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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?

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    $\begingroup$ I don't think I understand your notation, so I dont' guarentee this is on point. I f you look at the bound Castelnuovo provides for the geometric genus of a curve in projective space it involves understanding the hypersurfaces of a given degree d containing the curve and exact sequences similar to yours occur. You could look in 'Geometry of Algebraic Curves' for this Castelnuovo result. $\endgroup$
    – meh
    Dec 2, 2015 at 22:27

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