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.

  • $\begingroup$ What does "income" refer to here? Next, are you sure that you have written down the formula in 2) correctly? It would make more sense for the second map to be the map $\mathcal{O}_Y^{\otimes 2} \to \mathcal{O}_Y(-E)$ from 1) (or at least for one of the $\mathcal{O}(E)$ to be a $\mathcal{O}(-E)$)). Finally, $(\alpha,\beta)I^k$ is just the product of the ideals $(\alpha,\beta)$ and $I^k$ inside $\mathcal{O}_{P,X}$. $\endgroup$ – KReiser Feb 13 at 0:25
  • $\begingroup$ What does "income" refer to here? Sorry, I've already corrected the question title. Second, Why the map $$\mathcal{O}_{Y}^{\otimes 2} \longrightarrow \mathcal{O}_{Y}(-E)$$ is a surjection? Third, Thank you very much for clarifying the notation. $\endgroup$ – Emanuell Feb 13 at 0:36
  • $\begingroup$ The word "income" appears in the first line under the block-quoted section - I did not see it in the title. $\endgroup$ – KReiser Feb 13 at 0:39
  • $\begingroup$ I apologize. I edited the question again. $\endgroup$ – Emanuell Feb 13 at 0:43

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