I have seen variations of the following exact sequence referred throughout the literature as the Castelnuovo sequence:
$$0\longrightarrow \mathscr I_{X:H}(-d)\longrightarrow \mathscr I_X\longrightarrow \mathscr I_{X\cap H,H}\longrightarrow 0$$ where $X\subset\mathbb P^n$ is some scheme and $H$ is a hypersurface of degree $d$. The proof is straightforward from the definition of ideal quotients.
In what context did it first appear, and why is it named after Castelnuovo?