Definition of canonical pair

Let $$(X,D)$$ be a pair and $$f:Y\rightarrow X$$ a log resolution. Write $$K_Y + \widetilde{D} = f^{*}(K_X) + \sum_{i}a_iE_i$$ where $$\widetilde{D}$$ is the strict transform of $$D$$. I found the following definition:

the pair $$(X,D)$$ is $$(t,c)$$ is $$X$$ is terminal and $$(X,D)$$ is canonical meaning that $$a_i\geq 0$$ for all $$i$$.

Now, assume that $$X$$ is smooth and $$D = D_1 + \dots + D_r$$, where the $$D_i$$ are prime divisors, is simple normal crossing. Then the identity is a log resolution and hence $$(X,D)$$ is $$(t,c)$$. Is this correct? Or must one interpret the absence of exceptional divisors as the possibility of having arbitrarily negative discrepancies?

I am asking since this does not seem to match the arguments in a paper that I am reading. Are there different definitions of $$(t,c)$$ pair?

For instance is $$\mathbb{P}^2$$ with the divisor $$D = \{xyz=0\}$$ a canonical pair?

Thanks a lot.

• What is the paper you are reading whose arguments this does not match? Feb 8 at 14:07
• I prefer to avoid doing that until I understand if it really does not match. Feb 8 at 14:30

You are using the wrong equation to compute discrepancies. It should be $$K_Y = f^*(K_X + D) + \sum a_E(X,D) E$$ where the $$E$$ are not all necessarily exceptional.
For example if $$(X,D)$$ is already smooth and simple normal crossing then $$K_X = \operatorname{id}_X^*(K_X + D) - D$$ and so every irreducible component $$D_i\subset D$$ has discrepancy $$a_{D_i}(X,D)=-1$$. Thus the simple normal crossing pair $$(X,D)$$ is log canonical, and not canonical.
• Indeed, I took my definition from Kollár's book (Notation 2.6, line 2). In any case, I can still find an exceptional divisor of discrepancy $-1$: just blow up a stratum of $D$ (e.g. in your case of $(\mathbb P^2,D)$ blow up a node of $D$). The problem with ignoring the non-exceptional divisors in this case is that we don't have the "if it holds for one resolution, then it holds for all resolutions" statement. Feb 8 at 16:57
• (However you make a good point and, to qualify what I wrote, Kollár's definition implies that a simple normal crossing pair $(X,D)$ is canonical if $X$ and $D$ are smooth, but only log canonical if $D$ has more than one component.) Feb 8 at 17:05