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?

Thanks a lot.

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.