Let $(0\in X)$ be the germ of a normal surface singularity and let $f: Y \to X$ be the minimal resolution.


**Questions>**


(1) How can I define a map $f\_\*\mathcal{O}\_Y(K\_Y)\hookrightarrow \mathcal{O}\_X(K\_X)$?


(2) If $f\_\*\mathcal{O}\_Y(K\_Y)=\mathcal{O}\_X(K\_X)$ and $K\_X$ is Catier, then $f^\*\mathcal{O}\_X(K\_X)=\mathcal{O}\_Y(K\_Y)$. Why?



canonical divisors of a resolution of a normal surface singularity
Concerning (1), the map in question can be described in more down-to-earth terms as follows. Let $E = \sum E\_i \subset Y$ be the exceptional divisor of $f$. A section of $f\_\* \mathcal O\_Y(K\_Y)$ is by definition nothing but a section of $\mathcal O\_Y(K\_Y)$, which you can restrict to $Y \setminus E$. As $f$ restricts to an isomorphism $Y \setminus E \to X \setminus \{ 0 \}$, you get a section $\sigma$ of $\mathcal O\_X(K\_X)$ on $X \setminus \{ 0 \}$. Since $\mathcal O\_X(K\_X)$ is reflexive (as is any Weil divisorial sheaf on a normal variety), you can extend $\sigma$ to all of $X$. This defines a map $f\_\* \mathcal O\_Y(K\_Y) \to \mathcal O\_X(K\_X)$. (This actually works for normal varieties of arbitrary dimension.)


Regarding (2), as Donu pointed out, the condition $f\_\* \mathcal O\_Y(K\_Y) = \mathcal O\_X(K\_X)$ is equivalent to $X$ having rational singularities. (Beware, this only holds in dimension two. In general you additionally need $X$ to be Cohen-Macaulay, which is automatic for normal surface singularities.)


If $f\_\* \mathcal O\_Y(K\_Y) = \mathcal O\_X(K\_X)$ and $K\_X$ is Cartier, then essentially by definition $X$ has canonical singularities, also known as the Du Val or ADE singularities. Now if you write down the ramification formula
$$ K\_Y = f^\* K\_X + \sum a\_i E\_i $$
(which makes sense because $K\_X$ is Cartier), then having canonical singularities exactly means that all $a\_i \ge 0$. On the other hand, since $f$ is the minimal resolution, $K\_Y$ is $f$-nef and then so is $\sum a\_i E\_i$. By the Negativity Lemma, all $a\_i \le 0$. Hence all $a\_i = 0$, and $K\_Y = f^\* K\_X$. (Obviously this fails if we blow up $Y$ further.)


A general reference for these things is Kollár-Mori, "Birational geometry of Algebraic Varieties", Chapters 4 and 5.


