Discrepancy non log canonical singularities

Question

Suppose that $Y$ is a normal variety such that its canonical class $K_Y$ is $\mathbb{Q}$-Cartier. , and let $f:X \to Y$ be a resolution of the singularities of $Y$. Then $$ K_X= f^*(K_Y)+\sum_i a_iE_i $$ we can take the minimum of the $a_i$ for all the resolutions and all the divisors. This is call the discrepancy of the singularity. If the singularity is not log canonical, then discrepancy is $-\infty$. I had not been able to understand this fact..Some one can help me?

Answer 1

Yes, this is a tricky one. It's an example somewhere in Chapter 2 of Kollár--Mori, so you can look there for details. Let me explain how to get a divisor of discrepancy $2a+2$, and then you can see where to go from there.

For simplicity of notation, let's assume we're in dimension 2, and that $K_X = f^* K_Y +aE$ with $a<-1$. First blow up a smooth point on $E$ to get $g: W \rightarrow Y$, and write $\widetilde{E}$ for the proper transform of $E$, and $F$ for the new exceptional divisor. Then we have $$K_W=g^*K_Y + a \widetilde{E} + (1+a)F.$$ Now the trick is to blow up the intersection point $\widetilde{E} \cap F$ to get $h: T \rightarrow Y$. The usual calculation, done carefully, gives $$K_T=h^*K_Y+a \widetilde{\widetilde{E}} + (1+a) \widetilde{F} + (2a+2)G$$ where now $G$ is the new exceptional divisor. Now continue in this way: blow up the intersection point of $\widetilde{\widetilde{E}}$ and $G$ to get a divisor of discrepancy $3a+3$, and so on.