Computing a canonical divisor

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.

• 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. – Tabes Bridges Aug 3 at 19:29