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On a normal variety (possibly singular) one can make sense of the canonical class $K_X$ by computing it on the smooth locus and then extending over the singular part. I'm wondering how to actually do this in practice: for example, on the singular cone $C$ in $\mathbb P^3$ given by $y^2-xz=0$, what is the canonical class? What's a good example of a variety where $K_X$ is not Cartier (but preferably is $\mathbb Q$-Cartier) and I can still compute it easily?

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The class group of $C$ is $\mathbb{Z}+\mathbb{Z}/2$ where the first $\mathbb{Z}$ is the hyperplane $H$ at infinity. $C$ is singular, but it is still Gorenstein, so the adjunction formula applies and should give you $K_C=-2H$. For your secound question, cones are a good idea to try. –  Bonanza Aug 30 '11 at 1:39

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