# Hartshorne's proof of Castelnuovo's theorem

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.

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 $$\mathcal{L}(H+iY)$$.