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.