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


<http://en.wikipedia.org/wiki/AF%2BBG_theorem>


In the book of Fulton, page 60 


<http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf>


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) 



Max Noether's residual intersection theorem (Fundamentalsatz): importance and applications
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](http://www.msri.org/~de/papers/pdfs/1996-001.pdf), Bull. Amer. Math. Soc. **33** (1996). 


