Varieties as schemes
Some questions about schemes and varieties, one really basic. I follow the definitions as given in Hartshorne.
Firstly, my main question. I understood that Grothendiecks introduction of schemes revolutionized the subject. Just out of curiosity, could you give me some examples of theorems of techniques about varieties that would be hard or impossible to prove without the language of schemes?
Then some more basic questions. I learned about schemes first and then about varieties (which is really weird, i know), so i kind of missed the natural process of seeing it as an enlargement of the category. Of course affine varieties are given as schemes by the Spec functor. Now for a first question, am i correct in the following reasoning? Take a projective variety $V \subset \mathbb{P}^n$, given by an ideal $I(V) \subset k[x_0,\ldots, x_n]$, then the corresponding scheme is $\text{Proj}(k[x_0,\ldots, x_n]/I(V))$, right?
It seems right to me but it feels kind of slippery. So if somebody could say yes or no with some short explanation or background that would be great.
By the way i did notice that unlike the affine case where there is an equivalence between varieties and rings, here rings that are not isomorphic can give isomorphic projective varieties, right? ($\mathbb{C}[x,y]$ and $\mathbb{C}[x,y,z]/(xz - y^2)$ i guess, certainly not isomorphic rings, but every plane conic is isomorphic to $\mathbb{P}_{\mathbb{C}}^1$)
Now assuming this to be true, Spec and Proj basically give all varieties, because the quasi affine and quasi projective varieties are just open subschemes.
So i was wondering if we could in fact get all schemes this way, by allowing general rings in the above. But this is probably hopelessly naive. I was thinking of an example, and it seemed that the affine line with two origins (say over an algebraically closed field), derived by gluing two copies of $\mathbb{A}^1$ to each other everywhere except at the origin is a nice one. I cannot imagine this being embedded in some affine or projective space.
To put this into a question: given a generic scheme, could you give an intuitive "probability" of whether the scheme is actually the Spec or Proj of some ring/graded ring? As in, how "large" is this subset of schemes? (i guess its either almost everything or almost nothing..)
Also, there's the question: if the affine line is a scheme derived by gluing, why don't we allow such schemes to be varieties? In other words, why don't we define varieties as locally ringed spaces that are locally isomorphic to affine varieties? I recall reading that Weil actually defined them like this. Is there an obvious reason why Hartshorne did not follow this approach? It's probably a matter of taste, but it seems weird to me to define schemes by some "locally affine" property, while not following the same approach in the subcategory of varieties! In fact the approach for varieties is the opposite of local, there's always some ambient space!
I must say i did not read every page of Hartshorne at all, so i might have missed something.
As you must have noticed by now, my questions mostly concern motivation and background, except the one of Proj of a ring. Any help would be greatly appreciated!
Edit: Since it's quite a long story, i'll summarize the questions unanswered so far.
Is it in fact true that a projective variety with ideal $I$ is given as a scheme by $\text{Proj}(k[x_0,\ldots, x_n]/I)$? So from this it follows that every variety is either Proj of Spec of some ring, or an open subscheme of a scheme obtained in this way?
To what extent does the same hold for schemes, as in "how many" schemes are either Proj or Spec of some ring, or an open subscheme of one of those? (of course i just require an intuitive answer and expect no rigorous math) Or can we actually characterize those schemes, are they for example always separated?
It seems weird to me to define a scheme by a local property (locally affine), but a variety in the old language as a subset of some ambient space. Is there a good reason to do so, or is it done differenlty somewhere else and to do you have a reference for this?
Joachim
For your question about projective varieties and Proj, the answer is yes.
Spec of a ring, and Proj of a graded ring, are always separated, and being separated is inherited by open subschemes, so non-separated schemes give examples of schemes that are neither quasi-affine (i.e. open in an affine scheme) nor open in a Proj. [As an aside, note that for varieties, quasi-affine (open in an affine variety) implies quasi-projective (open in a projective variety), because affine space itself is open in projective space of the same dimension, while for any ring $A$, if we make $A[T]$ a graded ring by putting $A$ in degree $0$ and $T$ in degree $1$, then Proj $A[T] = $ Spec $A$, so being open in a Proj subsumes the possibility of being open in an affine scheme.]
As you wrote, Weil in fact introduced the concept of "abstract algebraic variety", which is an object which is locally an affine variety. In Hartshorne Ch. II, he similarly defines a variety over $k$ to be a separated integral finite type $k$-scheme, so quasi-projectivity is not assumed. It is convenient in Ch. I to impose quasi-projectivity just because it lets you get to some non-trivial examples and theorems without having to spend forever on the foundations. Also, it takes some effort to write down non-quasi-projective varieties; you don't tend to stumble upon them in beginners' excercises and constructions. [Weil introduced the concept because from his original definition/construction of the Jacobian of a curve over $k$, it wasn't clear that the Jacobian was projective.]
The constructions of the spectrum $Spec(A)$ of a graded ring and of $Proj (A)$ are more related than one might think.
For example suppose $A=k[X_1,...,X_n]$ for some algebraically closed field $k$.
Then to a relevant homogeneous ideal $I\subset A$ you can associate a projective subvariety $X=V_+( I)\subset Proj(A)= \mathbb P^{n-1}_k$ but also the associated affine cone $C(X)=V(I)\subset Spec(A)=\mathbb A^n_k$.
Geometrically the closed points of $C(X)$ are obtained by taking the union of all lines $\overline {Op}$ joining the origin $O=(X_1,...,X_n)\in \mathbb A^n_k$ to a closed point $p\in X\subset \mathbb P^{n-1}_k$, where $\mathbb P^{n-1}_k$ is seen as the hyperplane at infinity of $\mathbb A^n_k$.
What you have to carefully keep in mind is that the projective variety $X$ and its cone $C(X)$ are defined by the exact same ideal $I$, but interpreted differently.
And (to answer one of your questions) , yes, the projective subscheme $X=V_+( I)\subset Proj(A)= \mathbb P^{n-1}_k$ is isomorphic to the abstract scheme $Proj(k[X_0,X_1,...,X_n]/I)$.
A concrete example
Consider the conic $xy=z^2$ in $\mathbb P^2(\mathbb C)$ and its associated cone $xy=z^2$ in $\mathbb A^3(\mathbb C)$: the equation (or the ideal $I=(xy-z^2)$) is the same but the interpretation is different.
And, to illustrate the consideration of cones, notice that the closed point $p=(x,z)\in X$ (= classically the point with homogeneous coordinates $[0:1:0]$) gives rise to the line $\overline Op $ given by $x=z=0$ in $ \mathbb A^3_\mathbb C$, a ruling of the affine cone $C(X)$