try to clarify if anyone doesn't understand what I'm asking.

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When you have a graded ring $A$ finitely generated in degree 1 as a $A_0$-algebra, then what you'll have is a surjection of graded rings $A_0[X_1,\ldots,X_n] \twoheadrightarrow A$. This is what gives you the closed embedding $\mathrm{Proj} A \hookrightarrow \mathbb{P}^n_{A_0}$.

You can then also realize $A$ as a quotient of $A_0[X_1,\ldots,X_n]$ (so really cut out by equations!). Under this embedding, $A_1$ remembers the restriction of linear forms to $A$.

So I think this also answers your second question about the Veronese embedding. Composing with the Veronese embedding ensures now that your degree 1 component is equal to the linear forms on the ambient projective space restricted to your variety.

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  • $\begingroup$ Great answer. Possible minor typo, unless I've misunderstood something: I think you have $\mathbb{P}^{n}_{A}$ written where you should have $\mathbb{P}^{n}_{A_{0}}$. $\endgroup$ – Alex Wertheim Feb 23 at 2:35
  • $\begingroup$ @AlexWertheim You're absolutely right! Thanks! $\endgroup$ – loch Feb 23 at 2:45

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