I am reading Chapter 3 of the book "Singularities of the minimal model program", and I have some problems on calculating certain quantities.


Let $X$ be a projective variety, $L$ be an ample line bundle on $X$, and $K\_X \sim\_{\mathbb{Q}} rL$ for some $r\in \mathbb{Q}$. Then one can define the affine cone ($\S$ 3.8 loc. cit)


$$C\_a(X,L):={\rm Spec}\_k\sum\_{m \geq 0}H^0(X, L^m),$$ and its partial resolution


$$p: BC\_a(X,L):= Spec\_X\sum\_{m \geq 0}L^m \to C\_a(X,L).$$


Let $E$ be the exceptional divisor of $p$, and $\pi: BC\_a(X,L) \to X$ be the $\mathbb A^1$-bundle which identifies $E$ with $X$.


My problems are the following (it appears in Proposition 3.14, loc. cit):


(1) Why $\mathcal{O}\_{BC\_a(X,L)}(E)|\_E = L^{-1}$?


(2) Why $\pi^\*(K\_X) \sim\_{\mathbb Q}-rE$?


The (2) is especially confusing me: suppose $r>0$, then $K\_X$ is ample, how come $\pi^\*(K\_X)$ is not effective (i.e.$\sim\_{\mathbb Q}-rE$)?



Some calculations on cones
Let's use the following notation $C=C\_a(X,L)$ and $B=BC\_a(X,L)$.


(1) This is the usual statement about the normal bundle of the exceptional divisor restricted to the exceptional. You can look at the computation at the end of II.8 in [Hartshorne] to see that this restriction is always $\mathscr O(-1)$. In this situation, $L$ is your $\mathscr O(1)$. (I know that the proof I am referring to is for "ordinary" blow-ups, but the computation is the same).


(2) I am guessing that your confusion comes from thinking projectively, but $B$ is **not** projective! In particular, this fact does not mean that $\pi\*K\_X$ is not effective. Here is an example: Let $M=-P$ denote the divisor consisting of $-1$-times a single point $P\in \mathbb A^1$. Then $M$ is not only effective, but it is actually ample.


Now in the situation with which you are struggling, $C$ is affine with trivial Picard group and $B$ is a blow-up of that, which means that all non-trivial line bundles on $B$ can be understood by their behavior once restricted to $E$. 


More specifically, as explained at the start of the proof of 3.14, the projection $\pi:B\to X$ induces an isomorphism on the Picard groups. You have $\pi^\*: \mathrm{Pic} X\to \mathrm{Pic} B$ giving one direction of that isomorphism and the restriction to $E\simeq X$ giving the other: $\mathrm{Pic} B\to \mathrm{Pic} E\simeq \mathrm{Pic} X$. Now, (1) implies that via this $-L$ corresponds to $E$ and hence if $K\_X\sim\_{\mathbb Q} rL$, then $\pi^\*K\_X\sim\_{\mathbb Q} -rE$


