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I am trying to understand canonical divisors better by computing some examples.

Let $R = k[u,v,w]/(v^2 - uw)$, and set $X = \text{Spec}(R) \subseteq \mathbb{A}^3$. Let $\tilde{X}$ be the blow-up of $X$ at the origin, which naturally sits inside of $Y$, the blow-up of $\mathbb{A}^3$ at the origin, as a divisor. I want to compute the canonical divisor of $\tilde{X}$.

Because the blow-up map $\tilde{X} \to X$ is an isomorphism away from $0$ and $X$ is normal Gorenstein the canonical divisor is supported on the exceptional divisor $E$ of the blowup, i.e. $K_{\tilde{X}} = k E$ for some $k \in \mathbb{Z}$. The question is then: what is $k$?

My idea was: we know that $K_Y = 2 E'$ where $E'$ is the exeptional divisor on $Y$ (general fact about the canonical of a blow-up of a smooth variety along a smooth subvariety). As $\tilde{X}$ is a smooth divisor on $Y$, by the adjunction formula we get $K_{\tilde X} = (2 E' + \tilde{X})|_{\tilde{X}}$, but here I am stuck. Can anyone point me to how to proceed? Or maybe there is a better way of computing the canonical divisor on this surface? Do you know other examples that are easy to compute by hand? Thank you in advance.

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  • $\begingroup$ Adjunction is not so useful unless you can identify the class groups of your varieties and the restriction map between them. We have $E'|_{\tilde{X}} = E$, which is a conic in $E' \cong \mathbb P^2$, but until you identify the class of $\tilde X$ so that you can "restrict it to itself" you can't really get anywhere. $\endgroup$ – Tabes Bridges Aug 3 at 19:29

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