# Famous papers in algebraic geometry

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.

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In response to your edit: that's why I made it a comment. Really, if you look at Wikipedia or MacTutor biographies the big papers will be mentioned and I don't see why you wouldn't want to read those. For Serre, FAC (Faisceaux Algébriques Cohérents) and GAGA (Géometrie Algébrique et Géométrie Analytique) are two obvious targets and you could really spend a semester-long course trying to understand the implications of either. – Dylan Moreland Jun 20 '12 at 21:39
@Dylan: Thanks for the titles. A quick Google search gave me this: mathoverflow.net/questions/14404/serres-fac-in-english (FAC and GAGA both translated into English.) – Fredrik Meyer Jun 20 '12 at 21:46
Ah, neat. Hadn't seen that. And it depends on how much you care about arithmetic, but Deligne's two papers La conjecture de Weil I and II are good and will give you a good sense of how one uses étale cohomology. – Dylan Moreland Jun 20 '12 at 21:48
For reference purpose, here is the Wikipedia link – Sniper Clown Jun 21 '12 at 5:39

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 $\mathcal O (k)$ 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.

Edit 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.

Second Edit
Here is a translation of FAC into English.

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Georges, I think your mentioned theorem of FAC on coherent sheaves is almost trivial if the base scheme is Noetherian. I cannot think of usefulness of the theorem on non-Noetherian schemes. Perhaps the theorem is useful on non-scheme spaces(e.g. analytic spaces). – Makoto Kato Jun 21 '12 at 0:37
Dear Makoto, the theorem proved in Serre is not in the context of schemes (which didn't exist!) but in that of general ringed spaces.The concept of coherent sheaf was introduced implicitly by Oka and explicitly by H. Cartan in the context of complex geometry, as you suggest (cf. Levi problem, Cousin problems I and II, Theorems A and B,..) It was an act of remarkable audacity of Serre to dare think that coherent sheaves and their cohomology could be of any use in algebraic geometry, where the coarseness of the Zariski topology seemed to exclude such tools . – Georges Elencwajg Jun 21 '12 at 1:17

There's EGA by Grothendieck and SGA by Grothendieck et al. Also related to your question this question and this question on mathoverflow.

Finally wikipedia has a page on important publications in mathematics.

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Those qualify as papers? – Adrián Barquero Jun 20 '12 at 21:26
@AdriánBarquero I would say they are a series of papers. – Eugene Jun 20 '12 at 21:28
Some parts [the good parts, I'd say] of SGA were written by Deligne, and there are contributions from Raynaud, Artin, Verdier, Demazure, Illusie, Katz, ... – Dylan Moreland Jun 20 '12 at 21:28
@DylanMoreland I'll edit my answer to avoid misleading then. – Eugene Jun 20 '12 at 21:30
There's always the cop-out of "et. al.", which I would encourage in this case. – Dylan Moreland Jun 20 '12 at 21:33

I'm going to interpret this question in the way that you wrote it and not in the way people are answering it. Sure EGA, SGA, GAGA, etc are great works by masters, but in practice I haven't met very many people who have actually "read" these. The fact that you have "should" and "could" in the question rules all three of those out for me (also, how many people have read and understood Weil II?).

Probably my favorite paper in algebraic geometry is Schlessinger's famous Functors of Artin Rings. Maybe after three chapters of Hartshorne it will be hard to appreciate its importance, but there probably isn't a single branch of modern AG that doesn't rely on it in some sense. The level of generality is beautiful because it makes the statement and proof more accessible than if it were specific, and it allows you to use it all over the place. It is quite readable and in my opinion ought to be more widely read.

Another really good paper is Mumford's Picard Groups of Moduli Problems. Again, in modern AG it is hard to think of a single branch that doesn't consider moduli problems important. This paper really spells out in great detail the moduli of elliptic curves and how to do some computations with it. It is a great way to learn about moduli spaces (certainly with some more up-to-date references too) as well as serving as an introduction to and motivation for the definition of a stack.

I have other favorites, but they are seriously specialized, so I wouldn't recommend them to everyone.

