Exact sequence of sheaves for an hypersurface

Question

I am reading Supplement to "On the Inverse of Monoidal Transformation" by Shigeo Nakano, but i have trouble understanding proposition 9'. I hope someone will help.

Let $V$ be a complex manifold of dimension $n$ and $S\subset V$ a complex submanifold of codimension 1. The subvariety $S$ is isomorphic to $D\times \mathbb{P}^{r-1}$, $D$ open subset of $\mathbb{C}^m$ and $m+r=n$.

Consider the projection $\pi:D\times \mathbb{P}^{r-1}\rightarrow \mathbb{P}^{r-1}$ and let $E:=\pi^*(\mathcal{O}_{\mathbb{P}^{r-1}}(1))$. Suppose in addition that $\mathcal{O}_S(S)\simeq E^{-1}$.

Finally suppose that there is $W\subset V$ open subset such that $W\cap S\simeq D'\times \mathbb{P}^{r-1}$, $D'$ open subset of $\mathbb{C}^m$.

Nakano writes that we have the exact sequence of sheaves

$0\rightarrow \mathcal{O}_V(S)^{-2}\rightarrow \mathcal{O}_V(S)^{-1}\rightarrow_{\beta} E\rightarrow 0$

Question: I can't understand the morphism $\beta$: I imagine it is the restriction to $S$, but how come it is suriective and its kernel is $\mathcal{O}_V(-2S)$?

Answer 1

Let $\iota: S\to V$ be the inclusion. I presume you are familiar with the short exact sequence: $$0\to \mathcal{O}_V(-S)\to \mathcal{O}_V\to \iota_*\mathcal{O}_S\to 0$$ which we may twist by $\mathcal{O}(-S)$ to obtain

$$0\to \mathcal{O}_V(-2S)\to \mathcal{O}_V(-S)\to \iota_*\mathcal{O}_S(-S)\to 0.$$

Thus to obtain the sequence you describe, we must identify $\iota_*\mathcal{O}_S(-S)$ with $\iota_*E$. But you assume that $\mathcal{O}_S(S)\simeq E^{-1}$; dualizing gives the required claim.