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I would like to ask a couple of naive question about the following theorem of Max Noether:

In the book of Fulton, page 60

this theorem is called Max Noether's fundamental theorem.

Question 1. Are there some nice applications of this theorem apart from the simple ones that a given in the book of Fulton?

Question 2. Why this theorem is called fundamental? Is this for some historical reason? Is this theorem a part or a starting point of some modern theory?

(Unfortunately I was not able to get answers to these questions by simple googling)

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notice in fulton's book that the fundamental theorem implies the residue theorem which implies the riemann roch theorem. that seems like a significant application. – roy smith Jan 16 '12 at 5:52
up vote 7 down vote accepted

Perhaps one of most famous consequences of Noether "AF+BG Theorem" is Cayley-Bacharach Theorem, that I state below.

Theorem (Cayley-Bacharach). Let $X_1, X_2 \subset \mathbf{P}^2$ be two plane curves of degree $d$ and $e$, respectively, meeting in a collection of $d \cdot e$ distinct points $\Gamma$. If $C \subset \mathbf{P}^2$ is any plane curve of degree $d+e-3$ containing all but one point of $\Gamma$, then $C$ contains all of $\Gamma$.

When $d=e=3$ one has Chasles Theorem: if $\Gamma$ is a collection of $9$ points in $\mathbf{P}^2$ which are complete intersection of two cubics, then any cubic $C$ passing through $8$ of the points of $\Gamma$ contains the remaining point as well (this is essentially Proposition 3, page 63 in Fulton's book).

For a nice discussion of these results and their relation with Noether's theorem see the paper by Eisenbud, Green and Harris Cayley-Bacharach Theorems and Conjectures, Bull. Amer. Math. Soc. 33 (1996).

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Francesco, thank you for the answer and for the link! – aglearner Jan 16 '12 at 15:37
I will accept this answer, since the link that you give is really great (it is also useful for the course that I am supposed to give :)...) – aglearner Jan 18 '12 at 12:41

I do not know the history of the glorious title of this theorem and when I first learned it I wondered pretty much about the same things. It is not obvious that this statement deserves such a serious title. However, you have to keep in mind that when Noether proved it, the general understanding of these things was at a completely different level.

I would say to

Q1 that those applications are not simple. Especially Proposition 3 is a very important and at first perhaps surprising statement. This is completely unique to cubics and this statement is essentially equivalent to the group law on cubic curves. If you start there, isn't that an unbelievably beautiful statement that if you take a smooth cubic curve then you can define a group law just by drawing lines? Try to prove associativity without Proposition 3!

Q2 Probably this theorem today would not get the adjective "fundamental" and it would probably be hard to publish it in a good journal, but you must know other examples of fundamental theorems that now seem almost trivial. You have to place these in the appropriate era. I am not a historian so I will not do that, but let me point out that even if the proof does not seem hard at all, the statement is probably one of the first in a long line of important local-to-global theorems. In algebraic geometry we often have to rely on local data and observations. For instance, if you work on proper (e.g., projective) schemes you will not have any non-constant global regular functions. Sometimes you won't even have non-trivial line bundles. Hence it is important to be able to make global predictions based on local observations. Noether says that in this particular question you can do that.
As an exercise, that perhaps helps to appreciate the statement more, try to formulate what it means for the $\mathscr O_X$-module generated by the two homogenous polynomials.

Remark: see also Roy's comment about the Residue theorem and RR.

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Dear Sandor, I must respectfully disagree with your comment that Proposition 3 is "completely unique to cubics". It is actually the easiest case of Cayley-Bacharach theorem, that holds in any degree. The classical proof of this results also uses in an essential way Noether AF+BG theorem – Francesco Polizzi Jan 16 '12 at 11:19
Dear Francesco, I don't think we have a disagreement, but a misunderstanding. If you take $d=e=d+e-3$ in Cayley-Bacharach, then $d=e=3$. So, if in Proposition 3 you replace all appearances of the word "cubic" with "degree $d$" you do not get a theorem unless $d=3$. This is what I meant by unique to cubics. Of course, your point that Noether's theorem is used more generally is valid. My point was that I think that already the applications in Fulton's book are non-trivial. – Sándor Kovács Jan 16 '12 at 11:51
Dear Sandor, ok. Now I understand your comment, thanks for the clarification. – Francesco Polizzi Jan 16 '12 at 11:56
Sandor, thank you for your detailed answer! – aglearner Jan 16 '12 at 15:38

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