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It is well-known that log terminal singularities are rational, but log canonical singularities are not. On the other hand, rational singularities are not necessarily $\mathbb Q$-Gorenstein, so there are rational singularities that are not even log canonical (and hence not log terminal). On the third hand (if someone has one), rational Gorenstein singularities are canonical and hence log terminal and hence log canonical.

So that leads to the question: Is there a rational singularity that is $\mathbb Q$-Gorenstein, but not log canonical?

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Just a quick comment, log terminal/canonical singularities can be made sense of even in the absense of the Q-Gorenstein hypothesis. In particular, one can say that $X$ is log terminal/canonical if there exists a Q-divisor $\Delta \geq 0$ such that $K_X + \Delta$ is Q-Cartier and $(X, \Delta)$ is log terminal/canonical. de Fernex and Hacon wrote a paper about 5 years ago where they explained why this definition is actually quite reasonable (and appears say in other natural contexts). –  Karl Schwede Mar 7 '13 at 22:04
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Actually, if you look at my answer to your previous question, the same computation gives you an example for this as well. The example there is $\mathbb Q$-Gorenstein and if you choose the $n_i$ such that the intersections matrix is negative definite and $$ \frac 1{n_1}+\frac 1{n_2}+\frac 1{n_3}< 1 $$ holds, then you get a rational $\mathbb Q$-Gorenstein singularity that is not log canonical. I don't feel like computing determinants, but I am pretty certain that if you choose the $n_i$ large enough then they will work.

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