In the book Birational Geometry OF Algebraic Varieties (János Kollár) there is a corollary as stated
Corollary 4.14. Let $f : Y \longrightarrow (P \in X)$ be a resolution of the germ a surface singularity such that $R^{1}f_{*}\mathcal{O}_{Y} = 0$ and let $I \subset \mathcal{O}_{P, X}$ be the maximal ideal. Then $I\mathcal{O}_{Y} = \mathcal{O}_{Y}(-E)$ for some effective Cartier divisor $E \subset Y$ and $I^{\nu} = f_{*}\mathcal{O}_{Y}(-\nu E)$ for all $\nu > 0$.
In proof of this result, there are two passages that I cannot understand, so I come here, humbly asking if anyone can help me clarify these doubts. To namely
1) Let $\alpha, \beta \in I$ be general elements pulling back to global sections $f^{*}\alpha$, $f^{*}\beta$ of $\mathcal{O}_{Y}(-E)$ such that $(f^{*}\alpha = 0) \cap (f^{*}\beta = 0) = \emptyset $. Then $$\phi : = (f^{*}\alpha, f^{*}\beta) : \mathcal{O}_{Y}^{\otimes 2} \longrightarrow \mathcal{O}_{Y}(-E)$$ is a surjection. Why?
2) We have an exact sequence $$0 \longrightarrow \mathcal{O}_{Y}(E) \longrightarrow \mathcal{O}_{Y}^{\otimes 2} \longrightarrow \mathcal{O}_{Y}(-E) \longrightarrow 0$$ by comparing the determinants. Why? I did not understand by comparing the determinants.
3) I am also not understanding the following notation $f_{*}\mathcal{O}_{Y}(-(k+1)E) = (\alpha, \beta)I^{k}$. What does $ (\alpha, \beta)I^{k}$ mean?
Thank you very much.