I’m reading the Mathoverflow thread “Do you read the masters?“, and it seems the answer is a partial “yes”.
Some “masters” are mentioned, for example Riemann and Zariski. In particular, a paper by Zariski is mentioned, but not its title nor where it was published, so I have been unable to locate it (on “simple points”).
What are some famous papers by the masters that should (and could) be read by a student learning algebraic geometry? I’m currently at the level of the first three chapters of Hartshorne (that is, I know something about varieties, schemes and sheaf cohomology).
Edit: I should probably add that I’d like specific titles. The advice “anything by Serre” is unfortunaly not very helpful, considering Serre’s productivity.
Serre’s Faisceaux Algébriques Cohérents (=FAC) has the unique status of being:
a) Arguably the most important article in 20-th century algebraic geometry : it introduced sheaf-theoretic methods into algebraic geometry, including their cohomology, characterization of affine varieties by vanishing of said cohomology for coherent sheaves, twisting sheaves O(k) on projective varieties,…
Dieudonné and Grothendieck write in their Introduction to EGA that Chapters I and II of their treatise (and the the first two paragraphs of chapter III) can essentially be considered as easy transpositions (“transpositions faciles”) of Serre’s results in FAC (and of his posterior GAGA article).
b) Still very readable. Quoting Grothendieck and Dieudonné again “sa lecture peut constituer une excellente préparation à celle de nos Eléments” (reading it may constitute an excellent preparation to reading our Eléments)
And do not think that modern books or articles are necessarily simpler:
I remember M.S. Narasimhan (a pioneer in the construction of moduli spaces for vector bundles) explaining to students (admittedly some time ago) that FAC was still the best place to look for a proof that if in a short exact sequence two sheaves were coherent, so was the third.
I have just checked that the result above on coherent sheaves is not proved in EGA (which refers to FAC), nor in Hartshorne (who doesn’t even give the general definition of coherent), nor in Iitaka, nor in most books on algebraic geometry.
Actually the only such book I can think of that proves the result is Miyanishi’s Algebraic Geometry. (There are also books on complex geometry that prove it)
I’m not claiming that this theorem on sheaves is especially important, but want to emphasize how relevant FAC still is.
Here is a translation of FAC into English.