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You haven't met people who've read (or tried reading) EGA? Seriously? A lot of people I've met are very tempted by it. There are quite a few threads on MO discussing EGA/SGA. But I'm more on the arithmetic side of geometry. – Eugene Jun 20 '12 at 22:30
@Eugene Certainly people read parts of all the things I mentioned. Namely the parts they need for something. I highly recommend using them as references. But read all of EGA? Seriously? The top two comments on the thread you linked (by very successful algebraic geometers I might add) talk about why its a bad idea to try to read it. – Matt Jun 20 '12 at 23:06
Oh you meant all of it. You're probably right there. They are incredibly long. – Eugene Jun 20 '12 at 23:09

I was among those who discussed Zariski's paper on simple points on the MO thread. Here is the link.

One landmark paper is that of Deligne and Mumford on moduli spaces of curves. (It appeared in Publications IHES, and would be easy to track down.) It will need more than Hartshorne Chapters I, II, and III, but could well provide an incentive to learn that little bit more.

As I've mentioned in other threads on this topic, I think that Mumford's book Lectures on curves on algebraic surfaces is fantastic. (It is longer than a paper, but it is devoted to the proof of a single result. Along the way, it develops a lot of fantastic material and intuitions.)

Serre's GAGA paper is another classic.

Finally (until I think of more must-adds!) there is the paper of Clemens and Griffiths, The intermediate Jacobian of the cubic threefold. Since this may seem a little specialized, let me exlain why I think it deserves classic status: a smooth cubic curve in the plane is not rational (it has genus one); a smooth cubic surface in space is rational — it is $\mathbb P^2$ blown up at six points. A smooth cubic threefold in $\mathbb P^4$ was classically known to be unirational, but (before this paper) it was not known whether or not it was rational; this paper shows that it is not rational. Questions of rationality are fundamental in algebraic geometry, and this paper is a fundamental contribution; it also marks Griffiths's introduction of Hodge-theoretic ideas (the modern point of view on periods of integrals as studied by Abel and Picard, and later Lefschetz) as key tools in the study of concrete geometric questions. Note that the problem of rationality of cubic fourfolds remains open.

Okay; some more classics that came to mind while I was writing: Atiyah's paper on Vector bundles over an elliptic curve (one should first read Grothendieck's paper on vector bundles on $\mathbb P^1$), and (to give a more recent example) the paper of Graber–Harris–Starr, proving that the total space of a family of rationally connected varieties over a rationally connected base is rationally connected.

More: Variations on a theorem of Abel (I think this is the right title), by Griffiths. If you want to understand what the Abel–Jacobi theorem (and hence what Hodge theory and much else in modern algebraic geometry) might really be about, in concrete geometric terms, this is a paper you must read.

Deligne's note Théorie de Hodge I and his paper Théorie de Hodge II are also fantastic. (There is also part III, but it is more technical, since it deals with singular varieties.) There is a precursor, something like On a criterion for the degeneration of spectral sequences (but in French). These papers, like those of Griffiths that I've mentioned, mark the introduction of Hodge theory into modern algebraic geometry as a fundamental tool. Deligne's style is very different to Griffiths's; it is harder to see the concrete meaning of what he is doing than in Griffiths's work. But they are both masters, introducing ideas that are as fundamental and influential as any that I can think of in geometry.

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I think the Griffiths article is titled "Variations on a theorem of Abel". – wildildildlife Jun 21 '12 at 10:03
Thanks to t.b. for adding links to all of the papers. I found the papers of Deligne mentioned in the last paragraph to be very formal but the cleanliness of the presentation does count for something. – Dylan Moreland Jun 21 '12 at 19:30
@DylanMoreland: Dear Dylan, I agree that the papers have a formal aspect, but ... consider the theorem of the fixed part in Hodge II; that is very deep, and far from formal. How does the formalism that come beforehand lead to such deep conclusions ...? That is the mystery and wonder of Deligne's mathematics. Cheers, – Matt E Jun 22 '12 at 6:55

Not an expert here, but here's a fairly accessible paper I'm currently reading that I'm enjoying very much and which nobody seems to have mentioned yet: Deligne-Illusie's 1987 paper on lifting mod $p^2$. What's amazing about this paper (among other things) is that it gives a purely algebraic proof of the Kodaira-Nakano-Akizuki vanishing theorem (which is classically proved by bounding below Laplacian-type operators) by reduction to characteristic $p$ and use of the Frobenius(!).

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