For those of us who have forgotten, Castelnuovo's theorem is the following:
Theorem: If $Y$ is a curve on a surface $X$ with $Y \simeq \mathbb{P}^1$ and $Y^2 = -1$, then there is a morphism $f: X \to X_0$ to a smooth projective surface $X_0$ such that $X$ is the blow up of $X$ at some point and $Y$ is the exceptional divisor.
A proof of this theorem is given on page 414 of Hartshorne. To formulate my question, note that we will construct $X_0$ using the image of $X$ under a suitable map to projective space. In more detail, our aim is to show that the invertible sheaf $\mathcal{M} : = \mathcal{L}(H + kY)$ is semi-ample. Here, $H$ is some very ample divisor such that $H^1(\mathcal{L}(H))=0$ and $k = H.Y$ is assumed to be $\geq 2$.
Problem: Consider the following sequence of sheaves $$0 \longrightarrow \mathcal{L}(H + (i-1)Y) \xrightarrow{ \ \alpha \ } \mathcal{L}(H+iY) \xrightarrow{ \ \beta \ } \mathcal{O}_Y \otimes \mathcal{L}(H + i Y) \to 0.$$ Hartshorne claims this sequence is exact. In trying to verify this, I am finding that I am not sure of how $\alpha$ and $\beta$ are defined.
Best Answer
This exact sequence is just $0\to \mathcal{I}_Y\cong\mathcal{O}_X(-Y)\to \mathcal{O}_X\to \mathcal{O}_Y\to 0$ tensored with the line bundle $\mathcal{L}(H+iY)$.