This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow A^2/\mu_3$, the affine plane quotient the cyclic group of order 3, arises to the conclusion that the exceptional divisor is $E\sim P^1$, (no problems there) and $\mathcal{O}_E(-E)\sim \mathcal{O}(3)$ (problems here).

Given a variety $Y$ and an effective Cartier divisor $D$ on it, there seems to be a pretty standard exact sequence:

$$0 \longrightarrow \mathcal{O}_Y \longrightarrow \mathcal{O}_Y(D) \longrightarrow\mathcal{O}_D(D)\longrightarrow 0 $$ As far as I understand, if $U$ is an open set in $S$ and $D\cap U = div(g)_U$ (for $D$ a hypersurface, if you want, and extend by linearity), then

$$ \mathcal{O}_Y(D)(U)= \{g \in \mathcal{O}_Y(U) \vert div(g)\geq D \}$$

or equivalently $g/f$ is regular. The first map must be something like $g\rightarrow gf$ maybe with some order. A good answer to my question would include:

- Is this correct?
- What is $\mathcal{O}_D(D)$?
- What is the second map?
- What does $\mathcal{O}_D(-D)$ mean
- Why $\mathcal{O}_E(-E) \sim \mathcal{O}(3)$? I understood the RHS is generated by polynomials of degree 3?

I am aware this is a simple question and probably everyone knows why, but I could not find a proper answer for it.