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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.

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

1 Answer 1

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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.

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  • $\begingroup$ This would explain everything but for instance in Kollar's book "Singularities of the minimal model program" Definition 2.8 canonical singularities are defined just in terms of the discrepancies of the exceptional divisors. $\endgroup$
    – Puzzled
    Feb 8 at 16:43
  • $\begingroup$ 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. $\endgroup$
    – Tom Ducat
    Feb 8 at 16:57
  • $\begingroup$ (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.) $\endgroup$
    – Tom Ducat
    Feb 8 at 17:05

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