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).
