Some calculations on cones

Question

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.$\sim_{\mathbb Q}-rE$)?

Answer 1

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 $\pi^*K_X\sim_{\mathbb Q} -rE$