Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Projective geometry was introduced by fifteenth century Renaissance painters (like Alberti, da Vinci and Dürer) in the guise of perspective theory, although one could argue that Pappus was already doing projective geometry in the fourth century A.D.
Projective geometry was formalized as a synthetic geometric theory by Desargues around 1640, but coordinate projective geometry was introduced by Plücker only in 1830 in his article Über ein neues Coordinatensystem in the recently created Journal für reine und angewandte Mathematik.
In 1955 Serre described, in his ground-breaking article Faisceaux Algébriques Cohérents, a remarkable technique associating to a graded algebra $S$ a projective variety $Proj(S)$ and to a graded $S$-module $M$ a quasi-coherent sheaf $\tilde M$.

Here is my question:
Since Serre's technique is a priori so counterintuitive, I was wondering if there are previous publications hinting at the link between graded rings and projective algebraic geometry.
I'm hoping for some precursor, analogous to the introduction by Gel'fand of the spectrum of a $C^*$-algebra announcing Grothendieck's construction of an affine scheme associated to a completely arbitrary commutative ring.

Edit
After reading the comments I think I should clarify what I'm after:
it was indeed probably well known before Serre that one could associate to a projective variety $X\subset \mathbb P^n$embedded in $\mathbb P^n$ its graded coordinate ring $S(X)$: this is pretty easy once one has the concept of graded ring.

What I find much more difficult is to associate to a graded module of finite type over $S(X)$ a coherent sheaf and in particular to see line bundles on $\mathbb P^n$ as obtained from the twistings of a polynomial ring.
I am sure that nobody did that before Serre because nobody before him had even considered coherent sheaves on algebraic varieties endowed with the Zariski topology!
So my question is not whether someone found that correspondence before Serre (nobody did) but whether someone was, with the benefit of hindsight!, on the right track.

Also I would be grateful for precise references: with schemes and sheaves taught today more or less in Kindergarten it is easy to underestimate the audacity of some constructions at a time when sheaves had just been discovered (even their definition was in constant flux) and when homological algebra was a brandnew technique.

share|improve this question
5  
In a sense it is obvious, because the relationship between homogeneuous ideals and projective varieties is very well understood provided that the relation between affine varieties and ideals is well understood: we can always think of a projective variety as a conic affine variety. But I don't have any precise reference to something before Serre's FAC or Grothendieck's EGA II. –  Mauro Porta Jul 15 at 9:26
2  
I understand, that's why I just wrote a comment and not an answer. With "obvious" I meant that in my opinion, once one understands that homogeneuous ideals are the key to projective geometry and once one masters Zariski's approach to algebraic geometry, the construction of $\mathrm{Proj}$ is within reach. In any case, doing that in 1955 was surely amazing. –  Mauro Porta Jul 15 at 13:36
1  
Maybe Serre was the first to go from graded rings to projective varieties, but going from projective varieties to graded rings was quite popular. –  Ben Wieland Jul 15 at 17:56
1  
Dear @Ben: I'll edit my question in order to clarify what I'm after. –  Georges Elencwajg Jul 15 at 18:48

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.